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Geometric effects on bilayer convection in cylindrical containers

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Geometric effects on bilayer convection in cylindrical containers
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Johnson, Duane, 1970-
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vii, 199 leaves : ill. ; 29 cm.

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Aspect ratio ( jstor )
Convection ( jstor )
Flow distribution ( jstor )
Fluids ( jstor )
Index numbers ( jstor )
Liquids ( jstor )
Rayleigh number ( jstor )
Silicones ( jstor )
Temperature gradients ( jstor )
Velocity ( jstor )
Chemical Engineering thesis, Ph.D ( lcsh )
Dissertations, Academic -- Chemical Engineering -- UF ( lcsh )
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theses ( marcgt )
non-fiction ( marcgt )

Notes

Summary:
Liquid encapsulated crystal growth.
Thesis:
Thesis (Ph. D.)--University of Florida, 1997.
Bibliography:
Includes bibliographical references (leaves 194-198).
Additional Physical Form:
Also available online.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Duane Johnson.

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GEOMETRIC EFFECTS ON BILAYER CONVECTION
IN CYLINDRICAL CONTAINERS

















BY

DUANE JOHNSON







A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF
PHILOSOPHY



UNIVERSITY OF FLORIDA


1997














ACKNOWLEDGMENTS


To begin I would like to thank Dr. Ranga Narayanan, my advisor. Throughout my

thesis he has offered his support in every conceivable manner. His professionalism and

light-heartedness has made my experience both highly educational and enjoyable.

Many thanks, love and admiration go to my wife, Jody, whose love and support has

made all of this possible.

I would be remiss not to thank Dr. Ray Skarda and Dr. J.C. Duh at the NASA Lewis

Research center for there help, advice, and especially for there assistance in obtaining the

IR camera.. It is also necessary for me to thank those on my advisory board, Dr. L.

Johns, Dr. U. Kurzweg, and especially Dr. Ruby Krishnamurti and Dr. Jorge Vifials for

driving all the way from Tallahassee. My gratitude is also extended to Ken Reed for his

help in designing and constructing the experiment and Dr. A. Zaman for his help with

some of the calculations and the viscosity measurements of the silicone oil

I would also like to thank a few of the undergraduate students who have assisted me

in many ways: Chris Birdsall for his help in constructing the second version of the

experiment and Bryon Stackpole for his contribution to the control program and writing

the experiment manual.

Final acknowledgments go to the many graduate students and faculty at the

University of Florida. The numerous conversations and advice given was an essential

part of my progress.
ii








This work was supported by a fellowship from the NASA Graduate Student

Research Program, grant number NGT 3-52320 and NGT 51242 grants and from the

National Science Foundation, grant numbers CTS 95-00393 and CTS 93-07819.














































iii















TABLE OF CONTENTS


ACKN OW LEDGM ENTS ........................................................ .................................. ii

ABSTRA CT....................................................................................................................... vi

CHAPTERS


1. PHYSICS AND HISTORICAL PERSPECTIVE ...............................................

Introduction .....................................................................................................
Physics ...................................................................................................................2
Rayleigh Convection ................................................. .............................. 3
M arangoni Convection ................................................... .......................... 5
Pattern Selection ......................................... ................. ............................. 7
Bilayer Convection......................................................................................... 9
History ........................................................................................................... 14
Single Layers ................................................................................................ 14
Bilayers ..................................................................................................... 19

2. M ATHEM ATICAL M ODELING ....................................................................22

Linear M odel ................................................................................................. 22
Num erical ...................................................................................................... 32
Unfolding.......................................................................................................40
Nonlinear Analysis ........................................ ............... .......................... 46
Adjoint ......................................................................................................48
Galerkin-Eckhaus Expansion........................................ .......................... 50

3. EXPERIMENTAL APPARATUS AND PROCEDURE.....................................58

Apparatus.......................................................................................................... 59
Infrared Im aging System ..................................... .........................................59
Test Section .............................................................................................. 63
Heating and Cooling................................................... ............................ 67
Electronic Hardware Unit...............................................................................68
Procedure ................................................... ................................... ................. 72



iv









4. RESULTS AND DISCUSSION ........................................................................77

Introduction ................................................................................................... 77
Codimension-Two points ...................................... .............. ........................ 78
Effects of Air Height on Bilayer Convection................................... ........... 83
Observations from calculations ...................................................................84
Observations from experiments.................................................................... 95
Changes in Convection Coupling and Interfacial Structures .......................... 104
Changes in convection coupling................................................................ 106
Changes in interfacial structures ................................................................ 113
Other Observations in Convection-Coupling and Interfacial Structure .......115
Nonlinear Analysis ........................................ ................ ............................121
Case 1......................................................................................................123
Case 2................................................................. ...................................124
Case 3......................................................................................................125
Case 4...................................................................................................... 127

5. FUTURE SCOPE ...............................................................................................140

Experiments ....................................................................................................... 140
Nonlinear Analysis ....................................................................................... 141
Numerical Calculations ........................................................... ..................142


APPENDICES ...........................................................................................................143


A COM PUTER PROGRAM S ................................... .......................................143

B DRAW INGS AND DIAGRAM S .................................................................... 183



REFERENCES ..........................................................................................................194

BIOGRAPHICAL SKETCH ......................................................................................199













v














Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

GEOMETRIC EFFECTS ON BILAYER CONVECTION
IN CYLINDRICAL CONTAINERS



By

Duane Johnson

December, 1997



Chairman: Dr. Ranganathan Narayanan
Major Department: Chemical Engineering


The study of convection in two immiscible fluid layers is of interest for reasons

both theoretical as well as applied. Recently, bilayer convection has been used as a

model of convection in the earth's mantle. It is also an interesting system to use in the

study of pattern formation. Bilayer convection also occurs in a process known as liquid

encapsulated crystal growth, which is used to grow compound semiconductors. It is the

last application which motivates this study.

To analyze bilayer convection, theoretical models, numerical calculations and

experiments were used. One theoretical model involves the derivation of the Navier-

Stokes and energy equation for two immiscible fluid layers, using the Boussinesq

approximation. A weakly nonlinear analysis was also performed to study the behavior of

vi








the system slightly beyond the onset of convection. Numerical calculations were

necessary to solve both models. The experiments involved a single liquid layer of

silicone oil, superposed by a layer of air. The radius and height of each fluid layer were

changed to observe different flow patterns at the onset of convection.

From the experiments and theory, two major discoveries were made as well as

several interesting observations. The first discovery is the existence of codimension-two

points particular aspect ratios where two flow patterns coexist in cylindrical

containers. At these points, dynamic switching between different flow patterns was

observed. The second discovery was the effect of air convection on the flow pattern in

silicone oil. Historically, air has been considered a passive medium that has no effect on

the lower fluid. However, experiments were done to show that for large air heights,

convection in the air can cause radial temperature gradients at the liquid interface. These

temperature gradients then cause surface tension gradient-driven flows.

It was also shown that changing the radius of the container can change the driving

force of convection from a surface tension gradient-driven to buoyancy-driven and back

again. Finally, the weakly nonlinear analysis was able to give a qualitative description of

codimension-two points as well as the change in flow patterns due to the convecting air

layer.











vii













CHAPTER 1
PHYSICS AND HISTORICAL PERSPECTIVE





1.1. INTRODUCTION

The motivation for this research comes from a technique known as liquid-

encapsulated crystal growth. Liquid-encapsulated crystal growth is a process for growing

semiconductor crystals from bulk, liquid melts. Some examples of crystals grown using

this technique are gallium arsenide and gallium selenide, which are used in





Inert Gas



Liquid Encapsulant




Liquid Melt


Solid


Figure 1-1. Schematic of a liquid encapsulated crystal grower a system of three
convecting fluid layers. Convection in the GaAs liquid influences the quality of the
GaAs solid.

1





2

communications, lasers, as well as the next generation of computer processors. These

applications require that the material be of the highest purity and that the crystalline

structure be nearly flawless.

Take gallium arsenide (GaAs) for example. When solid gallium arsenide is melted,

the arsenic has a tendency to escape. This decomposition destroys the necessary

stoichiometric ratio of the crystal, diminishing its quality. Additionally, arsenic is highly

toxic and a serious hazard to humans. To prevent this decomposition, a lighter,

immiscible, viscous liquid, such as boron oxide (B203) is placed on top of the gallium

arsenide. This limits the transport of arsenic into the upper layer. To prevent arsenic gas

from bubbling through the encapsulant layer, an inert gas, such as argon, is pumped in at

a high pressure on top of the boron oxide. To grow the crystal, these three fluid layers are

typically placed into a cylindrical crucible. The crucible is then lowered into a Bridgman

furnace (Schwabe, 1981; Muiller, 1988), which is hot on top and cool enough at the

bottom to solidify only the gallium arsenide. This configuration creates a system full of

interesting physics and we will discuss some of these next.




1.2. PHYSICS

Although there are many different phenomena that can be studied in this system,

such as the morphological instability (Mullins and Sekerka, 1964; McFadden et al., 1984;

Glicksman et al., 1986; Davis, 1990) and double diffusion (Turner, 1985), this thesis will

concentrate on studying buoyancy-driven and interfacial tension-driven convection.

Morphological instability occurs when the solidification velocity the growth rate of the





3

solid is faster than some critical value, generating compositional undercooling. For

large growth rates, the flat, planar solid-liquid interface begins to deflect. These

deflections can be as small as a few microns ultimately growing into dendrites. Double

diffusive convection only occurs when there is more than one species in the liquid melt.

This typically occurs in crystal growth when a dopant is added to the semiconductor

compound.

One often assumes that the solidification is quasi-static. That is, the growth rate is

much slower than the time scale of the convection and slower than the critical growth rate

necessary for the morphological instability to occur. Additionally, only immiscible fluids

will be considered in this study precluding the possibility of double diffusion.




1.2.1. Rayleigh Convection

Buoyancy-driven convection, often referred to as natural convection or Rayleigh

convection, occurs as a result of the variation of density with respect to temperature under

a gravitational field. Imagine a layer of liquid bounded vertically by two horizontal rigid

Buoyancy-Driven Convection Interfacial Tension-Driven Convection

Cold Cold
gas gas



"- liquid 1 liquid c Iv ctio




Figure 1-2. Physics of Rayleigh and Marangoni convection.





4

plates, with the lower plate at a temperature greater than the upper plate. As density

typically decreases with an increase in temperature, the fluid near the top plate is heavier

than the fluid at the bottom plate, creating a gravitationally unstable system. However, if

the temperature difference across the layer of liquid is sufficiently small, then the fluid

simply conducts heat from the lower plate to the upper plate, creating a linear temperature

drop across the fluid. When the fluid is quiescent, a precarious balance exists between

the pressure gradient and buoyancy forces. For large depths, thermal expansivity and

gravity tend to upset this balance while kinematic viscosity and thermal diffusivity tend

to reinstate the balance. When the balance is upset by disturbances, the fluid is set into

motion which under certain circumstances will continue unhindered. This fluid motion is

called buoyancy-driven convection.

The extent of buoyancy-driven convection (if any) is given by the dimensionless

Rayleigh number, Ra.

ag- gATd3
Ra = (1.1)
VK

Here, a is the negative thermal expansion coefficient, g is gravity, AT is the vertical

temperature difference across the fluid layer, d is the depth of the fluid, v is the kinematic

viscosity, and K is the thermal diffusivity. If the temperature difference is increased

beyond what will be referred to as the critical temperature difference, then the

gravitational instability overcomes the viscous and thermal damping effects and the fluid

is set into motion, causing buoyancy-driven convection.





5


1.2.2. Marangoni Convection

Surface tension gradient-driven convection, unlike buoyancy-driven convection, can

occur in a fluid without a gravitational field. Imagine a layer of fluid which is bounded

below by a rigid plate and whose upper surface is in contact with a passive gas (Figure 1-

2). Above the passive gas is another rigid plate. A passive gas is a gas which conducts

heat like a solid, yet has no viscosity, so that it does not impart momentum to the liquid.

For the sake of consistency, allow the lower plate to be at a temperature greater than the

upper plate's temperature. Now, imagine that the interface between the lower liquid and

the passive gas is momentarily disturbed. The regions of the interface which are pushed

up experience a cooler temperature. Likewise, the regions of the interface which are

pushed down, increase in temperature. Typically, surface tension decreases with an

increase in temperature. Therefore, the regions of the interface which are pushed up

increase in surface tension, which pulls on the interface, while the regions of the interface

which are pushed down, decrease in surface tension. When the fluid is pulled along the

interface, warmer fluid from the bulk replaces the fluid at the interface enhancing the

surface tension-induced flow. If the temperature difference across the liquid is

sufficiently small, then the thermal diffusivity of the fluid will conduct away the heat or

the dynamic viscosity will resist the flow causing the surface to become flat and the

surface tension to become constant. As was the case in buoyancy-driven convection,

there exists a critical temperature difference where the surface tension gradient-driven

flow is not dampened by the thermal diffusivity or viscosity, and the fluid is set into






6


motion. Surface tension gradient-driven convection is characterized by the dimensionless

Marangoni number, Ma.


o ^Td
Ma (1.2)


Where aI is the change in the surface tension with respect to the temperature, and gi is the

dynamic viscosity.

The extent of either Rayleigh convection or Marangoni convection is primarily a

function of the fluid depths. By examining equations (1.1) and (1.2) we notice that

Rayleigh convection is proportional to the cube of the fluid depth and that Marangoni

convection is directly proportional to the fluid depth. From these scaling arguments, we

can conclude that for deeper fluids, buoyancy-driven convection is more prevalent, and



95


I
S........ m=2 .. -.. m=3

S80 \
cn
75

S\70
-70 \F "5-

65
U 60--- -

55
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Aspect Ratio


Figure 1-3. Plot of the critical Marangoni number versus the aspect ratio of a cylinder.
The mode, m, with the smallest Marangoni number at a given aspect ratio, is the mode
or flow pattern at the onset of convection.





7


for shallower depths, surface tension gradient-driven convection is more prevalent.

However, it has been show (Nield, 1964) that Rayleigh and Marangoni convection

reinforce one another. Therefore, at intermediate fluid depths, both Rayleigh and

Marangoni convection can occur.

There is another phenomena associated with surface tension gradient-driven

convection, often called the long wavelength Marangoni instability (Davis, 1983). This

instability typically occurs when either the surface tension or the depth of the fluid layer

is very small. The initiation of this instability is similar to the description given above.

However, in the long wavelength scenario, the convection cells are much larger than the

regular, or short wavelength, Marangoni convection. As the convection propagates, it

causes large scale deformation in the interface which can actually cause the interface to

rupture; that is the interface deforms to such an extent that it comes in contact with the

lower plate. This phenomenon occurs in the drying of films and coating processes. In all

of the cases examined in this thesis, both the surface tension and the liquid depths were

sufficiently large to avoid this instability.




1.2.3. Pattern Selection

In a fluid of infinite horizontal extent, there is no limit on the size or the number of

convection cells. The size of the convection cell is proportional to the wavelength, which

is inversely proportional to the wavenumber. However, in a bounded, finite-sized

container, only a finite number of convection cells may exist. Physically, this means that

at the onset of convection in a bounded container, only one flow pattern will usually





8


exist. As the aspect ratio (radius divided by the height) of the container increases, more

convection cells will appear (see Figure 1-3).

In a bounded cylinder, each flow pattern has associated with it an azimuthal and

radial mode, m and n, respectively. For example, at an aspect ratio of 1.0 in Figure 1-3,

there is an m = 1, n = 1 flow pattern, where m is the azimuthal mode and n is the radial

mode (see Figure 1-4). For an aspect ratio of 1.5, there exists an m = 0, n = 1 flow

pattern. The azimuthal mode is the number of times the azimuthal component of velocity

goes to zero, and the radial mode is the number of times the radial component of velocity

goes to zero starting from the center, for a given vertical cross-section. The azimuthal

and radial modes will be defined more precisely in the mathematical modeling section.

As the aspect ratio increases, the flow pattern switches from one flow pattern to

another. Aspect ratios where two different flow patterns can coexist are called

codimension-two points. Physically speaking, these are aspect ratios where the energy

states of each flow pattern are equal.



x O
m=0 m =2

toroidal x x bimodal
x




m l = 1 m=
n 1 n=2
unicellular double toroid


Figure 1-4. Schematic of three different flow patterns. Circles represent fluid flowing up
and X's represent fluid flowing down.





9

Lower Viscous Thermal Upper
Dragging Mode Coupling Coupling Dragging Mode










Figure 1-5. Schematic of the different types of convection-coupling. From the lower
dragging mode to the upper dragging mode, the buoyancy force in the upper layer is
increased and the dragging exerted by the lower layer decreases.




1.2.4. Bilayer Convection

We now change our thought experiment to include a viscous, less dense, immiscible

layer of fluid above the lower layer of fluid. Here the lower layer is bounded below by a

rigid, conducting plate and the upper layer is bounded above by another rigid, conducting

plate. Once again the temperature of the lower plate is greater than the upper plate. The

interface between the two fluids is allowed to deform and is capable of transporting heat

and momentum from one layer to the other. We will now consider the various types of

convection that can occur in a bilayer of two fluids.

In order to distinguish the various convective mechanisms, phrases such as

"convection initiating in one layer or another" are introduced. Clearly in a mathematical

sense, there is only a single condition for the onset of convection and this onset must

occur simultaneously in both layers. The notion of convection "initiating" in one layer or

another is ultimately a physical one and is perhaps best explained qualitatively. To






10





a b
Lower c d

Upper 0
layer




SII





0 0 0 0


Figure 1-6. Typical plots of the vertical component of velocity (top row) and the
temperature perturbations (bottom row) versus the fluid depths. The vertical dashed
line represents the interface, which separates the lower and upper fluid.

understand the statement in the context of convection with liquid bilayers, we consider

only Rayleigh convection and assume momentarily that the Marangoni effect is absent.

When one of the fluid layers is said to "initiate" convection prior to the other fluid, what

is meant is that its Rayleigh number has reached its critical value before the other layer.

The critical value for each fluid layer, in this situation, is compared to a different

problem. For the lower layer, the critical Rayleigh number is the critical Rayleigh

number it would have as if it were the only active fluid layer, superposed by a passive

fluid which only conducted away heat. For the upper layer, the critical Rayleigh number

is the one it would have if it was bounded above by a rigid conducting plate and bounded

below by a passive fluid that only conducts heat away.

Turning now to various convective mechanisms, consider Figure 1-5. Suppose that

convection initiates in the lower layer. The upper layer responds by being dragged,








generating counter rolls at the interface. Hot fluid flows up in the lower layer and down

in the upper layer. The upper layer is not buoyant enough and moves by a combination of

viscous drag and the Marangoni effect. This is seen in Figure 1-5a. This can also been

seen by plotting the velocity and temperature perturbations versus the fluid depth (Figure

1-6). Observe in Figure 1-6a that the sign of the velocity switches and the maximum

absolute value of the lower layer velocity is much greater than the maximum absolute

value of the velocity of the upper layer.

When the buoyancy in the upper layer increases and the upper layer begins to

convect, one of two things can happen. The first possibility is that the two fluids are

viscously coupled. Physically this can be shown in Figure 1-5b as counter-rotating rolls

in the two fluids. This can also be denoted by the vertical component of velocity

switching sign at the interface (Figure 1-6b), while the temperature perturbations may

switch sign in either the upper layer or the lower layer. If the temperature perturbation

switches sign in the upper fluid, then the lower layer is more buoyant. If the temperature

perturbation switches sign in the lower layer, then the upper layer is more buoyant.

Marangoni convection, for fluids whose surface tension decreases with an increase in

temperature, encourages this mode of convection.

The second possibility is thermal coupling where the rolls are co-rotating. Here hot,

rising fluid from the lower layer causes hot fluid in the upper layer to flow up. The

maximum of the vertical component of velocity and the temperature perturbations have

the same sign in each fluid layer (Figure 1-6c). Strictly speaking, the transverse

components of velocity should be zero at the interface. However, thermal coupling is





12

sometimes referred to the case when a small roll develops in one of the layers so as to

satisfy the no-slip condition at the interface.

As the buoyancy continues to increases in the upper layer, convection initiates in only

the upper layer and the lower layer is viscously dragged (Figure 1-5d). This situation

only occurs when the upper fluid is a liquid, as gases are very tenuous and wouldn't exert

enough shear. The vertical component of velocity in this case (Figure 1-6d) switches sign

and the maximum absolute value of the vertical component of velocity in the upper fluid

is much greater than the maximum absolute value of the vertical component of velocity in

the lower fluid.

Another indicator of what is occurring in bilayer convection can be inferred from the

fluid-fluid interface instead of the bulk convection. In a paper by Zhao et al. (1995), four

different interfacial structures were identified for any given convecting bilayer with a

deflecting interface. Each of these structures depends upon whether fluid was flowing

into or away from the trough or the crest, and whether the fluid was hotter or cooler at the

trough or the crest of the interface. Hot fluid flowing into a trough defines the first

interfacial structure. The second interfacial structure has hot fluid flowing into a crest.


I III


t It Hot Flow

II IV J i Cold Flow



Figure 1-7. The four possible interfacial structures at a fluid-fluid interface. Each
structure can give information about the driving force of the convection.






13

The third structure has hot fluid flowing away from a crest and the fourth structure has

hot fluid flowing away from a trough. Each of these four scenarios is given in Figure 1-7.

One of the important factors to consider in interfacial structures, is the direction of the

flow along the interface. As surface tension is usually inversely proportional to

temperature, at cooler regions of the interface, the surface tension will be higher and will

pull on the interface. Where the interface is hotter, the surface tension will be lower

causing the fluid to move away from warmer regions. Another important factor is the

direction of the flow into or away from a crest or a trough. One reason the interface

deflects is due to bulk convection, caused by buoyancy effects, pushing against the

interface. Consider two fluids whose dynamic viscosities are equal. If buoyancy-driven

convection is occurring mostly in the lower layer, then the fluid will flow up from the

lower layer into a crest. If the fluid flows down from the top layer into a trough, then one

would argue that buoyancy-driven convection occurs mostly in the upper fluid

In each of the four cases, the interfacial structure can be used to indicate the driving

force of the convection. In the first interfacial structure, the dominating driving force is

surface tension gradient-driven convection. This is seen as the cold fluid, with the higher

surface tension pulls the fluid up into the crest. The first interfacial structure can also

occur by buoyancy-driven convection in the upper layer, when the density of the upper

layer increases with an increase in temperature. In the second interfacial structure,

buoyancy drives convection in the lower phase. The hot, rising fluid pushes the interface

upwards. As the fluid moves along the interface, it cools and eventually sinks back

down. The third interfacial structure is dominated by buoyancy-driven convection in the

upper phase, or by surface tension gradient-driven convection where the surface tension





14


increases with respect to temperature. The fourth interfacial structure only occurs when

the lower fluid has a positive thermal expansion coefficient. In other words, the density

increases with an increase in the temperature, causing the cooler, lower fluid to flow up

into a crest.


1.3. HISTORY

Rayleigh-Marangoni convection is one of the classic problems in fluid mechanics,

dating back to the beginning of this century (Benard, 1900). From its initial roots, the

problem has split off into many different branches. For this reason, it is difficult to give a

comprehensive review of all of the aspects of this fascinating phenomena. Instead, only

the aspects which are relevant to the historical background of this thesis will be covered.

This section is divided into two major categories: single layers and bilayers. The

work on convection in single liquid layers is by far more comprehensive, with

concentrated efforts on the bilayer problem occurring only recently. For a comprehensive

review of Rayleigh-Marangoni convection, refer to the book Binard Cells and Taylor

Vortices by E. L. Koschmieder (1993). The following subsection on single layers, is a

review of the most relevant facts within Koschmieder's book, and several other works

that he dismissed.




1.3.1. Single Layers

Pearson (1958) was the first researcher to look at the dynamics of two laterally

unbounded fluid layers, where the upper fluid was considered to be an inviscid gas. He





15

assumed that the interface between the two fluids was non-deformable and found a

critical Marangoni number around 80 with a critical wave number of 2.0. Nield (1964)

looked at the combined effects of buoyancy and surface tension gradient-driven

convection in a single liquid layer. In this work, he performed a linear stability analysis

using a normal mode expansion. The single liquid layer was bounded below by a rigid

conductor and bounded above by a passive gas. The interface was assumed to be flat and

non-deformable. The dimensionless heat transfer at the free surface was modeled by the

Biot number.

9ao
+ BiO =0 (1.3)


for Bi = hdlk. Here, 0 is the dimensionless temperature, h is the heat transfer coefficient,

d and k are the depth and thermal conductivity of the liquid, respectively, and z is the

coordinate pointing out of the fluid into the passive gas layer. The Biot number can also

be written as:

Bi= kgasdliquid
Bi = (1.4)
kliquiddgas

Nield found that buoyancy and surface tension gradient-driven convection reinforce

one another. He also investigated the effect of the Biot number on the critical Rayleigh

number, and critical wave number. He found that decreasing the Biot number decreases

both the critical Rayleigh number and critical wave number.

The modeling of the heat transfer from the lower liquid to the passive gas was

improved upon by Normand et al. (1977). In their review, they allowed the temperature





16


of the passive gas to become perturbed, yet still considered the gas as being mechanically

passive. By doing this, a new formula for the Biot number was arrived at.


Bi = coqui cothO djgj (1.5)
liquid \ dliquid}

where o is the wave number of the gas perturbations. Equation (1.5) is equal to equation

(1.4) in the limit as o goes to zero. Although they did not calculate the effect of this new

Biot number on the critical Rayleigh number and the critical wave number, it is easy to

see the difference. For a fixed wave number and depths of the liquid and gas, the Biot

number in equation (1.5) is always greater than the Biot number in equation (1.4). By

taking the results from Nield's analysis, the critical Rayleigh number and the critical

wave number is seen to increase compared to Nield's results for the same liquid and gas

depths.

The effect of a deflecting surface was introduced by several researchers, but is

probably best described by Davis (1983) and later reviewed by Davis as well (1987). In

the review, he notes that the surface deflections destabilize the system when surface

tension gradients dominate and stabilize the system when buoyancy-drive convection is

dominant. He also notes, that in buoyancy-driven convection, the fluid flows up into a

surface elevation and that in surface tension gradient-driven convection, the fluid flows

up into a surface depression. Fluid flowing up into a depression was first noted in the

original experiments of B6nard (1900) and later confirmed by Cerisier et al. (1984).

Davis (1983) also noted that for vary shallow layers surface tension gradient-driven

convection leads to a long wavelength instability. He developed a nonlinear evolution of





17

the surface deflections by adding the contribution of VanderWaal's forces. Therefore, his

model is only valid for extremely thin layers.

We next move to the effects of bounded containers on the flow pattern in a single

liquid layer. Numerous papers have been written on pattern formation in Rayleigh and

Rayleigh-Marangoni convection for large aspect ratio containers. Among these papers,

several researchers have investigated the effects of boundaries on these patterns. As this

thesis will only concentrate on relatively small aspect ratios, this group of papers will be

neglected. The interested reader is refereed to Cross and Hohenberg (1993). Instead, we

will begin with a series of three papers Rosenblat (1982), Rosenblat et al. (1982a), and

Rosenblat et al. (1982b). Of these papers, we will concentrate on Rosenblat etal.

(1982a), which deals with cylindrical containers. In this paper, a weakly nonlinear

analysis was performed on the pure Marangoni problem, using the Galerkin-Eckhaus

expansion (Eckhaus, 1965; Manneville, 1990).

The most relevant result came from their analysis of codimension-two points, where

two different flow patterns coexist. For one of the codimension-two points, they were

able to show that the solution branched off to a secondary Hopf bifurcation. Physically

this means that the different linear modes could interact with each other (and/or

themselves) to give a dynamic equilibrium solution. Curiously, this Hopf bifurcation was

only seen when the aspect ratio was slightly greater than the aspect ratio of the

codimension-two point. When the aspect ratio was decreased to the other side of the

codimension-two point, the Hopf bifurcation disappeared.

The first systematic experimental investigation of the effects of bounded geometries

was conducted by Koschmieder and Prahl (1990). In their paper, they observed the flow





18

pattern in rectangular and cylindrical containers using aluminum particle tracers in

silicone oil. They report that the number of cells that are observed increases

monotonically as the aspect ratio increases.

The weakly nonlinear analysis in a single layer was later extended by Dauby and

Lebon (1996), who replaced the unrealistic vorticity-free boundary conditions with

realistic no-slip conditions. Their analysis was able to show that the patterns that

Koschmieder observed (Koschmieder and Prahl, 1990) are only visible in the weakly

nonlinear regime.

Another weakly nonlinear analysis was conducted by Echebarria et al. (1997) In their

paper, they took into consideration the rotational symmetry of the cylindrical geometry,

which allowed them to find solutions where the pattern would rotate in the cylinder. By

looking at only a single, highly resonant codimension-two point, they also found

solutions where a secondary Hopf bifurcations could occur. These bifurcations were

identified as a heteroclinic orbit between four different flow patterns, two of which were

the same as the other two, rotated by 900.

In all of the previous papers stated earlier, either an infinite horizontal fluid was

considered or the fluid was confined in a bounded cylinder using unrealistic vorticity-free

side-walls. In the paper by Zaman and Narayanan (1996), and later by Dauby et al.

(1997), a linear, three dimensional solution was found for Rayleigh-Marangoni

convection in a cylinder. Both papers assumed that the interface was flat and that the

side-walls of the cylinder were no-slip. One of the most interesting observations in these

two papers, is that the progression of modes was not the same as the vorticity-free

calculations (Rosenblat et al., 1982a). That is, the flow pattern predicted at the onset of





19


convection for a given aspect ratio is different for no-slip and vorticity-free side-walls.

The vorticity-free calculations contradict the results of Koschmieder and Prahl (1990).

Additionally, the different progression of modes changes the codimension-two points

analyzed by Rosenblat et al. (1982a) and Echebarria et al. (1997). For example, in

Rosenblat et al. interaction of a unicellular, m = 1, and an m = 2 flow was analyzed and a

secondary Hopf bifurcation was found for an aspect ratio slightly greater than the

codimension-two point. This interaction could not even occur according to the linear, no-

slip calculations. Therefore, the existence of the Hopf bifurcation in Rosenblat et al.'s

paper and the heteroclinic orbit found in Echebarria's paper, is in question.




1.3.2. Bilayers

Some of the earliest work done on bilayer convection was a series of linear stability

analyses. Smith (1966) improved upon the single layer problem by allowing the interface

to deform and did not assume a passive gas above. However, he ignored the effects of

buoyancy, and only allowed the surface tension to vary with respect to temperature. In

addition to a linear stability analysis, he also performed a long wavelength analysis. The

long wavelength analysis was able to show that surface deflections are important and can

lead to instabilities in very shallow fluid depths.

Experimental and theoretical work was performed later by Zeren and Reynolds

(1972). In their paper, the effect of buoyancy driven convection was included. They

were able to find three different instabilities: buoyancy-dominated, surface tension-





20


dominated, and "surface deflection dominated" convection. Their linear model was later

improved upon by Ferm and Wollkind (1982).

Interest in bilayer convection increased when a discontinuity in the density of the

earth's mantle was discovered. It was hypothesized that the earth's mantle was composed

of two, chemically distinct layers (Richter and Johnson, 1974). Today this hypothesis is

in general acceptance. The first record of the different types of bilayer convection-

coupling was mentioned in a paper by Honda (1982). Honda used a linear stability

analysis and a finite amplitude analysis to describe three different methods of convection

between the two fluid layers: thermal coupling, viscous coupling, and a dragging of one

fluid by the other.

In the analysis performed by Honda and later by Cserpes and Rabinowicz (1985)

and Ellsworth and Schubert (1988), the mechanical coupling mode was shown to be more

prevalent at and near the onset of convection. It was shown that thermal coupling is more

predominant when the ratio of viscosities is large (more than a factor of 100). However,

laboratory experiments performed with silicone oil and glycerol (Nataf et al., 1988,

Cardin et al., 1991) exhibited that thermal coupling was more stable than mechanical

coupling. This contradicted the earlier analytical results. While effects of interfacial

tension and interfacial deformation were unable to explain the discrepancy between the

analytical and experimental studies (Nataf et al., 1988), Cardin et al. (1991) were able to

show that the interfacial viscosity helped to explain why thermal coupling was more

stable. Additionally, the onset of oscillatory convection was seen to diminish for large

interfacial viscosities. Numerical and experimental work performed by Prakash and

Koster (1996) showed that when the driving forces for buoyancy driven-convection in





21

both layers are approximately equal, then thermal coupling is preferred, whereas,

mechanically coupled flow was observed when these driving forces were very different.

Unlike single liquid layers, a bilayer of two fluids can oscillate at the onset of

convection (Gershuni and Zhukovitskii, 1982; Rasenat et al., 1989). These oscillations

are caused by the interaction of the thermal and mechanical coupling modes. For

example, by changing the two fluid depths, it is possible to cause the thermal coupling

and viscous coupling modes to become simultaneously unstable at different wave

numbers. As it is impossible to have a superposition of these two modes, the system

oscillates between the equal energy states.

The oscillations between the thermal and viscous coupling in a horizontally infinite

bilayer of two fluids was analyzed by Colinet and Legros (1994). They showed that the

oscillations would appear as a traveling wave. This analysis was later verified

experimentally by Andereck et al. (1996).

Oscillatory onset of convection can also occur by the Rayleigh-Taylor instability.

The Rayleigh-Taylor instability occurs when a heavier fluid lies on top of a lighter fluid.

This typically occurs in systems when a liquid with a slightly smaller density lies on top

of another liquid. When the bilayer is heated, the lower liquid density decreases and

becomes smaller than the upper fluid's density. As the upper fluid sinks, it feels the

warmer fluid, heats up, and becomes more buoyant. This instability is avoided when two

fluids with reasonably different densities are considered.














CHAPTER2
MATHEMATICAL MODELING


This chapter includes all of the equations, derivations and numerical techniques

used to analyze a system of two immiscible fluids. The modeling consists of four major

sections:

A linear model of convection in two immiscible fluids, which are infinite in
the horizontal direction.

A numerical calculation of the linear model equations using a Chebyshev
spectral tau method.

A transformation and unfolding method used to map the results from the
infinite, unbounded calculations into a bounded cylinder.

A weakly nonlinear analysis of convection of two immiscible fluids in a
cylinder using a Galerkin-Eckhaus expansion.

The results from the first three sections are necessary to perform the weakly nonlinear

analysis in the fourth section. However, the results from each section can be used to

elucidate certain details of the problem.


2.1 LINEAR MODEL

The derivations of the linear model start by recreating the work of Ferm and

Wollkind (1982). This work considers two immiscible fluids bounded above and below

by rigid, thermally conductive plates. The temperature of the lower plate is always

assumed to be greater than the upper plate and the interface between the two fluids is



22





23


T = t Cold z = d2

fluid #2
z = r (x ,t*)
T* = Tm z =0

fluid #1

T* = T z = -d,


Figure 2-1. Schematic of the linear model. A bilayer of two immiscible fluids, bounded
by rigid, thermally conducing plates.

allowed to deflect. To simplify the calculation, the two fluids will be unbounded in the

horizontal direction (see Figure 2-1). The equations which determine the velocity,

pressure and temperature for each fluid are the familiar Boussinesq equations.

V.v = 0 (2.1)

(8v7 ** vT *
pi +vi Vv = pV2v -Vp, +Pig (2.2)


S *+ VT* = kV2 (23)
p'Cpi--+v=kVTi (2.3)

where pi is the density, uti is the dynamic viscosity, Ci is the specific heat, ki is the

thermal conductivity, vi = (ui, wi)T is the velocity vector, pi is the pressure, Ti is the

temperature and g is the gravitational vector. The asterix (*) above each variable denotes

that the variable is unscaled. The subscript i = 1, 2 represents the lower fluid (i = 1) and

the upper fluid (i = 2).





24

The major assumptions made in equation (2.1) through (2.3) are that the viscosity is

constant, the fluids do not generate heat through viscous dissipation, and the relative

change in the density is very small, that is Ap/p <<1. We will also assume that the

gravitational vector is constant and only points opposite the z direction. We will further

assume that the density, as well as the surface tension, vary only with respect to the

temperature.


,= Po,i -a ( Tref,)] (2.4)


= GOo E-I Tref ,)] (2.5)

1 p
where ac --- is the thermal expansion coefficient, Trefi is the reference
Pi qTi Trf
Pi &77

1 8a
temperature for fluid layer i, ao is the constant surface tension, and c I =- -*


The reference temperature for the lower and upper fluid layer's density is the temperature

at the interface, T,. The reference temperature for the surface tension will be Tb, where

i = 1 in equation (2.5).

For the analysis of this problem, we first assume that the fluid is at rest and only

conducts heat from the lower plate to the upper plate. Mathematically, this is realized by

letting v*i = 0 in (2.1) through (2.3). Furthermore, we will substitute equation (2.4) into

(2.2). The result is the following equation for the temperature.

d2Ti*
= 0 (2.6)
dz2 -*





25

with the boundary conditions:

Ti*(z*= -dl)= T T;'(z*= d2) T
dT* dT*
T(z* = 0)= T2 (z* = 0) = T,, k = k at z = 0

The solution for the temperature profile is:

Tb -Tm, Tt Tb Bi
Ti = T z Biz* +T, + T(T Tb) (2.7)
d, d (1I+ Bi) 1+ Bi

T* Tb-2, k= b T Tb Bi
T = T z*= t i)*+T + -&(T, T) (2.8)
d, k2 d2 t+ B 1+Bi

k2dl
where Bi is the Biot number.
kid2

The next step in the procedure is to make equations (2.1) through (2.3)

dimensionless. The length, velocity, time, and pressure are scaled with dl, K l/d,

d2 /K and p. K1 /d respectively. The non-dimensional temperature is defined as

T -T
i for i=1,2 (2.9)
Tb -Tm

The following symbols will be used for the ratio of thermophysical properties:

ca=a2/aI, l=d2/ld, k=k2/k,, K=K2/Ki, P=P2/91, and Ip=p.2/,91. Here

vi = pC/Pi is the kinematic viscosity, and K, = k,/p,C, is the thermal diffusivity. The

equations for the lower fluid become:

V-vi =0 (2.10)

1 vI + .VvI -V +V2 +Ra01 (2.11)
Pr at





26


at
^t+vl "V8, = e^ 20 (2.12)

and the equations for the upper fluid become.

V-'V2 =0 (2.13)

1 1v2
Pr v2 Vv2) = --V2 + V2v2 +aRaO2 z (2.14)
Pr at p p

02
+ v2 V02 = KV202 (2.15)
at

where, : is the modified pressure, Ra = gaI ATd3 /Ki v1 is the Rayleigh number, and

Pr = v,/K1 is the Prandtl number,

Moving on to the boundary conditions, we start with assuming the upper and lower

plates are rigid, no-slip boundaries at a constant temperature.

wl=u =0 and T* = T atz=-1 (2.16)

w2 = 0 andT2* = at z = (2.17)

We introduce the variable, l = l(x, t), which represents the surface deflections from the

initially flat interface, z = 0. For a deflecting surface, the unit normal, n, becomes.

(-x, 1)T
n = x 1) (2.18)


and the vector tangential to the surface is nt, where:


t (= aT (2.19)
S 1+(/





27

and n, can be shown to be orthogonal to n. Assuming the temperature and the heat flux

across the interface are continuous, we get:

,* = 2* (2.20)

n.klVTi* =n-k2VT2 (2.21)

As the fluids are immiscible, there is no penetration of one fluid into the other.

Furthermore, we assume that the fluids do not slip past one another at the interface.

Therefore:

v* = V2 at z = (2.22)

There also exists the kinematic condition of the interface.

af* ,a *
t* + uI ax* w (2.23)

The last two of the thirteen boundary conditions that are needed come from the tangential

and normal components of the stress balance.

n n n -n= c(V n) (2.24)

n- nt n. 2 -nt = nt .-Vs (2.25)

where T is the stress tensor and Vs is the surface gradient operator. Substituting the

dimensionless variables, T, n, and nt, into equation (2.16) through (2.25) and dropping the

asterix (*) gives:

w, = ==0 atz=-l (2.26)

w2 =u2 =0 atz=l (2.27)

9, =1 atz=-1 (2.28)















T-, +1 + -x---+ +Z
(9^") f__ 0X ze


x' ( xe -e Lxe xge ze ]I









x' x
(OcZ') (' = z -of z (-' j n + d-
Ni z e xg Ing Zc In 1

__ I 10z1 r1 Mg)I


(0E.Z) Ltz x x ze x_ x g e xe
gexe zL (XT Zeg






Q EZ) LL = z 1oj Zn In i )Q\XQ

(O1) LL==z oj IM_=_ n +
Ite lie

(6Z'Z) I=zlt Y~ -=z

8Z






29


Ma, given in (2.34) and (2.35), is the Marangoni number, Ma = cIATd,/Ki ,1. G is the


Weber number, where G = (P P2)gd and C is the Crispation number, where
CC-

C1 K1
Go di

We finally arrive at the essence of the linear model section, which is the normal

mode expansion of the variables. Basically, each variable is expanded in a series about

some parameter, 8, which is a measure of the deviation from the base state (conduction

state) of the system. Further, each variable at order e is again expanded in a Fourier series

in the 'x' direction and exponentially in time.

"u,(x,z,t) 0 U 1u,(z)"
w,(x,z,t) 0 Wi(z)
(x,z,t) = po, + s n1(z) ei'x e"' + O(2) (2.36)
O9(x,z,t) Oo,i (z)
I (x,t) I 0 I) I 1

for i = 1, 2. Note that a here represents the growth constant. The temperature solution to

the base state is given in dimensional form in equations (2.7) and (2.8). The

dimensionless form of the base state solution is:

vo,1 = VO,2 = 0 rI = 0 (2.37)
Ra
0o, = -z Po, = --- 2 for z < 0 (2.38)

00,2 = z PO,2 -z2 for z > 0 (2.39)
After substituting (2.36) into (2.10) through (2.15), the equations to the first order in

E for the lower phase are:

DW +icoU1 = 0 (2.40)





30


(D2 _o 2 1 -io1 = U1 (2.41)

(D2 2 )W -DT1 + Ral = oW1 (2.42)

(D2 29 + W = 0 (2.43)

and for the upper phase.

DW2 +icoU2 = 0 (2.44)


(D)-2 2-2 0 2 = U2 (2.45)


1
(D2 2 2 W DI2 +0a RaE2 =W2 (2.46)
P P


(D2 _2 2 2 = 0 (2.47)

d




example, take some arbitrary dependent variable, A. Then expand A in terms of E.

dA
w Ar = A. dE O(2
e=o

dA
where Ao= A(c = 0) and in general, A = A(rl(s),e). The term is the total derivative

of A, which can be written as:

dA 8A aA a a9A aA 8z rl
+ a+ as az
dE de 8r1 Be de 8z 1"1 Be





31

8z orn
The derivative 1, as z = j. We define tr, and A1 as follows, rll, and,

OA
A1. Therefore we have:
8 a


A(Tl(8),s)=Ao+ A +-s r = s+O(2) (2.48)


Equation (2.48) is then applied to the velocity, pressure and temperature of each

phase and substituted into the boundary conditions, equation (2.25) through (2.35).

Again, the terms of order e2 or higher are neglected.

U, = W, = 1 = 0 at z = -1 (2.49)

U2=W2=02=0 atz=/ (2.50)

W2 =W, =0 atz=0 (2.51)

U2 =U1 atz=0 (2.52)

01 = 2+ 11(1- ) atz=0 (2.53)

kDO2 = D1 at z = 0 (2.54)


I-2-1 + rl + 2(DW DW2)= 0 atz=0 (2.55)


(DU + iOW,)- p(DU2 + ioW2)= io Ma (n 0) atz=0 (2.56)

The next step in the procedure is to solve for the Rayleigh number, Ra, in an

eigenvalue problem, where the velocity, pressure and temperature of each phase are the

eigenvectors. As have been noted by previous workers, the Marangoni number, Ma, and

the Rayleigh number, Ra, are not independent of each other for a given experiment. The





32


ratio Ma/Ra = F, is a constant, which depends upon the thermophysical properties of

the fluid and the height of the lower layer. The equation Ma = FRa replaces Ma in

equation (2.56). Additionally, the growth rate a will be assumed to be zero. This

assumption precludes the possibility of finding oscillatory onset of convection. However,

if the latent root, Ra, becomes complex, then it is an indicator that a = 0 is not a solution

to the problem and indeed, a is imaginary. The final result is a plot of the Rayleigh

number versus the wave number. The procedure for finding this plot will be given in

detail in the next section.




2.2 NUMERICAL METHOD

The objective of the numerical methods was to solve the set of linear ordinary

differential equations (2.40) through (2.47) with the boundary conditions of (2.49)

through (2.56). The method of choice was the Chebyshev spectral tau method for three

reasons. The first reason is that the spectral tau method, in general, requires very few

number of terms to converge to the answer, resulting in a fast and efficient solution

technique. The second reason is that the tau method easily incorporates complicated

boundary conditions. The third reason is that the spectral method yields a as the latent

root if it is so desired and one may then search for the onset of oscillatory convection.

This section will briefly describe the details of the Chebyshev spectral tau method,

and how it was applied to this problem. For a more comprehensive review of spectral

methods, the reader is referred to Canuto et al. (1988) and Gottlieb and Orszag (1986). A






33


tutorial on the application of the Chebyshev spectral tau method to eigenvalue problems

is given by Johnson (1996).

Spectral methods are a particular numerical scheme for solving differential

equations. It is a discretization scheme developed from the method of weighted residuals

(Finlayson and Scriven, 1966). The tau method is one of the three most popular

techniques in spectral methods. These three techniques are the Galerkin, collocation and

the tau. However, only the tau technique will be used here. Before describing the

application of the Chebyshev spectral tau method to the problem, a brief review of the

theory behind the method of weighted residuals in order.

Suppose you were given the problem

au
+ Lu = XAu
at (2.57)
Bu=O

where L and A are linear operators, B is a linear boundary operator, and X is the

eigenvalue. Now express u in terms of an infinite series of trial functions. Here we choose

the Chebyshev polynomial as the trial functions.

CO
u(x,t)= a,, (tn (x) (2.58)
n=O

The function is then approximated by truncating the number of terms to some finite

value, N.

N
u(x,t)= uN(x,t) = Za,(t)Tn(x) (2.59)
n=O

The approximation error accrued by truncating the infinite series is given by SN.





34


au auN
-t -t +Lu-LuN Au+Au = EN (2.60)


The tau part of the spectral method is simply an easy way of handling the boundary

conditions. Note that UN in equation (2.59) must explicitly satisfy the boundary operator

B. This is not always an easy exercise. To accommodate the number of boundary

equations, say -, simply add T more equations to N.

N+T
UN(x,t)= Ia,(t)T(x) (2.61)
n=O

By adding t more variables, we need r more equations. These equations come from the

boundary conditions.

N+T
a,,(t)BT,(x) = 0 (2.62)
n=0

In fact, this is how the tau method gets its name.

The objective of the method of weighted residuals is to minimize EN by choosing a

test function which is orthogonal to the trial function in some inner product space with

respect to some weighting function. The Chebyshev polynomial is orthogonal to itself in


the integral from 1 < x < 1 with respect to the weighting function (I x2)-Y


T,(),TM,(x))= fT,(x')7T(x')(1 x')2 d= Cn6nm (2.63)
-1 2

where

2 n=0 F
cn = n > 0 and = n m (2.64)
0 n<0






35


Substitute equation (2.59) into (2.57) then take the inner product (2.63) of the result.

After simplification the result is:

da N+T N+
S+ an,(tXT,(x),LTm(x)) = X a, (tXT(x),AT,(x)) (2.65)
dt =0 n=o

To evaluate the inner products in equation (2.65), we need to know certain relations

between the Chebyshev polynomials and the result of the operation LTn(x). Suppose L is

a linear ordinary differential operator, which may or may not have constant coefficients.

For a simple example let L = d/ We want to know the relationship between the set of

an and bn for n = 0, 1, ..., N where:

N N
uN(x,t)= a,(t)T,(x) and LuN(x,t)= b,(t)T,(x) (2.66)
n=O n=O

A list of several linear operators are given in appendix A of Gottlieb and Orszag's book

(1986). A comprehensive discussion on how to find the relationship between an and bn is

given in Johnson (1996).

The first and second derivative operators are given below.


cb =2 pa, for Lu= d (2.67)
p=n+l
p+n odd


cAnb =2 p(p2 -n 2)a for Lu=d2 2 (2.68)
p=n+2
p+n even

To evaluate the Chebyshev polynomials on the boundary, the following formula can be

used.





36



Tn (+ 1)= ( n1)+ (2.69)
dx k=0 (2k + 1)

By using equation (2.67) or (2.68), the relationship between an and bn can be

expressed in matrix form. This method is much easier to implement in eigenvalue

problems and the details are given in Johnson (1996). For any derivative q, the qt

derivative coefficient an (q can be expanded in terms of the zeroth derivative coefficient

an by the following relationship.

a) = Eq a, (2.70)

where:

d a q)(t)Tn (x) (2.71)
dCXq n=O

and the matrix E is given by

E 2 n = 0,1,...,N-2
E =[ (n+)(m+) -m for (2.72)
cn m=n+1,n+3,...,N

for example, if N = 5

01030 5
00408 0
0 0 0 6 0 10
S0 0 0 0 8 0 (2.73)

0 0 0 0 0 10
00000 0

Before applying the Chebyshev spectral tau method to the problem, the equations

need to be modified. Typically the double curl of equations (2.11) and (2.14) is taken to

eliminate the pressure terms. This results in a fourth order derivative equation. However,

it was noted that finding the eigenvalues when the linear operator L contains a fourth





37

order derivative can cause difficulty in the convergence (Canuto et al., 1988; Gottlieb and

Orszag, 1986). This can be seen in equation (2.70). As the derivative q increases, the

entries in the matrix Eq increase in magnitude. As the entries in Eq increase, the problem

becomes "stiffer" and more difficult to solve numerically.

To avoid this numerical difficulty, equations (2.41), (2.42), (2.45) and (2.46) were

kept as second order derivatives. To remove the imaginary numbers in these four

equations, the divergence of the Navier-Stokes equation was taken, and the equation of

continuity substituted for Ui. This operation results in the following system of equations.

(D2 (w2) DII + RaE1 = 0
(D2 -o 2)I Ra D = 0 (2.74)
(D2 o2 + W = 0

(2 -D2 2 -)W2-DI2 +aRaO2 =0
P P
1(D2 2 Ra DO2 =0 (2.75)

K(D2 -2) 2+ W2I=0

The thirteen boundary conditions become:

W2 =W1 =0 atz=0
DW2 = DW, at z = 0

r2 f1 + o + 2(DW1 DW2)= 0 atz 0

(D2 +2)W 2 (D2 +c2)W =2 2Ma(rl -o) atz=0 (2.76)
kDO2=D DO atz=0
,= +2 +o1- k) atz=0
DW = W, = = 0 atz=-1
DW2 =W2 =02 =0 atz=/





38

We notice that the Chebyshev polynomial lies in the interval -1 < x < 1, whereas

the lower phase variables lie on the interval 1 < z < 0 and the upper phase variables lie

on the interval 0 < z l1. Before we can expand the dependent variables in the upper or

lower phase in terms of Chebyshev polynomials, we need to map each phase into the

Chebyshev space 1 < x < 1. This is accomplished by the two transformations.

x= 2z+1 for z<0 (2.77)

2
x2 =-z-1 for z>0 (2.78)

The change in independent variables requires the substitution.

d
d d x 2- for z<0
dz dx dz 2d (2.79)
Sfor z>0
1 dx

Now expand each dependent variable in terms of Chebyshev polynomials.

W, (z) N-_ a.
n,(z) = bj T (z) (2.80)



W2[ N-t 1 d
H2 (z) eJTj (z) (2.81)
[,(z)2 Z J=OJ

The surface deflection term, rl, is not expanded in terms of Chebyshev polynomials as it

is not a function of the domain variable, z. After equation (2.77) through (2.81) are

substituted in to (2.74) through (2.76), the inner product (2.63) is taken. This operation

results in a system of 6N+1 equations in the form.






39



1600

1500

1400

1300

1200

1100

1000
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
Wave Number


Figure 2-2. Plot of the Rayleigh number versus wave number.

A4 = RaB4 (2.82)


where (= (W,,i, 1,W2, 1-12, 2) The two matrices, A and B, contain the

Chebyshev coefficients, such as equation (2.73). Further details of these matrices are

given in the Matlab programs in Appendix A.

Equation (2.82) is simply an eigenvalue problem, which can be solved by several

standard software packages. The software package chosen was Matlab. All of the

relevant programs written in Matlab are given in appendix A. The construction of each

program was very similar. First, the matrices A and B were defined, then the eigenvalues

and eigenvectors for a fixed wave number, co, were solved. The wave number was

incremented and the eigenvalues and eigenvectors recomputed. At the end of the

program, the Rayleigh number versus wave number could be plotted as well as any of the





40


temperatures and vertical component of velocities versus the fluid depths, for a given

wave number.




2.3 UNFOLDING

In this section, a technique known as unfolding will be described. The unfolding

technique derives its name from the plot that it generates. When it is applied to a plot,

such as Figure 2-2 above, it takes the wave number from the unbounded geometry and

effectively "unfolds" into its discrete azimuthal and radial terms. The result is a series of

Rayleigh number versus aspect ratio curves on a single plot. This technique is used to

give a qualitative description of the flow pattern in a bounded geometry. Even though this

technique can be easily applied to rectangular geometries, only cylinders are considered

here.

We start our analysis by following closely the technique used by Rosenblat et al.

(1982a). Here we will consider a bilayer of two immiscible fluids confined in a cylinder.

The top and bottom of the cylinder consist of a rigid, no-slip thermally conductive plate,

where the lower plate is at a temperature greater than the upper plate. The interface

between the two fluids is flat and non-deformable. This restriction is not essential and

will be relaxed when certain examples are discussed later on. The scaling, at least

initially, will be the same as the scaling used in the linear model section. Scaling does not

affect the results unless approximations are made. Equations (2.1) through (2.3) will still

describe the nonlinear behavior of the two fluids. Again, we linearize equations (2.1)

through (2.3) and assume the onset of convection is steady, (o = 0).





41


V -v =0 (2.83)

V2V -V +Ra i=0 (2.84)

V201 +w1 =0 (2.85)

V*'V2 =0 (2.86)


V2 2 V2 12+ci RaO2 = 0 (2.87)
P P

1
KV 2 + 1-W2 = 0 (2.88)


where v1 = (uvi, )T ui is the azimuthal component of velocity, vi is the radial

component of velocity, and wi is the vertical component of velocity. The cylinder has

azimuthal coordinates, 0 < p < 2 radial coordinates, 0 < r a, and vertical

coordinates, 1 < z < 1.

Here, the boundary conditions are different because the surface is assumed to be

non-deformable and the fluid is bounded by the radial wall of the cylinder. We wish to

find solutions to (2.83) through (2.88) which are separable, in order to simplify the

calculations. The trick that achieves this is to force the radial wall, r = a, to be thermally

insulating and also restrict the vertical and tangential component of vorticity at the radial

wall to be zero. Mathematically, this is shown as:

o0, 9 ( )=9aw,
S-u = r -(i)= =0 at r=a for i= 1,2 (2.89)
9r r ar

The remainder of the boundary conditions are:

,0 = u, = vl = w, = 0 at z = -1 (2.90)






42


02 =2 =V2 =W2 =0 at z=/ (2.91)

w = w = 0 at z=O (2.92)

u1 = u2 at z = 0 (2.93)

v1 = v2 at z = 0 (2.94)

au, u 2
+M =0 at z=0 (2.95)
8z az ar

av, 8v2 1 a9
-a z +M -0 at z=O (2.96)
8z az r acp

1 = 02 at z=0 (2.97)

ao, 062
k at z=0 (2.98)
8z 8z

To find the solution to the system (2.83) through (2.98), we eliminate the pressure,

azimuthal velocity, ui, and radial velocity, vi, in favor of wi and Oi.

V4w1 + RaV2Oi = 0 (2.99)

V21 + w1 = 0 (2.100)


VV4w2 + aRaV2 2 = 0 (2.101)
P


KV202 + W2 =0 (2.102)


and the corresponding boundary conditions are.

8wl
06 = wi- 0 at z=-1 (2.103)
8z

8w2
02= 2 =w -0 at z=l (2.104)
8z





43


w1 =W2 =0 at z=0 (2.105)

aw, 8w2
Z at z=0 (2.106)
8z 8z

2W a 2 .2
W azw2 MV2 at z=0 (2.107)
8z 8z

01 =02 at z = 0 (2.108)

801 809
k =- at z=0 (2.109)
az 8z

80i 8wi
-0 at r=a for i=1,2 (2.110)
or or

Following Rosenblat et al., we try the solution.

W1(r,(pZ)= cosmq Jm(Xr)W,(z) (2.111)

01(r,p,z)= cosmp Jm(Xr)) (z) (2.112)

w2(r,p,z)= cosmp Jm (r)W2(z) (2.113)

2 (r, 9,z)= cosm( Jm(Xr)2 (z) (2.114)

where m = 0, 1, 2, ... is the azimuthal mode number. Jm is the Bessel function of order m

and k is determined by the boundary condition at the radial wall.


dJm(mna)= 0 (2.115)






44


Table 2-1. Table of the zeros of the derivative of the Bessel's Function.
Radial Mode

Azimuthal Mode 1 2 3
1 3.83 7.02 10.17
2 1.84 5.33 8.54
3 4.20 8.02 9.97


For each m, there exists an infinite number of radial modes, n = 0, 1, 2, ..., where (2.115)

holds. Table 2-1 gives the first few values of the zeros of the derivative of the Bessel's

function, smn = kmn a. These values are taken from Abramowitz and Stegun's Handbook

of Mathematical Functions (1966). The functions W,(z), 0,(z), W2(z), and 2(z) are the

solutions to the system (2.74) through (2.76) with a flat surface, r1 = 0. The significance

of the separation of variables lies in the relationship between kmn and the wave number,

o. By substituting in (2.111) through (2.114) into (2.83) through (2.88), we find a simple

relationship between the aspect ratio, a, and the wave number, co, for a fixed azimuthal

and radial mode.

Sm,n
a =- (2.116)


Figure 2-3 is an example of the application of equation (2.116) to the Rayleigh number

versus wave number plot.

Upon substituting equations (2.111) through (2.114) into the equations of continuity

and applying the definition of the Bessel's function, we find.





45


ui(r,(p,z)= 1mn cosmpJ m', mnr)DW,(z)

vi(r,q,,z)= -Xn2 sinm(pJm(mnr)DW,(z) (2.117)
mn (2.117)
w (r,9p,z)= cosm(pJ ,,mnr)W,(z)
6O(r,(p,z)= cosm pJm(mnr)O,(z)

for i= 1, 2. The pressure could also be determined from the Navier-Stokes equation,

however in the next section, the pressure term will not be needed. Notice that the

different coefficients in the domain (i.e. v and p) are accounted for by the z component of

the velocity and temperature, Wi(z) and Oi(z).

The unfolding technique allows us to find qualitative features of the flow field in a

cylinder using the relatively cheap and easy, one-dimensional calculations from the linear

model section. These results will be used later to show the existence of codimension-two

points and will also be used in the next section on nonlinear analysis.


1600

1500

1400-
l 'Y
1300 -

S1200

1100 m=1 m=2 m=0 m=3
n=l n=l n=l n=l
1000
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Aspect Ratio


Figure 2-3. Plot of the Rayleigh number versus aspect ratio. The plot is generated from
Figure 2-2 and equation (2.116)





46


2.3.1 Nonlinear Analysis

The limitations of the linear analysis will only allow us to predict which flow

pattern will occur at the onset of convection. It does not tell us the behavior of the fluid

flow when two flow patterns coexist, such as codimension-two points. Additionally, the

linear model does not guarantee that it will predict what we see in experiments, as the

experiments are always conducted, at least slightly, in the nonlinear regime.

The purpose of performing a nonlinear analysis on this problem is to provide some

theoretical insight into the complicated behavior of bilayer convection. The method of

choice is the Galerkin-Eckhaus expansion (Eckhaus, 1965). This method has shown some

success in predicting and describing certain nonlinear behavior. In the remainder of this

chapter, the several steps necessary to develop the nonlinear model will be developed.

The derivation will end with one or more amplitude equations which describe the

dynamic behavior of each of the various flow patterns.

We start the nonlinear analysis be rewriting the nonlinear Boussinesq equations

which describe the convection of two immiscible fluids in a cylinder. For the nonlinear

analysis, we will also assume the interface between the two fluids is non-deformable.

1 (va
Pr~ vi1 v VIV =-V;I +V2vl +Rao1 z
Pr at
V-V1 = 0 (2.118)

at
+v 9. 8 V2, +





47


1 (v2 +1 1 +
V2 V v2 =-v 2+ V 2v2 +aRa02
Pr at P p
Vv2 = 0 (2.119)
ae2 1
+vV2 'V02 =KV2 2 + W2
at k

The two velocities, w, and w2, appear in equation (2.118) and (2.119) because the base

state temperature is not zero. Remember, 0 is the perturbed temperature, therefore the

conductive temperature, To, needs to be subtracted from the equations. The boundary

conditions for the nonlinear problem are the same as those outlined in the previous

section.

v,=01=O at z=-I
v, = 82 = 0 at z = -1
V2=02=0 at z=l
w = w2 = 0 at z=0
u1 = u2 at z = 0
V1 = v2 at z = 0
61=02 at z=0
89, 892
k at z=0
oz dz
Bul u2 e31
-- +-M -=0 at z=0
oz 9z or
av, aV2 1 a9,
S + -M 0 at z=O
az az r q(p (2.120)

Next we linearize equation (2.118) and (2.119). Equation (2.120) need not be modified as

it is already linear.

V2Vp Vip + Rp, 0,
V.vlp =0 (2.121)
V2 01 + wlp = 0





48

1
vV2V2p --V' +aRR 02p
p
V.v2p =0 (2.122)

KV202p + 2p = 0

Here the subscript p represents the wave vector p= {m n j}, where m is the azimuthal

mode number, n is the radial mode number and j is the vertical wave number. For

example, j = 2 represents vertical stacking of the convection cells. Usually, vertical

stacking only occurs far into the nonlinear regime, except for very small aspect ratios.

Therefore no vertical stacking will be considered. We assumej = 1 always and neglect it

from here on out. We also note that there exists an infinite number of p's corresponding

to the infinite number of eigenvalues which satisfy (2.121) and (2.122).




2.3.2 Adjoint

The adjoint of a linear operator L is defined in some inner product space as.


(LypL) =(KL p,L)

where the asterix denotes the adjoint operator and \ is defined as:


Vp = (Ulp ,w i,1p ,U ,2p V2p2p, 2p)

We can define our linear operator, L from equation (2.121) and (2.122)

Li 044
L, 044= (2.123)
L= 044 L2(2.23)


where, 044 is a four by four matrix whose entries are zero. L, and L2 are defined as:






49


V2 0 0 0' Vv2 0 0 0
0 V2 0 0 0 vV2 0 0
L = and L
L 0 0 V2 Rp a 0 0 vV2 aR

0 0 1 V2 0 0 k KV2

Upon inspection we can find the adjoint operator L*


L=(L* L441 (2.123)


where L*, and L*2 are defined as:

V2 0 0 0 vV2 0 0 0
V2 0 0 vV2 0 0
LI= 0 0 V2 1 and L2 0 0 vV2

0 0 R V2 0 0 (aR KV2

It is obvious from (2.123) and (2.124) that L is not self-adjoint. Although it appears as if

it can be made self-adjoint by multiplying the fourth row of L, by Rp, and multiplying the

fourth row of L2 by aRpk. However, as we will see, the boundary conditions are not self-

adjoint.

Upon analyzing the terms of the bilinear concomitant, the adjoint boundary

condition can be found. For completeness, the linear adjoint problem is given below.

V2V -V +0 Z
Ip ip lp
V-v* =0 (2.125)

V;2* +R w =0
iP P ip

1 1*
vV2V -V2 + k2p Z

V-v* =0 (2.126)
2p
KV20* +cR* w* =0
2p p 2p





50


V1p = 01p = 0 at z = -1
2p= 2p =0 at z=1

Wp = W2p = 0 at z=0
pulp = *2p at z = 0

pVp = 2p at z=0
kp = K02p =0 at z=0

a0ep aw p a2p
aZ 2 M --- K z at z =0
8z 8z 02 dz

Sulp 8"2p
S- v at z=0
oz oz
(2.127)
Vlp ^2p
v- at z=0
dz oz

wip 1p
= v =0 at r=a for i=1,2
iP r 'P r az

The solution to the adjoint problem is found similarly to the non-adjoint solution.


u (r, ,z)= ,, cosm'pJ m (mnr)DWj(z)
ip
v (r,p,z)= -m sinm(pJ, ,,r )DW(z)
(2.128)
w* (r,(P,z)= 2 cosmpjm(,,r)Wi(z) for i = 1,2
ip mnn
O (r,(p,z)= cosmqpJm(mnr) (z)


2.3.3 Galerkin-Eckhaus Expansion

The Galerkin-Eckhaus expansion is one method for studying nonlinear problems

close to their linear state. The method involves the expansion of the nonlinear dependent

variables in terms of all of the linear modes, multiplied by an amplitude function, A.





51


V1 Mp

S-ZA(t) M p (2.129)
V2 p V2P
M
0M 2p

where, Mp = FRp is the pth eigenvalue of the linear problem and M is the Marangoni

number of the nonlinear problem. Again, p represents the infinite number of linear

solutions. Ap(t) is called the amplitude function, and in general, is a complex function of

both space and time. Here we will considered A to be real and only a function of time.

The eigenfunctions, Olp and 02p are multiplied by Mp/M to satisfy the nonlinear

boundary conditions.

The next step is often called the Liapunoz-Schmidt reduction (Stakgold, 1979) in

the mathematical literature. Usually it involves an expansion similar to (2.129) which is

substituted for the nonlinear dependent variables and then taking the inner product of the

nonlinear system. For some nonlinear operator N and some inner product (*,*), we have


N(y) ) = N Ap(t) l), y) (2.130)

For us, N is the nonlinear operator defined in (2.118) and (2.119), y is equation (2.129)

and Yp* is defined as:

/ = (u, l, Wl,,01,2,V2,W2,02 )

The inner product that will be defined here as:

2na -0 1
(ypv> J Jr f ilpdz+ Y2p 2pdz drdep
0 0 1-1 0




52

Substituting (2.118), (2.119), (2.129) and yp into (2.130) gives:

r{Kv ** / V2 /+ M {O* aO/+I P 2
Pr-1 It 2p a M lp at 2p at

v ,( v,- VI ))+ (v ,(v2V2 vV2))+

Mj{K; ,(v2ei.,))Ko ( ,,[cKV2 ))}I

Rv ,9zli +( V 2,
Pr,-{v*,vI .vv + v ,v2 *Vv2 + ,VI .v1 )+ ,V2 V02
(2.131)
We now let Q represent the linear terms in equation (2.131)

Q= (vV2V 1 VI)+ V2 V2
M v',(vP201+v ))+v* V 2,(V2v 0v V)2

M p 2p 02 + k W2 ) + R Vli) + vO ,2}
(2.132)
Before continuing, it can be shown that the pth eigenvalue of the adjoint problem is
equal to the pth eigenvalue of the linear problem, that is Rp* = Rp. Upon simplification
and substitution ofRp = Rp and equation (2.129), Q is simplified to:
M-Mp ( \ 1 / \
Q= M ii W, ( + w 2q Aq (2.133)

The number of terms in (2.131) may seem a bit daunting, particularly the nonlinear
terms. However, we can use the orthogonality conditions of the trigonometric functions
and the Bessel functions to simplify matters considerably. These orthogonality conditions
give us:





53


v pViq)= ( ,)= (,wq)=0 for peq and i=1,2 (2.134)

The number of nonlinear terms can be dramatically simplified by the following

formula.

27
2fi(my)f2)f( f3(p(p)d = O for (m +n+p)(m+n-p)(m-n+p)(m-n-p) 0
0

(2.135)

wherefl(x),f2(x), andf3(x) can be either sin(x) or cos(x).

With the orthogonality condition of (2.134), we substitute equation (2.129) and

(2.133) into (2.131). The ordinary differential equation of Ap can be neatly written as:

dAp
Ap d = sPPAP Y7pqrAq Ar (2.136)
q r

where:

a, p Pr-' {vp,Vl)p) V(,,V2p)}l)+( plo2p) p,02) (2.137)


PP w,e + w2p,)p (2.138)

M- M
Mp (2.139)


Spqr =r-1 {V p,Viq VVlr) + (Vp,V2q *VV2r)}+(OP,Vlq VOIr)+-(2p,V2q VO2r)

(2.140)

The parameter, Ep, is called the supercriticality parameter because it represents the

degree to which mode p has become supercritical (if at all). Equation (2.136) represents





54

the dynamic behavior of the infinite number of linear modes. To make this problem more

tractable, we need to decrease the number of amplitudes, Ap, to some finite set.

In order to determine whichp's to keep and which p's to ignore, we need to look at

the supercriticality parameter, sp. The value of ep is determined by the eigenvalues, Mp

or Rp, noting that Mp and Rp are interchangeable through the relationship Mp = FRp.

Assume we performed a linear calculation and found a large number of Mp, many more

than what we would use in the weakly nonlinear analysis. This finite set of Mp's is

called S.

The critical eigenvalue is defined as the smallest element in the set S.

M,=min(M,) VpcS (2.141)

We next define the parameter 6p, which is a measure of how stable the pth mode is

with respect to the critical mode, Mc.

M,-M,
P M- VpC S (2.142)
MP

Note that the value of 8p lies between 0 < 8p < -1. The parameter 6p is then used to

group the set S into three distinct sets: Su, Ss, and S {SU USs }. Su represents what will

be called the unstable set and Ss represents the stable set. The elements which do not fall

into any of these two categories are ignored. The two sets are defined as follows:

Su p 8p

Ss ,= I6{ p p < sP, -Su}} (2.144)





55

where 4u and s are arbitrary values (for example, C~ = 0.1 and s = 0.5). The set Su will

always contain at least one element, Mc, and may contain more. The number of elements

in Ss is determined by the cut-off value, 4s. This value is definitely not fixed, and will be

determined more by the experiments than by some numerical value. For example, in the

next section, we will see experiments where the modes (m = 2, n = 1), (m = 1, n = 1)

interact with the critical mode (m = 0, n = 2). Therefore, we would choose s such that

only the modes (m = 2, n = 1) and (m = 1, n = 1) are in the set Ss.

The set of equations (2.136) is further simplified by the procedure known as the

adiabatic reduction of the slave modes (Manneville, 1990). Pick a mode s which is an

element of the set Ss. If we assumed that we are only slightly nonlinear, M = Mc, then

the supercriticality parameter of the stable mode s is Es < 0 and <<1. Now suppose

that some finite disturbance causes the amplitude As to become non-zero. Initially, the

nonlinear terms involving As contributes little to the unstable modes. Additionally, the

linear contribution will not contribute significantly to the dynamic behavior of the

unstable modes, except for short times. Therefore we assume:

d =0 and N(As,A,)=0 Vs Ss, pe {SuUSs}
dt

When these two assumptions are made, equation (2.136) for all of the modes in the

set Ss, becomes:


A, =-E"7YsP ApAq Vs Ss and p,q Su (2.145)
P q Es

Substituting equation (2.145) into (2.136) gives:






56


dA p
p dt -- 1 pAp-l pqryAqAgr-1 Z~- ,s psr, pAqAr
q r s q r s
Vp,q,r eSU and s= Ss

(2.146)

Before leaving this chapter, I would like to clarify an often neglected topic in this

method. The topic involves the determination of Su and Ss. This determination is often

called the normal mode reduction. For simplicity, assume the eigenvalue in our linear

problem is the growth rate cp. Again, we use the index p in the same context as before

and we order the infinite set of cp from the largest (least stable) to the smallest (most

stable).

Here, R (or M) is just a parameter, and for every parameter, R, we determine a

unique set of Op's. Suppose we pick some R and perform our linear calculation to

determine the set of op. In general, = C p,r + iC p,i where op,r is the real part of ap

and p, i is the imaginary part of op. Now place each op into three groups. The first

group, called the unstable manifold, is for all op where cp,r > 0. The second group,

called the center manifold, is for all op where up,r = 0, or more generally

- C < p, < The last group is called the stable manifold where p, r < 0.

The next logical step, one would assume, would be to let all elements in the

unstable manifold to be in the set Su and let all elements in the center manifold to be in

the set Ss. The elements in the stable manifold are completely ignored. Following this

classification procedure, we would perform the adiabatic reduction of the elements in Ss

and arrive at equation (2.146). However, there is one important assumption that is made

in this technique; and that is whether the linear model is still valid for op > 0. If we go





57

back to the derivation of the linear model, we note that the linear model is only valid until

the first critical mode becomes unstable. The only thing we can say for certain is that this

unstable mode (or modes) grows exponentially, at least initially. Beyond this, we can not

guarantee that the linear model is still valid. Therefore, when a value of the Rayleigh

number, larger than the critical value, is used (R > Rc), the eigenvalues, (e.g. up) and

eigenfunctions (e.g. W,(z)), may be meaningless.

This argument is perhaps best explained in physical terms. In the course of

conducting a convection experiment, a temperature difference is applied. When this

temperature difference is less than the critical temperature difference necessary for the

onset of convection, the fluid is in a thermally conductive state. When the temperature

difference reaches its critical value, the fluid begins to flow. The linear model will

accurately predict the critical temperature difference and can tell what the flow will look

like at the onset of convection. Once the temperature difference increases beyond the

critical value, the linear model can not predict how the flow will interact with itself and

other flow patterns. This is most pronounced near codimension-two points.













CHAPTER 3
EXPERIMENTAL APPARATUS AND PROCEDURE




The objectives of the experiments were to observe the behavior of the fluid

convection at codimension-two points aspect ratios where two flow patterns coexist -

and at fluid depths where the initiation of convection switches from the lower layer to the

upper layer. To accomplish this, the test section was designed so that cylindrical inserts,

with different radii and heights, could be used interchangeably. The popular bilayer

system of silicone oil and air was chosen to both simplify the experiments and to generate

results which can be compared to previous experiments (Koschmieder and Biggerstaff,

1986). In order to cause buoyancy-driven convection in one or both fluids, the bilayer


IR camera Electronic Hardware Unit






TV and VCR
/1\\\




Test Section
Computer J p~]


Figure 3-1. Overall schematic of the experimental apparatus

58





59


system needed to be heated from below or cooled from above. The vertical temperature

difference applied across the test section was accurately monitored and controlled using a

computerized data acquisition system. When the temperature difference was large

enough to cause convection in the silicone oil, the change in the temperature field at the

silicone oil-air interface was detected using an infrared camera.

This chapter is split up into two parts, a description of the experimental apparatus

and a walk-through of the procedures taken when an experiment was performed. The

experimental apparatus is divided into four major parts: the infrared imaging system, the

test section, the heating and cooling system, and the electronic hardware unit which

includes a computer-controlled data acquisition program. For a very thorough

explanation of all aspects of the experimental apparatus, an instruction manual was

designed. This manual can be presented upon request.




3.1. APPARATUS


3.1.1. Infrared Imaging System

One of the first decisions that needed to be made was the type of flow visualization

technique to use. Predominantly, two different methods were well known at the time,

shadowgraphy and particle seeding. While both of theses methods were well-established,

each had its draw-back.

The shadowgraphic technique involved shining parallel light through the layer of

fluid, reflecting the light off a mirror at the bottom of the test section, then shining the





60



a b










Figure 3-2. An example of a shadowgraph picture (a) and an aluminum particle
experiment

reflected light onto a white background. The pattern that appears on the white

background works by the following principle. When a fluid is heated, its density

changes. The change in density causes a variation in the index of refraction of the fluid.

Therefore, as parallel light shines through the fluid, the light is either concentrated or

reflected. This concentration or reflection of light gives the flow pattern of the fluid. The

interested reader is referred to the references, Eckert and Goldstein (1976), Goldstein

(1983) and Koschmieder (1993).

In particle seeding, plate-like, reflective particles are added to the fluid of interest.

When the fluid flows horizontally, the particles lie flat and reflect light, causing the fluid

to appear brighter. When the fluid flows in the vertical direction, the thin side of the

particles point up and little light is reflected, causing the fluid to appear darker. From

this, the flow pattern can be observed.

Both of these methods, however, have their flaws. In shadowgraphy, the light has

to pass through the fluid twice before it appears on the white background. This

effectively averages the temperature throughout the fluid. This can cause a misguided

analysis of the flow pattern, particularly if most of the flow appears at the surface. In






61


particle seeding, the particles are often much denser than the fluid and quickly settle

during the course of the experiment. This was of particular concern as most of the

experiments lasted several hours to several days. Secondly, the addition of the particles

can unpredictably change the thermophysical properties of the fluid.

Another method which has previously received some attention is particle image

velocimetry (PIV) (Adria, 1991; Pline et al., 1991). Here again, particles are added to the

fluid. In this method, most of the particles are individually tracked every given time

period. From this, the velocity of the fluid at a point can be determined. Because the

software needed to calculate the numerous velocity vectors was still in its infancy, we

decided to forego this option.

The method that we chose to visualize the flow patterns, was an infrared (IR)

camera. The IR camera has two major advantageous. The first is that it is a non-intrusive

method of visualizing the flow, and secondly, it can be used with opaque fluids. The

ability to use the camera with opaque fluids is of particular interest in the application of

crystal growth, where opaque liquid metals are used. The IR camera that we used was an

Inframetrics model 760.

The basic feature of the IR camera is a single Mercury-Cadmium-Telluride

(HgxCdi.xTe) chip which measures infrared radiation in the 3 to 12 tm and the 8 to 12 tm

range. Each wavelength corresponds to an optimum temperature range. For our

experiments, the typical temperature range was around 25C to 450C. This temperature

range is best measured using the 8 to 12 pm wavelength. The 3 to 12 upm wavelength

works best for higher temperatures (around 1000C to 5000C). Additionally, the





62


transmission of infrared radiation through the atmosphere, is much better in the long

wavelength, 8 to 14 utm region. The field of view (FOV) is detected by scanners using

electromechanical servos (galvanometers), much the same as a standard television set.

For this camera, the FOV resolution is 640 X 480 pixels. Every other line (320 X 480

frame) is sampled 60 times every second, giving a full interlaced picture (640 X 480

frame) 30 times every second. This is the NTSC standard.

The accuracy of the temperature measurement is 0.20C for each pixel. The

majority of this error is due to noise, caused by random emissions of photons.

Fortunately, the camera has a built in feature which allows each pixel to be averaged over

2, 4, 8, or 16 frames. When the picture is averaged over 16 frames the error in each

measurement drops to 0.050C. For all of the experiments performed, the 16 frame

averager was used.

Additional features in the camera were used to improve the picture quality. The

simplest of which was a 3X magnification and a 12" close-up lens constructed of

magnesium. For each lens, the transmission of IR radiation of 3 to 12 pm is 99%. The

second feature, which is built into this model of IR camera, is the temperature "window".

The temperature window is the maximum temperature difference which the camera can

detect. The Inframetrics model 760 has temperature ranges of 2, 5, 10, 20, 50, and 1000C,

although only the 2C and 5C window were used. By selecting the minimum

temperature window, the temperature resolution can be maximized.

The infrared image, which is eventually seen, is a false-color image of the

temperature field. Every 1/30"' of a second, a 640 X 480 frame of IR radiation is






63


measured. The IR radiation is converted to a temperature by a value of the emissivity,

which must be entered by the user. The temperature is then mapped to one of several

available color palettes. For these experiments, the effective emissivity was found as

follows. The test section was constructed as if an experiment were to take place. The test

section was filled with silicone oil and allowed to equilibrate to room temperature. The

temperature was then accurately measured with a mercury thermometer. The emissivity

was changed until the reported temperature of the camera matched the temperature of the

thermometer.


3.1.2. Test Section

It was decided that the design of the test section should follow closely the design of



Table 3-1. Table of the thermophysical properties of the material and fluids used in the
experiment.
Parameter Units Dow Air (20C) Zinc Copper Lucite
Corning Oil Selenide

Density (g cm-3) 0.968 0.0012 5.27 5.96 1.19
Negative Thermal Expansion 9.6 33.3 0.078 0.501 7.3
(104 C-1)
Thermal Conductivity 1.59 0.262 180 4010 1.7
(10-4erg cm-' sec-' OC) -
Thermal Diffusivity 1.10 182 66.3 1160 -------
(103 cm2 sec-')
Kinematic Viscosity 0.692 0.157 ------- ------- -------
(stokes)
Interfacial Tension 20.9 ------- ------- ------- -------
(dyne cm-') (under air)
Negative Interfacial Tension 5.8 ------- ------- ------- -------
gradient (under air)
(10-2 dyne cm-' oC-)





64

previous experiments (Koschmieder and Biggerstaff, 1986; Koschmieder and Prahl,

1990). It was important to choose the two fluids and the material properties such that

they satisfied the assumptions made in the theoretical analysis. The theory did require

two conditions on the fluid boundaries, though. First, the upper plate at the top of the

upper fluid and the lower plate below the lower fluid, should be rigid and a perfect

conductor. For this, zinc selenide and copper respectively, with their high thermal

conductivity, were chosen. Secondly, the radial gradients should be minimized through

the side-walls of the cylinder. The material chosen here was lucite, whose thermal

conductivity is close to that of silicone oil. However, the lucite walls were thick enough

(greater than 3/8") so that any temperature perturbation from the outer walls was

minimized. The values of all known thermophysical properties are given in Table 3-1.

The two fluids chosen were a high viscosity, Dow Coring silicone oil with

nominal viscosities of 100 cS or 200 cS, and air. Other choices of gases, such as helium,

were discarded due to the difficulty of containing leaks. A second liquid layer was also

felt unnecessary as many unanswered questions were left for this simpler bilayer system

of a liquid and a gas.

The test section itself consisted of five separate pieces: a lower heating bath, a

liquid insert, an air insert, a clamp and an upper plate consisting of zinc selenide. All

pieces of the test section, except for the zinc selenide window, were constructed from

lucite. To maintain a flat, silicone oil-air interface, the liquid insert contained a "pinning

edge" and a reservoir (Figure 3-3). If additional silicone oil were added, the oil would

spill over into the reservoir. The pinning edge would then eliminate any menisci, keeping

the interface flat. After carefully filling the liquid insert with silicone oil, the air insert





65


Zinc Selenide Lens

I -i Lucite Clamp


air height
air height Air Height Insert

Pinning Edge

Sliquid height Liquid Insert

Heating Block


Figure 3-3. Cross-sectional view of the test section.


could be placed on top of the liquid insert. The clamp would fit on top of the air insert

and four screws, which ran through the clamp and into the lower heating bath, were

tightened to hold down the liquid and air inserts. The zinc selenide window was then

placed into a groove in the clamp. The clamp and lower bath were constructed so that

different liquid an air inserts could be used. Further details and drawings of the test

section are included in appendix B. Complete details of the experiment can be found in

the experiment instruction manual.

One of the most important considerations in designing the experiment was how well

the applied temperature difference across the oil-air bilayer, could be maintained. This

consideration was what led to the rather complicated design of the lower heating bath.

The lower heating bath consists of many parts but is primarily a continuously stirred, hot

water bath heated from below by an electric heater. The walls of the cylindrical bath are

constructed of 3" lucite. The top and bottom of the cylinder are capped with 3/16" thick

copper disks. Water and one magnetic stir bar are placed inside the bath. The bath is then





66

placed on top of a three inch diameter 2.5 W/in2 flexible heater. The heater is turn is

placed on top of a magnetic stirrer, which itself sits on a leveling plate.

The largest difficulty of the lower bath was preventing air bubbles from forming in

the bath, which was eventually eliminated by constructing an over-spill port. The over-

spill port was simply a hole in the side of the bath where a tube was inserted. The bath

would be over-filled with water such that additional water would spill out through the

over-spill port, and the end of the tube raised above the top of the bath. When the water

was heated and subsequently expanded, the excess water would flow into the tube. When

the bath cooled back down, the water in the tube would flow back into the bath, thus

eliminating any air bubbles. Without the over-spill port, the expansion of the water

would create too much pressure on the bath and the bath would eventually crack.

In addition to its availability, water has a high heat capacity, which makes it ideal

for temperature control. Although it takes longer to heat water to a certain temperature,

the high heat capacity will hold the temperature constant longer making it easier to

control. The stirring of the water by the magnetic stirrer helped to prevent any

temperature gradients from forming.

Because of the infrared imaging system, the top of the test section needed to be

heated by an infrared transparent medium. Here again, the simple choice of air was

made. The requirement of an infrared transparent material also dictated the use of the

5mm zinc selenide window. Zinc selenide has a high thermal conductivity and is greater

than 60% transmittive to infrared radiation between 0.7[tm and about 17um. Zinc

selenide is also slightly reflective to radiation in the 8 to 12 utm range. This reflection





67


caused problems with the imaging of the silicone oil-air interface. This problem was

resolved by coating the zinc selenide window with an anti-reflective polymer, which was

performed by II-VI incorporated.




3.1.3. Heating and Cooling System

The heating and cooling units consist of three parts: the electric heater for the lower

bath, the electric heater used for the upper plate and ambient air, and the cooling water

used to cool the ambient air. The objectives of the heating and cooling units were to add

or remove heat, when necessary, in order to maintain the constant temperature.

The entire test section, IR camera and air heating unit were enclosed in a clear,

lucite box. The temperature of the air inside the box was monitored by one of the

thermistors and controlled by an electric heater (hair dryer). The air was then stirred by a

fan to prevent temperature gradients from forming. Additionally, a radiator, in which

chilled water was pumped through, was used to continuously remove heat from inside the

box. This prevented the temperature of the air from becoming too high. The chilled

water was also kept at a constant temperature. There were two reasons why the test

section and IR camera were enclosed by the lucite box and the air inside kept at a

constant temperature. First, keeping the temperature exterior to the sides of the liquid and

air inserts constant, minimized heat from flowing through the side-walls. Secondly, the

absorption of infrared radiation by the atmosphere changes as a function of temperature.

To prevent any fluctuations in the transmission of infrared radiation between the IR

camera and the test section, the air was kept at a constant temperature.





68


Air, unlike water, has a low heat capacity. This caused difficulties in controlling the

upper temperature and could not be controlled as well as the lower bath's temperature

was controlled. The deviation of the upper temperature from its setpoint created the

largest error in the overall temperature control. Nonetheless, the overall temperature

control was quite good, with a standard deviation of 0.20C overall. We note here that

the lower temperature is read from the bottom of the lower copper plate and the upper

temperature is read from the top of the zinc selenide window. Due to the high thermal

conductivity of the copper and zinc selenide, small temperature perturbations occurring at

the top of the zinc selenide and the bottom of the copper plate, would be smoothed out

before they reached either of the two fluids. For this reason, the actual temperature across

the two fluids was probably even better than is reported here.




3.1.4. Electronic Hardware Unit

The objective of the electronic hardware was to link the temperature readings to the

computer-controlled program, and then transmit control decisions from the computer

program to the heaters. Additionally, the computer would control at which times the

VCR recorded the infrared images. For this experiment, three different temperatures

needed to be maintained at a constant setpoint: the lower bath, the temperature difference

across the bilayer of fluid, and the cooling water temperature. The temperatures were

then reported to the computer where the program would read the temperatures, and based

upon a given control algorithm, determine whether any heaters should be turned on or off.





69

Each of the temperatures were measured using a highly accurate thermistor. The

types of thermistors used were Omega, linear response, model OL-700 series,

thermistors. The thermistor located in the lower heating bath was a water-proof small

surface thermistor, the thermistor located on top of the zinc selenide plate was an

attachable surface mounted thermistor and the thermistor located in the cooling water

tank was a general purpose, water-proof thermistor. A thermistor is a temperature-

sensitive electrical resistor. As the temperature changes, the amount of resistance

changes thus changing the voltage drop across the thermistor leads. This voltage drop

can then be calibrated for a given range of temperatures. The thermistor was chosen over

the cheaper and more available thermocouple because the thermistor was more accurate

and the calibrations did not "drift" over time. This last feature is important as some of

the experiments could last up to three days. A specially designed, constant, 0.5V power

supply was applied across each thermistor. The resulting voltage drop across each

thermistor and the 0.5V from the power supply were then read into the computer through

a data acquisition board.

The data acquisition board was a DAS-1601 from Keithley Metrabyte. The DAS-

1601 has 16 analog input channels, with a sampling frequency of 0.1 MHz. Each analog

input channel is converted to a digital number using a 12 bit analog-to-digital converter

(ADC). As the range in voltage is from OV to 10V, the ADC conversion error is + 2.5mV

(10V divided by 212). The computer which housed the data acquisition board and ran the

control program was a PC compatible, Intel 486-66 MHz with an ISA motherboard. Data

was continually read from the data acquisition board by the control program.





70


The control program was written by myself in Visual Basic, version 3.0 and ran

under Microsoft Windows 3.1. Major revisions to the program were later performed by

Bryon Stakpole. The program read in the input (temperatures), and based on a

proportional-integral-derivative (PID) control algorithm, determined the value of the

output (whether to turn the heaters on or off). The parameters used in the PID control

algorithm were taking from Seborg et al. (1989). The temperature readings, setpoints,

output values, as well as other relevant data were displayed on the computer monitor.

The temperature readings and setpoints were recorded to a data file on the computer's

hard drive. A flow chart for the programming logic, is given in Figure 3-4.

After determining which heater should be turned on or off, the control program

would write the necessary data to the data acquisition board's output register. The board

would then send the digital signal to the electronic hardware unit. Inside the electronic

hardware unit were several circuit boards. Each circuit board consisted of several

channels which read each individual bit from the data acquisition boards register. If the

bit was on, the channel would trigger a transistor. Electricity would then flow through

the transistor to a sold state relay (SSR). When electricity flowed through the coil side of

the relay, the relay would close and allow electricity, at a higher current, to flow through

the other side of the SSR and into the heaters. This process of reading the temperatures,

performing a control decision and turning on the heaters (if necessary) was continually

performed as fast as the computer could execute the control program.

One of the automated features of the program was to change the temperature

difference (setpoint), after a given period of time. The duration of each setpoint called

a segment usually lasted two to four hours, and there were always several segments in








71



No-


S Initialization F
SStart Program -- Information --Yes- Overwrite?
Form 'Exists?
Form

No
Yes

S--== Set Up _
Add, Edit & Delete Data File
Fluid Parameters




Initialize Data
Acquisition Board





.-- ---- IR Program Form n---
Pause


Start/Pause Yes
Loop?


Start

Pause
Anag Pra --No-----. H-eaterControl
S Analog Program?
Data In

SNo On Off

Change
Seent Yes-- Segment Send Digital
Statistics SNumbee Over? Output To Heaters
(Data Averaging) -

SNo

El -- -u Ndw0
Experiment Shutdown Write Data Yes Writs Data
Update Screen Over? es Flag To File?
Variables

No -- No No


I std wn s Overheating? ,-No- VCR Control --Record- -

Prepare Prepare
X-Axis Y-Axis
Data Data
Pause -
SiSend Digital
--. -----. Update Graph Control Decision Output ToVCR




Shut Down
Experiment?


Yes


Send Digital Exit Program
Shutdown Signal Et Prog





Figure 3-4. Flow chart of the programming logic





72


each experiment conducted. For this reason, it was advantageous to automate the entire

experiment.

As was mentioned in the beginning of this chapter, the objective of the experiment

was to record the flow pattern at the onset of convection. The flow pattern was detected

with the IR camera and the image was sent to a VCR. Because of the duration of each

experiment, the VCR could not continuously record the IR images, for it would exceed

the limit of the VCR tape. To work around this, the VCR was controlled by the

computer. Every two minutes, the program would tell the VCR to record and after five

seconds, the program would send a signal to pause the VCR. As the fluid flow was very

slow, this interval would not miss any dynamic or transient behavior. Controlling the

VCR allowed an entire experiment to be conducted without any intervention, sometimes

over-night. At the end of the experiment, the program would shut off all power to the

experiment, including the computer.




3.2. PROCEDURE

This section will list, in chronological order, the procedures that were performed in

order to properly conduct an experiment.

For the sake of efficiency, the first operation was to turn on the IR camera. The

reason is that the infrared detector must operate at temperatures lower than 77 K. To

reach these temperatures, the IR camera has a built-in Sterling pump which removes heat.

This process usually took a couple of minutes. When the temperature was below the

minimum operating temperature, the pump slowed down. As the pump made a detectable






73


amount of noise, this shift in pump speed was an indicator that the IR detector could be


turned on and used to detect infrared images. Once the IR camera was running, it was


usually a good idea to check all of the operating parameters of the IR camera, to ensure


they are all correct.


The next step in the procedure was to load the test section. First, the liquid and air


insert of interest were chosen and screwed down with the lucite clamp. As the clamp had


a hole in the center for the zinc selenide window, the silicone oil could be added from the


top. Silicone oil (or whatever other fluid was being used) was added until it looked like




rogra Parameters V
Dat leNwle: [k] N:umber oSegments 7 | icTime
Numbr o (seh)
1 i r It o .. : i, I i, p-i Segment Time (min) I nterv l between
re Write to Date File Time (sec): 1 in 1.Ic... : f
..... .. recordings (se ) ,
PhysicalJ Parameters' Set Points
...Name Lowe Fp-r \u LowerBath Overall Temp- Difference .Water Tank
: :ate\ 1 Temperature (Lower Bath Upper Plate) Temperture.
Depth (cm .2 3 5 t 22
.2 355 5 5 Increment 22

Viscosity E-020.157 37 7 22
I :cm s)1 ,
erm 1.43E-03 2 Lower Fluid pper Fluid
DiffusiviIty 0.12 Temp. Diff. Temp Diff.
I 13 57 iGraph Length
Thermal 2 08E-04 3.33E-03 ,Time sec
'Expansion E03


Gain T1.ul TauD
2.00 ASpec t 1 0. 0


rDensity 1 9120E-03 -5 T ei tlizte o r

Cmments Bang Bang 4 [.-
o PI control [1i4--- -B1BE
PiD control





Figure 3-5. The initialization program window.





74


the section was full and the interface was flat. Conveniently, the flatness of the interface

could be checked using the IR camera. The reason is that silicone oil reflects a certain

amount of infrared radiation, much as the zinc selenide does. When the interface was not

flat, the silicone oil interface acted as a lens to infrared radiation. When the interface was

depressed in the center, IR radiation was concentrated and the center appeared warmer

than the edges, even though all of the silicone oil was at one, constant temperature.

When there was too much silicone oil, the interface was elevated at the center, dispersing

IR radiation, which made the interface appear cooler. This method of detecting the

flatness of the interface was very sensitive to the addition of even small amounts of

silicone oil.

Once the test section was filled, the level of the test section was checked and

adjusted, if necessary, with a leveling plate, which the test section sat on. The TV and

VCR were then turned on. To obtain the best image possible, the proper magnification

and close-up lens should be used such that the silicone oil-air interface filled most of the

TV screen. The focus of the IR camera was then adjusted to get a sharp picture. The

power to the computer and the electronic hardware unit were then turned on. This began,

among other things, the magnetic stirrer. The bath was then checked to ensure no air

bubbles have formed. If there were bubbles, water was added and the bubbles forced out

through the over-spill tube.






75



SInput Output: -- --i Parameters:
Voltage Temperature Set Point LowerBath Exp.Started: 11:59:55 am
Lower B 30.16 3000 Upper Plate 0 Exp. Ends: 6:59:55 pm
Upper Plate 377 15.1 30.WaerTank I Segment Began: :5:55 am
WaterTank 1. 22.00ment 12:59:55pm
Reference Time : 0 Segment Number 1 of 7
Voltage Date: VCR Status: Paused
Rayleigh Temperature Set Point Control Type: PID
Number Difference
Lower Fluid 1 29. 1 .472 1 .8Y-axis Adjusiment I
Upper Fluid 43b366 V 3684 LE EIi II IL I;E


Total Temperature Difference
10
8

Temp
Difference 4
2

360 720 1080 1440 1800
-2
Time



Figure 3-6. The Main program window


After the computer powered up, the control initialization program ran. Here, all of

the program parameters, physical parameters of the liquid and air inserts, control

parameters and the setpoints, were entered. A picture of the initialization program is

given in Figure 3-5. Once all of the necessary information was entered, the OK button

was clicked to go to the main program window (see Figure 3-6). Before clicking on the

Start button, all of the wires from the computer to the electronic hardware unit were

checked, and the connections to the thermistors were secured properly. This would avoid

receiving faulty temperature readings which could ruin the experiment. If everything

checked out, press the Start button. The program would begin to run.





76


The first temperature difference that was chosen, should be less than the critical

temperature difference necessary for the onset of convection. If the fluid began to

convect before it reached its first setpoint, then the program should be terminated and

restarted at a lower temperature difference. This was important, as the onset of

convection needed to be approached from the conductive state and the temperature must

be held steady, long enough for the fluid to reach equilibrium. Usually the step size for

each temperature difference was around 0.1 C across the silicone oil layer. Therefore, the

first temperature difference was applied and held constant for several hours. No flow was

observed. The temperature difference was then increased a little and the silicone oil

interface observed, to see if a flow pattern appeared. This was repeated until the

temperature field at the oil-air interface changed into a particular pattern. At this point,

the temperature difference and the flow pattern were recorded.














CHAPTER 4
RESULTS AND DISCUSSION





4.1 INTRODUCTION

In this fourth and final chapter, results from the linear stability analysis, weakly

nonlinear analysis and the experiments will be given. Each of these was used to shed

some light on different phenomena of bilayer convection. All of these events are a

function of the geometric parameters of bilayer convection in cylindrical containers:

aspect ratio (radius/height), the ratio of the fluid depths, and the total depth of both fluid

layers.

The results have been summarized into four major categories. The first topic is the

oscillations, or mode switching that occurs at certain codimension-two points. The second

topic will show that an increase in the air layer can affect, or even cause, fluid convection

in the lower fluid layer. The third topic deals with how the driving force for convection

(either buoyancy or interfacial tension) and the type of convection coupling (either

thermal or viscous) can switch as the radius of the container is increased, even though the

fluid depths are fixed. The fourth part will contain results from a weakly nonlinear

analysis on the effect of air height on bilayer convection.






77






78

4.2 CODIMENSION-TWO POINTS

As was mentioned in the pattern selection section of the Physics and Historical

Perspective chapter, there exists certain aspect ratios where two different flow patterns

can become simultaneously unstable. These aspect ratios are called codimension-two

points. To investigate these points, a series of experiments and linear calculations were

performed (Johnson and Narayanan, 1996). The experimental apparatus and procedures

are described in the Experimental Apparatus and Procedure chapter. The linear

calculations were performed by Zaman and Narayanan (1996).

Table 4-1 gives the critical Marangoni numbers for the azimuthal modes 0, 1, 2, and

3 for two different aspect ratios of 1.5 and 2.5. The Marangoni numbers were calculated

using a three dimensional model of the linearized Boussinesq equations in a cylinder. The

bottom and radial walls of the cylinder were assumed to be rigid, with a no-slip condition.

The gas above the liquid was assumed to be both mechanically and thermally passive.

The bottom of the cylinder was held at a constant temperature, while the radial walls were

assumed to be conductive and the liquid-gas interface was modeled with an effective heat

transfer coefficient. Finally, the liquid surface was assumed to be flat and non-

deformable. This was done to decrease the computational time and difficulty.



Table 4-1. Critical Marangoni number associated with each mode for aspect ratios of 1.5
and 2.5
Mode 1.5 Aspect Ratio 2.5 Aspect Ratio
0 90.45 69.4
1 101.3 70.8
2 112.0 70.4
3 129.8 73.0






79

There are two items of information to be obtained from Table 4-1. First, is that the

mode associated with the smallest Marangoni number, for a fixed aspect ratio, will be the

mode (flow pattern) present at the onset of convection. The second item of information is

the difference between the smallest and the next smallest Marangoni number. For

example, the difference between the first two Marangoni numbers for the 1.5 aspect ratio

is about 12%. The difference between the first mode (m = 0) and the second mode

(m = 2), for the 2.5 aspect ratio, is quite small, about 1.5%. The reason the modes are so

close is because the 2.5 aspect ratio is near a codimension-two point. This difference is

important experimentally when one tries to resolve which flow pattern will be present at

the onset of convection.

The first experiments used 86 cS silicone oil in a 5mm deep liquid insert with a 1.5

aspect ratio. From Table 4-1, the predicted flow pattern is the single toroid (m = 0). The

toroidal flow is depicted as fluid moving up the center of the cylinder, moving radially

across the top, then falling down along the sides. Indeed, as seen in Figure 4-1, the

infrared camera captured this flow pattern at the onset of convection. Further moderate

increases in the temperature difference did not change the flow pattern.

The second set of experiments used an aspect ratio of 2.5, which is close to a




Figure 4-1. An infrared image of the toroidal flow pattern in
a cylindrical container. The picture is taken looking down
onto the oil-air interface.





80

codimension-two point. Again a 5mm deep layer of 86 cS silicone oil was used. At the

onset of convection, a very faint m = 0, double toroidal pattern was seen. This agrees with

Table 4-1. However, when the temperature difference was increased by 0.05C across the

liquid layer, the flow pattern changed from the static double toroid to a dynamic mode

switching behavior. This flow pattern started with an m = 2, bimodal flow (Figure 4-2a).

One convection cell then increased in size forming a pattern resembling a combination of

the m = 1 unicellular flow and an m = 0 single toroidal flow (Figure 4-2b). When this cell

reached some critical size, it split into two cells (Figure 4-2c). Here the flow pattern was

the same as the first bimodal flow pattern rotated be 900 (Figure 4-2d). This process then

repeated itself (Figure 4-2e and 2f), returning to the original bimodal flow pattern.

This process of switching between different flow patterns repeated itself

approximately every twenty minutes. As long as the temperature difference remained

constant, this mode switching continued at a regular interval, although the exact period

















Figure 4-2. Time sequenced infrared images showing the switching between flow
patterns. The convection cells continuously oscillate between the different flow patterns
with a regular time interval as long as the temperature difference across the liquid is held
constant.






81

has never been accurately measured. This experiment was performed several times in a

somewhat sloppy manner and the oscillating behavior was seen every time. This was

done to verify that the oscillating behavior did not just occur for a small parameter range.

Additional experiments were also performed for a 2.6 aspect ratio with an 11.1 mm

air height. Here the flow pattern at the onset of convection was seen as a superposition of

a bimodal, m =2 and a double toroidal pattern. This experiment showed that a

codimension-two point did indeed exist near or at the 2.5 aspect ratio. However, as will

be explained later, the superposition of the two patterns may have been due to convection

in the air.

These set of experiments were able to prove that different linear modes can interact

with each other (and themselves) to yield dynamic nonlinear behavior. A similar

observation was seen for Rayleigh-Marangoni convection in square containers

(Ondarguhu et al., 1993). Although this work mentioned that the oscillating behavior was

a result of a Takens-Bogdanov (Golubitsky et al., 1988) bifurcation, which is associated

with codimension-two points, they did not prove that it was indeed a codimension-two

point. Secondly, the oscillations only occurred well into the supercritical region.

Codimension-two points were also studied in pure buoyancy flows (Zhao et al., 1995).

However, no oscillating behavior was seen for any of the aspect ratios investigated. From

these experiments, it appears that the free surface has something to do with the dynamic

behavior.

Several theoretical works describe weakly nonlinear behavior near codimension-two

points. Erneux and Reiss (1983) looked at supercritical bifurcations of two degenerate

eigenvalues (i.e. codimension-two points). They noted that when the supercritical






82

bifurcation was symmetric, and no imperfection was introduced, the steady solutions

would branch off into a steady secondary solution as the bifurcation parameter was

increased. However, when an imperfection to the base state was introduced, Hopf

bifurcations to a secondary solution were possible. This result could imply that the free

surface in Rayleigh-Marangoni convection acts to break the symmetry of the problem.

Rosenblat et al. (1982a) performed a weakly nonlinear analysis for the pure

Marangoni problem, neglected buoyancy effects. In their analysis they showed that for an

m = 1, m = 2 codimension-two point, it was possible for secondary Hopf bifurcations to

occur for aspect ratios slightly greater than the codimension-two point. However for the

m = 2, m = 0 codimension-two point, they did not find any Hopf bifurcations except for

small Prandtl numbers (less than 10). It is important to note the many differences between

their paper and the physical experiment. The most important being the lack of

gravitational effects and the assumption of an unphysical, vorticity-free side-wall

boundary condition. This latter condition will cause the modes to occur in a different

order than what is observed in the experiment. For example, the vorticity-free side-wall

condition generates m = 1, then m = 2, then m = 0 modes as the aspect ratio is increased,

whereas the no-slip side walls (Zaman and Narayanan, 1996; Dauby et al., 1997)

generates m = 1, then m = 0, then m = 2 modes as the aspect ratio is increased. Therefore

the no-slip side-walls will not have the m = 1, m = 2 codimension-two point.

Nonetheless, these theoretical works give qualitative evidence that the oscillations

seen in the experiments for the 2.5 aspect ratio, is a result of linear modes interacting.

Further verification of the experiments would require a linear calculation using no-slip

boundary conditions. The eigenfunctions from these calculations could then be used in






83


the nonlinear amplitude equations (equation 2.136). Here the three modes m = 0, m = 1,

and m = 2 would need to be simultaneously considered. In other words, these three modes

would need to be in the unstable set Su (see page 54). Such an analysis has been

conducted by Dauby et al. (1997), except in their paper only rectangular containers were

considered.

In these set of experiments, the existence of a codimension-two point was shown

definitively by observing two different flow patterns for aspect ratios near each other (2.5

and 2.6). Upon a slight increase in the temperature at the 2.5 aspect ratio, a dynamic

nonlinear interaction occurred. A qualitative explanation of this behavior is given by the

weakly nonlinear analysis of Rosenblat et al. (1982a).

The discovery of oscillating Rayleigh-Marangoni convection in cylindrical

containers at codimension-two points is important in the application of crystal growth.

The unsteady convection can lead to dislocations in the crystal or dopant stratifications,

both of which would yield a lower quality crystal. By understanding the existence of

codimension-two points, these particular aspect ratios could be avoided to improve the

crystal growth process.




4.3 EFFECTS OF AIR HEIGHT ON BILAYER CONVECTION

In all of the previous experiments and calculations performed to study convection in

a silicone-air system, the effects of air gave been neglected. It was thought that because

air had such a low viscosity, that any motion in the air would give negligible effect on the

convection in the silicone oil (or any fluid for that matter). In this section, evidence will






84

be given to show that convection in the air does indeed affect the convection in the

silicone oil. This statement is backed up be calculations from the unbounded linear model

from the Mathematical Modeling chapter, by bounded calculations for a single fluid

layer, and by several experiments.


4.3.1 Observations from calculations

Calculations were performed to determine the flow pattern at the onset of

convection. These computations involved linearized instability analysis for both laterally

unbounded as well as bounded geometries. The calculations assuming layers of

unbounded lateral extent were done in order to obtain qualitative features of the physics

of bilayer convection. Three features in particular were investigated. First, the effect of

the upper phase on the heat transfer resistance was studied. This was done by assuming

that the upper phase was either strictly passive, one that allowed thermal perturbations or

one that was both mechanically and thermally active. In each case, the effect of the air

height on the heat transfer resistance was established. The second feature that was

examined in the laterally unbounded geometry was the effect of the air height on the type

of convective coupling, thermal or mechanical. The third feature that was studied was the

effect of periodic lateral boundary conditions. This was done by imposing physically

unrealistic conditions on the side walls of the fluid bilayers. Thus the effect of side-walls

was obtained in bilayer convection at the expense of using unrealistic conditions. The

imposition of realistic no-slip conditions on the lateral walls for fluid bilayers with a

deflecting interface results in a complicated numerical computation. Consequently, the

laterally bounded layer model with no-slip rigid side-walls assumed a passive upper





85

phase and a non-deforming interface. All of the calculations used properties pertaining to

the silicone oil-air system as these were the fluids that were used in the experiments.

Turning to the first feature of the unbounded, bilayer calculations, three different

conditions of the heat transfer resistance in the air layer were considered. The first

condition assumes that the Biot number is constant and does not vary with the wave

number. The second condition assumes that the Biot number is a function of the wave

number, as demonstrated in the paper by Normand et al. (1977). This is equivalent to

allowing the air to have perturbations in its temperature profile, yet remain mechanically

passive. The third condition is reflected by a full bilayer calculation. In the third

condition, the air is allowed to convect and therefore includes both thermal and

mechanical perturbations.

The calculations using the constant Biot number were similar to those found in

Nield's paper, except here the surface was allowed to deflect. Despite this difference, the

results from these calculations are in close agreement with Nield's results. The reason for

this is the surface tension of silicone oil and air is quite large, therefore, the surface

deflections are small and contribute little to the critical Marangoni number. Table 4-2

gives a comparison of the results using the three different conditions on the heat transfer

resistance, with results from Nield's work.

Before examining the table of Rayleigh numbers, we pause to make a few

comments on the various assumptions of the air layer. Assuming that the Biot number is

constant is tantamount to pretending that the upper gas phase is truly passive and that no

perturbations, either thermal or mechanical are allowed. Consequently, the Biot number

is:






86


Bi = ka dl (4.1)
k,,il dair

where, kair is the thermal conductivity of air, koil is the thermal conductivity of the

silicone oil and dair and doil are the depths of the air and silicone oil, respectively. We

have observed earlier that a Biot number that changes with the wave number is equivalent

to letting only thermal perturbations in the gas phase. It is derived from the equations for

the bilayer given in the Mathematical Modeling chapter, where the velocity, pressure and

surface deflection perturbations are neglected. The temperature conditions for the lower

liquid layer, at the interface, are replaced by:

DO, + Bi1O = 0 (4.2)

with the Biot number as.


Bi(c) = "-o cotho d (4.3)
koil d,,,,

Note that the constant Biot number given in (4.1) which is used by several earlier

workers (Nield, 1964; Koschmieder, 1990) can be obtained from (4.3) by taking the limit

as the wave number goes to zero (the long wavelength assumption). The details of the

calculations when the upper layer is considered active, have been given earlier. A

comparison of the computed critical Rayleigh number and critical wave number for the

various cases produces some insight into the physics of the problem. Table 4-2 gives a

comparison of the critical Rayleigh number and the associated critical wave number, for

various air depths.





87

Two important points can be made from Table 4-2. First, the critical Rayleigh

number when the Biot number is given by (4.3), is always greater than the critical

Rayleigh number for the long wavelength Biot number (4.1). This is understandable as

the Biot number given in (4.3) is always greater than the Biot number given in (4.1). A

larger Biot number corresponds to a more conductive air layer, which more easily

dampens the perturbations. The critical wave number, however, differs very little between

these two cases. For small air heights, the critical Rayleigh number for the bilayer



Table 4-2. Critical Rayleigh number and wave number using four different conditions: a
single layer with equation (4.1) as the Biot number (Nield's Model), a single layer with a
deflecting interface using equation (4.1), a single layer with a deflecting interface using
equation (4.3) as the Biot number, and a bilayer calculation. The Rayleigh numbers of the
silicone oil and the air are defined with respect to their own thermophysical properties. In
each calculation, 4.2 mm of 100 cS silicone oil was assumed. The wave number of the
active air calculations is the same as the silicone oil.
Air Height Rayleigh Rayleigh Rayleigh Active Active
(mm) Number from Number using Number using Bilayer: Bilayer:
Nield's Model Bi from Bi from Rayleigh Rayleigh
equation (4.1) equation (4.3) number for number for
Silicone oil Air
0.1 513.3 514.2 514.5 526.4 1.11*10-5
(o_ = 2.55) (co = 2.55) (to = 2.55) (co = 2.57)
1 237.6 237.8 241.7 243.4 5.14*10-2
(_ = 2.18) (o = 2.18) (o = 2.16) (o = 2.16)
3 205.4 205.5 216.2 217.4 3.71
(o = 2.07) (( = 2.05) (co = 2.05) (o = 2.05)
5 198.6 198.6 213.7 213.2 28.12
(co = 2.04) () = 2.05) (co = 2.00) (co = 2.00)
7 195.7 195.7 213.3 201.6 102.1
(co = 2.03) (co = 2.04) (o = 2.00) (o = 1.85)
9 193.9 194.0 213.3 97.62 135.1
(co = 2.02) (o = 2.00) (o = 2.00) (co = 1.42)
14 191.8 191.9 213.2 17.01 137.9
(co = 2.01) (co = 2.00) (o = 2.00) (co = 0.92)





88

calculations is greater than either of the other two conditions. The increase in the critical

Rayleigh number can be attributed to allowing fluid motion in the air layer, therefore

removing more heat from the liquid and stabilizing the system. This is especially true for

smaller air heights.

The second important point that can be made from the table is when the air height

becomes large. For the Biot numbers in equations (4.1) and (4.3), the critical Rayleigh

number and the critical wave number reach an asymptotic value as the air height

increases. The active air layer calculations, on the other hand, show a dramatic decrease

in both the critical Rayleigh number of the liquid and the critical wave number. This can

also be explained by convection in the air layer, as follows. The magnitude of the

temperature drop in each layer in the conductive state depends upon the height and

conductivity of each layer. As the air layer increases in height, the temperature difference

across it will increase relative to the temperature difference across the lower liquid for a

fixed overall temperature drop. Indeed, as the overall temperature difference increases,

the fluid layers will begin to convect. Consequently, under critical conditions the

Rayleigh number of the lower liquid is small and only becomes smaller as the air height

increases. By contrast the Rayleigh number of the air becomes larger with an increase in

its height. Because the convection is dominant in the air layer, the liquid layer simply

responds to convection in the upper gas. While convection in both layers is simultaneous,

clearly the convection of air immediately sets up transverse temperature gradients in the

interface generating surface driven Marangoni and buoyancy convection in the liquid.

The decrease in the critical wave number must therefore be a signature of the pattern due

to dominant convection in the air layer.






89

The critical Rayleigh numbers corresponding to Table 4-2 were also calculated for

the case of a lower viscosity by reducing the value of this thermophysical property by

thirty percent. In the calculations for the long wavelength Biot number and varying Biot

number conditions, the critical Rayleigh number changed very little. However, depending

upon the air height, the Rayleigh numbers for the active air calculations changed

dramatically. For small air heights (0.1 mm and 1 mm) the Rayleigh number of the

silicone oil changed very little, but the Rayleigh number of the air decreased by over 40

percent. For large air heights (9 mm and 14 mm), the Rayleigh number for the air

changed very little but the Rayleigh number for the silicone oil increased dramatically.

The reason is, for small air heights, convection is "initiated" in the lower silicone oil

layer. Decreasing the viscosity does not change the critical Rayleigh number of the oil

significantly, but the silicone oil's critical temperature difference must decrease

corresponding to the viscosity decrease. The overall temperature difference must

therefore also decrease. Because for small air heights, air is nearly passive, it simply acts

like a conductor. A decrease in the overall temperature difference, therefore results in a

decrease in the temperature difference across the air, decreasing the air Rayleigh number.

For large air heights, convection in the air is dominant at onset. Decreasing the viscosity

of the silicone oil does not affect the air's Rayleigh number very much and therefore does

not change the overall temperature difference much either. The temperature difference

across the silicone oil virtually does not change and therefore a decrease in the oil's

viscosity increases the critical Rayleigh number of the oil.






90


An interesting note can be made about the effective Biot number of the bilayer

calculation. To find the effective Biot number, the Biot number in the long wavelength

Biot number calculations can be changed until the critical Rayleigh number is the same as

the Rayleigh number in the bilayer calculations. For small air heights (0.1 mm to 7 mm),

the Biot number must decrease with an increase in the air height to cause the critical

Rayleigh number to decrease. However, for large air heights (14 mm), the effective Biot

number will turn out to be negative. The reason for this peculiarity is that the convection

in the air layer causes the temperature perturbations to change signs. In other words, the

flow of heat into the air from the liquid decreases, although the net flow of heat into the

air is still positive.

Turning now to the second feature of the unbounded calculations, the vertical

components of velocity, or the eigenfunctions, W,(z) and W2(z), for various air heights

were calculated and are shown in Figure 4-3. Each graph represents calculations using

5 mm of 100 cS silicone oil. The vertical component of velocity is displayed at the




a b c
0 ...0 ..





5 mm 0 3 mm -5 mm 0 5 mm -5 mm 0 9 mm

Figure 4-3. Plot of the vertical component of velocity versus fluid depths, for 3 mm (a), 5
mm (b), and 9 mm (c) air heights. The liquid-gas interface is represented by he vertical
dotted line. For 3 mm, air is being dragged by the flowing silicone oil. For 5 mm, air is
convecting due to thermal coupling. For 9 mm, most of the convection occurs in the air
layer. Each calculation used 5 mm of 100 cS silicone oil.





91


critical wave number in each graph. For a small air height, one would expect that the

Rayleigh number in the oil, at onset, would be much greater than the Rayleigh number in

the air and we would say that the oil convects "first". Here motion in the air is caused by

the silicone oil dragging it. As the air height increases, the Rayleigh numbers in each

layer become comparable. In this scenario, we may still say that the convection is

comparable in both layers. The direction of the flow in the upper layer, depends upon its

thermophysical properties. If the flow in the upper layer is in the same direction as the

flow in the lower layer (co-rotating), then the convection is considered to be thermally

coupled. If the flow in the upper layer is in the opposite direction as the flow in the lower

layer (counter-rotating), then the convection is considered to be mechanically coupled.

For the calculations given in Figure 4-3b, convection is a little more dominant in the oil

and the air would be termed thermally coupled had it not been for a small counter roll

developed near the interface in the air layer. For a larger air height of 9 mm (Figure 4-3c)

the convection is almost entirely in the air layer, while the liquid layer appears mostly

passive. The onset of the strong motion in the air simultaneously causes tangential

gradients of temperature at the interface, inducing a weak (probably Marangoni driven)

motion in the oil.

The third feature of the laterally unbounded bilayer calculations is seen by

extending the results to give qualitative information on bounded containers. This is done

by relaxing the conditions on the side walls. In the Mathematical Modeling chapter, a

simple formula was given (2.116) to translate the calculations from a laterally unbounded

layer to a cylinder with insulating and vorticity-free side-walls. The formula is:






92



73.00



E 71.00
Z


^o \\
S69.00


m=1 m=2 m m=3 m=1 m=0
S67.00 n=1 n=1 n1 n=2 n=2
U


65.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Aspect Ratio


Figure 4-4. Critical Marangoni number versus aspect ratio plot. Calculations were done
using the bilayer, laterally unbounded model for 4.2 mm of 100 cS silicone oil. Equation
(4.4) was used to unfold the Marangoni number versus wave number plots to the
Marangoni number versus aspect ratio plot.

Sm,n
a = (4.4)
co

where, a is the aspect ratio, co is the wave number. smn are the zeroes of the derivative of

the Bessel's function, n is the radial mode, and m is the azimuthal mode. First, a graph of

the critical Marangoni number (or Rayleigh number) versus wave number is generated

from the laterally unbounded model. Using equation (4.4), each wave number translates

into an aspect ratio, for a particular radial and azimuthal mode. The result is given in

Figure 4-4. Two observations can be made from Figure 4-4. First, the critical Marangoni

number is not a monotonic function of the aspect ratio, but the minimum critical

Marangoni number is the same for each flow pattern. Secondly, the pattern changes as the






93


aspect ratio changes. Some of these observations carry over to the no-slip side-wall

calculations.

A comparison can be made between the vorticity-free and the no-slip side-wall

calculations. The vorticity-free side-wall calculations are shown in Figure 4-4 and the no-

slip calculations are shown in Figure 4-5. It can be observed that the minimum

Marangoni number for each mode is the same for the vorticity-free side-walls. For the no-

slip side-walls, at small aspect ratios, the minimum value of each mode is much greater

than the asymptotic minimum reached at aspect ratios greater than 4.0. The last

observation that can be made is that at larger aspect ratios, for no-slip calculations, the

modes quickly crowd together and become indistinguishable.

In Figure 4-5, at the aspect ratio of 2.0 for the Biot number of 0.30, the predicted

flow pattern is m = 0 (single toroid). However, it was shown in the paper by Dauby et al.,

1997 that past the minimum of the m = 0 line, a superposition of the single toroid and a

double toroid may be seen. Similarly, between the aspect ratios of 1.2 and 1.7 for a Biot

number of 3.0 in Figure 4-5, an m = 0 (single toroid) flow pattern will be seen. Past the

minimum of the m = 0 line, and between the aspect ratios of 1.7 and 2.0 a second toroid

will start to appear.

In both the no-slip calculations as well as the vorticity-free calculations, it may be

observed that certain aspect ratios correspond to a situation where two flow patterns

become simultaneously unstable. As was discussed earlier, such aspect ratios are called

codimension-two points and can be associated with oscillatory behavior in the immediate

post onset regime of flow (Rosenblat et al., 1982a; Johnson and Narayanan, 1996).




Full Text

PAGE 1

GEOMETRIC EFFECTS ON BILAYER CONVECTION IN CYLINDRICAL CONTAINERS BY DUANE JOHNSON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA I 1997

PAGE 2

ACKNOWLEDGMENTS To begin I would like to thank Dr. Ranga Narayanan, my advisor. Throughout my thesis he has offered his support in every conceivable manner. His professionalism and light-heartedness has made my experience both highly educational and enjoyable. Many thanks, love and admiration go to my wife, Jody, whose love and support has made all of this possible. I would be remiss not to thank Dr. Ray Skarda and Dr. J.C. Duh at the NASA Lewis Research center for there help, advice, and especially for there assistance in obtaining the IR camera.. It is also necessary for me to thank those on my advisory board, Dr. L. Johns, Dr. U. Kurzweg, and especially Dr. Ruby Krishnamurti and Dr. Jorge Viiials for driving all the way from Tallahassee. My gratitude is also extended to Ken Reed for his help in designing and constructing the experiment and Dr. A. Zaman for his help with some of the calculations and the viscosity measurements of the silicone oil I would also like to thank a few of the undergraduate students who have assisted me in many ways: Chris Birdsall for his help in constructing the second version of the experiment and Bryon Stackpole for his contribution to the control program and writing the experiment manual. Final acknowledgments go to the many graduate students and faculty at the University of Florida. The numerous conversations and advice given was an essential part of my progress. ii

PAGE 3

This work was supported by a fellowship from the NASA Graduate Student Research Program, grant number NGT 3-52320 and NGT 51242 grants and from the National Science Foundation, grant numbers CTS 95-00393 and CTS 93-07819. ui

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TABLE OF CONTENTS ACKNOWLEDGMENTS ii ABSTRACT vi CHAPTERS 1. PHYSICS AND HISTORICAL PERSPECTIVE 1 Introduction 1 Physics 2 Rayleigh Convection 3 Marangoni Convection 5 Pattern Selection 7 Bilayer Convection 9 History 14 Single Layers 14 Bilayers 19 2. MATHEMATICAL MODELING 22 Linear Model 22 Numerical 32 Unfolding 40 Nonlinear Analysis 46 Adjoint 48 Galerkin-Eckhaus Expansion 50 3. EXPERIMENTAL APPARATUS AND PROCEDURE 58 Apparatus 59 Infrared Imaging System 59 Test Section 63 Heating and Cooling 67 Electronic Hardware Unit 68 Procedure 72 IV

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4. RESULTS AND DISCUSSION 77 Introduction 77 Codimension-Two points 78 Effects of Air Height on Bilayer Convection 83 Observations from calculations 84 Observations from experiments 95 Changes in Convection Coupling and Interfacial Structures 104 Changes in convection coupling 106 Changes in interfacial structures 113 Other Observations in Convection-Coupling and Interfacial Structure 115 Nonlinear Analysis 121 Case 1 123 Case 2 124 Case 3 125 Case 4.. 127 5. FUTURE SCOPE 140 Experiments 140 Nonlinear Analysis 141 Numerical Calculations 142 APPENDICES 143 A COMPUTER PROGRAMS 143 B DRAWINGS AND DIAGRAMS 183 REFERENCES 194 BIOGRAPHICAL SKETCH 199

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy GEOMETRIC EFFECTS ON BILAYER CONVECTION IN CYLINDRICAL CONTAINERS By Duane Johnson December, 1997 Chairman: Dr. Ranganathan Narayanan Major Department: Chemical Engineering The study of convection in two immiscible fluid layers is of interest for reasons both theoretical as well as applied. Recently, bilayer convection has been used as a model of convection in the earth's mantle. It is also an interesting system to use in the study of pattern formation. Bilayer convection also occurs in a process knovm as liquid encapsulated crystal grov*?th, which is used to grow compound semiconductors. It is the last application which motivates this study. To analyze bilayer convection, theoretical models, numerical calculations and experiments were used. One theoretical model involves the derivation of the NavierStokes and energy equation for two immiscible fluid layers, using the Boussinesq approximation. A weakly nonlinear analysis was also performed to study the behavior of vi

PAGE 7

the system slightly beyond the onset of convection. Numerical calculations were necessary to solve both models. The experiments involved a single liquid layer of silicone oil, superposed by a layer of air. The radius and height of each fluid layer were changed to observe different flow patterns at the onset of convection. From the experiments and theory, two major discoveries were made as well as several interesting observations. The first discovery is the existence of codimension-two points particular aspect ratios where two flow patterns coexist in cylindrical containers. At these points, dynamic switching between different flow patterns was observed. The second discovery was the effect of air convection on the flow pattern in silicone oil. Historically, air has been considered a passive medium that has no effect on the lower fluid. However, experiments were done to show that for large air heights, convection in the air can cause radial temperature gradients at the liquid interface. These temperature gradients then cause surface tension gradient-driven flows. It was also shown that changing the radius of the container can change the driving force of convection from a surface tension gradient-driven to buoyancy-driven and back again. Finally, the weakly nonlinear analysis was able to give a qualitative description of codimension-two points as well as the change in flow patterns due to the convecting air layer. vn

PAGE 8

CHAPTER 1 PHYSICS AND HISTORICAL PERSPECTIVE 1.1. INTRODUCTION The motivation for this research comes from a technique known as liquidencapsulated crystal growth. Liquid-encapsulated crystal growth is a process for growing semiconductor crystals from bulk, liquid melts. Some examples of crystals grown using this technique are gallium arsenide and gallium selenide, which are used in Inert Gas Liquid Encapsulant Liquid Melt Solid Figure 1-1. Schematic of a liquid encapsulated crystal grower a system of three convecting fluid layers. Convection in the GaAs liquid influences the quality of the GaAs solid.

PAGE 9

communications, lasers, as well as the next generation of computer processors. These applications require that the material be of the highest purity and that the crystalline structure be nearly flawless. Take gallium arsenide (GaAs) for example. When solid gallium arsenide is melted, the arsenic has a tendency to escape. This decomposition destroys the necessary stoichiometric ratio of the crystal, diminishing its quality. Additionally, arsenic is highly toxic and a serious hazard to humans. To prevent this decomposition, a lighter, immiscible, viscous liquid, such as boron oxide (BjOj) is placed on top of the gallium arsenide. This limits the transport of arsenic into the upper layer. To prevent arsenic gas from bubbling through the encapsulant layer, an inert gas, such as argon, is pumped in at a high pressure on top of the boron oxide. To grow the crystal, these three fluid layers are typically placed into a cylindrical crucible. The crucible is then lowered into a Bridgman furnace (Schwabe, 1981; Miiller, 1988), which is hot on top and cool enough at the bottom to solidify only the gallium arsenide. This configuration creates a system full of interesting physics and we will discuss some of these next. 1.2. PHYSICS Although there are many different phenomena that can be studied in this system, such as the morphological instability (MuUins and Sekerka, 1964; McFadden et al., 1984; Glicksman et al., 1986; Davis, 1990) and double diffusion (Turner, 1985), this thesis will concentrate on studying buoyancy-driven and interfacial tension-driven convection. Morphological instability occurs when the solidification velocity the growth rate of the

PAGE 10

solid is faster than some critical value, generating compositional undercooling. For large growth rates, the flat, planar solid-liquid interface begins to deflect. These deflections can be as small as a few microns ultimately growing into dendrites. Double diffusive convection only occurs when there is more than one species in the liquid melt. This typically occurs in crystal growth when a dopant is added to the semiconductor compound. One often assumes that the solidification is quasi-static. That is, the growth rate is much slower than the time scale of the convection and slower than the critical growth rate necessary for the morphological instability to occur. Additionally, only immiscible fluids will be considered in this study precluding the possibility of double diffusion. 1.2.1. Rayleigh Convection Buoyancy-driven convection, often referred to as natural convection or Rayleigh convection, occurs as a result of the variation of density with respect to temperature under a gravitational field. Imagine a layer of liquid bounded vertically by two horizontal rigid Buoyancy-Driven Convection Cold gas Hot Interfacial Tension-Driven Convection Cold gas liquid warmer >^ v' r^,,.. -^£% 1. Hot Figure 1-2. Physics of Rayleigh and Marangoni convection.

PAGE 11

plates, with the lower plate at a temperature greater than the upper plate. As density typically decreases with an increase in temperature, the fluid near the top plate is heavier than the fluid at the bottom plate, creating a gravitationally unstable system. However, if the temperature difference across the layer of liquid is sufficiently small, then the fluid simply conducts heat from the lower plate to the upper plate, creating a linear temperature drop across the fluid. When the fluid is quiescent, a precarious balance exists between the pressure gradient and buoyancy forces. For large depths, thermal expansivity and gravity tend to upset this balance while kinematic viscosity and thermal diffusivity tend to reinstate the balance. When the balance is upset by disturbances, the fluid is set into motion which under certain circumstances will continue unhindered. This fluid motion is called buoyancy-driven convection. The extent of buoyancy-driven convection (if any) is given by the dimensionless Rayleigh number, Ra. Ra = ^ (1.1) VK ^ ^ Here, a is the negative thermal expansion coefficient, g is gravity, AT is the vertical temperature difference across the fluid layer, d is the depth of the fluid, v is the kinematic viscosity, and k is the thermal diffusivity. If the temperature difference is increased beyond what will be referred to as the critical temperature difference, then the gravitational instability overcomes the viscous and thermal damping effects and the fluid is set into motion, causing buoyancy-driven convection.

PAGE 12

1.2.2. Marangoni Convection Surface tension gradient-driven convection, unlike buoyancy-driven convection, can occur in a fluid without a gravitational field. Imagine a layer of fluid which is bounded below by a rigid plate and whose upper surface is in contact with a passive gas (Figure 12). Above the passive gas is another rigid plate. A passive gas is a gas which conducts heat like a solid, yet has no viscosity, so that it does not impart momentum to the liquid. For the sake of consistency, allow the lower plate to be at a temperature greater than the upper plate's temperature. Now, imagine that the interface between the lower liquid and the passive gas is momentarily disturbed. The regions of the interface which are pushed up experience a cooler temperature. Likewise, the regions of the interface which are pushed down, increase in temperature. Typically, surface tension decreases with an increase in temperature. Therefore, the regions of the interface which are pushed up increase in surface tension, which pulls on the interface, while the regions of the interface which are pushed down, decrease in surface tension. When the fluid is pulled along the interface, warmer fluid from the bulk replaces the fluid at the interface enhancing the surface tension-induced flow. If the temperature difference across the liquid is sufficiently small, then the thermal diffusivity of the fluid will conduct away the heat or the dynamic viscosity will resist the flow causing the surface to become flat and the surface tension to become constant. As was the case in buoyancy-driven convection, there exists a critical temperature difference where the surface tension gradient-driven flow is not dampened by the thermal diffusivity or viscosity, and the fluid is set into

PAGE 13

motion. Surface tension gradient-driven convection is cliaracterized by the dimensionless Marangoni number, Ma. Ma = |J,K (1.2) Where ct, is the change in the surface tension with respect to the temperature, and p, is the dynamic viscosity. The extent of either Rayleigh convection or Marangoni convection is primarily a function of the fluid depths. By examining equations (1.1) and (1.2) we notice that Rayleigh convection is proportional to the cube of the fluid depth and that Marangoni convection is directly proportional to the fluid depth. From these scaling arguments, we can conclude that for deeper fluids, buoyancy-driven convection is more prevalent, and 0.0 0.5 ao 90 ^ V 1— 1 \ =5 85 2 \ m=0 — m-1 \ \ m=2 m=3 c 80 O D1 ^75nJ S 70 \ \ '^. \ -. \ \ \ ^^-^ Critical o en \ \ ^s — 55 i— : ] _j 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Aspect Ratio Figure 1-3. Plot of the critical Marangoni number versus the aspect ratio of a cylinder. The mode, m, with the smallest Marangoni number at a given aspect ratio, is the mode or flow pattern at the onset of convection.

PAGE 14

for shallower depths, surface tension gradient-driven convection is more prevalent. However, it has been show (Nield, 1964) that Rayleigh and Marangoni convection reinforce one another. Therefore, at intermediate fluid depths, both Rayleigh and Marangoni convection can occur. There is another phenomena associated with surface tension gradient-driven convection, often called the long wavelength Marangoni instability (Davis, 1983). This instability typically occurs when either the surface tension or the depth of the fluid layer is very small. The initiation of this instability is similar to the description given above. However, in the long wavelength scenario, the convection cells are much larger than the regular, or short wavelength, Marangoni convection. As the convection propagates, it causes large scale deformation in the interface which can actually cause the interface to rupture; that is the interface deforms to such an extent that it comes in contact with the lower plate. This phenomenon occurs in the drying of films and coating processes. In all of the cases examined in this thesis, both the surface tension and the liquid depths were sufficiently large to avoid this instability. 1.2.3. Pattern Selection In a fluid of infinite horizontal extent, there is no limit on the size or the number of convection cells. The size of the convection cell is proportional to the wavelength, which is inversely proportional to the wavenumber. However, in a bounded, finite-sized container, only a finite number of convection cells may exist. Physically, this means that at the onset of convection in a bounded container, only one flow pattern will usually

PAGE 15

exist. As the aspect ratio (radius divided by the height) of the container increases, more convection cells will appear (see Figure 1-3). In a bounded cylinder, each flow pattern has associated with it an azimuthal and radial mode, m and n, respectively. For example, at an aspect ratio of 1.0 in Figure 1-3, there is an ni = 1, n = 1 flow pattern, where m is the azimuthal mode and n is the radial mode (see Figure 1-4). For an aspect ratio of 1.5, there exists an m = 0, n= 1 flow pattern. The azimuthal mode is the number of times the azimuthal component of velocity goes to zero, and the radial mode is the number of times the radial component of velocity goes to zero starting from the center, for a given vertical cross-section. The azimuthal and radial modes will be defined more precisely in the mathematical modeling section. As the aspect ratio increases, the flow pattern switches from one flow pattern to another. Aspect ratios where two different flow patterns can coexist are called codimension-two points. Physically speaking, these are aspect ratios where the energy states of each flow pattern are equal. m = n=l rm toroidal an bimodal CT unicellular CX)00 double toroid Figure 1-4. Schematic of three different flow patterns. Circles represent fluid flowing up and X's represent fluid flowing down.

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Lower Dragging Mode Viscous Coupling TheiTnal Coupling Upper Dragging Mode Figure 1-5. Schematic of the different types of convection-coupHng. From the lower dragging mode to the upper dragging mode, the buoyancy force in the upper layer is increased and the dragging exerted by the lower layer decreases. 1.2.4. Bilayer Convection We now change our thought experiment to include a viscous, less dense, immiscible layer of fluid above the lower layer of fluid. Here the lower layer is bounded below by a rigid, conducting plate and the upper layer is bounded above by another rigid, conducting plate. Once again the temperature of the lower plate is greater than the upper plate. The interface between the two fluids is allowed to deform and is capable of transporting heat and momentum from one layer to the other. We will now consider the various types of convection that can occur in a bilayer of two fluids. In order to distinguish the various convective mechanisms, phrases such as "convection initiating in one layer or another" are introduced. Clearly in a mathematical sense, there is only a single condition for the onset of convection and this onset must occur simultaneously in both layers. The notion of convection "initiating" in one layer or another is ultimately a physical one and is perhaps best explained qualitatively. To

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10 a r\ a V / Lower layer b Upper layer b d d Figure 1-6. Typical plots of the vertical component of velocity (top row) and the temperature perturbations (bottom row) versus the fluid depths. The vertical dashed line represents the interface, which separates the lower and upper fluid. understand the statement in the context of convection with liquid bilayers, we consider only Rayleigh convection and assume momentarily that the Marangoni effect is absent. When one of the fluid layers is said to "initiate" convection prior to the other fluid, what is meant is that its Rayleigh number has reached its critical value before the other layer. The critical value for each fluid layer, in this situation, is compared to a different problem. For the lower layer, the critical Rayleigh number is the critical Rayleigh number it would have as if it were the only active fluid layer, superposed by a passive fluid which only conducted away heat. For the upper layer, the critical Rayleigh number is the one it would have if it was bounded above by a rigid conducting plate and bounded below by a passive fluid that only conducts heat away. Turning now to various convective mechanisms, consider Figure 1-5. Suppose that convection initiates in the lower layer. The upper layer responds by being dragged,

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11 generating counter rolls at the interface. Hot fluid flows up in the lower layer and down in the upper layer. The upper layer is not buoyant enough and moves by a combination of viscous drag and the Marangoni effect. This is seen in Figure l-5a. This can also been seen by plotting the velocity and temperature perturbations versus the fluid depth (Figure 1-6). Observe in Figure l-6a that the sign of the velocity switches and the maximum absolute value of the lower layer velocity is much greater than the maximum absolute value of the velocity of the upper layer. When the buoyancy in the upper layer increases and the upper layer begins to convect, one of two things can happen. The first possibility is that the two fluids are viscously coupled. Physically this can be shown in Figure l-5b as counter-rotating rolls in the two fluids. This can also be denoted by the vertical component of velocity switching sign at the interface (Figure l-6b), while the temperature perturbations may switch sign in either the upper layer or the lower layer. If the temperature perturbation switches sign in the upper fluid, then the lower layer is more buoyant. If the temperature perturbation switches sign in the lower layer, then the upper layer is more buoyant. Marangoni convection, for fluids whose surface tension decreases with an increase in temperature, encourages this mode of convection. The second possibility is thermal coupling where the rolls are co-rotating. Here hot, rising fluid from the lower layer causes hot fluid in the upper layer to flow up. The maximum of the vertical component of velocity and the temperature perturbations have the same sign in each fluid layer (Figure l-6c). Strictly speaking, the transverse components of velocity should be zero at the interface. However, thermal coupling is

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12 sometimes referred to the case when a small roll develops in one of the layers so as to satisfy the no-slip condition at the interface. As the buoyancy continues to increases in the upper layer, convection initiates in only the upper layer and the lower layer is viscously dragged (Figure l-5d). This situation only occurs when the upper fluid is a liquid, as gases are very tenuous and wouldn't exert enough shear. The vertical component of velocity in this case (Figure l-6d) switches sign and the maximum absolute value of the vertical component of velocity in the upper fluid is much greater than the maximum absolute value of the vertical component of velocity in the lower fluid. Another indicator of what is occurring in bilayer convection can be inferred from the fluid-fluid interface instead of the bulk convection. In a paper by Zhao et al. (1995), four different interfacial structures were identified for any given convecting bilayer with a deflecting interface. Each of these structures depends upon whether fluid was flowing into or away from the trough or the crest, and whether the fluid was hotter or cooler at the trough or the crest of the interface. Hot fluid flowing into a trough defines the first interfacial structure. The second interfacial structure has hot fluid flowing into a crest. i '" t J If I f Hot Flow Ij I IV ^ "it V ^^'^ P'^ Figure 1-7. The four possible interfacial structures at a fluid-fluid interface. Each structure can give information about the driving force of the convection.

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13 The third structure has hot fluid flowing away from a crest and the fourth structure has hot fluid flowing away from a trough. Each of these four scenarios is given in Figure 1-7. One of the important factors to consider in interfacial structures, is the direction of the flow along the interface. As surface tension is usually inversely proportional to temperature, at cooler regions of the interface, the surface tension will be higher and will pull on the interface. Where the interface is hotter, the surface tension will be lower causing the fluid to move away from warmer regions. Another important factor is the direction of the flow into or away from a crest or a trough. One reason the interface deflects is due to bulk convection, caused by buoyancy effects, pushing against the interface. Consider two fluids whose dynamic viscosities are equal. If buoyancy-driven convection is occurring mostly in the lower layer, then the fluid will flow up from the lower layer into a crest. If the fluid flows down from the top layer into a trough, then one would argue that buoyancy-driven convection occurs mostly in the upper fluid In each of the four cases, the interfacial structure can be used to indicate the driving force of the convection. In the first interfacial structure, the dominating driving force is surface tension gradient-driven convection. This is seen as the cold fluid, with the higher surface tension pulls the fluid up into the crest. The first interfacial structure can also occur by buoyancy-driven convection in the upper layer, when the density of the upper layer increases with an increase in temperature. In the second interfacial structure, buoyancy drives convection in the lower phase. The hot, rising fluid pushes the interface upwards. As the fluid moves along the interface, it cools and eventually sinks back down. The third interfacial structure is dominated by buoyancy-driven convection in the upper phase, or by surface tension gradient-driven convection where the surface tension

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14 increases with respect to temperature. The fourth interfacial structure only occurs when the lower fluid has a positive thermal expansion coefficient. In other words, the density increases with an increase in the temperature, causing the cooler, lower fluid to flow up into a crest. 1.3. HISTORY Rayleigh-Marangoni convection is one of the classic problems in fluid mechanics, dating back to the beginning of this century (Benard, 1900). From its initial roots, the problem has split off into many different branches. For this reason, it is difficult to give a comprehensive review of all of the aspects of this fascinating phenomena. Instead, only the aspects which are relevant to the historical background of this thesis will be covered. This section is divided into two major categories: single layers and bilayers. The work on convection in single liquid layers is by far more comprehensive, with concentrated efforts on the bilayer problem occurring only recently. For a comprehensive review of Rayleigh-Marangoni convection, refer to the book Benard Cells and Taylor Vortices by E. L. Koschmieder (1993). The following subsection on single layers, is a review of the most relevant facts within Koschmieder' s book, and several other works that he dismissed. 1.3.1. Single Layers Pearson (1958) was the first researcher to look at the dynamics of two laterally unbounded fluid layers, where the upper fluid was considered to be an inviscid gas. He

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15 assumed that the interface between the two fluids was non-deformable and found a critical Marangoni number around 80 with a critical wave number of 2.0. Nield (1964) looked at the combined effects of buoyancy and surface tension gradient-driven convection in a single liquid layer. In this work, he performed a linear stability analysis using a normal mode expansion. The single liquid layer was bounded below by a rigid conductor and bounded above by a passive gas. The interface was assumed to be flat and non-deformable. The dimensionless heat transfer at the free surface was modeled by the Biot number. ae + 5/6 = (1.3) for Bi = ^yi^ Here, 9 is the dimensionless temperature, h is the heat transfer coefficient, d and k are the depth and thermal conductivity of the liquid, respectively, and z is the coordinate pointing out of the fluid into the passive gas layer. The Biot number can also be written as: „. ^gas^ liquid '^ liquid^ gas Nield found that buoyancy and surface tension gradient-driven convection reinforce one another. He also investigated the effect of the Biot number on the critical Rayleigh number, and critical wave number. He found that decreasing the Biot number decreases both the critical Rayleigh number and critical wave number. The modeling of the heat transfer from the lower liquid to the passive gas was improved upon by Normand et al. (1977). In their review, they allowed the temperature

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16 of the passive gas to become perturbed, yet still considered the gas as being mechanically passive. By doing this, a new formula for the Biot number was arrived at. k r \ Bi = CO 7 coth liquid d gas ^1 V "-liquid J (1.5) where co is the wave number of the gas perturbations. Equation (1.5) is equal to equation (1.4) in the limit as co goes to zero. Although they did not calculate the effect of this new Biot number on the critical Rayleigh number and the critical wave number, it is easy to see the difference. For a fixed wave number and depths of the liquid and gas, the Biot number in equation (1.5) is always greater than the Biot number in equation (1.4). By taking the resuhs from Nield's analysis, the critical Rayleigh number and the critical wave number is seen to increase compared to Nield's results for the same liquid and gas depths. The effect of a deflecting surface was introduced by several researchers, but is probably best described by Davis (1983) and later reviewed by Davis as well (1987). In the review, he notes that the surface deflections destabilize the system when surface tension gradients dominate and stabilize the system when buoyancy-drive convection is dominant. He also notes, that in buoyancy-driven convection, the fluid flows up into a surface elevation and that in surface tension gradient-driven convection, the fluid flows up into a surface depression. Fluid flowing up into a depression was first noted in the original experiments of Benard (1900) and later confirmed by Cerisier et al. (1984). Davis (1983) also noted that for vary shallow layers surface tension gradient-driven convection leads to a long wavelength instability. He developed a nonlinear evolution of

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17 the surface deflections by adding the contribution of Vander Waal's forces. Therefore, his model is only valid for extremely thin layers. We next move to the effects of bounded containers on the flow pattern in a single liquid layer. Numerous papers have been written on pattern formation in Rayleigh and Rayleigh-Marangoni convection for large aspect ratio containers. Among these papers, several researchers have investigated the effects of boundaries on these patterns. As this thesis will only concentrate on relatively small aspect ratios, this group of papers will be neglected. The interested reader is refereed to Cross and Hohenberg (1993). Instead, we will begin with a series of three papers Rosenblat (1982), Rosenblat et al. (1982a), and Rosenblat etal. (1982b). Of these papers, we will concentrate on Rosenblat etal. (1982a), which deals with cylindrical containers. In this paper, a weakly nonlinear analysis was performed on the pure Marangoni problem, using the Galerkin-Eckhaus expansion (Eckhaus, 1965; Manneville, 1990). The most relevant result came from their analysis of codimension-two points, where two different flow patterns coexist. For one of the codimension-two points, they were able to show that the solution branched off to a secondary Hopf bifurcation. Physically this means that the different linear modes could interact with each other (and/or themselves) to give a dynamic equilibrium solution. Curiously, this Hopf bifurcation was only seen when the aspect ratio was slightly greater than the aspect ratio of the codimension-two point. When the aspect ratio was decreased to the other side of the codimension-two point, the Hopf bifurcation disappeared. The first systematic experimental investigation of the effects of bounded geometries was conducted by Koschmieder and Prahl (1990). In their paper, they observed the flow

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18 pattern in rectangular and cylindrical containers using aluminum particle tracers in silicone oil. They report that the number of cells that are observed increases monotonically as the aspect ratio increases. The weakly nonlinear analysis in a single layer was later extended by Dauby and Lebon (1996), who replaced the unrealistic vorticity-free boundary conditions with realistic no-slip conditions. Their analysis was able to show that the patterns that Koschmieder observed (Koschmieder and Prahl, 1990) are only visible in the weakly nonlinear regime. Another weakly nonlinear analysis was conducted by Echebarria et al. (1997) In their paper, they took into consideration the rotational symmetry of the cylindrical geometry, which allowed them to find solutions where the pattern would rotate in the cylinder. By looking at only a single, highly resonant codimension-two point, they also found solutions where a secondary Hopf bifurcations could occur. These bifurcations were identified as a heteroclinic orbit between four different flow patterns, two of which were the same as the other two, rotated by 90. In all of the previous papers stated earlier, either an infinite horizontal fluid was considered or the fluid was confined in a bounded cylinder using unrealistic vorticity-free side-walls. In the paper by Zaman and Narayanan (1996), and later by Dauby et al. (1997), a linear, three dimensional solution was found for Rayleigh-Marangoni convection in a cylinder. Both papers assumed that the interface was flat and that the side-walls of the cylinder were no-slip. One of the most interesting observations in these two papers, is that the progression of modes was not the same as the vorticity-free calculations (Rosenblat et al., 1982a). That is, the flow pattern predicted at the onset of

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19 convection for a given aspect ratio is different for no-slip and vorticity-free side-walls. The vorticity-free calculations contradict the results of Koschmieder and Prahl (1990). Additionally, the different progression of modes changes the codimension-two points analyzed by Rosenblat etal. (1982a) and Echebarria etal. (1997). For example, in Rosenblat et al. interaction of a unicellular, m = 1, and an m = 2 flow was analyzed and a secondary Hopf bifurcation was found for an aspect ratio slightly greater than the codimension-two point. This interaction could not even occur according to the linear, noslip calculations. Therefore, the existence of the Hopf bifurcation in Rosenblat et al.'s paper and the heteroclinic orbit found in Echebarria' s paper, is in question. 1.3.2. Bilayers Some of the earliest work done on bilayer convection was a series of linear stability analyses. Smith (1966) improved upon the single layer problem by allowing the interface to deform and did not assume a passive gas above. However, he ignored the effects of buoyancy, and only allowed the surface tension to vary with respect to temperature. In addition to a linear stability analysis, he also performed a long wavelength analysis. The long wavelength analysis was able to show that surface deflections are important and can lead to instabilities in very shallow fluid depths. Experimental and theoretical work was performed later by Zeren and Reynolds (1972). In their paper, the effect of buoyancy driven convection was included. They were able to find three different instabilities: buoyancy-dominated, surface tension-

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20 dominated, and "surface deflection dominated" convection. Their linear model was later improved upon by Perm and WoUkind (1982). Interest in bilayer convection increased when a discontinuity in the density of the earth's mantle was discovered. It was hypothesized that the earth's mantle was composed of two, chemically distinct layers (Richter and Johnson, 1974). Today this hypothesis is in general acceptance. The first record of the different types of bilayer convectioncoupling was mentioned in a paper by Honda (1982). Honda used a linear stability analysis and a finite amplitude analysis to describe three different methods of convection between the two fluid layers: thermal coupling, viscous coupling, and a dragging of one fluid by the other. In the analysis performed by Honda and later by Cserpes and Rabinowicz (1985) and Ellsworth and Schubert (1988), the mechanical coupling mode was shovm to be more prevalent at and near the onset of convection. It was shown that thermal coupling is more predominant when the ratio of viscosities is large (more than a factor of 1 00). However, laboratory experiments performed with silicone oil and glycerol (Nataf et al., 1988, Cardin et al, 1991) exhibited that thermal coupling was more stable than mechanical coupling. This contradicted the earlier analytical results. While effects of interfacial tension and interfacial deformation were unable to explain the discrepancy between the analytical and experimental studies (Nataf et al., 1988), Cardin et al. (1991) were able to show that the interfacial viscosity helped to explain why thermal coupling was more stable. Additionally, the onset of oscillatory convection was seen to diminish for large interfacial viscosities. Numerical and experimental work performed by Prakash and Koster (1996) showed that when the driving forces for buoyancy driven-convection in

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21 both layers are approximately equal, then thermal coupling is preferred, whereas, mechanically coupled flow was observed when these driving forces were very different. Unlike single liquid layers, a bilayer of two fluids can oscillate at the onset of convection (Gershuni and Zhukovitskii, 1982; Rasenat etal., 1989). These oscillations are caused by the interaction of the thermal and mechanical coupling modes. For example, by changing the two fluid depths, it is possible to cause the thermal coupling and viscous coupling modes to become simultaneously unstable at different wave numbers. As it is impossible to have a superposition of these two modes, the system oscillates between the equal energy states. The oscillations between the thermal and viscous coupling in a horizontally infinite bilayer of two fluids was analyzed by Colinet and Legros (1994). They showed that the oscillations would appear as a traveling wave. This analysis was later verified experimentally by Andereck et al. (1996). Oscillatory onset of convection can also occur by the Rayleigh-Taylor instability. The Rayleigh-Taylor instability occurs when a heavier fluid lies on top of a lighter fluid. This typically occurs in systems when a liquid with a slightly smaller density lies on top of another liquid. When the bilayer is heated, the lower liquid density decreases and becomes smaller than the upper fluid's density. As the upper fluid sinks, it feels the warmer fluid, heats up, and becomes more buoyant. This instability is avoided when two fluids with reasonably different densities are considered.

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CHAPTER 2 MATHEMATICAL MODELING This chapter includes all of the equations, derivations and numerical techniques used to analyze a system of two immiscible fluids. The modeling consists of four major sections: • A linear model of convection in two immiscible fluids, which are infinite in the horizontal direction. • A numerical calculation of the linear model equations using a Chebyshev spectral tau method. • A transformation and unfolding method used to map the results from the infinite, unbounded calculations into a bounded cylinder. • A weakly nonlinear analysis of convection of two immiscible fluids in a cylinder using a Galerkin-Eckhaus expansion. The results from the first three sections are necessary to perform the weakly nonlinear analysis in the fourth section. However, the results from each section can be used to elucidate certain details of the problem. 2.1 LINEAR MODEL The derivations of the linear model start by recreating the work of Perm and WoUkind (1982). This work considers two immiscible fluids bounded above and below by rigid, thermally conductive plates. The temperature of the lower plate is always assumed to be greater than the upper plate and the interface between the two fluids is 22

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T =Tt T -T. T -T. raTT-T' 23 iT in ^4 ""i >'>i1fl%f 'V Cold fluid #2 fluid #1 z = ri (x ,t ) z =0 z = -d, Figure 2-1. Schematic of the hnear model. A bilayer of two immiscible fluids, bounded by rigid, thermally conducing plates. allowed to deflect. To simplify the calculation, the two fluids will be unbounded in the horizontal direction (see Figure 2-1). The equations which determine the velocity, pressure and temperature for each fluid are the familiar Boussinesq equations. V-v =0 /" ^ P; d\ dt ^-hv -Vv / / HV'v,. -Vp,. +p,.g (2.1) (2.2) Pi^pj T + V, -VT: dt = /t.V'T* (2.3) where p/ is the density, \ii is the dynamic viscosity, Cp/ is the specific heat, kf is the thermal conductivity, v/ = (w/, wif is the velocity vector, pj is the pressure, T/ is the temperature and g is the gravitational vector. The asterix (*) above each variable denotes that the variable is unsealed. The subscript i=\,2 represents the lower fluid (/ = 1) and the upper fluid (f = 2).

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24 The major assumptions made in equation (2.1) through (2.3) are that the viscosity is constant, the fluids do not generate heat tlirough viscous dissipation, and the relative change in the density is very small, that is Ap/p 1 We will also assume that the gravitational vector is constant and only points opposite the z direction. We will further assume that the density, as well as the surface tension, vary only with respect to the temperature. P/ = Po,; f a (^i T,^^ )] (2.4) ^ = ^o\-^^^:-T,ef,^ (2.5) 1 5p, where a, = p, 57; is the thermal expansion coefficient, ^ref,i is the reference Tref j 1 ^ temperature for fluid layer i, a„ is the constant surface tension, and a, = ^ '^n/J The reference temperature for the lower and upper fluid layer's density is the temperature at the interface, T,„. The reference temperature for the surface tension will be Tt, where /= 1 inequation (2.5). For the analysis of this problem, we first assume that the fluid is at rest and only conducts heat from the lower plate to the upper plate. Mathematically, this is realized by letting y-/^ in (2.1) through (2.3). Furthermore, we will substitute equation (2.4) into (2.2). The result is the following equation for the temperature. d'T* ^ = (2.6)

PAGE 32

25 with the boundary conditions: The solution for the temperature profile is: "' d, J, (1 + 50 l + Bi^^''^"^ ^^-^^ '" d, /t/ J2O + 5/) '1 + 5/^^' ^'^ ^^-^^ where 5z = -^ is the Biot number. The next step in the procedure is to make equations (2.1) through (2.3) dimensionless. The length, velocity, time, and pressure are scaled with dj, K.Jd
PAGE 33

26 -^+v,-ve,=v2e, (2.12) and the equations for the upper fluid become. V-V2=0 (2.13) Pr -— V,'2+-VV2 +ai?ae2Z (2.14) dt + y,-Ve,=KV^Q^ (2.15) where, :' is the modified pressure, i?a = ga, ATd^^Ki ^i is the Rayleigh number, and Pr = v,/Kj is the Prandtl number, Moving on to the boundary conditions, we start with assuming the upper and lower plates are rigid, no-slip boundaries at a constant temperature. wi=wi=0 and 7]* = r^ at z = -1 (2.16) W2 = W2 = and Tj = T^ at z = / (2.17) We introduce the variable, r\ = r[(x,t), which represents the surface deflections from the initially flat interface, z = 0. For a deflecting surface, the unit normal, n, becomes. (-%, ij and the vector tangential to the surface is n„ where: (l 'YeJ t i ^ = (2.19)

PAGE 34

27 and n^ can be shown to be orthogonal to n. Assuming the temperature and the heat flux across the interface are continuous, we get: T^' = T^ (2.20) n-yt,Vr;=iri-^2Vr; (2.21) As the fluids are immiscible, there is no penetration of one fluid into the other. Furthermore, we assume that the fluids do not slip past one another at the interface. Therefore: v,=V2 at z* = x\ (2.22) There also exists the kinematic condition of the interface. ^ + ".^ = >-. (2.23) The last two of the thirteen boundary conditions that are needed come from the tangential and normal components of the stress balance. n-T*-ii-ii-X2-n = a(v^-n) (2.24) n-Ti-iij-n-Tj-n, =nfV5a (2.25) where x is the stress tensor and V, is the surface gradient operator. Substituting the dimensionless variables, T, n, and n„ into equation (2.16) through (2.25) and dropping the asterix (*) gives: w, =M, =0 atz = -l (2.26) ^2 = "2 = at z = / (2.27) e, =1 atz = -l (2.28)

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28 G2=-X atz = / 5r| 5ri —+ U, — — = w, for z = ri ot dx dr\ — (w2-w,)=w, -M2 for z = r| dx (2.29) (2.30) (2.31) — (W]-W2J=W2-Wi /or z = ri 0, = 02 atz = T) ^502 aria02^ 501 5r|50| dz dx dx dz dx dx at z = -q (2.32) (2.33) (2.34) -Pi+2 5u5x /'a ^ v5xy 2 r du^ dwdz dx dr[ 5w2 dx dz dr\ ^ 1 + \dxj 5u, dx dx) 5U[ 9w,^ 9z dx dy\ Swj 9x 9z \dxj + (%> + (Ma0->/) 5x^ 1 + 5xy (2.35) 2)1 '^9w2 9u2^ 5z dx ) 9x 2 9u2 5w2 + 5z dx ( f^ ^2^ 19ri dx) 9w, 9ui 9z dx ) 5ri j_ 9x 2 9ui 5w, ^ 9z 9x ^Sri^^^ V v9xy ^ + M3f 90, 9r[a0i^ 9x 9x dz 1 + OT] Kdx) (2.36)

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29 Ma, given in (2.34) and (2.35), is the Marangoni number. Ma = cj, Arc/, /k, |.x G is the Weber number, where G (p\-P2)gd{ and C is the Crispation number, where CWe finally arrive at the essence of the linear model section, which is the normal mode expansion of the variables. Basically, each variable is expanded in a series about some parameter, s, which is a measure of the deviation from the base state (conduction state) of the system. Further, each variable at order 8 is again expanded in a Fourier series in the 'x' direction and exponentially in time. fu^(x,z,t)^ ( 0^ f^l(^)] ^i(x,z,t) W,(z) ,:':(X,Z,0 = Po,i + s n,(z) e,(x,z,o %J 0,(Z) V r](xj) J V ) ^ ^0 J e'™^ e" '+0(e') (2.36) for / = 1, 2. Note that cr here represents the growth constant. The temperature solution to the base state is given in dimensional form in equations (2.7) and (2.8). The dimensionless form of the base state solution is: Vo,l = Vo,2 Qo,i = -^ Tlo=0 Ra J Pq,\=—Y^ forz<0 (2.37) (2.38) 0,2 apRa J P0,2 =-— ^ -^ for z > 2^ ^^^^^ (2.39) Af^er substituting (2.36) into (2.10) through (2.15), the equations to the first order in s for the lower phase are: Z)W, +/coU, =0 (2.40)

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30 (Z)2-')U, -/con, =aU, (2.41) (d^-(o2)w, -Dni+i?a0, =aWi (2.42) (i)^-co^>,+W,=0 (2.43) and for the upper phase. DW2+/coU2=0 (2.44) k(d2-2)82+--W2 =0 K (2.45) (2.46) (2.47) where, D = — dz The expansion of the variables on the surface requires some more care. For example, take some arbitrary dependent variable, A. Then expand A in terms of s. dA ^ = ^"^.s s + E=0 O(s0 dA where A„= A(s = 0) and in general, A = A(n(s),s). The term — is the total derivative de of A, which can be written as: dA_dA dAd^ dA ^P^^^^^ de as dr\ de ~ ds dz dx] de

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31 8z An The derivative ^ = 1, as z = r\. We define rii and A, as follows, — = rii, and, dA -z— = A, Therefore we have: 5s a(ti(s),£)=A„ + A,+5A ^1 8=0 J s + o) (2.48) Equation (2.48) is then applied to the velocity, pressure and temperature of each phase and substituted into the boundary conditions, equation (2.25) through (2.35). Again, the terms of order s" or higher are neglected. U, =W, =01=0 atz = -l (2.49) U2=W2=02=O atz = / (2.50) W2=Wi=0 atz = (2.51) U2=Ui atz = (2.52) 0, =02+il,(l-X) atz = ^Z)0, =D0, atz = n2-n,+ c Tli+2(Z)W, -^DW2)=0 atz = (2.53) (2.54) (2.55) (z)Ui+/coW,)-^(z)U2+icoW2)=/o)Ma(r|, -0i) atz = (2.56) The next step in the procedure is to solve for the Rayleigh number, Ra, in an eigenvalue problem, where the velocity, pressure and temperature of each phase are the eigenvectors. As have been noted by previous workers, the Marangoni number, Ma, and the Rayleigh number, Ra, are not independent of each other for a given experiment. The

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32 ratio MaiRa = T is a constant, which depends upon the thermophysical properties of the fluid and the height of the lower layer. The equation Ma = YRa replaces Ma in equation (2.56). Additionally, the growth rate a will be assumed to be zero. This assumption precludes the possibility of finding oscillatory onset of convection. However, if the latent root, Ra, becomes complex, then it is an indicator that a = is not a solution to the problem and indeed, a is imaginary. The final result is a plot of the Rayleigh number versus the wave number. The procedure for finding this plot will be given in detail in the next section. 2.2 NUMERICAL METHOD The objective of the numerical methods was to solve the set of linear ordinary differential equations (2.40) through (2.47) with the boundary conditions of (2.49) through (2.56). The method of choice was the Chebyshev spectral tau method for three reasons. The first reason is that the spectral tau method, in general, requires very few number of terms to converge to the answer, resulting in a fast and efficient solution technique. The second reason is that the tau method easily incorporates complicated boundary conditions. The third reason is that the spectral method yields a as the latent root if it is so desired and one may then search for the onset of oscillatory convection. This section will briefly describe the details of the Chebyshev spectral tau method, and how it was applied to this problem. For a more comprehensive review of spectral methods, the reader is referred to Canuto et al. (1988) and Gottlieb and Orszag (1986). A

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33 tutorial on the application of the Chebyshev spectral tau method to eigenvalue problems is given by Johnson (1996). Spectral methods are a particular numerical scheme for solving differential equations. It is a discretization scheme developed from the method of weighted residuals (Finlayson and Scriven, 1966). The tau method is one of the three most popular techniques in spectral methods. These three techniques are the Galerkin, collocation and the tau. However, only the tau technique will be used here. Before describing the application of the Chebyshev spectral tau method to the problem, a brief review of the theory behind the method of weighted residuals in order. Suppose you were given the problem du ^— + Lw = XAu ot (2.57) Bw=0 where L and A are linear operators, B is a linear boundary operator, and A, is the eigenvalue. Now express u in terms of an infinite series of trial functions. Here we choose the Chebyshev polynomial as the trial functions. 00 u{x,t)=Y,a„{t)TSx) (2.58) =o The function is then approximated by truncating the number of terms to some finite value, N. N u{x,t)=u^{x,t)= Y.^„it)T„{x) (2.59) The approximation error accrued by truncating the infinite series is given by Syy.

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34 du duj^ -z,-~z— + 'Lu~'Lu;^-XAu + XAu^=Sj^ (2.60) The tau part of the spectral method is simply an easy way of handling the boundary conditions. Note that wjv in equation (2.59) must explicitly satisfy the boundary operator B. This is not always an easy exercise. To accommodate the number of boundary equations, say x, simply add t more equations to N. N+T "ivfeO= H^n(t)T„{x) (2.61) =0 By adding x more variables, we need x more equations. These equations come from the boundary conditions. N+T I;„(0B7;(x) = (2.62) 11=0 In fact, this is how the tau method gets its name. The objective of the method of weighted residuals is to minimize g/v by choosing a test function which is orthogonal to the trial function in some inner product space with respect to some weighting function. The Chebyshev polynomial is orthogonal to itself in the integral from 1 < x < 1 with respect to the weighting function (l x^)~ 1 (t„ (x), T; (x)) = JT, (x%„ (x')(l x'^ )"^ dx' = ^ c„5„„ (2.63) -1 ^ where 12 n = c„ = [ n^ m 1 n = m \ n>Q and 5,„„ = S (2.64) n<0

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35 Substitute equation (2.59) into (2.57) then take the inner product (2.63) of the resuh. After simpHfication the resuh is: da '^^'^ ^-^"^ ^+ i:^nitXTSx),LT,M) = ^i:%itXTXx),AT,M) (2.65) n={) „=0 To evaluate the inner products in equation (2.65), we need to know certain relations between the Chebyshev polynomials and the result of the operation Lr„(x). Suppose L is a linear ordinary differential operator, which may or may not have constant coefficients. For a simple example let L = ^ We want to know the relationship between the set of an and ^„ for n = 0, 1, ., N where: N N u^{xj)=Ya„itX{x) and W(x,0= Z^(0?;(^) (2.66) A list of several linear operators are given in appendix A of Gottlieb and Orszag's book (1986). A comprehensive discussion on how to find the relationship between a„ and bn is given in Johnson (1996). The first and second derivative operators are given below. CnK=^ Y.Pap for Lu = du dx (2.67) p=n+l p+n odd ^nb„=2 £p(/-'K f^ Li/ = ^'y^^2 (2.68) p=n+2 p+n even To evaluate the Chebyshev polynomials on the boundary, the following formula can be used.

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36 ^—T (+i)=(+irmx! IJ (2.69) By using equation (2.67) or (2.68), tlie relationship between „ and b^ can be expressed in matrix form. This method is much easier to implement in eigenvalue problems and the details are given in Johnson (1996). For any derivative q, the ^"' derivative coefficient a„<^^ can be expanded in terms of the zeroth derivative coefficient af2 by the following relationship. where: (2.70) dx'' = Ya%)T,Xx) n=0 (2.71) and the matrix E is given by 2 n = 0,l,...,N-2 ^~p("+i)(m+i)J=~ for example, if N = 5 (2.72) E = ro 1 3 5^ 4 8 6 10 8 10 Oy (2.73) Before applying the Chebyshev spectral tau method to the problem, the equations need to be modified. Typically the double curl of equations (2.11) and (2.14) is taken to eliminate the pressure terms. This results in a fourth order derivative equation. However, it was noted that finding the eigenvalues when the linear operator L contains a fourth

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37 order derivative can cause difficulty in the convergence (Canuto et al, 1988; Gottlieb and Orszag, 1986). This can be seen in equation (2.70). As the derivative q increases, the entries in the matrix E^ increase in magnitude. As the entries in E9 increase, the problem becomes "stiffer" and more difficult to solve numerically. To avoid this numerical difficulty, equations (2.41), (2.42), (2.45) and (2.46) were kept as second order derivatives. To remove the imaginary numbers in these four equations, the divergence of the Navier-Stokes equation was taken, and the equation of continuity substituted for U/. This operation results in the following system of equations. (d^-co^)w, -Z)n,+/?a0j =0 (£)^-co^)li-i?aD0i =0 (2.74) (d^-co^)=), +Wi=0 -(z)2-co^)W2--Dn2+ai?a02 =0 -(D^-G)^)T2-ai?a£)02 =0 (2.75) k(d^ -(0^)02 +-W2=0 The thirteen boundary conditions become: Wj = W; = at z = DW2 = DW, at z = n2-n,+^— ^Jti„+2(dW, -^i)W2)=0 atz = (d^+co^)w, -1^(0^+ CO ^)W2 = a ^ Ma (ri„01 ) atz = (2.76) A:Z)02=D0, atz = 0, =02+Ti„(l-)^) atz = Z)Wj=Wi=0, =0 atz = -l DWj = W2 = 02 = at z = /

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38 We notice that the Chebyshev polynomial lies in the interval 1 < x < 1 whereas the lower phase variables lie on the interval 1 < z < and the upper phase variables lie on the interval < z < / Before we can expand the dependent variables in the upper or lower phase in terms of Chebyshev polynomials, we need to map each phase into the Chebyshev space 1 < x < 1 This is accomplished by the two transformations. X, = 2z + 1 for z < (2.77) 2 Xj =— z-1 for z> The change in independent variables requires the substitution. (2.78) d d dz dx dx ydz d 2 — for z < dx 2 d — — for z > ./ dx (2.79) Now expand each dependent variable in terms of Chebyshev polynomials. n,(z) .0.(4 W^iz) naCz) .02(4 N-T a T; iz) (2.80) KCjj N-T /, T, (z) (2.81) VVyV The surface deflection term, r|, is not expanded in terms of Chebyshev polynomials as it is not a function of the domain variable, z. After equation (2.77) through (2.81) are substituted in to (2.74) through (2.76), the inner product (2.63) is taken. This operation results in a system of 6N+1 equations in the form.

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39 1600 2.5 3,0 Wave Number 3.5 4.0 4.5 Figure 2-2. Plot of the Rayleigh number versus wave number. A^ = RaB^ (2.82) where ^ = (w,,ni,0, W2,n2,02,r|y The two matrices, A and B, contain the Chebyshev coefficients, such as equation (2.73). Further details of these matrices are given in the Matlab programs in Appendix A. Equation (2.82) is simply an eigenvalue problem, which can be solved by several standard software packages. The software package chosen was Matlab. All of the relevant programs written in Matlab are given in appendix A. The construction of each program was very similar. First, the matrices A and B were defined, then the eigenvalues and eigenvectors for a fixed wave number, co, were solved. The wave number was incremented and the eigenvalues and eigenvectors recomputed. At the end of the program, the Rayleigh number versus wave number could be plotted as well as any of the

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40 temperatures and vertical component of velocities versus the fluid depths, for a given wave number. 2.3 UNFOLDING In this section, a technique knovra as unfolding will be described. The unfolding technique derives its name from the plot that it generates. When it is applied to a plot, such as Figure 2-2 above, it takes the wave number from the unbounded geometry and effectively "unfolds" into its discrete azimuthal and radial terms. The result is a series of Rayleigh number versus aspect ratio curves on a single plot. This technique is used to give a qualitative description of the flow pattern in a bounded geometry. Even though this technique can be easily applied to rectangular geometries, only cylinders are considered here. We start our analysis by following closely the technique used by Rosenblat et al. (1982a). Here we will consider a bilayer of two immiscible fluids confined in a cylinder. The top and bottom of the cylinder consist of a rigid, no-slip thermally conductive plate, where the lower plate is at a temperature greater than the upper plate. The interface between the two fluids is flat and non-deformable. This restriction is not essential and will be relaxed when certain examples are discussed later on. The scaling, at least initially, will be the same as the scaling used in the linear model section. Scaling does not affect the results unless approximations are made. Equations (2.1) through (2.3) will still describe the nonlinear behavior of the two fluids. Again, we linearize equations (2.1) through (2.3) and assume the onset of convection is steady, (a = 0).

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41 ^•V|=0 (2.83) V^,-V.:+i?ae,z=^0 (2.84) V^Qi+Wi=0 (2.85) "^•^2=0 (2.86) ^V\,--Vr,+aRaQ,z-0 (2.87) kV%+-W2=0 (2.88) where Vj = (w^ v,. w,. ) w/ is the azimuthal component of velocity, v/ is the radial component of velocity, and wj is the vertical component of velocity. The cylinder has azimuthal coordinates, 0<(p<27T, radial coordinates, 0
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44 Table 2-1. Table of the zeros of the derivative of the Bessel's Function. Radial Mode Azimuthal Mode 1 1 3.83 1.84 4.20 7.02 5.33 8.02 10.17 8.54 9.97 For each m, there exists an infinite number of radial modes, n = 0, 1, 2, ..., where (2.1 15) holds. Table 2-1 gives the first few values of the zeros of the derivative of the Bessel's function, Synn = '^mn a. These values are taken from Abramowitz and Stegun's Handbook of Mathematical Functions (1966). The functions W,(z), 0](z), W2(z), and e^iz) are the solutions to the system (2.74) through (2.76) with a flat surface, ti = 0. The significance of the separation of variables lies in the relationship between Xfy^ and the wave number, CO. By substituting in (2.1 1 1) through (2.1 14) into (2.83) through (2.88), we find a simple relationship between the aspect ratio, a, and the wave number, co, for a fixed azimuthal and radial mode. a CO (2.116) Figure 2-3 is an example of the applicafion of equafion (2.116) to the Rayleigh number versus wave number plot. Upon substitufing equations (2. 11 1) through (2.1 14) into the equations of continuity and applying the definition of the Bessel's function, we find.

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45 u^(r,(p,z)= y. cos(pj',„(^„„/)DW,.(z) / mn ^ v.(r,(?,z)=-y2 sinm(pJ„(x,„„r)DW,(z) w, (r,(p,z)= cosm
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46 2.3.1 Nonlinear Analysis The limitations of the Unear analysis will only allow us to predict which flow pattern will occur at the onset of convection. It does not tell us the behavior of the fluid flow when two flow patterns coexist, such as codimension-two points. Additionally, the linear model does not guarantee that it will predict what we see in experiments, as the experiments are always conducted, at least slightly, in the nonlinear regime. The purpose of performing a nonlinear analysis on this problem is to provide some theoretical insight into the complicated behavior of bilayer convection. The method of choice is the Galerkin-Eckhaus expansion (Eckhaus, 1965). This method has shown some success in predicting and describing certain nonlinear behavior. In the remainder of this chapter, the several steps necessary to develop the nonlinear model will be developed. The derivation will end with one or more amplitude equations which describe the dynamic behavior of each of the various flow patterns. We start the nonlinear analysis be rewriting the nonlinear Boussinesq equations which describe the convection of two immiscible fluids in a cylinder. For the nonlinear analysis, we will also assume the interface between the two fluids is non-deformable. = -V:'i+v\i+i?fle,z f 7i ^ av, ^^1=0 (2.118)

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1 fdy, ^ 47 P P Pr V-V2=0 (2.119) dt ^ /t The two velocities, Wj and w,, appear in equation (2.118) and (2.119) because the base state temperature is not zero. Remember, is the perturbed temperature, therefore the conductive temperature, T^^ needs to be subtracted from the equations. The boundary conditions for the nonlinear problem are the same as those outlined in the previous section. V] =0, =0 at z = -\ V2 = 62 = at z = l w, = W2 = at z = ^1=^2 ^^ z = Vj = V2 at z = 61=62 at z = 56, dQ, k^ = —^ at z = oz oz ^-li-^+M— ^ = at z = oz dz dr 5vi 5v2 1 90, ^' ^' '^"^ (i.m) Next we linearize equation (2.1 18) and (2.1 19). Equation (2.120) need not be modified as it is already linear. V\,p-V,^+i?^0,^z "^^1^=0 (2.121) V20,^ + W,^=O

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48 vVSp--V'2p+ai?^e2^z "^•^2^=0 (2.122) Here the subscript/? represents the wave vector /?= {m nj}, where m is the azimuthal mode number, n is the radial mode number and j is the vertical wave number. For example, ./ = 2 represents vertical stacking of the convection cells. Usually, vertical stacking only occurs far into the nonlinear regime, except for very small aspect ratios. Therefore no vertical stacking will be considered. We assume 7 = 1 always and neglect it from here on out. We also note that there exists an infinite number ofp's corresponding to the infinite number of eigenvalues which satisfy (2.121) and (2.122). 2.3.2 Adjoint The adjoint of a linear operator L is defined in some inner product space as. {lm/,,l*) = (lV,,l) where the asterix denotes the adjoint operator and v|/ is defined as: We can define our linear operator, L from equation (2.121) and (2.122) L = 44 VO44 L2J (2.123) where, O44 is a four by four matrix whose entries are zero. L, and L, are defined as:

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49 L, = V' v^ v^ R. 1 v^J and L2 = f V72 vV Upon inspection we can find the adjoint operator L' L = J-i U44 lo 44 T where L*. and L*, are defined as: (2.123) l; = ^V^ 0^ VVI R; V^y and L2 = ^v^ 1 vV^ vV^ 1 k V ai?; kV^J It is obvious from (2.123) and (2.124) that L is not self-adjoint. Although it appears as if it can be made self-adjoint by multiplying the fourth row of L, by Rp, and multiplying the fourth row of L2 by aRpk. However, as we will see, the boundary conditions are not selfadjoint. Upon analyzing the terms of the bilinear concomitant, the adjoint boundary condifion can be found. For completeness, the linear adjoint problem is given below. V^' V +e z Ip 1/; V-v =0 1/' (2.125) 2p p 2p ^ 2p V-V* =0 KV^e* +aR* w* =0 ip p ip (2.126)

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50 Vip=0ip=O at z = -l V2p = e2p = at z = l Wjp = W2p = at z = P"i*p = "2p at z = pvi*p = V2p at z = ^ip = kGjp =0 at z = 5Glp az + X'M^ dz dz aw*p 5^2p at z = 5z -^ dz 5^i"p az ^V2p = "az at z = ij a / : 1 rv \ sw,; ae; at z = (2.127) arv-p/ ar az =^ "' ''^^ ^""^ = ^'^ The solution to the adjoint problem is found similarly to the non-adjoint solution. "*, (^^'^)= ^m cosmcp j',„(x,„„r)z)W,.(z) v*^(r,(p,z)=-'%;sini.?9J^(:^^„r)z)W,(z) (2.128) \ (r,q>,^)= ^'„„ cosmcp J,„(:^™/)w,(z) for / = 1, e* (r,(p,z)= cosw(pJ,„(:V,„„r)0,.(z) 2.3.3 Galerkin-Eckhaus Expansion The Galerkin-Eckhaus expansion is one method for studying nonlinear problems close to their linear state. The method involves the expansion of the nonlinear dependent variables in terms of all of the linear modes, muhiplied by an amplitude function, A.

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51 f ,, \ Sa/O ^e M^'^ \^2J 2p M.. (2.129) where, Mp = TRp is the pth eigenvalue of the linear problem and M is the Marangoni number of the nonlinear problem. Again, p represents the infinite number of linear solutions. Kp{t) is called the amplitude function, and in general, is a complex function of both space and time. Here we will considered A to be real and only a function of time. The eigenfunctions, Qjp and Q2p are multiplied by M^jM to satisfy the nonlinear boundary conditions. The next step is often called the Liapunoz-Schmidt reduction (Stakgold, 1979) in the mathematical literature. Usually it involves an expansion similar to (2.129) which is substituted for the nonlinear dependent variables and then taking the inner product of the nonlinear system. For some nonlinear operator N and some inner product (*,*) we have / ^w^; = a^iajo \ V p J 'M^. (2.130) For us, N is the nonlinear operator defined in (2.1 18) and (2.1 19), \^ is equation (2.129) and M^p is defined as: V = (w,,Vi,W,,ei,W2,V2,W2'02y The inner product that will be defined here as: V;,'Vp -1 dr d(p

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52 Substituting (2.1 18), (2.1 19), (2.129) and \\ip into (2.130) gives: \\ \p dt I \ '-P dt 1] Mp[\ ^p dt / \ 2p' dt \,.(^^^.-^0)-(v;.(vv^.-iv,)). ^{(v;.e,z)+a(v;,e,i)}Pr-{(v;.v,.Vv,) + (v;,v,.Vv,)} + -^{(9;.v,.Ve,) + (e;,v,.V9, (2.131) We now let Q represent the linear terms in equation (2.131) fi = {v;.(v^v,-V,)).(v;,(vVS-iV,|. (2.132) Before continuing, it can be shown that thep^'^ eigenvalue of the adjoint problem is equal to the p^'^ eigenvalue of the linear problem, that is Rp Rp. Upon simplification and substitution of i?p = Rp and equation (2. 129), Q is simplified to: e = ^{("-;)^i("^-8;)}A, (2.133) The number of terms in (2.131) may seem a bit daunting, particularly the nonlinear terms. However, we can use the orthogonality conditions of the trigonometric functions and the Bessel functions to simplify matters considerably. These orthogonality conditions give us:

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2% 53 vjp,v,.^) = (e;^,e,.^) = (e;^,w,.^) = for p^^q and / = 1,2 (2.134) The number of nonlinear terms can be dramatically simplified by the following formula. '.% j/i(m(p)/2((p)/3(;?(p)'ip.v,^) + (v;^,v2^)}f(e;^,e,^) + (e;^,e2^) (2.137) pqr ^P^{^xpK) + ^{^2p,Q*2p) (2.138) M-Mp ^P^^r~ (2.139) p (2.140) The parameter, Zp, is called the supercriticality parameter because it represents the degree to which mode/? has become supercritical (if at all). Equation (2.136) represents

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54 the dynamic behavior of the infinite number of hnear modes. To make this problem more tractable, we need to decrease the number of amplitudes, A^,, to some finite set. In order to determine which /j's to keep and which />'s to ignore, we need to look at the supercriticality parameter, Zp. The value of Zp is determined by the eigenvalues, Mp or Rp, noting that Mp and Rp are interchangeable through the relationship Mp = TRp. Assume we performed a linear calculation and found a large number of Mp, many more than what we would use in the weakly nonlinear analysis. This finite set of MpS is called S. The critical eigenvalue is defined as the smallest element in the set S. M, smin(M^) ^p^S (2.141) We next define the parameter 5^, which is a measure of how stable the /?* mode is with respect to the critical mode, Mq. M,-M^ ^p^~T4 "^P^^ (2.142) p Note that the value of 5^, lies between < 5p < -1 The parameter 5^ is then used to group the set S into three distinct sets: 5^, S^, and 5 {5^ U^s }• '^u represents what will be called the unstable set and ^s represents the stable set. The elements which do not fall into any of these two categories are ignored. The two sets are defined as follows: S\i^^p 5^<;u,/7g5| (2.143) ^s-f^p| |§J<;s,/'e{S-5u}} (2.144)

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55 where ^^ and C,^ are arbitrary values (for example, Cu = 0-1 and Cs = 0.5). The set S^j will always contain at least one element, M^, and may contain more. The number of elements in Ss is determined by the cut-off value, C,^. This value is definitely not fixed, and will be determined more by the experiments than by some numerical value. For example, in the next section, we will see experiments where the modes (m = 2, n= 1), (m= 1, n= 1) interact with the critical mode (m = 0, n = 2). Therefore, we would choose C,^ such that only the modes (m = 2, n = 1) and (m = 1, n = 1) are in the set S^. The set of equations (2.136) is further simplified by the procedure known as the adiabatic reduction of the slave modes (Manneville, 1990). Pick a mode s which is an element of the set S^. If we assumed that we are only slightly nonlinear, M=M^, then the supercriticality parameter of the stable mode 5 is s^. < and |sj 1 Now suppose that some finite disturbance causes the amplitude A^ to become non-zero. Initially, the nonlinear terms involving A^ contributes little to the unstable modes. Additionally, the linear contribution will not contribute significantly to the dynamic behavior of the unstable modes, except for short times. Therefore we assume: -^ = and A^(a,,A^)=0 V^ e^s,;^ e {^^ U^s} When these two assumpfions are made, equation (2.136) for all of the modes in the set S^, becomes: ^s=-LL^—'^p^q yseS^ and p,q eS^ (2.145) Substituting equation (2.145) into (2.136) gives:

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a 56 P „ D A V^ X^ A X"' V^ V^ sqr dt q r s q r ^ s \/p,q,r sSu and seS^ (2.146) Before leaving this chapter, I would like to clarify an often neglected topic in this method. The topic involves the determination of S^j and 5*5. This determination is often called the normal mode reduction. For simplicity, assume the eigenvalue in our linear problem is the growth rate Op. Again, we use the index p in the same context as before and we order the infinite set of Gp from the largest (least stable) to the smallest (most stable). Here, R (or M) is just a parameter, and for every parameter, R, we determine a unique set of OpS. Suppose we pick some R and perform our linear calculation to determine the set of Op. In general, a ^ = a ^^ + /a ^ ^ where o^ ,is the real part of Op and Gpi is the imaginary part of Gp. Now place each Gp into three groups. The first group, called the unstable manifold, is for all Gp where Gp^r > 0The second group, called the center manifold, is for all Gp where cjp,r = 0, or more generally hi <^p,r < h\ The last group is called the stable manifold where Gp^r < 0The next logical step, one would assume, would be to let all elements in the unstable manifold to be in the set 5^ and let all elements in the center manifold to be in the set S^. The elements in the stable manifold are completely ignored. Following this classification procedure, we would perform the adiabatic reduction of the elements in S^ and arrive at equation (2.146). However, there is one important assumption that is made in this technique; and that is whether the linear model is still valid for Gp > 0. If we go

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57 back to the derivation of the linear model, we note that the linear model is only valid until the first critical mode becomes unstable. The only thing we can say for certain is that this unstable mode (or modes) grows exponentially, at least initially. Beyond this, we can not guarantee that the linear model is still valid. Therefore, when a value of the Rayleigh number, larger than the critical value, is used {R > Re), the eigenvalues, (e.g. Op) and eigenfunctions (e.g. W,(z)), may be meaningless. This argument is perhaps best explained in physical terms. In the course of conducting a convection experiment, a temperature difference is applied. When this temperature difference is less than the critical temperature difference necessary for the onset of convection, the fluid is in a thermally conductive state. When the temperature difference reaches its critical value, the fluid begins to flow. The linear model will accurately predict the critical temperature difference and can tell what the flow will look like at the onset of convection. Once the temperature difference increases beyond the critical value, the linear model can not predict how the flow will interact with itself and other flow patterns. This is most pronounced near codimension-two points.

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CHAPTER 3 EXPERIMENTAL APPARATUS AND PROCEDURE The objectives of the experiments were to observe the behavior of the fluid convection at codimension-two points aspect ratios where two flow patterns coexist and at fluid depths where the initiation of convection switches from the lower layer to the upper layer. To accomplish this, the test section was designed so that cylindrical inserts, with different radii and heights, could be used interchangeably. The popular bilayer system of silicone oil and air was chosen to both simplify the experiments and to generate results which can be compared to previous experiments (Koschmieder and Biggerstaff, 1986). In order to cause buoyancy-driven convection in one or both fluids, the bilayer IR camera Electronic Hardware Unit Test Section Figure 3-1. Overall schematic of the experimental apparatus 58

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59 system needed to be heated from below or cooled from above. The vertical temperature difference applied across the test section was accurately monitored and controlled using a computerized data acquisition system. When the temperature difference was large enough to cause convection in the silicone oil, the change in the temperature field at the silicone oil-air interface was detected using an infrared camera. This chapter is split up into two parts, a description of the experimental apparatus and a walk-through of the procedures taken when an experiment was performed. The experimental apparatus is divided into four major parts: the infrared imaging system, the test section, the heating and cooling system, and the electronic hardware unit which includes a computer-controlled data acquisition program. For a very thorough explanation of all aspects of the experimental apparatus, an instruction manual was designed. This manual can be presented upon request. 3.1. APPARATUS 3.1.1. Infrared Imaging System One of the first decisions that needed to be made was the type of flow visualization technique to use. Predominantly, two different methods were well known at the time, shadowgraphy and particle seeding. While both of theses methods were well-established, each had its draw-back. The shadowgraphic technique involved shining parallel light through the layer of fluid, reflecting the light off a mirror at the bottom of the test section, then shining the

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60 Figure 3-2. An example of a shadowgraph picture (a) and an aluminum particle experiment reflected light onto a white background. The pattern that appears on the white background works by the following principle. When a fluid is heated, its density changes. The change in density causes a variation in the index of refraction of the fluid. Therefore, as parallel light shines through the fluid, the light is either concentrated or reflected. This concentration or reflection of light gives the flow pattern of the fluid. The interested reader is referred to the references, Eckert and Goldstein (1976), Goldstein (1983) and Koschmieder (1993). In particle seeding, plate-like, reflective particles are added to the fluid of interest. When the fluid flows horizontally, the particles lie flat and reflect light, causing the fluid to appear brighter. When the fluid flows in the vertical direction, the thin side of the particles point up and little light is reflected, causing the fluid to appear darker. From this, the flow pattern can be observed. Both of these methods, however, have their flaws. In shadowgraphy, the light has to pass through the fluid twice before it appears on the white background. This effectively averages the temperature throughout the fluid. This can cause a misguided analysis of the flow pattern, particularly if most of the flow appears at the surface. In

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61 particle seeding, the particles are often much denser than the fluid and quickly settle during the course of the experiment. This was of particular concern as most of the experiments lasted several hours to several days. Secondly, the addition of the particles can unpredictably change the thermophysical properties of the fluid. Another method which has previously received some attention is particle image velocimetry (PIV) (Adria, 1991; Pline et al., 1991). Here again, particles are added to the fluid. In this method, most of the particles are individually tracked every given time period. From this, the velocity of the fluid at a point can be determined. Because the software needed to calculate the numerous velocity vectors was still in its infancy, we decided to forego this option. The method that we chose to visualize the flow patterns, was an infrared (IR) camera. The IR camera has two major advantageous. The first is that it is a non-intrusive method of visualizing the flow, and secondly, it can be used with opaque fluids. The ability to use the camera with opaque fluids is of particular interest in the application of crystal growth, where opaque liquid metals are used. The IR camera that we used was an Inframetrics model 760. The basic feature of the IR camera is a single Mercury-Cadmium-Telluride (Hg^Cdi.Je) chip which measures infrared radiation in the 3 to 12 ^lm and the 8 to 12 ^m range. Each wavelength corresponds to an optimum temperature range. For our experiments, the typical temperature range was around 25C to 45C. This temperature range is best measured using the 8 to 12 |um wavelength. The 3 to 12 |j,m wavelength works best for higher temperatures (around lOOT to SOOT). Additionally, the

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62 transmission of infrared radiation through the atmosphere, is much better in the long wavelength, 8 to 14 \xm region. The field of view (FOV) is detected by scanners using electromechanical servos (galvanometers), much the same as a standard television set. For this camera, the FOV resolution is 640 X 480 pixels. Every other line (320 X 480 frame) is sampled 60 times every second, giving a full interlaced picture (640 X 480 frame) 30 times every second. This is the NTSC standard. The accuracy of the temperature measurement is 0.2C for each pixel. The majority of this error is due to noise, caused by random emissions of photons. Fortunately, the camera has a built in feature which allows each pixel to be averaged over 2, 4, 8, or 16 frames. When the picture is averaged over 16 frames the error in each measurement drops to 0.05C. For all of the experiments performed, the 16 frame averager was used. Additional features in the camera were used to improve the picture quality. The simplest of which was a 3X magnification and a 12" close-up lens constructed of magnesium. For each lens, the transmission of IR radiation of 3 to 12 ^im is 99%. The second feature, which is bulk into this model of IR camera, is the temperature "window". The temperature window is the maximum temperature difference which the camera can detect. The Inframetrics model 760 has temperature ranges of 2, 5, 10, 20, 50, and lOO^C, although only the 2C and 5C window were used. By selecting the minimum temperature window, the temperature resolution can be maximized. The infrared image, which is eventually seen, is a false-color image of the temperature field. Every 1/30"' of a second, a 640X480 frame of IR radiation is

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63 measured. The IR radiation is converted to a temperature by a value of the emissivity, which must be entered by the user. The temperature is then mapped to one of several available color palettes. For these experiments, the effective emissivity was found as follows. The test section was constructed as if an experiment were to take place. The test section was filled with silicone oil and allowed to equilibrate to room temperature. The temperature was then accurately measured with a mercury thermometer. The emissivity was changed until the reported temperature of the camera matched the temperature of the thermometer. 3.1.2, Test Section It was decided that the design of the test section should follow closely the design of Table 3-1. Table of the thermophysical properties of the material and fluids used in the experiment. Parameter Units Dow Corning Oil Air (ZCC) Zinc Selenide Copper Lucite Density (g cm"') 0.968 0.0012 5.27 5.96 1.19 Negative Thermal Expansion (io'c-') 9.6 33.3 0.078 0.501 7.3 Thermal Conductivity (10-^ergcm-'sec"'C') 1.59 0.262 180 4010 1.7 Thermal Diffusivity (10' cmsec') 1.10 182 66.3 1160 Kinematic Viscosity (stokes) 0.692 0.157 Interfacial Tension (dyne cm'') 20.9 (under air) Negative Interfacial Tension gradient (10-^ dyne cm"' C') 5.8 (under air)

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64 previous experiments (Koschmieder and Biggerstaff, 1986; Koschmieder and Prahl, 1990). It was important to choose the two fluids and the material properties such that they satisfied the assumptions made in the theoretical analysis. The theory did require two conditions on the fluid boundaries, though. First, the upper plate at the top of the upper fluid and the lower plate below the lower fluid, should be rigid and a perfect conductor. For this, zinc selenide and copper respectively, with their high thermal conductivity, were chosen. Secondly, the radial gradients should be minimized through the side-walls of the cylinder. The material chosen here was lucite, whose thermal conductivity is close to that of silicone oil. However, the lucite walls were thick enough (greater than 3/8") so that any temperature perturbation from the outer walls was minimized. The values of all known thermophysical properties are given in Table 3-1. The two fluids chosen were a high viscosity, Dow Corning silicone oil with nominal viscosities of 100 cS or 200 cS, and air. Other choices of gases, such as helium, were discarded due to the difficulty of containing leaks. A second liquid layer was also feh unnecessary as many unanswered questions were left for this simpler bilayer system of a liquid and a gas. The test section itself consisted of five separate pieces: a lower heating bath, a liquid insert, an air insert, a clamp and an upper plate consisting of zinc selenide. All pieces of the test section, except for the zinc selenide window, were constructed from lucite. To maintain a flat, silicone oil-air interface, the liquid insert contained a "pinning edge" and a reservoir (Figure 3-3). If additional silicone oil were added, the oil would spill over into the reservoir. The pinning edge would then eliminate any menisci, keeping the interface flat. After carefially filling the liquid insert with silicone oil, the air insert

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65 air height 1 1 1 1 n tj];~: Uquid height Zinc Selenide Lens Lucite Clamp Air Height Insert Pinning Edge Liquid Insert Heating Block Figure 3-3. Cross-sectional view of the test section. could be placed on top of the liquid insert. The clamp would fit on top of the air insert and four screws, which ran through the clamp and into the lower heating bath, were tightened to hold down the liquid and air inserts. The zinc selenide window was then placed into a groove in the clamp. The clamp and lower bath were constructed so that different liquid an air inserts could be used. Further details and drawings of the test section are included in appendix B. Complete details of the experiment can be found in the experiment instruction manual. One of the most important considerations in designing the experiment was how well the applied temperature difference across the oil-air bilayer, could be maintained. This consideration was what led to the rather complicated design of the lower heating bath. The lower heating bath consists of many parts but is primarily a continuously stirred, hot water bath heated from below by an electric heater. The walls of the cylindrical bath are constructed of y4" lucite. The top and bottom of the cylinder are capped with 3/16" thick copper disks. Water and one magnetic stir bar are placed inside the bath. The bath is then

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66 placed on top of a three inch diameter 2.5 W/in" flexible heater. The heater is turn is placed on top of a magnetic stirrer, which itself sits on a leveling plate. The largest difficulty of the lower bath was preventing air bubbles from forming in the bath, which was eventually eliminated by constructing an over-spill port. The overspill port was simply a hole in the side of the bath where a tube was inserted. The bath would be over-filled with water such that additional water would spill out through the over-spill port, and the end of the tube raised above the top of the bath. When the water was heated and subsequently expanded, the excess water would flow into the tube. When the bath cooled back down, the water in the tube would flow back into the bath, thus eliminating any air bubbles. Without the over-spill port, the expansion of the water would create too much pressure on the bath and the bath would eventually crack. In addition to its availability, water has a high heat capacity, which makes it ideal for temperature control. Although it takes longer to heat water to a certain temperature, the high heat capacity will hold the temperature constant longer making it easier to control. The stirring of the water by the magnetic stirrer helped to prevent any temperature gradients from forming. Because of the infrared imaging system, the top of the test section needed to be heated by an infrared transparent medium. Here again, the simple choice of air was made. The requirement of an infrared transparent material also dictated the use of the 5mm zinc selenide window. Zinc selenide has a high thermal conductivity and is greater than 60% transmittive to infrared radiation between 0.7|um and about 17^m. Zinc selenide is also slightly reflective to radiation in the 8 to 12 )am range. This reflection

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67 caused problems with the imaging of the siUcone oil-air interface. This problem was resolved by coating the zinc selenide window with an anti-reflective polymer, which was performed by II-VI incorporated. 3.1.3. Heating and Cooling System The heating and cooling units consist of three parts: the electric heater for the lower bath, the electric heater used for the upper plate and ambient air, and the cooling water used to cool the ambient air. The objectives of the heating and cooling units were to add or remove heat, when necessary, in order to maintain the constant temperature. The entire test section, IR camera and air heating unit were enclosed in a clear. Incite box. The temperature of the air inside the box was monitored by one of the thermistors and controlled by an electric heater (hair dryer). The air was then stirred by a fan to prevent temperature gradients from forming. Additionally, a radiator, in which chilled water was pumped through, was used to continuously remove heat from inside the box. This prevented the temperature of the air from becoming too high. The chilled water was also kept at a constant temperature. There were two reasons why the test section and IR camera were enclosed by the Incite box and the air inside kept at a constant temperature. First, keeping the temperature exterior to the sides of the liquid and air inserts constant, minimized heat from flowing through the side-walls. Secondly, the absorption of infrared radiation by the atmosphere changes as a function of temperature. To prevent any fluctuations in the transmission of infrared radiation between the IR camera and the test section, the air was kept at a constant temperature.

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68 Air, unlike water, has a low heat capacity. This caused difficulties in controlling the upper temperature and could not be controlled as well as the lower bath's temperature was controlled. The deviation of the upper temperature from its setpoint created the largest error in the overall temperature control. Nonetheless, the overall temperature control was quite good, with a standard deviation of 0.2C overall. We note here that the lower temperature is read from the bottom of the lower copper plate and the upper temperature is read from the top of the zinc selenide window. Due to the high thermal conductivity of the copper and zinc selenide, small temperature perturbations occurring at the top of the zinc selenide and the bottom of the copper plate, would be smoothed out before they reached either of the two fluids. For this reason, the actual temperature across the two fluids was probably even better than is reported here. 3.1.4. Electronic Hardware Unit The objective of the electronic hardware was to link the temperature readings to the computer-controlled program, and then transmit control decisions from the computer program to the heaters. Additionally, the computer would control at which times the VCR recorded the infrared images. For this experiment, three different temperatures needed to be maintained at a constant setpoint: the lower bath, the temperature difference across the bilayer of fluid, and the cooling water temperature. The temperatures were then reported to the computer where the program would read the temperatures, and based upon a given control algorithm, determine whether any heaters should be turned on or off.

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69 Each of the temperatures were measured using a highly accurate thermistor. The types of thermistors used were Omega, Unear response, model OL-700 series, thermistors. The thermistor located in the lower heating bath was a water-proof small surface thermistor, the thermistor located on top of the zinc selenide plate was an attachable surface mounted thermistor and the thermistor located in the cooling water tank was a general purpose, water-proof thermistor. A thermistor is a temperaturesensitive electrical resistor. As the temperature changes, the amount of resistance changes thus changing the voltage drop across the thermistor leads. This voltage drop can then be calibrated for a given range of temperatures. The thermistor was chosen over the cheaper and more available thermocouple because the thermistor was more accurate and the calibrations did not "drift" over time. This last feature is important as some of the experiments could last up to three days. A specially designed, constant, 0.5V power supply was applied across each thermistor. The resulting voltage drop across each thermistor and the 0.5V from the power supply were then read into the computer through a data acquisition board. The data acquisition board was a DAS1601 from Keithley Metrabyte. The DAS1601 has 16 analog input channels, with a sampling frequency of 0.1 MHz. Each analog input channel is converted to a digital number using a 12 bit analog-to-digital converter (ADC). As the range in voltage is from OV to lOV, the ADC conversion error is 2.5mV (lOV divided by 2'-^). The computer which housed the data acquisition board and ran the control program was a PC compatible, Intel 486-66 MHz with an ISA motherboard. Data was continually read from the data acquisition board by the control program.

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70 The control program was written by myself in Visual Basic, version 3.0 and ran under Microsoft V^/indows 3.1. Major revisions to the program were later performed by Bryon Stakpole. The program read in the input (temperatures), and based on a proportional-integral-derivative (PID) control algorithm, determined the value of the output (whether to turn the heaters on or off). The parameters used in the PID control algorithm were taking from Seborg et al. (1989). The temperature readings, setpoints, output values, as well as other relevant data were displayed on the computer monitor. The temperature readings and setpoints were recorded to a data file on the computer's hard drive. A flow chart for the programming logic, is given in Figure 3-4. After determining which heater should be turned on or off, the control program would write the necessary data to the data acquisition board's output register. The board would then send the digital signal to the electronic hardware unit. Inside the electronic hardware unit were several circuit boards. Each circuit board consisted of several channels which read each individual bit from the data acquisition boards register. If the bit was on, the chaimel would trigger a transistor. Electricity would then flow through the transistor to a sold state relay (SSR). When electricity flowed through the coil side of the relay, the relay would close and allow electricity, at a higher current, to flow through the other side of the SSR and into the heaters. This process of reading the temperatures, performing a control decision and turning on the heaters (if necessary) was continually performed as fast as the computer could execute the control program. One of the automated features of the program was to change the temperature difference (setpoint), after a given period of time. The duration of each setpoint called a segment usually lasted two to four hours, and there were always several segments in

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( start Program Initialization Information Fomi Add. Edit & Delete Fluid Parameters 71 -NoExists? No .+_ Setup Data File -Yes*Ovenwrite? Pause Start/Pause Loop? Initialize Data Acquisition Board IR Program Form M — No Start Analog Data In Statistics (Data Averaging) Update Screen Variables Prepare X-Axis \ Data Prepare Y-Axis Data I Ctiange Segment Number lYes Pause Program? Segment Over? Experiment Over? Shutdown Flag (Yes Overheating? Update Graph Control Decision Stiutdown Flag Heater Control 0fl Oft Send Digital Output To Heaters Tfm -vH' w^'^'^ No NoVCR Control 1 -Record Pause 1 Send Digital Output roVCR Shut Down Experiment? Send Digital Shutdown Signal Exit Program Figure 3-4. Flow chart of the programming logic

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72 each experiment conducted. For this reason, it was advantageous to automate the entire experiment. As was mentioned in the beginning of this chapter, the objective of the experiment was to record the flow pattern at the onset of convection. The flow pattern was detected with the IR camera and the image was sent to a VCR. Because of the duration of each experiment, the VCR could not continuously record the IR images, for it would exceed the limit of the VCR tape. To work around this, the VCR was controlled by the computer. Every two minutes, the program would tell the VCR to record and after five seconds, the program would send a signal to pause the VCR. As the fluid flow was very slow, this interval would not miss any dynamic or transient behavior. Controlling the VCR allowed an entire experiment to be conducted without any intervention, sometimes over-night. At the end of the experiment, the program would shut off all power to the experiment, including the computer. 3.2. PROCEDURE This section will list, in chronological order, the procedures that were performed in order to properly conduct an experiment. For the sake of efficiency, the first operation was to turn on the IR camera. The reason is that the infrared detector must operate at temperatures lower than 77 "K. To reach these temperatures, the IR camera has a built-in Sterling pump which removes heat. This process usually took a couple of minutes. When the temperature was below the minimum operating temperature, the pump slowed down. As the pump made a detectable

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73 amount of noise, this shift in pump speed was an indicator that the IR detector could be turned on and used to detect infrared images. Once the IR camera was running, it was usually a good idea to check all of the operating parameters of the IR camera, to ensure they are all correct. The next step in the procedure was to load the test section. First, the liquid and air insert of interest were chosen and screwed down with the Incite clamp. As the clamp had a hole in the center for the zinc selenide window, the silicone oil could be added from the top. Silicone oil (or whatever other fluid was being used) was added until it looked like Initialization Information Program Parameters Data File Name: lltesttx! Directory Name: C:\lrprog2 Number of Segments I Segment Time (min) Write to Data File Time (sec): Record Time (sec) Interval between pi20 recordings (sec) Et Physical Parameters Lower Fluid Set Points Upper Fluid Depth (cm) Kinematic Viscosity (cmVs) TTiermal Diffusivi^ (cm*/s) Thermal Expansion O/'C) Thermal Conductivity (W/cm-C) Density (g/cm*) 0.998 0.157 0.182 2.62E+03 Delete Lower Bath Overall Temp Difierence Water Tank Temperature (Lower Both Upper Plate) Temperature 35 + 5 T 35.5 5.S 1 1 increment 36 6 1 1 36.5 6.5 |o.i 1 37 7 + 22 tj 22 "" 22 — 22 22 Lower Fluid Upper Fluid Temp. Diff. Temp. Diff. l./I.H ) 57 \.n .^ I M, 2.00 i. 3.5? t 3.03 "" A.2<.) .... AM sm f* Graph Length Jiqqq Time (sec) '— — — J Comments: ? "^ Aspect Ratio &. — Control Parameters OBong Bang O PI control PID control Gain Taul TauD 1 1 1 4 1.4 A .1 .02 .1 Figure 3-5. The initialization program window.

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74 the section was full and the interface was flat. Conveniently, the flatness of the interface could be checked using the IR camera. The reason is that silicone oil reflects a certain amount of infrared radiation, much as the zinc selenide does. When the interface was not flat, the silicone oil interface acted as a lens to infrared radiation. When the interface was depressed in the center, IR radiation was concentrated and the center appeared warmer than the edges, even though all of the silicone oil was at one, constant temperature. When there was too much silicone oil, the interface was elevated at the center, dispersing IR radiation, which made the interface appear cooler. This method of detecting the flatness of the interface was very sensitive to the addition of even small amounts of silicone oil. Once the test section was filled, the level of the test section was checked and adjusted, if necessary, with a leveling plate, which the test section sat on. The TV and VCR were then turned on. To obtain the best image possible, the proper magnification and close-up lens should be used such that the silicone oil-air interface filled most of the TV screen. The focus of the IR camera was then adjusted to get a sharp picture. The power to the computer and the electronic hardware unit were then turned on. This began, among other things, the magnetic stirrer. The bath was then checked to ensure no air bubbles have formed. If there were bubbles, water was added and the bubbles forced out through the over-spill tube.

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75 IR Program Version 3.0 1 — input Upper Plate Water Tank RefcrencG Voltage Lower Fluid Upper Fiuid Voltage |.3E4 { Temperature ]30.18 1 Set Point 1 30.00 1 |.377 1 J2B.01 1 1 30.00 1 ].387 1 21.32 Temperature Difference 22.00 1 Set Point ].S17 1 Rayleigit Number { 2596. I 1 1.472 1 {l.8S { I 3_3B6 1 {3.E84 { |4.64 1 1 — Output: 1 Lower Bath 1 g i Upper Plate Lqi JU Water Tank | m J^ Time; Date: 12:09:58 pm 8/5/97 |M^H I — Parameters: Exp. Started: Exp. Ends: | B:59:55 pm 11:59:55 am Segment Began: 11:53:55 am Segment I 12:59:55 pm Ends: — Segment Number: 1 of 7 VCR Status: Paused Control Type: PID / — (Si e. Pause VCR Exit Total Temperature Difference Temp Difference 720 1080 1440 Time 1800 •.I'lMMMWian.Figure 3-6. The Main program window After the computer powered up, the control initialization program ran. Here, all of the program parameters, physical parameters of the liquid and air inserts, control parameters and the setpoints, were entered. A picture of the initialization program is given in Figure 3-5. Once all of the necessary information was entered, the OK button was clicked to go to the main program window (see Figure 3-6). Before clicking on the Start button, all of the wires from the computer to the electronic hardware unit were checked, and the connections to the thermistors were secured properly. This would avoid receiving faulty temperature readings which could ruin the experiment. If everything checked out, press the Start button. The program would begin to run.

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76 The first temperature difference that was chosen, should be less than the critical temperature difference necessary for the onset of convection. If the fluid began to convect before it reached its first setpoint, then the program should be terminated and restarted at a lower temperature difference. This was important, as the onset of convection needed to be approached from the conductive state and the temperature must be held steady, long enough for the fluid to reach equilibrium. Usually the step size for each temperature difference was around O.rC across the silicone oil layer. Therefore, the first temperature difference was applied and held constant for several hours. No flow was observed. The temperature difference was then increased a little and the silicone oil interface observed, to see if a flow pattern appeared. This was repeated until the temperature field at the oil-air interface changed into a particular pattern. At this point, the temperature difference and the flow pattern were recorded.

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CHAPTER 4 RESULTS AND DISCUSSION 4.1 INTRODUCTION In this fourth and final chapter, resuhs from the hnear stability analysis, weakly nonlinear analysis and the experiments will be given. Each of these was used to shed some light on different phenomena of bilayer convection. All of these events are a fimction of the geometric parameters of bilayer convection in cylindrical containers: aspect ratio (radius/height), the ratio of the fluid depths, and the total depth of both fluid layers. The results have been summarized into four major categories. The first topic is the oscillations, or mode switching that occurs at certain codimension-two points. The second topic will show that an increase in the air layer can affect, or even cause, fluid convection in the lower fluid layer. The third topic deals with how the driving force for convection (either buoyancy or interfacial tension) and the type of convection coupling (either thermal or viscous) can switch as the radius of the container is increased, even though the fluid depths are fixed. The fourth part will contain results from a weakly nonlinear analysis on the effect of air height on bilayer convection. 77

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78 4.2 CODIMENSION-TWO POINTS As was mentioned in the pattern selection section of the Physics and Historical Perspective chapter, there exists certain aspect ratios where two different flow patterns can become simultaneously unstable. These aspect ratios are called codimension-two points. To investigate these points, a series of experiments and linear calculations were performed (Johnson and Narayanan, 1996). The experimental apparatus and procedures are described in the Experimental Apparatus and Procedure chapter. The linear calculations were performed by Zaman and Narayanan (1996). Table 4-1 gives the critical Marangoni numbers for the azimuthal modes 0, 1,2, and 3 for two different aspect ratios of 1.5 and 2.5. The Marangoni numbers were calculated using a three dimensional model of the linearized Boussinesq equations in a cylinder. The bottom and radial walls of the cylinder were assumed to be rigid, with a no-slip condition. The gas above the liquid was assumed to be both mechanically and thermally passive. The bottom of the cylinder was held at a constant temperature, while the radial walls were assumed to be conductive and the liquid-gas interface was modeled with an effective heat transfer coefficient. Finally, the liquid surface was assumed to be flat and nondeformable. This was done to decrease the computational time and difficulty. Table 4-1. Critical Marangoni number associated with each mode for aspect ratios of 1.5 and 2.5 Mode 1.5 Aspect Ratio 2.5 Aspect Ratio 90.45 69.4 1 101.3 70.8 2 112.0 70.4 3 129.8 73.0

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79 There are two items of information to be obtained from Table 4-1. First, is that the mode associated with the smallest Marangoni number, for a fixed aspect ratio, will be the mode (flow pattern) present at the onset of convection. The second item of information is the difference between the smallest and the next smallest Marangoni number. For example, the difference between the first two Marangoni numbers for the 1.5 aspect ratio is about 12%. The difference between the first mode (m = 0) and the second mode (m == 2), for the 2.5 aspect ratio, is quite small, about 1.5%. The reason the modes are so close is because the 2.5 aspect ratio is near a codimension-two point. This difference is important experimentally when one tries to resolve which flow pattern will be present at the onset of convection. The first experiments used 86 cS silicone oil in a 5mm deep liquid insert with a 1.5 aspect ratio. From Table 4-1, the predicted flow pattern is the single toroid (m = 0). The toroidal flow is depicted as fluid moving up the center of the cylinder, moving radially across the top, then falling down along the sides. Indeed, as seen in Figure 4-1, the infrared camera captured this flow pattern at the onset of convection. Further moderate increases in the temperature difference did not change the flow pattern. The second set of experiments used an aspect ratio of 2.5, which is close to a Figure 4-1. An infrared image of the toroidal flow pattern in a cylindrical container. The picture is taken looking down onto the oil-air interface.

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80 codimension-two point. Again a 5mm deep layer of 86 cS silicone oil was used. At the onset of convection, a very faint m = 0, double toroidal pattern was seen. This agrees with Table 4-1. However, when the temperature difference was increased by 0.05C across the liquid layer, the flow pattern changed from the static double toroid to a dynamic mode switching behavior. This flow pattern started with an m = 2, bimodal flow (Figure 4-2a). One convection cell then increased in size forming a pattern resembling a combination of the m = 1 unicellular flow and an m = single toroidal flow (Figure 4-2b). When this cell reached some critical size, it split into two cells (Figure 4-2c). Here the flow pattern was the same as the first bimodal flow pattern rotated be 90 (Figure 4-2d). This process then repeated itself (Figure 4-2e and 2f), returning to the original bimodal flow pattern. This process of switching between different flow patterns repeated itself approximately every twenty minutes. As long as the temperature difference remained constant, this mode switching continued at a regular interval, although the exact period Figure 4-2. Time sequenced infrared images showing the switching between flow patterns. The convection cells continuously oscillate between the different flow patterns with a regular time interval as long as the temperature difference across the liquid is held constant.

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has never been accurately measured. This experiment was performed several times in a somewhat sloppy manner and the oscillating behavior was seen every time. This was done to verify that the oscillating behavior did not just occur for a small parameter range. Additional experiments were also performed for a 2.6 aspect ratio with an 11.1 mm air height. Here the flow pattern at the onset of convection was seen as a superposition of a bimodal, m = 2 and a double toroidal pattern. This experiment showed that a codimension-two point did indeed exist near or at the 2.5 aspect ratio. However, as will be explained later, the superposition of the two patterns may have been due to convection in the air. These set of experiments were able to prove that different linear modes can interact with each other (and themselves) to yield dynamic nonlinear behavior. A similar observation was seen for Rayleigh-Marangoni convection in square containers (Ondar9uhu et al, 1993). Although this work mentioned that the oscillating behavior was a result of a Takens-Bogdanov (Golubitsky et al., 1988) bifurcation, which is associated with codimension-two points, they did not prove that it was indeed a codimension-two point. Secondly, the oscillations only occurred well into the supercritical region. Codimension-two points were also studied in pure buoyancy flows (Zhao et al, 1995). However, no oscillating behavior was seen for any of the aspect ratios investigated. From these experiments, it appears that the free surface has something to do with the dynamic behavior. Several theoretical works describe weakly nonlinear behavior near codimension-two points. Erneux and Reiss (1983) looked at supercritical bifurcations of two degenerate eigenvalues (i.e. codimension-two points). They noted that when the supercritical

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82 bifurcation was symmetric, and no imperfection was introduced, the steady solutions would branch off into a steady secondary solution as the bifurcation parameter was increased. However, when an imperfection to the base state was introduced, Hopf bifurcations to a secondary solution were possible. This result could imply that the free surface in Rayleigh-Marangoni convection acts to break the symmetry of the problem. Rosenblat et al. (1982a) performed a weakly nonlinear analysis for the pure Marangoni problem, neglected buoyancy effects. In their analysis they showed that for an m = 1, m = 2 codimension-two point, it was possible for secondary Hopf bifurcations to occur for aspect ratios slightly greater than the codimension-two point. However for the m = 2, m = codimension-two point, they did not find any Hopf bifurcations except for small Prandtl numbers (less than 10). It is important to note the many differences between their paper and the physical experiment. The most important being the lack of gravitational effects and the assumption of an unphysical, vorticity-free side-wall boundary condition. This latter condition will cause the modes to occur in a different order than what is observed in the experiment. For example, the vorticity-free side-wall condition generates m = 1, then m = 2, then m = modes as the aspect ratio is increased, whereas the no-slip side walls (Zaman and Narayanan, 1996; Dauby et al, 1997) generates m = 1, then m = 0, then m = 2 modes as the aspect ratio is increased. Therefore the no-slip side-walls will not have the m = 1, m = 2 codimension-two point. Nonetheless, these theoretical works give qualitative evidence that the oscillations seen in the experiments for the 2.5 aspect ratio, is a result of linear modes interacting. Further verification of the experiments would require a linear calculation using no-slip boundary conditions. The eigenfunctions from these calculations could then be used in

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83 the nonlinear amplitude equations (equation 2.136). Here the three modes m = 0, m = 1, and m = 2 would need to be simultaneously considered. In other words, these three modes would need to be in the unstable set S^J (see page 54). Such an analysis has been conducted by Dauby et al. (1997), except in their paper only rectangular containers were considered. In these set of experiments, the existence of a codimension-two point was shown definitively by observing two different flow patterns for aspect ratios near each other (2.5 and 2.6). Upon a slight increase in the temperature at the 2.5 aspect ratio, a dynamic nonlinear interaction occurred. A qualitative explanation of this behavior is given by the weakly nonlinear analysis of Rosenblat et al. (1982a). The discovery of oscillating Rayleigh-Marangoni convection in cylindrical containers at codimension-two points is important in the application of crystal growth. The unsteady convection can lead to dislocations in the crystal or dopant stratifications, both of which would yield a lower quality crystal. By understanding the existence of codimension-two points, these particular aspect ratios could be avoided to improve the crystal growth process. 4.3 EFFECTS OF AIR HEIGHT ON BILAYER CONVECTION In all of the previous experiments and calculations performed to study convection in a silicone-air system, the effects of air gave been neglected. It was thought that because air had such a low viscosity, that any motion in the air would give negligible effect on the convection in the silicone oil (or any fluid for that matter). In this section, evidence will

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84 be given to show that convection in the air does indeed affect the convection in the silicone oil. This statement is backed up be calculations from the unbounded linear model from the Mathematical Modeling chapter, by bounded calculations for a single fluid layer, and by several experiments. 4.3.1 Observations from calculations Calculations were performed to determine the flow pattern at the onset of convection. These computations involved linearized instability analysis for both laterally unbounded as well as bounded geometries. The calculations assuming layers of unbounded lateral extent were done in order to obtain qualitative features of the physics of bilayer convection. Three features in particular were investigated. First, the effect of the upper phase on the heat transfer resistance was studied. This was done by assuming that the upper phase was either strictly passive, one that allowed thermal perturbations or one that was both mechanically and thermally active. In each case, the effect of the air height on the heat transfer resistance was established. The second feature that was examined in the laterally unbounded geometry was the effect of the air height on the type of convective coupling, thermal or mechanical. The third feature that was studied was the effect of periodic lateral boundary conditions. This was done by imposing physically unrealistic conditions on the side walls of the fluid bilayers. Thus the effect of side-walls was obtained in bilayer convection at the expense of using unrealistic conditions. The imposition of realistic no-slip conditions on the lateral walls for fluid bilayers with a deflecting interface results in a complicated numerical computation. Consequently, the laterally bounded layer model with no-slip rigid side-walls assumed a passive upper

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85 phase and a non-deforming interface. All of the calculations used properties pertaining to the silicone oil-air system as these were the fluids that were used in the experiments. Turning to the first feature of the unbounded, bilayer calculations, three different conditions of the heat transfer resistance in the air layer were considered. The first condition assumes that the Biot number is constant and does not vary with the wave number. The second condition assumes that the Biot number is a function of the wave number, as demonstrated in the paper by Normand et al. (1977). This is equivalent to allowing the air to have perturbations in its temperature profile, yet remain mechanically passive. The third condition is reflected by a full bilayer calculation. In the third condition, the air is allowed to convect and therefore includes both thermal and mechanical perturbations. The calculations using the constant Biot number were similar to those found in Nield's paper, except here the surface was allowed to deflect. Despite this difference, the results from these calculations are in close agreement with Nield's results. The reason for this is the surface tension of silicone oil and air is quite large, therefore, the surface deflections are small and contribute little to the critical Marangoni number. Table 4-2 gives a comparison of the results using the three different conditions on the heat transfer resistance, with results from Nield's work. Before examining the table of Rayleigh numbers, we pause to make a few comments on the various assumptions of the air layer. Assuming that the Biot number is constant is tantamount to pretending that the upper gas phase is truly passive and that no perturbations, either thermal or mechanical are allowed. Consequently, the Biot number is:

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86 n cur oil _. ^^ = 17^ (4.1) ml air where, kair is the thermal conductivity of air, koU is the thermal conductivity of the silicone oil and dair and doU are the depths of the air and silicone oil, respectively. We have observed earlier that a Biot number that changes with the wave number is equivalent to letting only thermal perturbations in the gas phase. It is derived from the equations for the bilayer given in the Mathematical Modeling chapter, where the velocity, pressure and surface deflection perturbations are neglected. The temperature conditions for the lower liquid layer, at the interface, are replaced by: D0,+5z0i=O (4.2) with the Biot number as. k 5/(co) = 7^^cocoth f ^\ (4.3) Note that the constant Biot number given in (4.1) which is used by several earlier workers (Nield, 1964; Koschmieder, 1990) can be obtained from (4.3) by taking the limit as the wave number goes to zero (the long wavelength assumption). The details of the calculations when the upper layer is considered active, have been given earlier. A comparison of the computed critical Rayleigh number and critical wave number for the various cases produces some insight into the physics of the problem. Table 4-2 gives a comparison of the critical Rayleigh number and the associated critical wave number, for various air depths.

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87 Two important points can be made from Table 4-2. First, the critical Rayleigh number when the Biot number is given by (4.3), is always greater than the critical Rayleigh number for the long wavelength Biot number (4.1). This is understandable as the Biot number given in (4.3) is always greater than the Biot number given in (4.1). A larger Biot number corresponds to a more conductive air layer, which more easily dampens the perturbations. The critical wave number, however, differs very little between these two cases. For small air heights, the critical Rayleigh number for the bilayer Table 4-2. Critical Rayleigh number and wave number using four different conditions: a single layer with equation (4.1) as the Biot number (Nield's Model), a single layer with a deflecting interface using equation (4.1), a single layer with a deflecting interface using equation (4.3) as the Biot number, and a bilayer calculation. The Rayleigh numbers of the silicone oil and the air are defined with respect to their own thermophysical properties. In each calculation, 4.2 mm of 100 cS silicone oil was assumed. The wave number of the active air calculations is the same as the silicone oil. Air Height (mm) Rayleigh Number from Nield's Model Rayleigh Number using Bi from equation (4.1) Rayleigh Number using Bi from equation (4.3) Active Bilayer; Rayleigh number for Silicone oi! Active Bilayer: Rayleigh number for Air 0.1 513.3 (CO = 2.55) 514.2 (co = 2.55) 514.5 (co = 2.55) 526.4 (co = 2.57) 1.11*10-' 1 237.6 (CO = 2.18) 237.8 (co = 2.18) 241.7 (co = 2.16) 243.4 (CO = 2.16) 5.14*10-2 3 205.4 (CD = 2.07) 205.5 ( = 2.05) 216.2 (CO = 2.05) 217.4 (co = 2.05) 3.71 5 198.6 (co = 2.04) 198.6 (03 = 2.05) 213.7 (co = 2.00) 213.2 (CD = 2.00) 28.12 7 195.7 (co = 2.03) 195.7 (OD = 2.04) 213.3 (co = 2.00) 201.6 (CO = 1.85) 102.1 9 193.9 ( = 2.02) 194.0 (co = 2.00) 213.3 (co = 2.00) 97.62 (co = 1.42) 135.1 14 191.8 ( = 2.01) 191.9 (co = 2.00) 213.2 (co = 2.00) 17.01 (co = 0.92) 137.9

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88 calculations is greater than either of the other two conditions. The increase in the critical Rayleigh number can be attributed to allowing fluid motion in the air layer, therefore removing more heat from the liquid and stabilizing the system. This is especially true for smaller air heights. The second important point that can be made from the table is when the air height becomes large. For the Biot numbers in equations (4.1) and (4.3), the critical Rayleigh number and the critical wave number reach an asymptotic value as the air height increases. The active air layer calculations, on the other hand, show a dramatic decrease in both the critical Rayleigh number of the liquid and the critical wave number. This can also be explained by convection in the air layer, as follows. The magnitude of the temperature drop in each layer in the conductive state depends upon the height and conductivity of each layer. As the air layer increases in height, the temperature difference across it will increase relative to the temperature difference across the lower liquid for a fixed overall temperature drop. Indeed, as the overall temperature difference increases, the fluid layers will begin to convect. Consequently, under critical conditions the Rayleigh number of the lower liquid is small and only becomes smaller as the air height increases. By contrast the Rayleigh number of the air becomes larger with an increase in its height. Because the convection is dominant in the air layer, the liquid layer simply responds to convection in the upper gas. While convection in both layers is simultaneous, clearly the convection of air immediately sets up transverse temperature gradients in the interface generating surface driven Marangoni and buoyancy convection in the liquid. The decrease in the critical wave number must therefore be a signature of the pattern due to dominant convection in the air layer.

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89 The critical Rayleigh numbers corresponding to Table 4-2 were also calculated for the case of a lower viscosity by reducing the value of this thermophysical property by thirty percent. In the calculations for the long wavelength Biot number and varying Biot number conditions, the critical Rayleigh number changed very little. However, depending upon the air height, the Rayleigh numbers for the active air calculations changed dramatically. For small air heights (0.1 mm and 1 mm) the Rayleigh number of the silicone oil changed very little, but the Rayleigh number of the air decreased by over 40 percent. For large air heights (9 mm and 14 mm), the Rayleigh number for the air changed very little but the Rayleigh number for the silicone oil increased dramatically. The reason is, for small air heights, convection is "initiated" in the lower silicone oil layer. Decreasing the viscosity does not change the critical Rayleigh number of the oil significantly, but the silicone oil's critical temperature difference must decrease corresponding to the viscosity decrease. The overall temperature difference must therefore also decrease. Because for small air heights, air is nearly passive, it simply acts like a conductor. A decrease in the overall temperature difference, therefore results in a decrease in the temperature difference across the air, decreasing the air Rayleigh number. For large air heights, convection in the air is dominant at onset. Decreasing the viscosity of the silicone oil does not affect the air's Rayleigh number very much and therefore does not change the overall temperature difference much either. The temperature difference across the silicone oil virtually does not change and therefore a decrease in the oil's viscosity increases the critical Rayleigh number of the oil.

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90 An interesting note can be made about the effective Biot number of the bilayer calculation. To find the effective Biot number, the Biot number in the long wavelength Biot number calculations can be changed until the critical Rayleigh number is the same as the Rayleigh number in the bilayer calculations. For small air heights (0.1 mm to 7 mm), the Biot number must decrease with an increase in the air height to cause the critical Rayleigh number to decrease. However, for large air heights (14 mm), the effective Biot number will turn out to be negative. The reason for this peculiarity is that the convection in the air layer causes the temperature perturbations to change signs. In other words, the flow of heat into the air from the liquid decreases, although the net flow of heat into the air is still positive. Turning now to the second feature of the unbounded calculations, the vertical components of velocity, or the eigenfunctions, W,(z) and W2(z), for various air heights were calculated and are shown in Figure 4-3. Each graph represents calculations using 5 mm of 100 cS silicone oil. The vertical component of velocity is displayed at the b 5 mm 3 mm -5 mm 5 mm -5 mm 9 mm Figure 4-3. Plot of the vertical component of velocity versus fluid depths, for 3 mm (a), 5 mm (b), and 9 mm (c) air heights. The liquid-gas interface is represented by he vertical dotted line. For 3 mm, air is being dragged by the flowing silicone oil. For 5 mm, air is convecting due to thermal coupling. For 9 mm, most of the convection occurs in the air layer. Each calculation used 5 mm of 100 cS silicone oil.

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91 critical wave number in each graph. For a small air height, one would expect that the Rayleigh number in the oil, at onset, would be much greater than the Rayleigh number in the air and we would say that the oil convects "first". Here motion in the air is caused by the silicone oil dragging it. As the air height increases, the Rayleigh numbers in each layer become comparable. In this scenario, we may still say that the convection is comparable in both layers. The direction of the flow in the upper layer, depends upon its thermophysical properties. If the flow in the upper layer is in the same direction as the flow in the lower layer (co-rotating), then the convection is considered to be thermally coupled. If the flow in the upper layer is in the opposite direction as the flow in the lower layer (counter-rotating), then the convection is considered to be mechanically coupled. For the calculations given in Figure 4-3b, convection is a little more dominant in the oil and the air would be termed thermally coupled had it not been for a small counter roll developed near the interface in the air layer. For a larger air height of 9 mm (Figure 4-3 c) the convection is almost entirely in the air layer, while the liquid layer appears mostly passive. The onset of the strong motion in the air simultaneously causes tangential gradients of temperature at the interface, inducing a weak (probably Marangoni driven) motion in the oil. The third feature of the laterally unbounded bilayer calculations is seen by extending the results to give qualitative information on bounded containers. This is done by relaxing the conditions on the side walls. In the Mathematical Modeling chapter, a simple formula was given (2.1 16) to translate the calculations from a laterally unbounded layer to a cylinder with insulating and vorticity-free side-walls. The formula is:

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92 73.00 65.00 Figure 4-4. Critical Marangoni number versus aspect ratio plot. Calculations were done using the bilayer, laterally unbounded model for 4.2 mm of 100 cS silicone oil. Equation (4.4) was used to unfold the Marangoni number versus wave number plots to the Marangoni number versus aspect ratio plot. a = CO (4.4) where, a is the aspect ratio, co is the wave number. s„„ are the zeroes of the derivative of the Bessel's function, n is the radial mode, and m is the azimuthal mode. First, a graph of the critical Marangoni number (or Rayleigh number) versus wave number is generated from the laterally unbounded model. Using equation (4.4), each wave number translates into an aspect ratio, for a particular radial and azimuthal mode. The result is given in Figure 4-4. Two observations can be made from Figure 4-4. First, the critical Marangoni number is not a monotonic function of the aspect ratio, but the minimum critical Marangoni number is the same for each flow pattern. Secondly, the pattern changes as the

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93 aspect ratio changes. Some of these observations carry over to the no-sHp side-wall calculations. A comparison can be made between the vorticity-free and the no-slip side-wall calculations. The vorticity-free side-wall calculations are shown in Figure 4-4 and the noslip calculations are shown in Figure 4-5. It can be observed that the minimum Marangoni number for each mode is the same for the vorticity-free side-walls. For the noslip side-walls, at small aspect ratios, the minimum value of each mode is much greater than the asymptotic minimum reached at aspect ratios greater than 4.0. The last observation that can be made is that at larger aspect ratios, for no-slip calculations, the modes quickly crowd together and become indistinguishable. In Figure 4-5, at the aspect ratio of 2.0 for the Biot number of 0.30, the predicted flow pattern is m = (single toroid). However, it was shown in the paper by Dauby et al, 1997 that past the minimum of the m = line, a superposition of the single toroid and a double toroid may be seen. Similarly, between the aspect ratios of 1.2 and 1.7 for a Biot number of 3.0 in Figure 4-5, an m = (single toroid) flow pattern will be seen. Past the minimum of the m = line, and between the aspect ratios of 1.7 and 2.0 a second toroid will start to appear. In both the no-slip calculations as well as the vorticity-free calculations, it may be observed that certain aspect ratios correspond to a situation where two flow patterns become simuhaneously unstable. As was discussed earlier, such aspect ratios are called codimension-two points and can be associated with oscillatory behavior in the immediate post onset regime of flow (Rosenblat et al., 1982a; Johnson and Narayanan, 1996).

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94 0.0 95 90 85 -c 80 75 re .y 70 u 65 -. 60 55 0.5 Biot number = 3.0 lOU 170 1 '\ 1 o ^ 160 _m=0 i_ \\ \ ..m=1 O \ \ ^ m=2 01150 c re \___W-.,\ _m=3 ml40\ \ \ 2 \ \ '^ re130u \ "*• "^ 4-* \ "^^ C120 \ "^^-U 110 \ s.^ ^ "-^'^^"'i^ -^^._. 100 90 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Aspect Ratio Biot number = 0.30 4.5 5.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Aspect Ratio Figure 4-5. Plot of the bounded, linear calculations using an insulating, no-slip side-walls. In each graph, 5 mm of 100 cS silicone oil is used. The Biot number in the first graph is 3.0 and the Biot number in the second graph is 0.3. An increase in the Biot number causes the curves to shift to smaller aspect ratios.

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95 Increasing the air height causes a decrease in the wave number at the onset of convection and this translates into an increase of the aspect ratio where a codimension-two point occurs. These observations will be recalled as the results from experiments are discussed in the next section. 4.3.2 Observations from experiments A series of careful experiments were carried out to investigate the effect of the upper layer height on the pattern formation at the onset of convection and also to verify whether convection in the upper air layer could drive convection in the lower layer. The details of the experimental apparatus and procedures were provided earlier. Two different samples of Dow Corning silicone oil were used. From earlier experience (Zhao et al., 1995) most of the thermophysical properties of the oil, except for the dynamic viscosity, could be assumed to be constant within the temperature range studied. Dow Corning silicone oils are a blend of poly-methylsiloxanes and the viscosities are strong functions of temperature. The viscosities of two separate samples of silicone oil were measured with a Paar, cone and plate viscometer over a temperature range. The functional relationship of viscosity with temperature for each sample is reported below. 11^ = -2.9 T + 1050 ^j=-1.5T + 560 ^^"^^ where the subscripts a and b refer to the two samples, the temperature T is in degrees Kelvin and is valid between 25C and 50''C. Sample a had a lower nominal viscosity than sample b. If the upper surface is flat and the upper phase is passive.

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96 viscosity conveniently scales out of the problem and the bounded layer model results, given in Figure 4-5, are independent of the value of viscosity. For the first set of experiments, a liquid layer insert, 5.0 mm deep and 20 mm in diameter (2.0 aspect ratio) was used. Three different air heights of 3 mm, 14 mm and 20 mm for silicone oils from samples a and b were used. The second set of experiments used an insert that was 4.2 mm deep with a 21 mm diameter (2.5 aspect ratio) with sample a as the test fluid. Air heights of 3 mm, and 20 mm were used with this aspect ratio. The first set of experiments was done to study how convection, which initiates in the air, can affect the flow pattern in the silicone oil. The second set of experiments was conducted to see how changing the air height could affect oscillatory behavior in the silicone oil. Figure 4-6 displays the different flow patterns observed at the onset of convection for the 2.0 aspect ratio. First the 3 mm air and sample a silicone oil were used. An m = 3 mm Sample a Sample b 14 mm 20 mm Figure 4-6. Infrared images of the flow pattern for different air heights and different viscosities of silicone oil. (a) tlirough (c) uses sample a silicone oil. (d) through (f) uses sample b silicone oil. (a) and (d) had a 3 mm air height, (b) and (e) had a 14 mm air height, and (c) and (f) had a 20 mm air height.

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97 (double toroid), pattern was observed (Figure 4-6a). The critical temperature difference across the liquid layer was 0.9C (8.4C overall). This flow pattern agrees with the noslip, three-dimensional, linear calculations in a bounded cylinder. Next, a 14 mm, then a 20 mm air insert were used; again using sample a silicone oil. This time the flow patterns seen at the onset of convection were much different than predicted. For the 14 mm air height (Figure 4-6b), the double toroidal pattern became more skewed, or lop-sided, on one side. For the 20 mm air height (Figure 4-6c), the pattern changed even further and an m = 1 pattern was seen. The critical temperature difference across the liquid was 0.9C (16.2C overall) and 0.35C (8.7C overall) for the 14 mm and 20 mm air inserts, respectively. The flow pattern for sample a silicone oil with a 20 mm air height does not agree with the bounded, linear calculations, even after the change in Biot number is considered. This observation calls for an explanation. The unicellular and the lop-sided double toroidal flow can, however, be predicted by convection in the air layer, which the bounded layer model with a passive gas layer does not predict. The aspect ratios for the air layer, when the 14 mm and 20 mm air inserts were used, were 0.71 and 0.50, respectively. In the paper by Hardin et al. (1990), calculations were performed in a bounded cylinder using conducting and insulating sidewalls and bounded on the top and the bottom by rigid plates. One could imagine that the surface tension between the oil and the air is sufficiently large such that the oil acts as a rigid plate to the air. The heat transfer from the air to the liquid is given proportionally by the inverse of the Biot number given by equation (4.1). This inverse is 17 for 14 mm and 24 for the 20 mm air height. Therefore, as far as the air is concerned the liquid acts

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98 Table 4-3. Temperature differences across the silicone oil and overall temperature differences for the six experiments given in Figure 4-6 Air height Sample a Sample b 3 mm 0.9C (8.4T overall) 1.5C(14.5C overall) 14 mm 0.9C(16.2''C overall) 0.8C (14.4C overall) 20 mm 0.35C (S.VC overall) 0.4C (9.7C overall) effectively like a conductor. This assumption is substantiated by calculations given by Sparrow et al. (1963). Additionally, table I in Hardin et al.'s paper gives a list of Rayleigh numbers for different aspect ratios. For an aspect ratio of 0.5 (20 mm air) and conducting side-walls, the Rayleigh number is 8012. The Rayleigh number of air is 14.2AT(i\ Using the thermophysical properties of air, which are reliably recorded in the literature, the critical temperature difference of air can be calculated as 8.8C. This is in reasonable agreement with the measured critical temperature difference across the air layer, which is 8.35C given in Table 4-3. The total temperature difference, for the 20 mm air height experiment, was 8.7C. Similarly for a 0.71 aspect ratio (14 mm air) the temperature difference across the air layer calculated from table I in Hardin et al.'s paper is 15.3C. The overall temperature difference in the experiment for 14 mm was 16.2C which works out to be 15.3C across the air layer and equal to the theoretical prediction. It may be noted though, that as the air layer increases, the Biot number decreases. A decrease in the Biot number causes the critical wave number to decrease. As the aspect ratio for a particular flow pattern is inversely proportional to the wave number, the curves in a Rayleigh number (or Marangoni number) versus aspect ratio plot shift to the right. In other words, for a fixed aspect ratio, the change in the Biot number may cause the change

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99 in the observed flow pattern. To answer this question, each experiment was conducted again using sample b silicone oil whose viscosity is higher than sample a by about 60%. As the Rayleigh and Marangoni number are inversely proportional to the viscosity, increasing the viscosity of the silicone oil from sample a to sample b should have proportionally increased the observed critical temperature difference, provided that the observed flow pattern is indeed the result of the onset of convection in the lower layer. Figure 4-6d shows the double toroidal flow pattern observed at the onset of convection for the 2.0 liquid aspect ratio, 3 mm air insert and sample b silicone oil. The critical temperature difference this time was 1.5C (14.5C overall); 67% more than the same experiment using sample a silicone oil confirming the hypothesis that convection initiates in the lower layer. Figure 4-6e shows a flow pattern that looks like a combination of an m = and an m = 1 flow for the 14 mm air height. Figure 4-6f shows the m = 1, unicellular flow pattern for the 20 mm air height. More importantly, the critical temperature difference for the 14 mm air height was 0.8C (14.4C overall) and the critical temperature difference for the 20 mm air height was 0.4''C {9.TC overall). For the 20 mm air height, after the viscosity had increased by 60%, the flow pattern and the critical temperature difference across the liquid had changed slightly. In the 14 mm air height however, using sample b silicone oil, the flow pattern changed from a skewed double toroid to a superposition of m = and m = 1 For the 20 mm air height, the reason the critical temperature difference changed very little is because changing the viscosity of the silicone oil did not change the critical temperature difference for the air. The results from the 14 mm air height were more complicated, but can still be explained by the convection in the air. For sample a.

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100 which had the lower viscosity and with a 14 mm air height, the temperature difference in the air was 15.3C. As was pointed out earlier, the calculation of Hardin for pure buoyancy convection gives a temperature drop across the air of 15.3 C. An increase in the viscosity caused the temperature drop in the air to decrease to 13.2C, still indicating a substantial amount of convection in the air. The side-walls for the air are lucite and in numerical simulations using a finite volume code it could be inferred that they nearly acted as conducting walls. For the larger viscosity sample b silicone oil with 14 mm air height, the convection in the air at onset was predominant so that the air convected Tirst' and only the air flow pattern was primarily seen at the onset of convection as expected. Table 4-4. Comparison of the critical temperature differences in the experiments with the critical temperature difference calculated from Hardin et al.'s paper. The experimental values reported are for the sample a silicone oil. The thermophysical properties of air, used to calculate the critical temperature difference from Hardin et al.'s paper, are reported in Table 4-5 Air height Experiments (C) Hardin's Conducting side-walls fC) Hardin's Insulating side-walls (C) 14 mm 15.3 15.3 9.1 20 mm 8.4 8.8 4.2 In Figure 4-6e, a superposition of an axisymmetric m = 0, single toroid and a unicellular, m = 1 flow pattern was seen at the onset of convection. This flow pattern can be beautifully explained using the resuhs from Hardin et al.'s paper. In figure 5 of Hardin et al.'s paper, a graph of the critical Rayleigh number versus aspect ratio, for various azimuthal modes are given. For the 0.50 aspect ratio (20 mm air height), a purely unicellular flow pattern is predicted. At a 0.71 aspect ratio (14 mm air height) a codimension-two point is predicted and both the m = 1 and m = modes are equally

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101 likely to occur at the onset of convection. Figure 5 in Hardin at al.'s paper is in agreement with the results of the experiment. The ratio of the temperature differences for the 1.5 and 2.0 aspect ratios (for a 3 mm air height and 5mm liquid depth) can be compared to the computed ratio of the Marangoni number for the same aspect ratios. Taking the ratio eliminates errors in the measurements of the thermophysical properties, such as k, and v,. The experimental temperature difference ratio was 1.2/0.9 = 1.33, and the ratio of the corresponding Marangoni numbers was 69.6/62.3 = 1.12, less than an 18% difference. One explanation is that the air layer, even for small air heights, still affects the observed temperature difference across the liquid. The measured temperature difference was the overall temperature difference across both the silicone oil and the air layers. The temperature difference across the silicone oil was calculated from the long wavelength Biot number (4.1). If the air layer was convecting by any mechanism, the temperature difference across the air layer would be less than that predicted by using the long wavelength Biot number. In fact this is also seen from Table 4-2. Therefore, the actual temperature difference across the silicone oil would then be greater than the temperature difference backed out from the experiments. This would increase the experimental Marangoni number and make the two ratios closer. There is however one more reason why a discrepancy of 18% between the two ratios can occur even if one were to assume that the upper gas is passive for the 3mm air height experiments. A very small change in the liquid depth of 0.25mm changes the aspect ratio by 5% and this results in raising the aspect ratio near 1.5. On inspection of Figure 4-5, one can conclude that the calculated Marangoni number will

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102 increase by about 12%. This problem will however be obviated at the larger aspect ratios. Indeed, the ratio of the measured critical temperature drops at aspect ratios 2.0 and 2.5 was within 20% of the calculated ratio. Changing the air height can also dramatically affect the flow behavior in the liquid layer for other aspect ratios. In the Codimension-two point section, a dynamic behavior termed "mode switching" was discovered for an aspect ratio of 2.5, which was near a codimension-two point. However, as was discussed above, changing the air height not only can cause the air to convect first, but it can also shift the codimension-two point as well. In fact, convecting upper air will shift the codimension-two point to the right. This can be seen by the decrease of the wave number in Table 4-2 for larger air heights. In Figure 4-7 infrared images of the 2.5 aspect ratio are given for 3 mm and 20 mm air heights. For the 3 mm air height, the mode switching disappeared and a broken double toroid or "c" pattern replaced it. This agrees with Figure 4-6. When the air height decreases, the Biot number increases and the codimension-two point moves to the left. For the new Biot number, the flow pattern shifted from a codimension-two point to an m Figure 4-7. Infrared images of the flow pattern. In (a) and (b), the liquid aspect ratio was 2.5, with a 4.2 mm liquid height. In (a), a 3 mm air height was used. In (b) a 20 mm air height was used.

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103 = (double toroid). For the 20 mm air height, again the mode switching has disappeared. Here, as was the case for the 2.0 aspect ratio, the flow pattern is the unicellular, m = 1. The overall temperature difference here was S-OT. The aspect ratio of the air was 0.525. The critical temperature difference for this aspect ratio from Hardin et al.'s paper is 8.3C. Again, for the 20 mm air height, the experiments and calculations are in close agreement. Observe that the no-slip calculations give a codimension-two point near an aspect ratio of 2.2 for the Biot number of 0.3. In the experiments reported in the Codimensiontwo point section, the codimension-two point was found at an aspect ratio of 2.5. As mentioned above, an active air layer will move the codimension-two point closer to 2.5 By using both calculations and experiments, a new convection-coupling mechanism was discovered. This mechanism occurs when the upper fluid initiates convection be buoyancy forces. The upper convecting fluid then created temperature gradients across the fluid-fluid interface. These temperature gradients in turn cause surface tension-driven flow in the lower fluid. It was also shown that changing the upper fluid's depth can significantly change the heat transfer from the lower fluid layer. A sufficiently large change in the heat transfer can cause the flow pattern to change. It is hoped that these results will lead to a better understanding of the role gases play in liquid encapsulated crystal growth. Perhaps the gas height can be changed at certain stages of the crystal growth to avoid regions where oscillations convection occurs in the liquid.

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104 4.4 CHANGES IN CONVECTION COUPLING AND INTERFACIAL STRUCTURES In this section, the convection-coupling mechanism and the interfacial structure will be studied for two different bilayer systems. In each case, the fluid depths are changed to observe different interfacial structures or different convection-coupling mechanisms near the minimum of the Rayleigh number versus wave number plots. When this occurs and the plot is subsequently unfolded, the interfacial structure and the convection-coupling mechanism will change as the aspect ratio of the container increases. Because of the large number of dimensionless groups the main ideas in this paper are exemplified by calculations for two, bilayer systems. These are the silicone oil-air system and the glycerol-silicone oil system. Their properties are shown in Table 4-5. Note the different signs of the interfacial tension gradient, a,, in each of the bilayer systems. Table 4-5. Thermophysical properties of the two bilayer systems used in the calculations: 1) silicone oil and air and 2) glycerol and silicone oil Bilayer System 1 Bilayer System 2 Property (units) Silicone Oil Air Glycerol Silicone Oil Pi (g / cm') 0.968 0.0012 1.26 0.97 a/ClO'^C') 9.6 33.3 4.9 9.45 k/ (10-' erg/cm sC) 1.59 0.262 2.94 1.6 K/(10-'cm^/s) 1.10 182 0.89 1.16 vi (Stokes) 0.692 0.157 7.45 4.99 G„ (dyne/cm) 20.9 — 25 — Qi (dyne/cm C) 0.05 — -0.13 —

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105 Table 4-6. Possible combinations of the ratio of the temperature perturbations to surface deflection, and the ratio of the derivative of the vertical component of velocity to the surface deflections. The four different combinations predict different behavior in the bulk fluid. 0.(O)-Ti, Til DW, (0) 111 Case I negative positive Case II positive negative Case III positive positive Case IV negative negative The linear, unbounded model, described in the Mathematical Modeling chapter, will be used to describe the effect that varying the aspect ratio has on both the driving force for convection and the type of convection coupling. To determine whether the bilayer convection is thermally or viscously coupled, one need only look at the vertical component of velocity and the temperature profiles (Johnson et al., 1997; Rasenat et al., 1989). To find out which interfacial structure is present, two different ratios need to be calculated. The first ratio is the perturbed temperature evaluated at the interface, 0i(O) rii, divided by the interfacial deflection, ri,. The second ratio is the derivative of the velocity evaluated at the interface, DW|(0), divided by the surface deflection. For example, if the first ratio is positive, then the fluid must be warmer at a crest. If the second ratio is positive, then the fluid must flow away from a crest. Table 4-6 gives the values of each ratio for each of the four possible interfacial structures (see Figure 1-7). As was described in the first chapter, the interfacial structures indicate the driving force for convection.

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106 4.4.1 Changes in Convection-Coupling The first system considered was the popular silicone oil and air system. As was noted in the previous section (Johnson et al, 1997), when the air layer is large, convection initiates in the air. This convection then creates a temperature difference across the liquid interface simultaneously causing surface tension induced convection. Continuing this reasoning, various depths of silicone oil and air were considered where convection is equally likely to initiate in either the lower or the upper layer. As an example, a depth of 2 mm of silicone oil, for a variety of air heights, was chosen. For each air height, a plot of the Rayleigh number versus wave number was calculated (Figure 4-9). For small air heights (0.01 mm) the critical wave number is 2.55 and gradually decreases as the air height increases. For example, the critical wave number for a 3 mm air height is 2.00. The critical wave number is the value of the wave number at the minimum of the Rayleigh number versus wave number curve, for a fixed air height. At an air height greater than 5 mm, the critical wave number drastically shifts to lower values. This occurs due to the dominant convection in the air layer. As the air height W W^ 0000 0000 0000 0000 W v^ ( M M a) Lower b) Viscous Dragging Mode Coupling c) Thermal Coupling d) Upper e) Surface Tension Dragging Mode Induced Figure 4-8. Schematic of the different types of convection-coupling. From the lower dragging mode to the upper dragging mode, the buoyancy force in the upper layer is increased and the dragging exerted by the lower layer decreases. Pure thermal coupling with surface-driven flow is caused by the upper fluid buoyantly convecting and simuhaneously inducing surface tension-or buoyancy driven convection in the lower layer, near the interface.

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107 TO ) Lower ^-—-^ layer : / \ \^^ /[ Upper \ layer i Si : / \^^ m / ---^ 1^ ^ 1 Cj o O 1H3 O ^ 1 ^ 1) E E s O o (N ^ O 00 ^ X3 m 1 — ( r? N c/5 II 00 -f-i 6 E 3 en II id o rji 03 JO o 3 tio (U o 'o -. 1) O .— 1 ;^ -I-' ^ aj N ^ ^ Ul .2 E E o o ^ s !=l O on O ^ JO >r) td ^-H o ^^ 3 o o rxi ^ W "ti rrt U< s T3 o O 1 1 t4-( I— 1 c ri E E o in o o fi fi ri, Oh c3 03 ? N ON -4 00 O O B E (U CI c -a fc s H O o r-l L_j f J r) V:; OB tH CN t/1 o 13

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108 increases, the buoyancy effect in the air layer increases and eventually becomes greater than the buoyancy effect in the silicone oil. Before the effect of the aspect ratio on the convection mechanism is explained, it would be instructive to make some more comments on these mechanisms. To do this efficiently, consider Figure 4-8. As noted earlier, Figure 4-8a is a situation where the convection is dominant in the lower layer and the upper layer responds by being dragged. An example of this is shown in Figure 410a for the silicone oil-air system. Observe that the sign of the velocity switches from the lower to upper layer and that the maximum of the lower layer velocity is generally much greater than the maximum of the upper layer velocity in magnitude. The corresponding situation in Figure 4-8b is depicted in Figure 410b. Here, the magnitudes of the lower and upper velocities are of comparable order and the velocity and the temperature change sign from the lower to upper layer. Additionally, 160 140 Qj 120 I 100 j= 80 ai '^ 60 >• m Qi 40 20 \ \ \ '• 3 mm \ \\ \ \ \ \ \ 5mm ^ \\ • \ \ \ \ \ 5.5 mm" """^ 6 mm — \ 7mm ~ \ 10mm -_._-—0.0 0.5 1.0 0.01 mm 1.5 2.0 \Na\B Number 2.5 3.0 3.5 Figure 4-9. Plot of the Rayleigh number of silicone oil versus the wave number for various air heights. As the air height increases, the critical wave number decreases.

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109 hot fluid flowing up towards the interface in the lower layer is combined with hot fluid flowing down in the upper layer at the interface. This appears to contradict our view taken earlier that the upper fluid is also buoyant. On ftirther inspection of the numbers that generate the temperature perturbation plot, an isotherm, 0(z) = 0, is observed in the domain of the upper fluid very near the interface and to the right of the vertical dotted line, which represents the unperturbed interface. In other words, hot fluid does flow up in the upper layer but not at the interface. Figure 4-8c can be nearly depicted by Figure 410c. Observe that the velocities in both the upper and lower layer show comparable minima and a small counter-roll has developed in the air layer to preserve the no-slip condition between the fluids. It is possible to obtain a situation where no counter-roll develops in the upper layer. In this situation, this would be called pure thermal coupling as no motion in either layer is generated by viscous drag. In other words, it is possible to obtain a structure where the fluid depths are such that the thermal coupling in perfect, co-rotating rolls are obtained. In such a situation, the transverse components of velocity perturbations at the interface are zero. This can be seen later in this paper with liquid bilayers (see Figure 416a as an example). Figure 4-8d can not be depicted in a silicone oil-air system, because the air does not drag the silicone oil due to the very small ratio of dynamic viscosities. However, the calculations using glycerol-silicone oil system show this dragging effect well (Figure d). This situation is qualitatively the reverse of Figure 4-8a. The last convection mechanism. Figure 4-8e, is seen in calculations using silicone oil-air (Figure e). Notice that within the scale of the plot, the lower velocity is nearly zero. A closer look at

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110 w a W b -2 mm 5 mm -2 mm 5 mm Figure 4-11. Plot of the vertical component of velocity for a wave number of 1.9 (a) and 2.7 (b). The depth of the silicone oil was 2 mm and the depth of the air was 5 mm. In (a), CO = 1.9 and the air is mostly thermally coupled. In (b), co = 2.9 and the air is mostly viscously coupled. the actual numbers indicates that the velocity in the lower fluid has the same sign as the velocity in the upper fluid and is less than 1% of the maximum velocity in the upper fluid. Returning to the task of relating the aspect ratio to convection mechanisms, the vertical component of velocity for both fluids is plotted for two different wave numbers, for a 5 mm air height (Figure 4-1 1). At a wave number of co = 1.9, Figure 4-1 la shows mostly a thermal coupling of the silicone oil and air. For a larger wave number of CO = 2.7, Figure 4-1 lb shows mostly dragging of the air by the silicone oil. This feature is most enhanced after the plots are unfolded. The next step is to unfold Figure 4-9 using equation 2.116. The resuh is shown in Figure 4-12, for various azimuthal and radial modes. Each azimuthal and radial mode determines a different flow pattern at the onset of convection. The same plots in Figure 411 now represent the vertical component of velocity for various aspect ratios. For example, the wave number of 2.7 converts to an aspect ratio of 0.68 for the unicellular

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Ill 100 95 90 85 3 Pi 80 -I 75 70 65 60 55 50 m=2 m=0 n=1 n=1 m=1 n=2 m=0 n=2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Aspect Ratio 3.5 4,0 4.5 5.0 Figure 4-12. Plot of the Rayleigh number of the silicone oil versus the cylindrical aspect ratio. This plot was generated from Figure 4-9 and the wave number to aspect ratio conversion formula (equation 2.1 16). A depth of 2 mm of silicone oil and a 5 mm air height was assumed. (m= 1, n= 1) flow. The wave number of 1.9 converts to an aspect ratio of 1.6 for a bimodal flow pattern (m = 2, n = 1). What does this exercise explain? It shows that for cylinders, (as well as rectangular geometries) the type of convection-coupling mechanism can change as the aspect ratio increases. There are two ways the aspect ratio of the liquid can change, either changing the radius or the height. In Figure 4-9, the height of the silicone oil is fixed at 2 mm. Therefore, Figure 4-12 corresponds to a situation where the radius is being changed. The next question that can be raised is: why does changing the radius of the cylinder affect the type of convection? ^

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112 The change in the convection-coupling as the aspect ratio increases, can best be explained by analyzing the Rayleigh number versus aspect ratio plots for each fluid layer. The Rayleigh number for each fluid is defined as: Ra, a,gATj(ii'' K,V, Ra,= ajgATjJj' K2V2 (4.8) (4.9) The temperature difference in each phase, AT, or ATj, is calculated from the linear, conduction state just prior to the onset of convection. As the width of the container increases, the aspect ratios of each layer increase. However, the energy required to convect each layer changes with the aspect ratio and this could either increase for both, decrease for both or increase for one and decrease for the other. This ambiguity occurs 200 180 ? 160 c sz 140 ^ 120 100 80 \ \ \ \ \ \ \ ~^ """ "'3.'5 mm ^-''' — ----'3'0 mm ^.,''' — ^'2.5 mm — \ \ Case I Case II 0.5 1.0 1.5 2.0 2.5 V\/a\e Number 3.0 3.5 Figure 4-13. Plot of the Rayleigh number of the silicone oil versus the wave number for three different depths of the silicone oil. The solid lines denote a Case I interfacial structure and the dotted lines denote a Case II interfacial structure. The air height is 6 mm.

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113 because the critical Rayleigh number is a non-mono tonic function of the aspect ratio. In other words, increasing the aspect ratio can easily cause the convection mechanism to change generating a situation like Figure 4-8b for one aspect ratio and Figure 4-8c for another. Other mechanisms (Figure 4-8a, d, e) are also possible and depend on the particular bilayer system being studied. 4.4.2 Changes in the Interfacial Structure In the next set of calculations, a silicone oil-air bilayer system was also chosen. In this exercise, the Rayleigh number versus the wave number plots are generated for 6 mm of air and various depths of the silicone oil: 2.5 mm, 3.0 mm, and 3.5 mm. In Figure 413, the solid lines denote a Case I interfacial structure, and the dotted lines denote a Case II interfacial structure. As was discussed in the introduction, Case I indicates surface tension-driven convection and Case II indicates buoyancy-driven convection in the lower layer. For 3.0 mm of silicone oil, the interfacial structure changes from Case I to Case II at the critical wave number. When the silicone oil layer increases to 3.5 mm, the buoyancydriven, Case II interfacial structure becomes more unstable. When the silicone oil height decreases to 2.5 mm, the surface tension-driven Case I interfacial structure becomes more unstable. This is in qualitative agreement with the physics. As the depth of the silicone oil layer increases, buoyancy forces become more dominant than surface tension forces. As the silicone oil layer decreases, surface tension forces become more dominant.

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114 When Figure 4-13 is unfolded, the dominating driving force for convection can change as the aspect ratio increases, and this is depicted in Figure 4-14. This is most pronounced at codimension-two points, where two flow patterns coexist. Typically, the change from one interfacial structure to the next is quite gradual. The surface deflections slowly flatten as the interfacial structure switches from Case II to Case I. This can be seen around an aspect ratio of 0.9. At codimension-two points, though, the switch from one interfacial structure to the next can be abrupt. That is, on one side of the codimension-two point, the fluid is buoyancy-driven with one spatial pattern, then switches to a surface tension-driven flow on the other side of the codimension-two point, with a different spatial pattern. This can be seen at codimension-two points with aspect ratios of 1.2 or 1.7. 250 225 (V Si E 3 200 ^ 175 0) > 150 a: 125 100 1 1 1 1 1 \ s / X m = 1 n = 1 'v.. 'fn = -'-m'=~l"'^""" m = ..^-" n=1 1 n = 1 n = 2 —J \ n = 2 \ — 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Aspect Ratio 3.5 4.0 4.5 5.0 Figure 4-14. Plot of the Rayleigh number of the silicone oil versus the aspect ratio. Solid lines denote a Case I interfacial structure. Dotted lines denote a Case II interfacial structure.

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115 1240 \ Case II Oscillating Switch from C to Case III ase I / ^ 1220 E 3 J 1200 ai >-1180 x ^ Region rA ^^ \ \ Case III 1160 Switch from Case II to Case I 1140 2.00 2.50 3.00 3.50 Wave Number 4.00 4.50 5.00 Figure 4-15. Plot of the Rayleigh nvimber of the glycerol versus the wave number. Heights of 4.15 cm for glycerol and 3.0 cm for silicone oil were assumed. 4.4.3 Other Observations in Convection-Coupling and Interfacial Structure In the last example the liquid-liquid bilayer glycerol-silicone oil is examined. The thermophysical properties, which are listed in Table 4-5, are taken from the paper by Cardin et al. (1991). In this system, switching between different convection mechanism and three different interfacial structures can be seen. Figure 4-15 again gives a plot of the Rayleigh number versus wave number for a glycerol-silicone oil bilayer. Glycerol, which has the larger density, lies below the layer of siUcone oil. The height of the glycerol is 4.15 cm and the height of the silicone oil is 3.0 cm. For the calculations performed, only steady, and not Hopf bifurcations were analyzed. The Rayleigh number was calculated and when the imaginary part of the Rayleigh number became non-zero, it was inferred that the onset of convection was oscillatory. While for real values of the Rayleigh number, the numbers are correct, for

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116 complex values, the real part of the Rayleigh number has no physical interpretation. The dotted lines in Figure 4-15 and Figure 4-17 are regions where oscillations occur. This does not change the qualitative discussion given below, and therefore it was felt that a full search for the Hopf bifurcations were uimecessary. A plot of the vertical component of velocity for co = 2.7 and co = 3.5 are given in Figure 4-16. For co = 2.7, the convection is nearly thermal coupled, with a small counterroll in the upper fluid. For co = 3.5, the convection is viscously coupled. Indeed, as indicated by previous researchers, the oscillations in Figure 41 5 are due to a competition between the thermal and viscous coupling. The call-outs in Figure 4-15 denote wave numbers at which the interfacial structure changes. For a wave number less than 3.8, the interfacial structure is Case IL Between CO = 3.8 and co = 3.9, the interfacial structure is Case I. For wave numbers greater than CO = 3.9, the interfacial structure is Case III. Here also the interfacial structure changes near the wave numbers where oscillatory onset of convection occurs. Both the interfacial structures and the convection-coupling can be used to describe what is occurring in the two liquids. For wave numbers smaller than the oscillatory region, the two fluids are thermally coupled and the interfacial structure is a Case II. As was described earlier, a Case II interfacial structure denotes buoyancy-driven convection occurring in the lower layer. The hot plumes of the lower layer rise up and drive convection in the upper liquid, creating a thermally coupled bilayer, with a small counterroll. For wave numbers much larger than the oscillatory region (co > 3.9), the convection is viscously coupled and the interfacial structure is a Case III flow structure. For a

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117 -4.15 cm 3 cm w /\ b vy -4.15 cm 3 cm Figure 4-16. Plot of the vertical components of velocity for 4.15 cm of glycerol and 3.0 cm of silicone oil. In (a), the two fluids are thermally coupled, with a small counter-roll in the upper layer, co = 2.7 In (b), the two fluids are viscously coupled, and co = 3.5. The vertical dotted line is the unperturbed interface. Case III interfacial structure, buoyancy-driven convection occurs mostly in the upper layer. The cold, sinking fluid in the upper layer pushes down and depresses the interface. In this study, the interfacial structure seems to indicate that viscous coupling occurs when the upper fluid "initiates" convection causing the lower fluid to flow and convect in a gear-like maimer. What seems somewhat peculiar is the Case II interfacial structure at wave numbers between co = 3.4 and co = 3.8, while the two fluids are viscously coupled. However, this can be explained by the differences in the dynamic viscosities of the two fluids. The dynamic viscosity of glycerol is about twice that of the dynamic viscosity of the silicone oil. Even though convection may begin to initiate in the silicone oil, the silicone oil must overcome the higher dynamic viscosity of the glycerol. Only when the silicone oil convection is much more vigorous, does it begin to deflect the interface down to give a Case III interfacial structure. The Case I interfacial structure is a transitional structure between Case II and Case III.

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118 Unfolding Figure 4-15 gives an extremely interesting and complicated plot of the Rayleigh number versus the wave number (see Figure 4-17). The dark lines represent viscous coupling, the thirmer lines represent thermal coupling, and the dotted lines represent oscillatory onset of convection. Four different flow patterns are given. At different aspect ratios, the flow can be either thermally coupled, viscously coupled, or oscillating between these two states, at the onset of convection. Some of the most interesting aspect ratios, however, occur at the codimension-two points. For example, the codimension-two point around an aspect ratio of 0.8 goes from a thermally coupled unicellular flow (m= 1, n= 1), to a viscously coupled bimodal flow (m = 2, n = 1). There is a high probability that at these codimension-two points, nonlinear interactions will be very dynamic. As was mentioned in the codimension-two point section, a dynamic switching between an axisymmetric m = flow and a bimodal m = 2 flow was experimentally found in the slightly supercritical region, near a codimensiontwo point. In that system, only a single layer of silicone oil was used. Andereck et al. (1996), found oscillations between viscous coupling and thermal coupling in a silicone oil-flourinert system. When these two phenomena are combined, a highly oscillatory state in the supercritical region may be possible. Even more interesting dynamics may occur at codimension-tv/o points such as the 1.1 aspect ratio. Here a codimension-two point is close to an oscillating bimodal flow and an oscillating axisymmetric flow.

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119 1260 1240 ^HI I 1220 gi 'v 1200 >n 1180 1160 Thermal Coupling Oscillations Viscous Coupling \ 0.25 0.50 0.75 1.00 1.25 Aspect Ratio 1.50 1.75 2.00 Figure 4-17. Plot of the Rayleigh number of the glycerol versus the aspect ratio. The plot is generated from Figure 4-15 and equation 2.1 16. As the aspect ratio changes, the convection switches from viscous coupling, to oscillatory flow, to thermal coupling One important point must be considered before these results are compared with experiments. As was mentioned earlier, the order in which the modes appear as the aspect ratio is increased, is different for vorticity-free side walls and no-slip side-walls. Additionally, the Rayleigh numbers for the unfolded plots are not the same as those in a bounded calculation, especially for smaller aspect ratios. While the differences between vorticity-free side-walls and no-slip side-walls can not be ignored, the effects of aspect ratios on convection mechanisms and interfacial structures with realistic side-wall conditions are expected to be qualitatively similar to the vorticity-free case. Calculations performed for bilayer convection in laterally unbounded geometries give a qualitative picture of the different ways in which convection can occur in a bounded cylinder. These different types of bilayer convection depend upon the layer in which the

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120 convection is tlie most dominant. The types of convection also depend upon how the layer that did not initiate convection, responds to the layer that did initiate convection. The hierarchy of convection mechanisms has been explained both by looking at interfacial structures and also by considering the perturbed temperature and velocity profiles tlirough the fluid layers. Specific examples of silicone oil-air and glycerolsilicone oil have been used to exemplify the arguments made. Having done this, the mechanisms of onset of convection in a bounded right circular cylinder was explained. Because the difficulty of the computations is determined by the side-wall conditions, it was assumed that the vertical and tangential component of vorticity vanished at the vertical side-walls. This assumption allows the results of the unbounded case and the qualitative features as a function of the aspect ratio to be determined. It was observed that the aspect ratio did indeed affect the nature of the onset of convection. As the depths of the fluids were assumed constant, it was apparent that the change in radius could affect the physics of the flow and flow structures. This unusual result is explained by the observation that a change in the radius, changes the aspect ratios of both fluid layers, and the energy required for each layer to convect changes differently with aspect ratio because of differing thermophysical properties. Moreover, the onset of oscillations and sudden pattern changes at codimension-two points were observed. All of this has an impact in the future studies where bilayer convection is of importance. For example, a nonlinear analysis of the bilayer systems should determine which codimension-two points give interesting oscillating behavior. The effects of bounded geometry on bilayer convection should lead to many exciting experiments.

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121 Experiments, where the onset of convection is unsteady and the Rayleigh numbers in each fluid layer are comparable, ought to show many of the phenomena discussed above. The discovery of many of these oscillations will have practical applications in liquid encapsulated crystal growth. As the lower liquid solidifies, the aspect ratio of the lower liquid and the depth ratios will change. Oscillatory convection ought to be seen when the lower liquid aspect ratio reaches a codimension-two point and when the liquid depth ratios are such that the buoyancy forces in each layer are equal. Oscillatory convection is of particular interest in crystal growth as the fluctuating temperature continually melts then solidifies the crystal, creating defects in the crystal. 4.5 NONLINEAR ANALYSIS In the final section of this chapter, results from the nonlinear analysis will be discussed. Four different cases are analyzed. The first two cases involved "simple points", where only one flow pattern exists at the onset of convection. The other two cases are codimension-two points where two flow patterns coexist at the onset of convection. At first, an attempt was made to recover the results for a single fluid layer from Rosenblat, Davis and Homsy (1982a), hence forth referred to as RDH. Unfortunately, I was unable to reproduce the values of the coefficients of the amplitude equations. However, every possible criterion for correctness was met by my calculations. This will be elaborated on later.

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122 In order to make some progress on the problem, the values of the coefficients were calculated for various air heights under three conditions. The first condition was for an infinite Prandtl number. The second condition used the value of the Prandtl number for 100 cS silicone oil (Pr = 909). The third condition used a different scaling for the upper fluid's dependent variables. Considering a finite Prandtl number made very little difference in the value of the coefficients. However, changing the way in which the problem is scaled not only changed the nonlinear behavior quantitatively, but qualitatively as well. To begin, the equations for the four different cases will be derived. The derivations are also given in detail by RDH, therefore only the major steps will be given here. The form of the amplitude equations are the same for both the single layer and the bilayer problem. ^p-^ = ^p^p-yp,A,^r (4.10) where the summation over q and r is implied. Here the values of the coefficients for the single layer are: a^=Pr-'(v;,v^) + (e;,e^) (4.11) M-M^ R-R„ T„, =Pr-'(v;,v,-Vv\ + (e;,v,.VeJ (4.13) and for two fluid layers:

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123 ap= -/ -ir^r-/ -7-\ (4-14) ^lp^^\p) + \^2p'^2p M-Mp R-Rp "^ P^l~ / A* \ / A* \ \^lp^^lp) + \W2p>^2p) (4.16) For all of the following cases a depth of 5mm of 100 cS silicone oil was used. Three different air heights of 0.1mm, 1mm, and 10mm were evaluated for the single fluid layer. For the bilayer calculations, five different air heights were considered: 0.1mm, 1mm, 5mm, 9mm, and 1 0mm 4.5.1 Case 1 The first case involves a single point where the m = 1 n == 1 curve in Figure 4-4 (say) is at a minimum. Therefore, for different air heights the aspect ratio must change. The set of unstable modes, S^, and the set of stable modes, S^, for case 1 is: 5t; = {ll} 55 = ^1,01,31,41,12} (4.17) However, due to the orthogonality condition (equation 2.135), only certain modes in S^ will be relevant. Keeping the orthogonality condition in mind, equation (4.10) for p= {11} becomes: <^ii 7. =^11^11 -^7111121 +Y 112111^^11-^21 ~\Yiiiioi +7110111^-^11^01 (418)

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124 As was discussed in the Mathematical Modeling chapter, the modes {21} and {01} are considered slave modes. After the adiabatic reduction is performed, the amplitude equations for these two modes become: Aj, = Af, (4.19) A„=^^^^A?, (4.20) ^01 Substituting equations (4.19) and (4.20) into equation (4.18) gives the complete amplitude equation for the single mode p= {\\}. (iA, •^u j^ ~^\\^i\~^\\^\\ (4-21) where: \Y211111 / \Yoiiii c,, =2 / Y211111 / V \Yuii2! +Y 112111 /T +\Yiinoi +Y 110111/' £21 ^ ^ £01 4.5.2 Case! Case 2 looks at the simple mode ;? = {01}. Again the aspect ratio was changed for different air heights so that the aspect ratio was always at the minimum of the Rayleigh number versus aspect ratio curve. The sets of unstable and stable modes are: S^j = {01} S, = {31, 21, 41, 12, 22, 02} (4.22) The amplitude equation is: ^01 r, = ^01 ^01 ~2yoioioiAoi -2^010102 + Yoio20iyAoiAo2 (4-23)

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125 The only stable modes to survive the orthogonality condition is/? = {02} '^Y 02 01 01 2 '02 Ao2=^ A^, (4.24) Substituting equation (4.24) into (4.23) gives: "01 *^oi J. ~ ^01 Aqi — 2y 01 01 01 Aq, Cq, Aq, (4.z5) where: / \f 02 01 01 ^01 ~ 2^010102 +7 01 02 01 y ~ (4.zo} ^02 4.5.3 Case 3 The third case looks at the first of two codimension-two points. This codimensiontwo point is the interaction /? = {11} with q^ {21}. The amplitude equations become more complex when the interactions between two unstable, resonant modes are considered. The set of stable and unstable modes are: 5„= {11,21} 5*, = {01,31,41,12,22} (4.27) The amplitude equation for /> = { 1 1 } is: ^'" r]t ~^"ii "villi 21 +71121 11/^11^21 "ymioi +7110111 /^iiAqi ~\7llI122 +7112211/^11^22 ~\yil2131 +7113121/^^21^31 (4.28) ~\7ll2112 +7ll 1221 /^2lA|2 and the amplitude equation foxp = {21 } is:

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126 a,. dt S21 A21 ^Y211111-'^ll y 211131 "'"7213111^^11-^31 "y 211112 +72112 11/^11-^12 ~\y 212101 "'" Y 21 01 21 /^21"^C IY21214! "*" Y21412I /^2I-^41 (4.29) The reduced equations for the five stable modes Aq,, A31, A41, A,,, and Aj, are: ^oi-'^oi ~" -^Yoin n-'^ii "*" ^Yoi2i2i-'^2i ^31-^31 ~ VJ3III2I "'" Y312I ll/^ll'^2! ^41-^41 ~ '^Y412121^21 ^12-^12 "Vj 12 1121 "'"Yl22111_/^1|-^21 S22A22 — 2Y 221111 Ai (4.30) (4.31) (4.32) (4.33) (4.34) Substituting the stable modes into equations (4.28) and (4.29) gives: a
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127 = (> ^2111 17211131 + Y213111 ^ y 311121 +7312111 J / W 121121 +Yl2211iy + \Y2illi2 +7211211^ ^12 + 2(7 + 7: 212101 ^ I 21012 J 01 11 11 -01 C21 — 2. (^ \7oi2121 / \7412121 J212IOI +7210121 J~~~ +Vy 212141 +7214121/7 •01 '41 4.5.4 Case 4 The last case looked at the codimension-two point of p = {21} and p = {01}. The set Sn and 5*5 are: 5j; = -^l,0l} 5, = {31,41,12,11,22,02} (4.37) The amplitude equations for the unstable modes are: dA a 01 01 dt ^01-^01 2yoio,oiAqi 2yo,2i2iA2i ^012122+7012221/^21^22 "\7 01 01 02 +70102 01/^01^02 dA a 21 21 dt — £-)i A'21^21 (7210121+7212101/^01^21 (7212141+7214121/^21^41 ~ \7 2121 02 +72102 21/^21^02 "VY 21 01 22 +72122 01/^01^22 The amplitude equations for the stable modes are: ^41-^41 '^7412121-^21 ^22A22 V7 222101 +722 0121/^21^01 ^02 02 ^7 02 01 01 -"^01 + ^7 02 21 21 21 (4.38) (4.39) (4.40) (4.41) (4.42) Upon substitution yields:

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128 dh ^m J^ ~ ^01 ^01 "^YoiOlOl^OI ~^Yoi2121^21 "§0121^01^21 "^01^01 V'-^-') dA^^ 2 3 *^21 ~~L ~ ^21 ^21 ~ ^2101'^01-^21 82101-^ 01-^ 21 ~ ^21-^21 (4.44) where: / \ Y 0221 21 / w 222101 +Y220121 J ^0121 2l|yoio2oi + Yoioi02/~~T + \Y 0121 22 +Y 01 22 21/" ^02 ^22 / \ Y^'' '^oi 2lyoio2oi +Y 01 01 02 r~l '02 ^2101 ~ Y212IOI "'" Y 2101 / 212101 / 210121 / \ Y 02 01 01 / \ (Y 222101 +Y22012iy ^2101 2|j212102 +Y2102 21/~ + ^Y 21 01 22 + Y2122 01^j ~ C21 2 ^02 ^22 ,Y 02 2121 / \Y 4121 21 ( Y 02 2121 / \ Y2I2IO2 +Y210221_/~ +VY21214I +Y2I4121/ ^02 ^41 The next step in the analysis was to calculate all of the coefficients (i.e. c,„) for the single layer and bilayer systems at the various air heights (Table 4-7, Table 4-8, Table and Table 4-10). The program that carried out the calculations was written in Maple. All of the programs are given in Appendix A. The Maple program would first read in all of the coefficients of the Chebyshev polynomials (the eigenfunctions) of the linear problem and its adjoint. Next, it would define all of the velocities and temperatures (equation 2.117) and their adjoints (equation 2.128). The velocity and temperatures of both fluid layers (when the upper fluid layer was considered) were normalized by the maximum of the absolute value of the lower fluid's vertical component of velocity. That is:

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129 ,. max Norm = ^ ^ WjU. (4gn(w,Q) (4.45) where Zg is the value of z at the maximum. Finally the program would take the volume integral of the products of the velocities and temperatures. One of the major advantages of using Maple, was that it could solve almost all of the integrations analytically, then evaluate the integrals at the boundary. The disadvantage of this method is that it is very slow. In fact, it would take several hours to calculate the coefficients for a single air height. The values of the coefficients for all four cases and for various air heights are given in Table 4-8, Table 4-9, and Table 4-10. Coefficients for a single layer of fluid with a Biot number of zero, were also calculated to compare with the results of RDH. Unfortunately, the coefficients were not equal. In fact they were orders of magnitude apart. To verify that the coefficients from the Maple program were indeed being calculated properly, several things were done. First the eigenvalues of the linear problem were compared with the eigenvalues from RDH. These agreed up to three significant figures. Secondly the first five eigenvalues from the linear problem were compared to the first five eigenvalues from its adjoint. As the theory predicts, these eigenvalues were the same up to four significant figures. Next, the biorthogonality condition was tested. ^>^p.e.; = \^>f^„HO for;;;^^ (4.46) The azimuthal and radial components are trivially satisfied so these were not considered. The first eigenfunction from the linear problem (the eigenfunction associated with the smallest eigenvalue) and the second eigenfunction of the adjoint were found. These were

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[30 Table 4-7. Rayleigh numbers for different modes at different air heights for different aspect ratios. All calculations were done for a 5mm depth of 100 cS silicone oil. Simple Mode (m=l Single Layer Bilayer d = 0.1 d=1.0 d = 10 d = 0.1 d=1.0 d = 5.0 d = 9.0 d = 10 AR=0.7 AR=0.9 AR=0.9 AR=0.7 AR=0.9 AR=0.9 AR=1.1 AR=1.3 m n Ra Ra Ra Ra Ra Ra Ra Ra 1 635.9 310.9 247.1 638.9 314.9 270.2 198.4 126.0 2 963.4 395.5 343.2 966.6 403.9 377.2 205.4 167.8 1464 550.8 491.2 1469 565.7 541.4 264.7 248.9 3 1786 652.4 586.9 1792 671.9 648.1 303.9 304.6 4 3058 1074 982.1 3076 1117 1094 440.4 499.7 1 2 3076 1080 988.0 3095 1123 1100 442.3 502.2 Simple Mode (m=0) AR=1.5 AR=1.8 AR=1.9 AR=1.5 AR=1.8 AR=1.9 AR=2.2 AR=2.6 m n Ra Ra Ra Ra Ra Ra Ra Ra 1 635.4 309.3 247.1 638.3 313.4 270.2 185.8 125.4 3 1 642.4 310.2 249.3 645.2 314.7 272.9 190.1 126.0 2 1 680.1 334.3 262.1 683.7 337.7 285.4 193.1 135.8 4 1 740.2 345.6 280.2 743.0 352.0 307.5 226.1 143.1 1 2 742.1 346.3 280.8 744.9 352.8 308.2 226.8 143.5 2 2 1002 446.1 361.9 1005 456.5 398.0 311.4 193.5 2 1081 476.7 386.6 1085 488.4 425.2 334.1 209.4 Codimension-Two Point (m=l, m=2) AR=0.9 AR=1.05 AR=1.15 AR=0.9 AR=1.05 R = 1.15AR=1.15 AR=1.55 m n Ra Ra Ra Ra Ra Ra Ra Ra 1 678.3 328.3 262.6 681.9 331.8 286.0 198.4 134.9 2 711.9 341.6 269.7 714.7 347.8 295.8 205.4 138.7 927.9 433.4 335.2 931.0 443.4 368.4 264.7 180.7 3 1077 496.6 380.4 1081 509.2 418.4 303.9 211.2 4 1725 768.3 575.9 1731 793.6 635.7 440.4 350.0 1 2 1735 772.5 578.9 1741 798.0 639.1 442.3 352.1 2 2 2927 1281 953 2943 1338 1060 682.7 555.8 Codimension-Two Point (m=2, m=0) AR=1.3 AR=1.55 AR=1.65 AR=1.3 AR=1.55 AR=1.7 AR=2.05 AR=2.25 m n Ra Ra Ra Ra Ra Ra Ra Ra 2 1 641.8 313.5 249.4 645 317.3 274.1 188.3 127.2 1 653.1 314.1 252.5 655.9 319.0 274.1 188.7 127.7 3 1 686.3 326.5 262.9 689.0 332.0 283.7 196.8 133.5 4 1 875.8 400.6 322.0 878.8 409.2 341.9 246.0 169.2 1 2 879.0 401.8 322.9 882.0 410.5 342.9 246.9 169.8 1 1 949.6 461.1 352.7 955.4 464.0 393.9 279.3 190.5 2 2 1298 566.6 452.0 1302 582.2 472.9 347.6 254.8 2 1422 615.2 490.1 1426 633 511.5 372.7 280.9

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131 Table 4-8. Amplitude coefficients for the weakly nonlinear analysis using both single and bilayer model. All calculations were done for a 5mm depth of 100 cS silicone oil. The Prandtl Number is assumed to be infinite. Simple Mode (m=l) Single Layer d = 0.1 d=1.0 d = 10 AR = 0.7 AR = 0.9 AR = 0.9 d = 0.1 AR = 0.7 d=1.0 AR = 0.9 Bilayer d = 5.0 AR = 0.9 d = 9.0 AR = 1.1 d=10 AR=1.3 an 0.0679 0.1303 0.1543 0.0680 0.1295 0.1440 0.0569 0.0321 C11 1.48E-05 8.82E-03 1.42E-02 1.90E-05 8.32E-03 1 .07E-02 5.03E-03 -7.90E-05 Simple Mode (m= AR = 1.5 =0) AR = 1.8 AR=1.9 AR=1.5 AR = 1.8 AR=1.9 AR=1.9 AR = 2.6 aoi 0.0699 0.1252 0.1570 0.0699 0.1240 0.1466 0.0520 0.0364 2yoioioi -2.55E-02 -1.07E-01 -1.49E-01 -2.55E-02 -1.05E-01 -1.33E-01 -3.66E-02 -9.50E-05 C01 3.21 E-03 1.14E-01 2.27E-01 3.22E-03 1.08E-01 1.72E-01 5.43E-02 -4.94E-04 Codimension-Two Point (m=l, m=2j AR = 0.9 AR = 1.05 AR = 1.15 AR = 0.9 AR = 1.05 AR = 1.15 AR = 1.15 AR = 1.55 an 0.0832 0.1510 0.2010 0.0832 0.1498 0.1873 0.0765 0.0442 fl121 -8.44E-03 -2.70E-02 -3.66E-02 -8.42E-03 -2.67E-02 -3.34E-02 -1 .OOE-02 -2.27E-04 gii2i -1.08E-04 2.14E-03 6.09E-03 -1.05E-04 2.00E-03 4.39E-03 1 .93E-03 7.09E-05 C11 3.13E-04 1 .34E-02 3.46E-02 3.15E-04 1 .26E-03 2.61 E-02 5.39E-03 1.21E-04 AR = 0.9 AR = 1.05 AR = 1.15 AR = 0.9 AR = 1.05 AR = 1.15 AR = 1.15 AR = 1.55 a21 0.0519 0.0861 0.1098 0.0519 0.0854 0.1028 0.0393 0.0175 Y2nni -6.54E-03 3.07E-02 5.98E-02 -6.46E-03 3.17E-02 5.08E-02 1 .64E-02 -1.26E-02 92111 3.51 E-04 1.75E-02 4.18E-02 3.58E-04 1 .63E-02 3.13E-02 7.78E-03 -1.57E-05 C21 -1 .60E-04 7.81 E-04 2.54E-03 -1.57E-04 7.11 E-04 1.69E-03 8.30E-04 3.94E-05 Codimension-Two Point (m=2, m=0^ AR = 1.3 AR = 1.55 AR = 1.65 1 AR = 1.3 AR=1.55 AR=1.7 AR = 2.05 AR = 2.25 aoi 0.061 0.106 0.132 0.061 0.105 0.128 0.049 0.028 2yoioioi -2.50E-02 -9.57E-02 -1.31E-01 -2.50E-02 -9.41 E-02 -1.20E-01 -3.52E-02 -1 .24E-03 27012121 2.15E-03 -3.97E-03 -8.69E-03 2.13E-03 -3.86E-03 -7.44E-03 -2.31 E-03 1.91 E-03 goi2i 6.32E-04 1 .50E-02 2.96E-02 6.35E-04 1.42E-02 2.55E-02 1.13E-02 3.22E-05 COI 2.40E-03 6.45E-02 1 .26E-01 2.42E-03 6.07E-02 1.08E-01 5.08E-02 5.00E-04 AR = 1.3 AR = 1.55 AR = 1.65 AR=1.3 AR=1.55 AR=1.7 AR = 2.05 AR = 2.25 a2i 0.075 0.135 0.173 0.075 0.134 0.167 0.064 0.036 f2101 1.69E-02 7.50E-02 1.06E-01 1.69E-02 7.39E-02 9.74E-02 2.95E-02 -4.43E-04 92101 1.38E-03 4.38E-02 8.86E-02 1 .39E-03 4.13E-02 7.67E-02 3.50E-02 1.14E-04 C21 -8.17E-05 4.16E-03 9.32E-03 -7.93E-05 3.91 E-03 8.42E-03 2.19E-03 2.61 E-05

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132 Table 4-9. Amplitude coefficients for the weakly nonlinear analysis using both single and biiayer model. All calculations were done for a 5mm depth of 100 cS silicone oil. The Prandtl number is finite. Simple Mode (m=l) Single Layer d = 0.1 d=1.0 d = 10 AR = 0.7 AR = 0.9 AR = 0.9 d = 0.1 AR = 0.7 d=1.0 AR = 0.9 Biiayer d = 5.0 AR = 0.9 d = 9.0 AR = 1.1 d = 10 AR=1.3 an 6.79E-02 0.1303 0.1543 0.0680 0.1295 0.1440 0.0532 0.0321 C11 1 .85E-05 8.88E-03 1 .42E-02 1 .94E-05 8.32E-03 1.07E-02 4.76E-03 -1.51E-04 Simple '. aoi Mode (m= AR = 1.5 =0) AR = 1.8 AR=1.9 AR=1.5 AR=1.8 AR = 1.9 AR=1.9 AR = 2.6 0.0699 0.1252 0.1571 0.0699 0.1244 0.1466 0.0537 0.0364 2yoioioi -2.53E-02 -1.06E-01 -1.48E-01 -2.55E-02 -1.05E-01 -1.33E-01 -3.12E-02 1.72E-02 C01 3.19E-03 1.14E-01 2.27E-01 3.25E-03 1.08E-01 1.72E-01 4.93E-02 1.75E-03 Codimension-Two Point (m=l, m=2' AR = 0.9 AR = 1.05 AR = 1.15 > AR = 0.9 AR = 1.05 AR = 1.15 AR = 1.15 AR = 1.55 an 0.0832 0.1505 0.2009 0.0832 0.1498 0.1873 0.0569 0.0442 fl121 -8.55E-03 -2.72E-02 -3.67E-02 -8.41 E-03 -2.67E-02 -3.34E-02 -8.05E-03 -3.52E-04 gii2i -1.11E-04 2.17E-03 6.15E-03 -1 .05E-04 2.00E-03 4.39E-03 1 .06E-03 5.94E-05 C11 3.18E-04 1.35E-02 3.47E-02 3.16E-04 1.26E-02 2.62E-02 4.39E-03 1 .24E-04 ail 0.0519 0.0861 0.1098 0.0519 0.0854 0.1028 0.0421 0.0175 Y2nni -6.33E-03 3.09E-02 6.00E-02 -6.45E-03 3.02E-02 5.09E-02 2.43E-02 -1.53E-02 92111 3.63E-04 1 .76E-02 4.20E-02 3.60E-04 1.63E-02 3.13E-02 9.21 E-03 -2.04E-04 C21 -1 .59E-04 8.11E-04 2.60E-03 -1.56E-04 7.13E-04 1.70E-03 6.16E-04 5.48E-05 Codimension-Two Point (m=2, m^O] AR = 1.3 AR = 1.55 AR = 1.65 I AR = 1.3 AR=1.55 AR = 1.7 AR = 2.05 AR = 2.25 aoi 0.061 0.106 0.132 0.061 0.105 0.128 0.049 0.028 2Y010101 -2.48E-02 -9.55E-02 -1.30E-01 -2.49E-02 -9.41 E-02 -1.20E-01 -2.96E-02 1 .29E-02 2yoi2i2i 2.07E-03 -4.07E-03 -8.80E-03 2.12E-03 -3.88E-03 -7.46E-03 -1.72E-03 3.93E-03 goi2i 6.34E-04 1.50E-02 2.96E-02 6.38E-04 1 .42E-02 2.55E-02 1 .08E-02 -2.74E-04 C01 2.38E-03 6.44E-02 1.26E-01 2.44E-03 6.09E-02 1.08E-01 4.82E-02 -1.50E-03 a2i 0.075 0.135 0.173 0.075 0.134 0.167 0.064 0.036 f2101 1.73E-02 7.53E-02 1.06E-01 1.69E-02 7.39E-02 9.74E-02 2.99E-02 7.36E-04 92101 1.33E-03 4.35E-02 8.82E-02 5.54E-01 4.14E-02 7.68E-02 3.36E-02 -1.21 E-03 C21 -8.21 E-05 4.23E-03 9.44E-03 -7.86E-05 3.92E-03 8.43E-03 2.14E-03 -9.67E-05

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133 Table 4-10. Amplitude coefficients for the weakly nonlinear analysis using both single and bilayer model. All calculations were done for a 5mm depth of 100 cS silicone oil. The Prandtl number is finite. The dependent variables variables of the upper fluid have nbeen scaled with respect to their own thermophysical properties. Simple Mode (m=l) Bilayer d = 0.1 AR = 0.7 d=1.0 AR = 0.9 d = 5.0 AR = 0.9 d = 9.0 AR=1.1 d=10 AR=1.3 an 0.0679 0.1294 0.1472 0.7422 2.6557 C11 1.94E-05 8.29E-03 1.08E-02 2.47E-02 3.39E-02 Simple Mode (m=0) AR = 1.5 AR=1.8 AR=1.9 AR=1.9 AR = 2.6 aoi 0.0699 0.1243 0.1499 0.7108 2.5967 27010101 -2.55E-02 -1.05E-01 -1.33E-01 -1.54E-01 2.66E-01 C01 3.25E-03 1.07E-01 1.73E-01 3.00E-01 2.77E-01 Codimension-Two Point (m=l, m=2) AR = 0.9 AR = 1.05 AR = 1.15 AR = 1.15 AR = 1.55 an 0.0832 0.1497 0.1920 0.7775 2.9284 fl121 -8.41 E-03 -2.67E-02 -3.35E-02 -3.46E-02 5.25E-02 gii2i -1.05E-04 1.99E-03 4.42E-03 4.78E-03 1.88E-02 C11 3.16E-04 1.25E-02 2.63E-02 2.20E-02 2.25E-02 a2i 0.0519 0.0853 0.1048 0.2495 1 .9394 7211111 -6.45E-03 3.01 E-02 5.10E-02 5.49E-02 -4.36E-01 92111 3.59E-04 1 .63E-02 3.15E-02 2.59E-02 1.59E-02 C21 -1.56E-04 7.10E-04 1 .70E-03 1.67E-03 3.81 E-03 Codimension-Two Point (m=2, m=0) AR = 1.3 AR=1.55 AR = 1.7 AR = 2.05 AR = 2.25 aOi 0.061 0.011 0.131 0.646 2.303 27010101 -2.49E-02 -9.40E-02 -1.20E-01 -1.46E-01 1.88E-01 27012121 2.12E-03 -3.87E-03 -7.49E-03 -7.33E-03 -2.03E-02 goi2i 6.38E-04 1 .42E-02 2.57E-02 5.49E-02 -2.03E-02 C01 2.43E-03 6.06E-02 1 .09E-01 2.42E-01 -8.51 E-02 a21 0.075 0.134 0.171 0.831 2.730 f2101 1 .69E-02 7.38E-02 9.79E-02 1.18E-01 -2.03E-01 92101 1.40E-03 4.12E-02 7.72E-02 1.71E-01 -3.22E-02 C21 -7.86E-05 3.90E-03 8.48E-03 1 .42E-02 3.76E-02

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134 then substituted into equation (4.46). The value of this integral for several different cases, was less than 1 0'^, well within acceptable error. How is it possible that an eigenfunction was found that satisfies the linear and adjoint problem as well as the biorthogonality condition, yet is incorrect? Ruling out this possibility, the normalization constant was considered. In RDH, the normalization constant for the velocity and temperature is done by setting the temperature at the surface equal to one, 0(z = l) = 1 Because I wanted the same normalization constant for both one and two layers of fluid, this constant could not be used, because 0,(z = l);^l necessarily. To eliminate the factor, the value of a,, was compared with RDH's equivalent coefficient. To simplify the comparison, a large value of the Prandtl number was assumed. lim (Qll'^ll/ .. .-,x Therefore, any normalization constant would cancel out. Secondly, by looking at the equations for vi-'p, 9p, and 9'p, we notice that the azimuthal and radial components also cancel out from the numerator and denominator. 2j[ 1 / t jcos' mcp c/cp j r J„', M^ dr |0(z)0(z) Jz \Q(z)@*(z)dz mn' mn 2% 1 A.„„r jcos^ mcp t/9 I r J^ I ^^ J dr |w(z)W* (z) dz |w(z)W* (z) dz Even in this simplified calculation the numbers did not compare. With these numerous checks, the values of the coefficients of RDH were definitely suspect. Additionally, in a private communication with Dr. Pierre Dauby, he mentioned that he was also unable to verify RDH's calculation.

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135 Before discussing the results of the calculations, a brief description of what exactly was done for the different scaling is in order. If you recall from the Mathematical Modeling section, the dependent variables in both fluid layers were scaled using the thermophysical properties of just the lower fluid. For example the dimensionless velocities in each phase are: w, where the prime denotes the dimensioned velocity, and k, and d, are the thermal difftisivity and depth of the lower fluid. This non-dimensionalization was chosen to be consistent with previous work by Perm and Wollkind (1982) hence forth referred to as FW. However, the choice of scaling is often determined by some factor within the domain of the variable (Lin and Segel, 1994). For example, the choice of J,/k, makes sense for the lower fluid's velocity because it is much larger than, say, d, /v, (see Table 4-5). But is the upper fluid's velocity accurately measured on the same scale? Probably not. A more appropriate choice of scaling for the upper fluid would be to scale the variables with respect to their own thermophysical properties. The thermophysical properties used for the lower fluid in the new scaling were the same as those in FW. The scaling for the upper fluid's variables used the same constants except they were the upper fluids constants, that is to say: V, = •"k,M "'-^^ ''=ZAr "•=Z (4.49) T-T e, = — ^ at; ^, = t' z' t/f/K, ^1 ~ ^ d, ^'~ ^T, h /' z' d;lK, ''-
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136 where AT, and ATj are the temperature differences across the lower and upper fluid respectively, and T„ is the temperature at the fluid-fluid interface. The choice of the thermal diffusivity as the scaling for the time and velocity of air was really quite arbitrary, because the Prandtl number of air is about one (Pr = 0.86). The kinematic viscosity could have been chosen but the choice of the thermal diffusivity simplified the notation, as we will see next. After rescaling, the nonlinear equation becomes: V V,. = Pi^^\^ + vrVv,J = V\-Vi?, + i?A2 (4.51) -^+v,-ve, = v2g, + w, for i = 1 and 2. It was not necessary to rescale and recompute the linear problem because the linear solutions are invariant to scaling. For example, given some scaling, and some linear operator, L, we can write: L(av) = aL(v) (4.52) Therefore, we need only rescale the solution to the linear problem L(v). To verify this, the linear problem was rescaled using the new scaling talked about previously. The solution, after being rescaled back to the original scaling, was exactly the same. However, nonlinear operators do not have this property. This makes it necessary to recompute the amplitude equations. Following a similar procedure as in the nonlinear analysis section of the Mathematical Modeling chapter, we arrive at a new amplitude equation.

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137 dh. ."^ = SA,-Y;„A,A, where: 1/. P = w. s = M-Mp R~Rp M. R, (4.53) (4.54) (4.55) prr'(v;;,vi,-vvi.)+Pr2-'(v;,,v,, vv2.)+(e;,,vi^ ve,J+ e;^,v2, -ve^, pqr yVXp,Qlp] + [W2p,Q2p (4.56) Notice that both a^ and yp^, have changed. Conveniently, the four cases previously given do not need to be rederived. The new coefficients of ap and Yp^, need only re recomputed and substituted back into the proper equations. To begin the discussion of the results, we will make some general comments about the coefficients themselves. Table 4-8 gives the coefficients of the amplitude equations using FW scaling and an infinite Prandtl number. For certain situations, the coefficients agree qualitatively with those of RDH. For example, the single layer and bilayer system A„ b -^ Figure 4-18. Bifurcation diagram for the m = simple mode case. The first diagram is for a 1mm air height and the second diagram is for a 10mm air height.

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138 both predict that the m = flow for depths of Imm will give a subcriticai bifurcation and that the amplitude of the subcriticai branch will be positive (Figure 418a). However, at an air depth of 0.1mm, the Cq, coefficient changes sign. This will cause the bifurcation curve to flip (Figure 418b). However, a negative coefficient for Cq, may indicate the higher order terms are necessary (i.e. A^q,). Another general statement that can be made is that the coefficients of the bilayer system always converge to the coefficients of the single layer system when the air height becomes small. This is in agreement with the physics, as the passive gas assumption of the single layer model is only valid for very small air heights. In fact the numbers from Table 4-8, Table 4-9, and Table 4-10 give an indication of exactly when this assumption begins to fail. We now analyze the effect of the infinite Prandtl number assumption. Looking at the bilayer coefficients for an air height of 10mm in Table 4-8 and Table 4-9, we notice that Co, switches sign for an infinite Prandtl number but does not for a finite value of the Prandtl number (Pr = 909). Just the opposite happens for the coefficients Cj, and Cq,, for an air height of 10mm. The coefficients do not switch sign between 9mm and 10mm heights for an infinite Prandtl number, but do switch sign for the finite Prandtl number. However, all of the coefficients for the single layer model and for the bilayer model change very little between Table 4-8 and Table 4-9. From this we may conclude that the Prandtl number only has an effect for very large air heights, where the convection in the air becomes substantial. This conclusion leads to questions of what effect the scaling might have on the coefficients, particularly when the inertial terms of the air are divided by the Prandtl number of the air and not the Prandtl number of the silicone oil.

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139 Each of the coefficients were recalculated using the new equations for a^ and Ypq^ in equations (4.54) and (4.56). These coefficients are given in Table 4-10. First it may be noted that the results from the single layer model will not change by rescaling the problem. Secondly, in light of the results of the Prandtl number in the previous paragraph, the Prandtl numbers were kept and not assumed to be infinite. Again the coefficients converge to the single layer model and the bilayer model with FW scaling, when the air height becomes small. For the air height of 0.1mm, Imm, 5mm, and 9mm in all of the cases, the qualitative behavior of the solution did not change. However, quantitatively the coefficients did change significantly. The qualitative behavior of the solutions changed from 9mm to 1 0mm air heights, where several coefficients changed sign again. In conclusion, the scaling and infinite Prandtl number assumption make little difference in the qualitative behavior of the solutions for small air heights. For small air heights, all of the different conditions (scaling and Prandtl number) converge to the same values. However, when the air height becomes large and convection in the air becomes significant, these assumptions are important. Unfortunately, the values of the coefficients of the amplitude equation did not agree with those from RDH even though several checks were performed. Also, the fact that the calculations do not predict the proper bifurcation diagrams indicates that higher order terms may be necessary in the amplitude equations. Future progress on this problem should involve a method to check the accuracy of these values.

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CHAPTER 5 FUTURE SCOPE This final chapter includes several ideas for future projects and areas into which I think the problem could be expanded. These ideas are broken into three major categories: experiments, nonlinear analysis, and numerical calculations. Each idea should lead to some interesting results and provide more insight into this fascinating phenomenon. 5.1 EXPERIMENTS Because of the large number of parameters in bilayer convection systems, there are many different experiments that could be performed. In this section I have concentrated on three main ideas. The first is a continuation of the single layer experiments. The second idea explores the richer interaction of two liquid layers. The last idea incorporates a solidifying lower liquid into the physics. A simple series of experiments could be performed to verify the work performed by Echebarria et al. (1997). In their paper, a graph was given which predicts that the m = 1, m = 2 codimension-two point should give several interesting patterns; among these are rotating flow patterns. In several coarse experiments, I have seen rotating flow patterns as well asm=l,m = 2, m = 3, and m = 4, flow pattern interactions. However, none of these experiments were performed in a careful and systematic manner. The incorporation of a second liquid layer should reveal many interesting flow patterns. The challenge to these experiments is finding a way to view both the upper and 140

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141 lower fluids in a cylindrical container. Assuming this obstacle can be overcome, one of the more interesting experiments to run would be depth ratios were the unbounded layer calculations predict oscillatory onset of convection. This has been seen in large rectangular containers by Andereck et al. (1996). However in small aspect ratios, the oscillatory onset of convection could occur at codimension-two points to give some really interesting dynamic behavior. To carry this problem further towards its application in crystal growth, it is necessary to look at the coupling of convection and solidification. This could be done experimentally with any solidifying liquid, although it would be easier to use a liquid that solidified near room temperature. Gallium, for example, fits this criterion quite well. One could carry out experiments that monitored the solid-liquid interface levels and the convective flow patterns simultaneously. The crystal could then be characterized after the experiment to investigate the effect of certain flow pattern son the crystal quality. Another interesting aspect of the problem would be to vary the crystal growth rate and model the interaction of heat being released from the solidifying crystal into the convecting liquid. 5.2 NONLINEAR ANALYSIS The most important next step of the nonlinear analysis is to verify the correctness of the coefficients. Once this is accomplished, there are several different cases one could analyze. Particularly one could continue the analysis of various depth ratios. Some of the questions that could be answered are "if the air begins to flow first, but the liquid becomes unstable soon after, which flow will persist?" The nonlinear analysis could also

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142 be tested to see if it predicts some of the experiments where the air layer convects first and causes flow in the silicone oil. A weakly nonlinear analysis could also be performed for two liquid layers. An interesting question would be how dot the modes interact with each other when the onset of convection is oscillatory. This is particularly true for codimension-two points. 5.3 NUMERICAL CALCULATIONS Finally, a series of numerical calculations could be very useful in explaining many questions. We have learned from the vorticity-free calculations and the no-slip calculations, that there are qualitative differences in predictions of these two models for a single fluid layer. Therefore, in order to accurately predict experiments of two fluid layers, a full tliree dimensional calculation will be necessary. The three dimensional calculation can be broken in to two separate classes: linear and nonlinear. Linear calculations similar to Dauby et al. (1997) could be performed to study the onset of convection in two liquid layers. These calculations should give accurate critical temperature data which could aid in the experiments. Nonlinear analysis could be performed to give useful examples of real liquid-encapsulated crystal growth. The nonlinear calculations could also be compared with the weakly nonlinear analysis. To date I know of no such comparison for any nonlinear system. Whether any of these ideas are followed up on, or other creative projects are designed, it is certain that this problem will yield many more fascinating phenomena.

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APPENDIX A COMPUTER PROGRAMS Appendix A includes all of the Matlab and Maple programs used in this thesis. Below is list of each program. Initialization File for Case 8 Case 10, Case 14 and Case 15 145 Initialization File for Case 12 and Case 13 147 Case 8 I49 Case 9 I54 Case 10 158 Case 12 152 Case 13 154 Case 14 156 Case 15 I59 Case 16 172 Case 17 I75 Case 19 I79 Case 20 Igl Case 8 through Case 15 are Matlab programs which find the eigenvalues (Rayleigh numbers) and eigenvectors (velocities and temperatures) for the single layer and bilayer systems. Below is a useful table which gives the number of fluid layers considered, whether or not a deflecting interface was considered, the scaling used in deriving the equations, whether it was the linear problem or the adjoint problem and the order of the differential equations. Case 16 calls Case 12 and its adjoint. Case 13, to find the stable and unstable sets for a given single fluid layer. Case 17 is similar to Case 16, where it calls Case 14 and its adjoint, Case 15, to find the stable and unstable sets of a bilayer calculation. 143

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144 Case 22 through Case 24 are Maple programs. Case 22 finds the amplitude coefficients, ap and yp^, for the single fluid layer. Case 23 finds the amplitude coefficients for a bilayer using the Perm and WoUkind scaling. Case 24 finds the amplitude coefficients using the scaling with respect to each fluid variable. Case Number of Deflecting Scaling Adjoint Order of Numbei" Layers Interface Equations ^ 8 2 yes F and W no 2 9 2 no FandW no 2 10 2 no new scaling no 2 12 1 no FandW no 4 13 1 no FandW yes 4 14 2 i^no ^^' no FandW no 4 15 2 FandW yes 4 Table A-1 Table of different Matlab programs

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145 Initialization File for Case 8 Case 10, Case 14 and Case 15 %Enter the dimension of the Chebyshev polynomial ***************** N=12; %Define the parameters in this problem ************************ % thermophys densityl = density2 = thermexpl = fluid thermexp2 = fluid thermcondl = thermcond2 = thermdiffl = thermdiff2 = kinviscl = kinvisc2 = sigma = 2 0.9 sigmal = 0.0 gravity = 98 depthl = 3 ; depth2 = 6 ; ical parameters in cgs units ********************** .968; %lower density .0012; %upper density 9.6e-4; %negative thermal expansion coefficient of lower 3 3 3 e 3 ; 1.59e4 2 .62e3 l.le-3 0.182; .6922; .15 7; 5; 0; tempdiff = 0.3; %negative thermal expansion coefficient of upper %Thermal conductivity of lower fluid %Thermal conductivity of upper fluid %Thermal diffusivity of lower fluid %Thermal diffusivity of upper fluid %kinematic viscosity of lower fluid %kinematic viscosity of upper fluid %surface tension %negative surface tension gradient %magnitude of gravity %depth of lower fluid %depth of upper fluid %Temperature difference across lower fluid % Define dimensionless number ****************************************** Ra = gravity thermexpl tempdiff depthl^3 / (thermdiffl kinviscl) ; Ma = sigmal tempdiff depthl / (thermdiffl kinviscl densityl) ; Gamma = sigmal/ (gravity thermexpl depthl^2 densityl); %Gamma = ratio of Ma to Ra G = densityl gravity depthl'^2 / sigma; %G = Weber number C = kinviscl densityl thermdiffl / (sigma depthl) ; %C = Crispation Number % Define ratio of thermophysical properties ***************************** thermexp = thermexp2 /thermexpl; depth = depth2 / depthl; thermcond = thermcond2 / thermcondl ; thermdiff = thermdiff2 / thermdiffl; density = density2 / densityl; vise = (kinvisc2 density2) / (kinviscl densityl) ; kinvisc = kinvisc2 / kinviscl; %Define the submatrices used in defining the full matrices A and B El=chdrlmat (N) ;

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U6 E2=chdr2mat (N) ; Iden=eye (N) ; %Define the boundary condition vectors vecO = chbvec(0, N, 1) ; vecOn = chbvec(0, N, -1) ; vecl = chbvec(l, N, 1) ; vecln = chbvecd, N, -1) ; vec2 = chbvec(2, N, 1) ; %Initiate A and B matrices A=zeros (6*N+1,6*N+1) ; B=zeros (6*N+1, 6*N+1) ; eigenvals=2eros (1, 6*N+l) ,clear avals; % Define the begining, ending and incremental values of the wave number omegaFirst = 2.0; omegaLast = 4.0; omegaStep = 0.1; omega = omegaFirst;

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147 Initialization File for Case 12 and Case 13 % casel2in.m % Nield's problem, fourth order deriv, no deflection %Enter the dimension of the chebyshev polynomial ***************** N = 10; %Define the parameters in this problem ************************ % thermophysical parameters in cgs units ********************** density = 0.968; %liquid density thermexp = 9.5e-4; %negative thermal expansion coefficient thermcond = 1.59e4; %Thermal conductivity thermcondgas = 2.62e3; %Thermal conductivity of the gas thermdiff = l.le-3; %Thermal diffusivity kinvisc = 1.0; %k;inematic viscosity sigma = 2 0.9; %surface tension sigmal = 0.05; %negative surface tension gradient gravity = 980; %magnitude of gravity depth = 0.50; %depth of liquid depthgas = 0.01; %depth of gas Biot = (2 62e3/thermcond) (depth/depthgas) ; %Biot = Biot Number Gamma = sigmal/ (gravity thermexp depth^2 density) ; %Gamma = ratio of Ma to Ra % Gamma = 0.0001; % Biot = 1; %Define the submatrices used in defining the full matrices A and B El=chdrlmat (N) ; E2=chdr2mat (N) ; E4=E2*E2; Iden=eye (N) ; %Define the boundary condition vectors vecO = chbvec{0, N, 1); vecOn = chbvec(0, N, -1); vecl = chbvecd, N, l) ; vecln = chbvecd, N, -1); vec2 = chbvec(2, N, 1); %Initiate A and B matrices A=zeros (2*N,2*N) ; B=zeros (2*N,2*N) ; eigenvals=zeros (1, 2*N) ; evals=0; % Define the begining, ending and incremental values of the wave number omegaFirst = 1.0;

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148 omegaLast = 4.0; omegas tep = 0.1; omega = omegaFirst;

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149 Cases % Case 8 is a Matlab file which attempts to do the same thing that % case3 through case6 of the MathCad files do. % This program returns a vector "avals" which is an order list of all % eigenvalues % It assumes steady state onset of convection and solves for the % Rayleigh number as the eigenvalue % Surface is allowed to deflect % In cases and case5, surface viscosity is added. % Bilayer of two arbitrary fluids %Define the A matrix *************************************************************** %Row 1 Column 1 A(1:N-2,1:N) = 4 *E2 (1 :N-2 : ) omega"2 *Iden (1 :N-2 : ) ; %Row 1 Column 2 A{1:N-2,N+1:2*N) = -2 *E1 {1 :N-2 : ) ; %Row 2 Column 2 A(N-1:2* (N-2) ,N+1:2*N) = 4 *E2 (1 :N-2 : ) omega'^2 *Iden (1 :N-2 : ) ; %Row 3 Column 1 A(2*N-3 :3* (N-2) 1:N) = Iden (1 :N-2 : ) ; %Row 3 Column 3 A(2*N-3 :3* (N-2) ,2*N+1:3*N) = 4 *E2 (1 :N-2 : ) omega^2 *Iden (1 :N-2 : ) ; %Row 4 Column 4 A(3*N-5:4* (N-2) ,3*N+1:4*N) = visc/density .* (4/depth"2 *E2 (1 :N-2 : ) omega'^2 *Iden (l :N-2 : ) ) ; %Row 4 Column 5 A(3*N-5:4* (N-2) ,4*N+1:5*N) = -2 / (density*depth) El(l:N-2,:); %Row 5 Column 5 A(4*N-7:5* (N-2) ,4*N+1:5*N) = (l/density) (4/depth^2 *E2 (1 :N-2 : ) omega''2 *Iden(l:N-2, : ) ) ; %Row 6 Column 4 A(5*N-9:6* (N-2) ,3*N+1:4*N) = (l/thermcond) *Iden (1 :N-2 : ) ; %Row 6 Column 5 A(5*N-9:6* (N-2) 5*N+1:6*N) = thermdif f (4/depth^2 *E2 (1 :N-2 : ) omega*2.*Iden(l:N-2, : ) ) ; % Begin Boundary Conditions *******************************************************

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150 %Row 7 Column 4 A{6*N-11,3*N+1:4*N) = vecOn ( : ) ; %Row 8 Column 1 A(6*N-10, 1:N) = vecO { : ) ; %Row 9 Column 1 A{6*N-9,1:N) = depth *vecl {:)' ; %Row 9 Column 4 A(6*N-9, 3*N+1:4*N) = -vecln ( : ) ; %Row 10 Column 1 A{S*N-8,1:N) = 4.*vecl( :) ; %Row 10 Column 2 A{6*N-8,N+1:2*N) = -vecO ( : ) ; %Row 10 Column 4 A(6*N-8,3*N+1:4*N) = {4*visc/depth) *vecln ( : ) ; %Row 10 Column 5 A(6*N-8,4*N+1:5*N) =vec0n(:)'; %Row 10 Column 7 A(6*N-8,6*N+1) = (G + omega^2)/C; %Row 11 Column 1 A(6*N-7,1:N) = 4 *vec2 { : ) + omega^2 *vec0 ( : ) ; %Row 11 Column 4 A(6*N-7,3*N+1:4*N) = -vise (4/depth''2 *vec2 ( : ) + omega^2 *vecO ( : ) ) ; %Row 12 Column 3 A(6*N-6,2*N+1:3*N) = -2 *vecl { : ) ; %Row 12 Column 6 A(6*N-6,5*N+1:6*N) = (2*thermcond/depth) *vecln ( : ) ; %Row 13 Column 3 A(6*N-5,2*N+1:3*N) = vecO ( : ) ; %Row 13 Column 6 A(6*N-5,5*N+1:6*N) = -vecOn{:)'; %Row 13 Column 7 A{6*N-5,6*N+1) = (1/thermcond 1) ; %Row 14 Column 4 A(6*N-4, 3*N+1:4*N) = vecl ( : ) ; %Row 15 Column 4 A{6*N-3,3*N+1:4*N) = vecO ( : ) ;

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151 %Row 16 Column 6 A(6*N-2, 5*N+1:6*N) = vecO ( : ) ; %Row 17 Column 1 A{6*N-1,1:N) = vecln(:)'; %Row 18 Column 1 A(6*N,1:N) = vecOn( : ) ; %Row 19 Column 3 A(6*N+1,2*N+1:3*N) = vecOn{:)'; % Define the B matrix %Row 1 Column 3 B(1:N-2,2*N+1:3*N) = -Iden (1 :N-2 : ) ; %Row 2 Column 3 B{N-1:2* (N-2) ,2*N+1:3*N) = 2 *El(l :N-2 : ) ; %Row 4 Column 6 B(3*N-5:4* (N-2) 5*N+1:6*N) = -thermexp *Iden (1 :N-2 : ) ; %Row 5 Column 6 B{4*N-7:5* (N-2) ,5*N+1:6*N) = (2*thermexp/depth) *E1{1 :N-2 : ) ; %Row 11 Column 3 B(6*N-7,2*N+1:3*N) = (omega ^2 Gamma) *vecO (:)' ; %Row 11 Column 7 B(6*N-7,6*N+1) = omega'^2 Gamma; % Find the eigenvalues [eigenvecs, eigenvals] = eig(A,B); % Find the eigenvalues and eigenvectors eigenvals = diag (eigenvals) ; % Find the entries which are not infinity or NaN's indexl = ; for index2 = 1 : length (eigenvals) if isnan( eigenvals (index2) ) % We don t want NaN values elseif isinf ( eigenvals (index2) ) % Nor the infinities elseif abs (eigenvals (index2) ) > 10*10 % Some values are finite but ridiculously large else indexl = indexl + 1; evals (indexl) = eigenvals (index2) ;

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152 if abs (imag (evals (indexl) ) ) < le-10 % Get rid of very small imaginary parts evals (indexl) = real (evals (indexl) ) ; end end end % Sort the eigenvalues (Rayleigh number) by smallest positive value first % then the smallest (in magnitude) of negative numbers. evals=sortl (evals) ; % Find the eigenvectors associated with the smallest eigenvalue for indexl = l : length (eigenvals) if real (evals (1) ) =real (eigenvals (indexl) ) Minlndex = indexl ; break; end end % Find the velocities, temperatures and surface deflection % Find the proper coefficients vellCoef = eigenvecs (1 :N, Minlndex) ; vel2Coef = eigenvecs (3*N+1 :4*N, Minlndex) ; templCoef = eigenvecs (2*N+1 : 3*N, Minlndex) ; temp2Coef = eigenvecs (5*N+1 : 6*N, Minlndex) ; sd = eigenvecs (S*N+1, Minlndex) ; % Surface deflection % Eliminate residual imaginary parts vellCoef = real (vellCoef) ; vel2Coef = real (vel2Coef ) ; templCoef = real (templCoef ) ; temp2Coef = real (temp2Coef ) ; sd = real (sd) ; % Use the coefficients to find the vector plots Z = -1:0.01:1; X = -depthl: (depthl + depth2) /2 00 :depth2 ; veil = tnfunc (vellCoef z) ; vel2 = tnfunc (vel2Coef, z) ; tempi = tnfunc (templCoef z) ; temp2 = tnfunc (temp2Coef, z) ; % Concantenate the lower and upper velocities and temperatures nl = length (veil) n2 = length (vel2) vel (linl) = veil ( : ) ; vel (nl+1 :nl+n2-l) = vel2(2:n2); temp(l:nl) = tempi (:) ; temp (nl+1 :nl+n2-l) = temp2(2:n2); % Normalize the velocity and temperature maxvel = max (abs (real (vel) ) ) ;

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153 maxtemp = max (abs (real (temp) ) ) ; vel = vel / maxvel ; temp = temp ./ maxtemp; % Find the type of interfacial structure tempatl = sum (templCoef ) ; % Find the temperature at the interface dvelatl =0; % Find the derivative of the velocity at the interface for index = 1:N dvelatl = dvelatl + vellCoef (index) (index 1)^2; end ratiol = dvelatl / sd; ratio2 = tempatl / sd; if (ratiol > 0) (ratio2 < 0) Case = 1 ; elseif (ratiol < 0) (ratio2 > 0) Case = 2 ; elseif (ratiol > 0) (ratio2 > 0) Case = 3 ; elseif (ratiol < 0) (ratio2 < 0) Case = 4 ; end % Find the other parameters Ral = real (evals (1) ) Ma = Gamma Ral ; Tempdiff = (thermdiffl kinviscl Ral) / (gravity thermexpl depthl^S) ; Ra2 = Ral (depth*4 thermexp) / (thermdiff kinvisc thermcond) ;

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154 Case 9 % This program returns a vector "evals" which is an order list of all eigenvalues % It assumes steady state onset of convection and solves for the Rayleigh number as the eigenvalue. % No surface deflections or surface viscosity. Scaling is the same as Farm and Wollkind % Bilayer of two arbitrary fluids %Define the A matrix *********************************************************** %Row 1 Column 1 A{1:N-2,1:N) = 4 *E2 (1 : N-2 : ) omega'"2 *Iden (1 :N-2 : ) ; %Row 1 Column 2 A(1:N-2,N+1:2*N) = -2 *E1 (1 : N-2 : ) ; %Row 2 Column 2 A(N-1:2* {N-2) ,N+1:2*N) = 4 *E2 {1 :N-2 : ) omega'^2 *Iden (1 :N-2 : ) ; %Row 3 Column 1 A(2*N-3:3* (N-2) ,1:N) = Iden (1 :N-2 : ) ; %Row 3 Column 3 A(2*N-3:3* {N-2) ,2*N+1:3*N) = 4 *E2 (1 :N-2 : ) omega^2 *Iden (1 :N-2 : ) ; %Row 4 Column 4 A(3*N-5:4* (N-2) 3*N+1:4*N) = visc/density .* {4/depth^2 *E2 (1 :N-2 : ) omega^2 *Iden (1 :N-2 : ) ) ; %Row 4 Column 5 A(3*N-5:4* (N-2) ,4*N+1:5*N) = -2 / {density*depth) El(l:N-2,:); %Row 5 Column 5 A(4*N-7:5* {N-2) ,4*N+1:5*N) = (l/density) {4/depth''2 *E2 (1 :N-2 : ) omega*2 *Iden(l :N-2, : ) ) ; %Row 6 Column 4 A(5*N-9:6* (N-2) ,3*N+1:4*N) = (l/thermcond) *Iden {1 :N-2 : ) ; %Row 6 Column 6 A(5*N-9:6* (N-2) 5*N+1:6*N) = thermdif f {4/depth^2 *E2 (1 :N-2 : ) omega^2.*Iden(l:N-2, : ) ) ,% Begin Boundary Conditions for A matrix **************************************** %Row 7 Column 1 A(6*N-11,1:N) = vecOn{:)';

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155 %Row 8 Column 1 A(6*N-10, 1:N) = vecln{:)'; %Row 9 Column 3 A(6*N-9,2*N+1:3*N) = vecOn ( : ) ; %Row 10 Column 4 A(6*N-8,3*N+1:4*N) = vecO ( : ) ; %Row 11 Column 4 A(6*N-7,3*N+1:4*N) = vecl ( : ) ; %Row 12 Column 6 A{6*N-6, 5*N+1:6*N) = vecO ( : ) ; %Row 13 Column 1 A(6*N-5,1:N) = vecO { : ) ; %Row 14 Column 4 A(6*N-4,3*N+1:4*N) = vecOn{:)'; %Row 15 Column 1 A(6*N-3,1:N) = depth. vecl (:)' ; %Row 15 Column 4 A(6*N-3,3*N+1:4*N) = -vecln ( : ) ; %Row 16 Column 3 A(6*N-2,2*N+1:3*N) = vecO ( : ) ; %Row 16 Column 6 A(6*N-2, 5*N+1:6*N) = -vecOn ( : ) ; %Row 17 Column 3 A(6*N-1,2*N+1:3*N) = vecl ( : ) ; %Row 17 Column 6 A(6*N-1, 5*N+1:6*N) = (thermcond/depth) *vecln ( : ) ; %Row 18 Column 1 A{6*N,1:N) = 4.*vec2 ( :) ; %Row 18 Column 4 A{6*N, 3*N+1:4*N) = (4 visc/depth''2) .* vec2n { : ) ; % Define the B matrix **************************************************************** %Row 1 Column 3 B(1:N-2,2*N+1:3*N) = Iden (1 : N-2 : ) ; %Row 2 Column 3 B(N-1:2* (N-2) ,2*N+1:3*N) = 2 *E1 (1 :N-2 : ) ;

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156 %Row 4 Column 6 B(3*N-5:4* (N-2) ,5*N+1:6*N) = -themexp *Iden (1 :N-2 : ) ; %Row 5 Column 6 B(4*N-7:5* (N-2) ,5*N+1:G*N) = {2*thermexp/depth) *E1(1 :N-2 : ) ; %Row 18 Column 3 B(6*N,2*N+1:3*N) = {omega ^2 Gamma) *vecO (:)' ; % Find the eigenvalues [eigenvecs, eigenvals] = eig(A,B); % Find the eigenvalues and eigenvectors eigenvals = diag (eigenvals) ; % Find the entries which are not infinity or NaN's indexl = ; for index2 = 1 : length (eigenvals) if isnan( eigenvals {index2) ) % We don't want NaN values elseif isinf{ eigenvals (index2) ) % Nor the infinities elseif abs (eigenvals {index2) ) > lO^'lO % Some values are finite but ridiculously large else indexl = indexl + 1; evals (indexl) = eigenvals (index2) ; if abs (imag (evals (indexl) ) ) < le-10 % Get rid of very small imaginary parts evals (indexl) = real (avals (indexl) ) ; end end end % Sort the eigenvalues (Rayleigh number) by smallest positive value first % then the smallest (in magnitude) of negative numbers. evals=sortl (evals) ; % Find the eigenvectors associated with the smallest eigenvalue for indexl = 1 : length (eigenvals) if real (evals (1) ) == real (eigenvals (indexl) ) Minlndex = indexl; break; end end % Find the other parameters Ral = real (evals (1) ) ; Ma = Gamma Ral; Tempdiff = (thermdiffl kinviscl Ral) / (gravity thermexpl depthl^3) ;

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157 Ra2 = Ral (depth^4 thermexp) / (thermdiff kinvisc thermcond) ;

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158 Case 10 % This program returns a vector "avals" which is an order list of all eigenvalues % It assumes steady state onset of convection and solves for the Rayleigh number as the eigenvalue. % No surface deflections or surface viscosity. % Scaling is with respect to the each fluid's thermophysical properties. % Bilayer of two arbitrary fluids %Def ine the A matrix ********************** + ************************************ Ql = zeros (N-2,N) Q2 = zeros (N-2,N) Ql = 4.*E2 {l:N-2, Q2 = 4.*E2 (l:N-2, %Row 1 Column 1 A(1:N-2,1:N) = Ql ; ) omega^2 .* Iden (1 :N-2 : ) ; ) (omega^2*depth^2) .* Iden (1 :N-2 : ) ; %Row 1 Column 2 A(1:N-2,N+1:2*N) = -2 *E1 (1 :N-2 : ) ; %Row 2 Column 2 A{N-1:2* (N-2) ,N+1:2*N) = Ql ; %Row 3 Column 1 A{2*N-3 :3* (N-2) ,1:N) = Iden (1 :N-2 : ) ; %Row 3 Column 3 A(2*N-3:3* (N-2) ,2*N+1:3*N) = Ql ; %Row 4 Column 4 A(3*N-5:4* (N-2) ,3*N+1:4*N) = Q2 ; %Row 4 Column 5 A(3*N-5:4* (N-2) ,4*N+1:5*N) = -2 *E1 (1 :N-2 : ) ; %Row 5 Column 5 A(4*N-7:5* (N-2) ,4*N+1:5*N) = Q2 ; %Row 6 Column 4 A(5*N-9:6* (N-2) ,3*N+1:4*N) = Iden (1 :N-2 : ) ; %Row 6 Column 6 A(5*N-9:6* (N-2) ,5*N+1:6*N) = Q2 ; % Begin Boundary Conditions

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159 %Row 7 Column 1 A(6*N-11,1:N) =vecOn(:)'; %Row 8 Column 1 A{6*N-10,1:N) = vecln(:)'; %Row 9 Column 3 A{S*N-9,2*N+1:3*N) = vecOn { : ) ; %Row 10 Column 4 A(6*N-8,3*N+1:4*N) = vecO ( : ) ; %Row 11 Column 4 A(6*N-7,3*N+1:4*N) = vecl { : ) ; %Row 12 Column 6 A{6*N-6,5*N+1:6*N) = vecO ( : ) ; %Row 13 Column 1 A(6*N-5,1:N) = vecO { : ) ; %Row 14 Column 4 A(6*N-4,3*N+1:4*N) =vecOn(:)'; %Row 15 Column 1 A(6*N-3,1:N) = vecl( : ) ; %Row 15 Column 4 A(e*N-3,3*N+l:4*N) = (thermdif f /depth^2 ) .* vecln ( : ) ; %Row 16 Column 3 A(6*N-2,2*N+1:3*N) = vecO ( : ) ; %Row 16 Column 6 A(6*N-2, 5*N+1:6*N) = -depth/thermcond. *vecOn ( : ) ; %Row 17 Column 3 A(6*N-1,2*N+1:3*N) = vecl ( : ) ; %Row 17 Column 6 A(6*N-1, 5*N+1:6*N) = -vecln (:)' ; %Row 18 Column 1 A(6*N,1:N) = 4 .* vec2 ( : ) ; %Row 18 Column 4 A{6*N,3*N+1:4*N) = (4 vise thermdiff / depth"3) .* vec2n(:)'; % Define the B matrix %Row 1 Column 3 B(1:N-2,2*N+1:3*N) = -Iden (1 :N-2 : ) ;

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160 %Row 2 Column 3 B(N-1:2* (N-2) ,2*N+1:3*N) = 2 *E1 (1 :N-2 : ) ; %Row 4 Column 6 B{3*N-5:4* {N-2) ,5*N+1:6*N) = -RaRat *Iden (1 :N-2 : ) ; %Row 5 Column 6 B(4*N-7:5* (N-2) ,5*N+1:6*N) = 2*RaRat *E1(1 :N-2 : ) ; %Row 18 Column 3 B(6*N,2*N+1:3*N) = (omega* 2 Gamma) *vecO (:)' ; % Find the eigenvalues ********************************************************** [eigenvecs, eigenvals] = eig(A,B); % Find the eigenvalues and eigenvectors eigenvals = diag (eigenvals) ; % Find the entries which are not infinity or NaN's indexl = ; for index2 = 1 : length (eigenvals) if isnan( eigenvals (index2) ) % We don't want NaN values elseif isinf ( eigenvals {index2) ) % Nor the infinities elseif abs (eigenvals (index2) ) > 10*10 % Some values are finite but ridiculously large else indexl = indexl + 1 ; evals (indexl) = eigenvals (index2) ; if abs (imag( evals (indexl) ) ) < le-10 % Get rid of very small imaginary parts evals (indexl) = real (evals (indexl) ) ; end end end % Sort the eigenvalues (Rayleigh number) by smallest positive value first % then the smallest (in magnitude) of negative numbers. evals=sortl (evals) ; % Find the eigenvectors associated with the smallest eigenvalue for indexl = 1 : length (eigenvals) if real (evals (1) ) == real (eigenvals (indexl) ) Minlndex = indexl; break ; end end % Find the other parameters Ral = real (evals (1) ) ;

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161 Ma = Gamma Ral; Tempdiff = (thermdiffl kinviscl Ral) / (gravity thermexpl depthl*3) ; Ra2 = Ral (depth^4 therraexp) / (thermdiff kinvisc thermcond) ;

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162 Case 12 % casel2 This is the fourth order Nield problem without surface deflections % casel2in.m needs to run first in order to initialize parameters. % Single liquid layer with passive gas above. No surface deflections %Define the A matrix *********************************************************** %Row 1 Column 1 A(1:N-4,1:N) = 16 *E4 (1 :N-4 : ) (8*omega^2) *E2 (1 :N-4 : ) + omega'^4 .* Iden{l:N-4, : ) ; %Row 2 Column 1 A(N-3:2*N-6,1:N) = Iden ( 1 :N-2 : ) ; %Row 2 Column 2 A{N-3:2*N-e,N+l:2*N) = 4 *E2 (1 :N-2 : ) omega''2 .* Iden (1 :N-2 : ) ; % Begin Boundary Conditions ***************************************************** %Row 4 Column 2 A{2*N-5,N+1:2*N) = 2 *vecl ( : ) + Biot *vecO ( : ) ; %Row 5 Column 1 A(2*N-4,1:N) = vecO ( : ) ; %Row 6 Column 1 A(2*N-3,1:N) = 4 .* vec2 ( : ) ; %Row 7 Column 1 A(2*N-2,1:N) = vecOn{:)'; %Row 8 Column 1 A{2*N-1,1:N) =vecln(:)'; %Row 9 Column 2 A(2*N,N+1:2*N) = vecOn ( : ) ; % Define the B matrix *********************************************************** %Row 1 Column 2 B{1:N-4,N+1:2*N) = omega"2 .* Iden (1 :N-4 : ) ; %Row S Column 2

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163 B(2*N-3,N+1:2*N) = (omega ^2 Gamma) .* vecO { : ) ; % Find the eigenvalues [eigenvecs, eigenvals] = eig{A,B); % Find the eigenvalues and eigenvectors eigenvals = diag (eigenvals) ; % Find the entries which are not infinity or NaN's indexl = ; for index2 = l : length (eigenvals) if isnan( eigenvals (index2) ) % We don't want NaN values elseif isinf( eigenvals (index2) ) % Nor the infinities elseif abs (eigenvals (index2) ) > 10*10 % Some values are finite but ridiculously large else indexl = indexl + 1 ,evals (indexl) = eigenvals (index2) ; if abs (imag (evals (indexl) ) ) < le-10 % Get rid of very small imaginary parts evals (indexl) = real (evals (indexl) ) ; end end end % Sort the eigenvalues (Rayleigh number) by smallest positive value first % then the smallest (in magnitude) of negative numbers. evals=sortl (evals) ; % Find the eigenvectors associated with the smallest eigenvalue for indexl = 1 : length (eigenvals) if real (evals (1) ) == real (eigenvals (indexl) ) Minlndex = indexl; break ; end end % Find the other parameters Ra = real (evals (1) ) ; Ma = Gamma Ra; Tempdiff = (thermdiff kinvisc Ra) / (gravity thermexp depth's) ;

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164 Case 13 % casel3 This is the adjoint of the Nield problem without surface deflections % casel3in.ni needs to run first in order to initialize parameters. % Single liquid layer with passive gas above. No surface deflections %Def ine the A matrix ********* + ************************************************* %Row 1 Column 1 A{1:N-4,1:N) = 16 *E4 (1 :N-4 : ) 8*omega'^2 .* E2{l:N-4,:) + omega''4 .* Iden(l:N-4, : ) ; %Row 1 Column 2 A(1:N-4,N+1:2*N) = -Iden (1 :N-4 : ) ; %Row 2 Column 2 A(N-3:2*N-6,N+1:2*N) = 4 *E2 (1 :N-2 : ) omega^2 .* Iden (1 :N-2 : ) ; % Begin Boundary Conditions ***************************************************** %Row 4 Column 2 A(2*N-5,N+1:2*N) = 2 *vecl( : ) + Biot *vecO ( : ) ; %Row 5 Column 1 A{2*N-4,1:N) = vecO { :) ; %Row 6 Column 1 A{2*N-3,1:N) = vec2 ( : ) ; %Row 7 Column 1 A{2*N-2,1:N) = vecOn ( : ) ; %Row 8 Column 1 A(2*N-1,1:N) = vecln(:)'; %Row 9 Column 2 A(2*N,N+1:2*N) =vecOn{:)'; % Define the B matrix *********************************************************** %Row 2 Column 1 B(N-3 :2*N-6, 1:N) = -omega'^2 .* Iden (1 :N-2 : ) ; %Row 4 Column 1 B(2*N-5,1:N) = (2 omega^2 Gamma) .* vecl( : ) ;

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165 % Find the eigenvalues *************************************************************** [eigenvecs, eigenvals] = eig(A,B); % Find the eigenvalues and eigenvectors eigenvals = diag (eigenvals) ; % Find the entries which are not infinity or NaN's indexl = ; for index2 = 1 : length (eigenvals) if isnan( eigenvals (index2) ) % We don't want NaN values elseif isinf( eigenvals (index2) ) % Nor the infinities elseif abs (eigenvals (index2) ) > IC^IO % Some values are finite but ridiculously large else indexl = indexl + 1; evals (indexl) = eigenvals (index2) ; if abs (imag (evals (indexl) ) ) < le-10 % Get rid of very small imaginary parts evals (indexl) = real (evals (indexl) ) ; end end end % Sort the eigenvalues (Rayleigh number) by smallest positive value first % then the smallest (in magnitude) of negative numbers. evals = sortl (evals) ,% Find the eigenvectors associated with the smallest eigenvalue for indexl = 1 : length (eigenvals) if real (evals (1) ) == real (eigenvals (indexl) ) Minlndex = indexl; break; end end % Find the other parameters Ra = real (evals (1) ) ; Ma = Gamma Ra; Tempdiff = (thermdiff kinvisc Ra) / (gravity thermexp depth" 3) ;

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166 Case 14 % casel4 This is the fourth order bilayer problem without surface deflections % casel4in.m needs to run first in order to initialize parameters. % Bilayer with no surface deflections. Perm and Wollkind scaling. %Define the A matrix %Row 1 Column 1 A(1:N-4,1:N) = 16 *E4 (1 :N-4 : ) (8*omega^2 ) *E2 (1 :N-4 : ) + omega'^4 .* Iden(l:N-4, : ) ; %Row 2 Column 1 A(N-3:2*N-6,1:N) = Iden (1 :N-2 : ) ; %Row 2 Column 2 A{N-3:2*N-6,N+1:2*N) = 4 *E2 (1 :N-2 : ) omega"2 .* Iden (1 :N-2 : ) ; %Row 3 Column 3 A(2*N-5:3*N-10,2*N+1:3*N) = kinvisc (16/depth^4 *E4 (1 :N-4 : ) (8/depth*2*omega^2) .*E2 (l:N-4, :) +omega^4 .* Iden (1 :N-4 : ) ) ; %Row 4 Column 3 A{3*N-9:4*N-12,2*N+1:3*N) = (l/thermcond) .* Iden (1 :N-2 : ) ; %Row 4 Column 4 A(3*N-9:4*N-12,3*N+1:4*N) =thermdiff .* (4/depth"2 *E2 (1 :N-2 : ) omega^2 .* Iden (1 :N-2, : ) ) ; % Begin Boundary Conditions %Row 5 Column 1 A(4*N-11,1:N) = 4 .* vec2 { : ) ; %Row 5 Column 3 A(4*N-11,2*N+1:3*N) = (4 vise / depth''2) .* vec2n ( : ) ; %Row 6 Column 1 A(4*N-10, 1:N) = depth .* vecl( : ) ; %Row 6 Column 3 A(4*N-10,2*N+1:3*N) = -vecln ( : ) ; %Row 7 Column 2 A(4*N-9,N+1:2*N) = depth .* vecl( : ) ;

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167 %Row 7 Column 4 A(4*N-9,3*N+1:4*N) = -thermcond .* vecln ( : ) ; %Row 8 Column 2 A(4*N-8,N+1:2*N) = vecO ( : ) ; %Row 8 Column 4 A(4*N-8, 3*N+1:4*N) = -vecOn ( : ) ; %Row 9 Column 1 A(4*N-7,1:N) = vecO ( : ) ; %Row 10 Column 3 A(4*N-6,2*N+1:3*N) = vecOn{:)'; %Row 11 Column 1 A{4*N-5,1:N) = vecOn ( : ) ; %Row 12 Column 1 A{4*N-4,1:N) = vecln (:)' ; %Row 13 Column 2 A(4*N-3,N+1:2*N) =vecOn{:)'; %Row 14 Column 3 A(4*N-2,2*N+1:3*N) = vecO ( : ) ; %Row 15 Column 3 A(4*N-1, 2*N+1:3*N) = vecl ( : ) ; %Row 16 Column 4 A(4*N,3*N+1:4*N) = vecO ( : ) ; % Define the B matrix %Row 1 Column 2 B(1:N-4,N+1:2*N) = omega^2 .* Iden (1 :N-4 : ) ; %Row 3 Column 4 B(2*N-5:3*N-10,3*N+1:4*N) = (thermexp omega''2) .* Iden (1 :N-4, : ) ; %Row 5 Column 2 B(4*N-11,N+1:2*N) = (omega "2 Gamma) .* vecO ( : ) ; % Find the eigenvalues ********************************************************** [eigenvecs, eigenvals] = eig(A,B); % Find the eigenvalues and eigenvectors eigenvals = diag (eigenvals) ; % Find the entries which are not infinity or NaN's indexl = ;

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168 for index2 = 1 : length (eigenvals) if isnan( eigenvals (index2) ) % We don t want NaN values elseif isinf ( eigenvals (index2) ) % Nor the infinities elseif abs (eigenvals (index2) ) > lO^lO % Some values are finite but ridiculously large else indexl = indexl + 1 ; avals (indexl) = eigenvals (index2) ; if abs (imag(evals (indexl) ) ) < le-10 % Get rid of very small imaginary parts evals (indexl) = real (evals (indexl) ) ; end end end % Sort the eigenvalues (Rayleigh number) by smallest positive value first % then the smallest (in magnitude) of negative numbers. evals=sortl (evals) ; % Find the eigenvectors associated with the smallest eigenvalue for indexl = 1 : length (eigenvals) if real (evals (1) ) == real (eigenvals (indexl) ) Minlndex = indexl; break ; end end % Find the other parameters Ral = real (evals (1) ) ; Ma = Gamma Ral; Tempdiff = (thermdiffl kinviscl Ral) / (gravity thermexpl depthl^3) ; Ra2 = Ral (depth^4 thermexp) / (thermdiff kinvisc thermcond) ;

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169 Case 15 % caselB. This is the fourth order bilayer problem without surface deflections % The adjoint problem to case 14. % caselBin.m needs to run first in order to initialize parameters. % Bilayer with no surface deflections. Ferm and Wollkind scaling. %Define the A matrix ***********•****************** + *************************•**** %Row 1 Column 1 A{1:N-4,1:N) = 16 *E4 (1 : N-4 : ) ( 8*omega"2 ) *E2 ( 1 :N-4 : ) +omega"4 .* Iden(l:N-4, :) ; %Row 1 Column 2 A(1:N-4,N+1:2*N) = -Iden (1 :N-4 : ) ; %Row 2 Column 2 A(N-3:2*N-6,N+1:2*N) = 4 *E2 (1 :N-2 : ) omega"2 .* Iden (1 :N-2 : ) ; %Row 3 Column 3 A{2*N-5:3*N-10,2*N+1:3*N) = kinvisc (16/depth*4 *E4 (1 : N-4 : ) {8/depth^2*omega^2) .*E2 (l:N-4, :) + omega^4 .* Iden (1 :N-4 : ) ) ; %Row 3 Column 4 A(2*N-5:3*N-10,3*N+1:4*N) = ( -l/thermcond) .* Iden(l:N-4, :) ; %Row 4 Column 4 A(3*N-9:4*N-12,3*N+1:4*N) =thermdiff .* (4/depth"2 *E2 (1 :N-2 : ) omega*2 .* Iden (1 :N-2 : ) ) ; % Begin Boundary Conditions %Row 5 Column 1 A(4*N-11,1:N) = depth^2 .* vec2 ( : ) ; %Row 5 Column 3 A(4*N-11,2*N+1:3*N) = -kinvisc .* vec2n ( : ) ; %Row 6 Column 1 A(4*N-10, 1:N) = visc*depth .* vecl ( : ) ; %Row 6 Column 3 A(4*N-10,2*N+1:3*N) = -kinvisc .* vecln(:)'; %Row 7 Column 2 A(4*N-9,N+1:2*N) = vecl { : ) ;

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[70 %Row 7 Column 4 A{4*N-9,3*N+1:4*N) = -thermdiff /depth .* vecln ( : ) ; %Row 8 Column 2 A(4*N-8,N+1:2*N) = thermcond .* vecO ( : ) ; %Row 8 Column 4 A(4*N-8,3*N+1:4*N) = -thermdiff .* vecOn(:)'; %Row 9 Column 1 A(4*N-7,1:N) = vecO ( :) ; %Row 10 Column 3 A(4*N-6,2*N+1:3*N) = vecOn ( : ) ; %Row 11 Column 1 A(4*N-5,1:N) = vecOn ( : ) ; %Row 12 Column 1 A(4*N-4,1:N) = vecln { : ) ; %Row 13 Column 2 A{4*N-3,N+1:2*N) = vecOn ( : ) ; %Row 14 Column 3 A(4*N-2,2*N+1:3*N) = vecO ( : ) ; %Row 15 Column 3 A(4*N-1,2*N+1:3*N) = vecl ( : ) ; %Row 16 Column 4 A{4*N, 3*N+1:4*N) = vecO { : ) ; % Define the B matrix %Row 2 Column 1 B(N-3 :2*N-e,l:N) = -omega'^2 .* Iden (1 :N-2 : ) ; %Row 4 Column 3 B(3*N-9:4*N-12,2*N+1:3*N) = (-thermexp omega^2) .* Iden (1 :N-2, : ) ; %Row 7 Column 1 B(4*N-9,1:N) = (omega ^2 Gamma) .* vecl ( : ) ; % Find the eigenvalues ********************************************************** [eigenvecs, eigenvals] = eig(A,B); % Find the eigenvalues and eigenvectors eigenvals = diag (eigenvals) ; % Find the entries which are not infinity or NaN's

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171 indexl = ; for index2 = 1 : length (eigenvals) if isnan{ eigenvals (index2) ) % We don t want NaN values elseif isinf ( eigenvals {index2) ) % Nor the infinities elseif abs (eigenvals (index2) ) > 10^10 % Some values are finite but ridiculously large else indexl = indexl + 1; evals (indexl) = eigenvals (index2) ; if abs (imag( evals (indexl) ) ) < le-10 % Get rid of very small imaginary parts evals (indexl) = real (evals (indexl) ) ; end end end % Sort the eigenvalues (Rayleigh number) by smallest positive value first % then the smallest (in magnitude) of negative numbers. evals=sortl (evals) ; % Find the eigenvectors associated with the smallest eigenvalue for indexl = l : length (eigenvals) if real (evals (1) ) == real (eigenvals (indexl) ) Minlndex = indexl; break; end end % Find the other parameters Ral = real (evals (1) ) ; Ma = Gamma Ral ; Tempdiff = (thermdiffl kinviscl Ral) / (gravity thermexpl depthl^3) ; Ra2 = Ral (depth^4 thermexp) / (thermdif f kinvisc thermcond) ;

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172 Case 16 % caselG.m is an m-file which finds the unstable and slave modes % for a given cylindrical aspect ratio % Calls case 12 and casel3 % List table of zeros of the derivative of the Bessel s function % lambda (m+l .n) where m is the azimuthal mode and n is the radial mode lambda = [3.8317059 7.0155867 10.1734681 13.3236919363 1.84118 5.33144 8.53632 11.70600 3.05424 6.70613 9.96947 13.17037 4.20119 8.01524 11.34592 14.58585 5.31755 9.28240 12.68191 15.96411]; CurrentPath = pwd; cd d: \users\johnd\projects\nonlin~l\casel2; casel2in; % Define the parameters used suTol = 0.1; ssTol = 0.8; index = ; [lastm, lastn] = size (lambda) ; Ravec = zeros (lastm, lastn); delta = zeros (lastm, lastn); anfiles = zeros (lastm lastn, 8); bnfiles = zeros (lastm lastn, 8); anafiles = zeros (lastm lastn, 9); bnafiles = zeros (lastm lastn, 9); an = zeros (N, 1) ; bn = zeros (N, 1) ; ana = zeros (N, 1) ; bna = zeros (N, 1) ; su = zeros (1,3) ; ss = zeros (1,3) ; % Ask the user for the aspect ratio errflag = 1; while errflag AspectRatio = input ('Enter the aspect ratio: '); errflag = 0; if isempty (AspectRatio) disp(' Error entering aspect ratio') errflag = 1; end end % Find Ra, velocity and temperature for all interesting m's and n's for m = 0:lastm-l for n = 1: lastn omega = lambda (m+l, n) / AspectRatio;

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173 % Calculate the linear problem cd d: \users\johnd\projects\nonlin~l\casel2 ; casel2 ; Ravec {m+l,n) = Ra; an = eigenvecs (1 :N,MinIndex) ; bn = eigenvecs (N+l :2*N,MinIndex) ; % Write the data to its own file aval ( [ cd CurrentPath] ) ; index = index + l; anfiles (index, : ) = [ 'an' int2str (m) int2str (n) txt ] ; eval ( [ save ', anfiles (index, :) an -ascii ] ) ; bnfiles (index, : ) = [ 'bn' int2str (m) int2str (n) .txt ] ; eval (['save ', bnfiles (index, :) bn -ascii ] ) ; end end % Clear the variables of Case 12 % clear casel2 cd d: \users\johnd\projects\nonlin~l\casel3 ; % casel3in clear A B eigenvals evals; A=zeros (2*N,2*N) ; B=zeros (2*N,2*N) ; eigenvals=zeros (1, 2*N) ; evals=0 ; index = ; % Find velocity and temperature vectors of the adjoint problem for m = :lastm-l for n = l:lastn omega = lambda (m+l,n) / AspectRatio; % Calculate the adjoint problem cd d: \users\johnd\projects\nonlin~l\casel3; casel3 ; ana = eigenvecs (1 :N,MinIndex) ; bna = eigenvecs (N+l :2*N,MinIndex) ; % Write the data to its own file eval ( [ cd CurrentPath] ) ; index = index + 1 ; anaf iles (index, : ) = [ 'ana' int2str (m) int2str (n) .txt ] ; eval ([' save ', anaf iles (index, :) ana ', -ascii ] ) ; bnafiles (index, : ) = [ 'bna' int2str (m) int2str (n) .txt ] ; eval ([' save ', bnafiles (index, :) bna ', -ascii ] ) ; end end % Find the critical Rayleigh number Rac = min ( min (Ravec) ) ; index = ;

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174 indexl = ; ind.ex2 = ; % Find the Ra s for the unstable and slave modes for m = 0:lastm-l for n = l:lastn delta(m+l,n) = (Ravec (m+l,n) Rac) / Ravec (m+1, n) ; if (delta (m+1, n) <= suTol) indexl = indexl + 1 ; su(indexl,l) = Ravec (m+l,n) • su (indexl, 2) = m; su ( indexl 3 ) = n ; elseif (delta (m+1, n) < ssTol) (delta (m+1, n) > suTol) index2 = index2 + l; ss(index2,l) = Ravec (m+1, n) ; ss (index2, 2) = m; ss (index2, 3) = n; end end end % Sort the set of unstable and slave modes su = sort2 (su) ; ss = sort2 (ss) ; % Write the unstable and slave modes to a file sufid = fopen ( su. txt 'w'); ssfid = f open (' ss txt 'w'); fprintf (sufid, '%6.3f %3.0f %3.0f\n', su'); fprintf (ssfid, '%S.3f %3.0f %3.0f\n', ss ) ; f close (sufid) ; fclose (ssfid) ; % Print the modes to the screen fprintf (1, \nThe unstable modes :\n') fprintfd, '%6.3f %3 Of %3.0f\n', su ) fprintf (1, \n\nThe slave modes :\n') fprintfd, '%S.3f %3 Of %3.0f\n', ss )

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175 Case 17 % caselV.m is an m-file which finds the unstable and slave modes % for a given cylindrical aspect ratio % Calls Casel4 and CaselS % List table of zeros of the derivative of the Bessel s function % lambda (m+1 .n) where m is the azimuthal mode and n is the radial mode lambda [3 8317059 7 0155867 10 1734681 13.3236919363 1 84118 5 33144 8 53632 11.70600 3 05424 6 70613 9 96947 13.17037 4 20119 8 01524 11 34592 14.58585 5 31755 9 28240 12 68191 15.96411] ; CurrentPath = pwd; cd d:\users\johnd\projects\nonlin~l\casel4; casel4in; % Define the parameters used suTol = 0.01; SSTol = 0.95; index = ; [lastm, lastn] = size (lambda) ; Ravec = zeros (lastm, lastn); delta = zeros (lastm, lastn); anlfiles = zeros (lastm lastn, 9); bnlfiles = zeros (lastm lastn, 9); anlafiles = zeros (lastm lastn, 10); bnlafiles = zeros (lastm lastn, 10); anl = zeros (N, 1) ; bnl = zeros (N,l); anla = zeros (N,l); bnla = zeros (N,l); an2files = zeros (lastm lastn, 9); bn2files = zeros (lastm lastn, 9); an2afiles = zeros (lastm lastn, 10); bn2afiles = zeros (lastm lastn, 10); an2 = zeros (N, 1) ; bn2 = zeros (N, 1) ; an2a = zeros (N,l); bn2a = zeros (N,l); su = zeros (1,3) ; ss = zeros (1,3) ; % Ask the user for the aspect ratio errflag = 1; while errflag AspectRatio = input (' Enter the aspect ratio of the lower fluid: ) ; errflag = 0;

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176 if isempty (AspectRatio) disp(' Error entering aspect ratio') errflag = 1; end end % Find Ra, velocity and temperature for all interesting m's and n's for m = 0:lastm-l for n = l:lastn omega = lambda {m+l,n) / AspectRatio; % Calculate the linear problem cd d: \users\johnd\projects\nonlin-l\casel4; casel4 ; Ravec(m+l,n) = Ral; anl = eigenvecs (1 :N,MinIndex) ; bnl = eigenvecs (N+l :2*N,iyiinIndex) ; an2 = eigenvecs {2*N+1 :3*N,MinIndex) ; bn2 = eigenvecs {3 *N+1: 4 *N,MinIndex) ; % Write the data to its own file aval ( [ cd CurrentPath] ) ; index = index + 1; anlf iles (index, : ) = [ anl int2str (m) int2str (n) txt eval { [ save ', anlf iles (index, :) anl ', -ascii bnlf iles (index, : ) = [ 'bnl int2str (m) int2str (n) txt eval ([' save ', bnlf iles (index, :) bnl ', -ascii an2files (index, : ) = [ 'an2 int2str (m) int2str (n) .txt eval ([' save ', an2f iles (index, :) an2 ', -ascii bn2files (index, : ) = [ 'bn2 int2str (m) int2str (n) .txt eval (['save ', bn2f iles (index, :) bn2 ', -ascii end end % Clear the variables of Case 14 % clear casel4 cd d: \users\johnd\projects\nonlin~l\casel5 ; % caselSin clear A B eigenvals avals, A=zeros (4*N,4*N) ; B=zeros (4*N,4*N) ; eigenvals=zeros (1, 4*N) ; evals=0 ; index = ; % Find velocity and temperature vectors of the adjoint problem for m = :lastm-l for n = l:lastn omega = lambda (m+l,n) / AspectRatio; % Calculate the adjoint problem cd d: \users\johnd\projects\nonlin~l\casel5; easels ; J ; ]) ; J / ]) ; J / ]) ; J / ]);

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177 anal = eigenvecs (1 :N,MinIndex) ; bnal = eigenvecs (N+l :2*N,MinIndex) ; ana2 = eigenvecs (2*N+1 : 3*N,MinIndex) ; bna2 = eigenvecs {3*N+1 :4*N,MinIndex) ; % Write the data to its own file eval ( [ cd CurrentPath] ) ; index = index + 1; analf iles (index, : ) = [ anal int2str (m) int2str (n) txt eval ([' save ', analf iles (index, :) anal ', -ascii bnalfiles (index, : ) = [ 'bnal int2str (m) int2str (n) .txt eval ([' save ', bnalfiles (index, :) bnal ', -ascii ana2files (index, : ) = [ ana2 int2str (m) int2str (n) .txt eval ([' save ', ana2f iles (index, :) ana2 ', -ascii bna2f iles (index, : ) = [ 'bna2 int2str (m) int2str (n) .txt eval (['save ', bna2f iles (index, :) bna2 ', -ascii end end % Find the critical Rayleigh number Rac = min ( min(Ravec) ) ; index = ; indexl = ; index2 = ; % Find the Ra s for the unstable and slave modes for m = 0:lastm-l for n = 1 : lastn delta(m+l,n) = (Ravec (m+l,n) Rac) / Ravec (m+l,n) ; if (delta (m+l,n) <= suTol) indexl = indexl + 1; su(indexl,l) = Ravec (m+1, n) ; su (indexl, 2) = m; su ( indexl 3 ) = n ; elseif (delta (m+1, n) < ssTol) (delta (m+1, n) > suTol) index2 = index2 + 1; ss(index2,l) = Ravec (m+1, n) ; ss (index2,2) = m; ss (index2 ,3) = n; end end end % Sort the set of unstable and slave modes su = sort2 (su) ; ss = sort2 (ss) ; % Write the unstable and slave modes to a file sufid = f open (' su. txt 'w'); ssfid = fopen( ss txt 'w'); fprintf (sufid, '%6.2f %3.0f %3.0f\n', su'); fprintf (ssfid, '%6.2f %3.0f %3.0f\n', ss ) ; f close (sufid) ; ) ; t ) ; ) ; I ) ;

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178 f close (ssfid) ; % Print the modes to the screen fprintfd, \n The unstable modes :\n') fprintf (1, \n Ra m n \n ) fprintf (1, %6.3f %3.0f %3.0f\n' su') fprintf (1, \n The slave modes :\n') fprintf (1, \n Ra m n \n' ) fprintfd, %6.3f %3.0f %3.0f\n' ss )

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179 Case 19 % casel9.m plots the Rayleigh number versus the aspect ratio % for a single fluid layer % Calls case 12 % List table of zeros of the derivative of the Bessel s function % lambda (m+1 .n) where m is the azimuthal mode and n is the radial mode lambda [3 .8317059 1.84118 3.05424 4.20119 5.31755 7.01558S7 5.33144 6.70613 8.01524 9.28240 10.1734681 8.53632 9.96947 11.34592 12 .68191 13 .3236919363 11.70600 13 .17037 14.58585 15.96411] ; % Find which directory we are currently in and initialize the parameters CurrentPath = pwd; cd d: \users\johnd\projects\nonlin~l\casel2; casel2in; % Define the parameters used in this program counter = ; NumSteps=ceil ( (omegaLast-omegaFirst) /omegaStep ) ; clear Ravec; Ravec = zeros (NumSteps 7) ; % Loop through the values of omega and find the corresponding Rayleigh numbers for omega = omegaFirst : omegaStep : omegaLast counter = counter + 1; casel2 ; Ravec {counter, 1) = Ra; Ravec (counter, 2) = omega; % Find the aspect ratio for each azimuthal and radial mode % Currently we are only interested in the following modes: % {m,n} = {1,1}, {2,1}, {0,1}, {3,1}, {1,2}, {4,1} Ravec (counter, 3) = lambda{2,l) / omega; % {m,n} = {l,l} Ravec (counter, 4) = lambda (3,1) / omega; Ravec (counter, 5) = lambda (1,1) / omega; Ravec (counter, 6) = lambda (4,1) / omega; Ravec (counter, 7) = lambda (5,1) / omega; end % {m,n} = {2,1} % {m,n} = {0,1} % {m,n} = {3,1} % {m,n} = {4,1} % Find the minimum Rayleigh number RaMin = min (Ravec {:, 1) ) ; GraphYmin = floor (RaMin / 10) 10; GraphYmax = ceil (RaMin 1.1 / 10) 10; xmin = 0.5; xmax = 2.5; textPosition = GraphYmin + (GraphYmax GraphYmin) /2 ; lineHeight = (GraphYmax GraphYmin) / 14; Plot the Rayleigh number versus aspect ratio for the

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180 % various azimuthalradial mode pairs % The syntax is plot(Ra, pairl, colorl, Ra, pair2, color2 ... fignum = figure (' Position' [360, 150, 600, 300]); orient landscape plot (Ravec( : ,3) Ravec(:,l), 'k', Ravec(:,4), Ravec {:,!), 'b', Ravec(:,5), Ravec {:,!), 'r', Ravec(:,6), Ravec (:,1), 'g', ... Ravec(:,7), Ravec (:,l), 'm') title ( 'Rayeligh Number versus Aspect Ratio'); xlabel ( 'Aspect Ratio' ) ; ylabel ( 'Rayleigh Number' ) ; axisScale = axis; axis([xmin, 2.5, GraphYmin, GraphYmax, ]); grid on legend ( '1,1' '2,1', '0,1', '3,1', '4,1', -l) text(xmax + 0.25, textPosition, ['depth = num2str (depth) ] ) text (xmax + 0.25, textPosition lineHeight, ['Biot = num2str (Biot) ] ) text (xmax + 0.25, textPosition 2*lineHeight ['Gamma = num2str (Gamma) ] ) % Get back to the original directory cd d: \users\johnd\projects\nonlin~l\casel9; % Ask user if they want to save the data response = input ('Do you wish to save the data? ', 's'); response = upper (response) ; if strcmp('Y', response (1)) fileName = input (' Enter file name: ', 's'); fid = f open (fileName, 'w'); fprintf(fid, '%6.3f, %5.2f, %5.3f, %5.3f, %5.3f, %5.3f, %5.3f\n' Ravec ) ; f close (fid) ; disp ( • ) ; disp(['Data written to file fileName]) disp ( ) ; end % End of the program disp (' Program casel9 completed')

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181 Case 20 % case2 0.m plots the Rayleigh number versus the aspect ratio % for a single fluid layer % Calls case 14 % List table of zeros of the derivative of the Bessel s function % lambda (m+1 .n) where m is the azimuthal mode and n is the radial mode lambda [3.8317059 1.84118 3.05424 4.20119 5.31755 7.0155867 5.33144 6.70613 8.01524 9.28240 10.1734681 8.53632 9.96947 11.34592 12.68191 13 .3236919363 11.70600 13 .17037 14.58585 15.96411] ; % Find which directory we are currently in and initialize the parameters CurrentPath = pwd; cd d: \users\ johnd\projects\nonlin~l\casel4 ; casel4in; % Define the parameters used in this program counter = ; NumSteps=ceil ( (omegaLast-omegaFirst) /omegaStep ); clear Ravec; Ravec = zeros (NumSteps, 7) ; % Loop through the values of omega and find the corresponding Rayleigh numbers for omega = omegaFirst : omegaStep : omegaLast counter = counter + 1 ; casel4; Ravec (counter, 1) = Ral; Ravec (counter, 2) = omega; % Find the aspect ratio for each azimuthal and radial mode % Currently we are only interested in the following modes: % {m,n} = {1,1}, {2,1}, {0,1}, {3,1}, {1,2}, {4,1} Ravec (counter, 3) = lambda (2,1) Ravec (counter, 4) = lambda (3,1) Ravec (counter, 5) = lambda (1,1) Ravec (counter, 6) = lambda (4,1) Ravec (counter, 7) = lambda (5,1) end / omega ; % {m,n} = {1 1} / omega ; % {m,n} = {2 1} / omega ; % {m,n} = (0 1} / omega ; % {m,n} = {3 1} / omega ; % {m,n} = {4 1} % Find the minimum Rayleigh number RaMin = min (Ravec ( : 1) ) ; GraphYmin = floor (RaMin / 10) 10; GraphYmax = ceil (RaMin l.l / 10) 10; xmin = 0.5; xmax = 2.5; textPosition = GraphYmin + (GraphYmax GraphYmin) /2 ; lineHeight = (GraphYmax GraphYmin) / 14; % Plot the Rayleigh number versus aspect ratio for the

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182 % various azimuthalradial mode pairs % The syntax is plot(Ra, pairl, colorl, Ra, pair2 color2, ... fignum = figure (' Position' [360, 150, 600, 300]); orient landscape plot (Ravec ( : ,3) Ravec(:,l), 'k', Ravec(:,4), Ravec ( : 1) 'b', ... Ravec(:,5), Ravec {:, 1) 'r', Ravec(:,6), Ravec (:,1), 'g', ... Ravec(:,7), Ravec (:,1), 'm') title { 'Rayeligh Number versus Aspect Ratio'); xlabel { Aspect Ratio ) ; ylabel { Rayleigh Number ) ; axisScale = axis; axis { [xmin, 2.5, GraphYmin, GraphYmax, ]); grid on legend ( '1,1' '2,1', '0,1', '3,1', '4,1', -1) text (xmax + 0.23, textPosition, ['lower depth = num2str (depthl) ] ) text (xmax + 0.23, textPosition lineHeight, ['upper depth = num2str (depth2) ] ) text (xmax + 0.23, textPosition 2*lineHeight ['Biot = num2str (Biot) ] ) text (xmax + 0.23, textPosition 3*lineHeight, ['Gamma = num2str (Gamma) ] ) % Get back to the original directory cd d: \users\johnd\projects\nonlin~l\case2 0; % Ask user if they want to save the data response = input ('Do you wish to save the data? ', 's'); response = upper (response) ; if strcmp('Y', response (1)) fileName = input ('Enter file name: ', 's'); fid = fopen( fileName, 'w'); fprintf(fid, '%6.3f, %5.2f, %5.3f, %5.3f, %5.3f, %5.3f, %5.3f\n', Ravec ) ; fclose (fid) ; disp ( ) ; disp(['Data written to file fileName]) disp ( ) ; end % End of the program disp (' Program case2 completed')

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APPENDIX B DRAWINGS AND DIAGRAMS Liquid Insert 184 Air Insert 185 Clamp 186 Rough Sketch of Program 187 Initialization Flow Chart 188 Data In Flow Chart 189 Display Flow Chart 190 Control Decision Flow Chart 191 Data Out Flow Chart 192 Overall Flow Chart of the Experiment Program 193 182

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184 Liquid Insert Liquid Insert Top View Drawing Notes: 1) Diameter "a" and the height "b" depend on the desired aspect ratio 2) The angle of the pinning edge is not 2) Drawing is not to scaie b b + 2 mm Cross-Sectional View Liquid Insert (Material: Lucite Drawn by: Duane Johnson Last Revised: August 11, 1997

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185 Air Insert Air Insert Cross-Sectional View Top View Drawing Notes: 1) The dimension "a" depends on tlie desired air aspect ratio 2) The dimension "b" depends on the desired height of the air layer 3) Drawing is not to scale Air Inserts Material: Lucite Drawn by: Duane Johnson Last Revised: August 11, 1997

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186 Ciamp Drawing Notes: 1) Drawing is not to scale Clamp Material: Lucite Drawn by: Duane Johnson Last Revised: August 11,1 997

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187 Rough Sketch of Program / Initialization \ Information ) Form !R Program Form

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[88 Initialization Flow Chart Start Program Initialization Information Form Add, Edit & Delete Fluid Parameters Initialize Data Acquisition Board To Data In

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189 Data In Flow Chart Pause Statistics (Data Averaging) To Display From Initialization Information Form IR Program Form From h* — Control Decision

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190 Display Flow Chart From Data In Update Screen Variables Prepare X-Axis Data Prepare Y-Axis Data To Control — Decision

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191 Control Decision Flow Chart Change Segment Number
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192 Data Out Flow Chart From Control Decision Send Digital Output To Heaters -No^ VCR Control Pause RecordSend Digital Output To VCR Yes To Exit Program i

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193 Overall Flow Chart of the Experiment Program -Nostart Program Initialization Information Form File Exists? -YesOverwrite? Yes ; Add, Edit & Delete Fluid Parameters Setup Data File Initialize Data Acquisition Board -* IR Program Form )< Pause • Start/Pause Loop? Start _^ Analog Data In Statistics (Data Averaging) / Update Screen \ \ Variables 1 Change Segment ^YesNumber Pause Program? t No Segment Over? Experiment Over? -NoHeater Control On I Send Digital j Output To Heaters Off i "] No 1 Shutdown ^,,, Overheating? / Prepare \, <: X-A)(is ) \ Data / Prepare Y-Axis Data \ Update Graph JL Control Decision Write Data To File? 1' NO V / utdown Flag -Yes^' Write Data 1 i \ [-NoDown VCR Control Pause L Record Send Digital Output To VCR Shut Experiment? Yes J Send Digital Shutdown Signal Exit Program )

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REFERENCES Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, National Bureau of Standards, Washington DC, 1966, fifth edition Adria, R.J., Particle Imaging Techniques for Experimental Fluid Mechanics, Ann. Rev. Fluid Mech., 23, 261, (1991) Andereck, CD., Colovas, P.W. and Degen, M.M., Advances in Multi-Fluid Flows, 3, (SIAM, Philadelphia, 1996). Benard, H., Les Tourbillons Cellulaires dans une Nappe Liquide, Rev. Gen. Sciences PureAppL, 11, 1261,(1990) Canuto, C, Hussaini, M.Y., Quarteroni, A. and Zang, T.A., Spectral Methods in Fluid Dynamics, SpringerVerlag, Berlin, 1988 Cardin, P., Nataf, H.C. and Dewost, P., Thermal Coupling in Layered Convection: Evidence for an Interface Viscosity Control from Mechanical Experiments and Marginal Stability Analysis, J. Phys. II 1, 599 (1991). Cerisier, P., Perez-Garcia, C, Jamond, C, and Pantaloni, J., Wavelength Selection in Benard-Marangoni Convection, Phys. Rev. A, 35, 1949 (1987) Cross, M.C., and Hohenberg, P.C, Pattern Formation Outside of Equilibrium, Rev. Mon. Phys., 65, 851,(1993) Cserpes, L. & Rabinowicz, X.X., Gravity and Convection in a Two-Layer Mantle, Earth Plan. Sci. Lett. 76, 193, (1985) Dauby, P. and Lebon, G., Benard-Marangoni Instabilities in Rigid Rectangular Containers, J. Fluid Mech., 329, 25 (1996). Dauby, P., Lebon, G. and Bouhy, E., to appear in Phys. Rev. E. (1997). Davis. S.H., Rupture of Thin Liquid Films, Waves on Fluid Interfaces: Proceedings of symposium Conducted by the Mathematics Research Center, p.291, 1983 Davis, S.H., Thermocapillary Instabilities, Ann, Rev, Fluid Mech. 19, 403, (1987) 194

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i 195 ,, Davis, S.H., Hydrodynamic Interactions in Directional Solidification, J. Fluid Mech., 212, 241 (1990) Dijkstra, H.A., On the Structure of Cellular Solutions in Rayleigh-Benard-Marangoni 1 Flows in Small-Aspect-Ratio Containers, J. Fluid Mech., 243, 73 1 (1 992) i Echebarria, B., Krmpotic, D., and Perez-Garcia, C, Resonant Interactions in Benard; Marangoni Convection in Cylindrical Containers, Physica D, 99, 487 (1997) 1 Eckert, E.R. G., Goldstein, R.J., Measurements in Heat Transfer, Hemisphere press, Washington D.C., p.241, 1976 Eckhaus, W., Studies in Non-Linear Stability Theory, SpringerVerlag, New York, 1965 Ellsworth, K. & Schubert, G., Numerical Models of Thermally and Mechanically Coupled Two-Layer Convection of Highly Viscous Fluids, Geophys. J. 93, 347, (1988) Erneux, T. and Reiss, EX., Singular Secondary Bifurcation, SIAM J. Appl. Math., 44, 463 (1984) Ferm, E.N. and Wollkind, D.J., Onset of Rayleigh-Benard-Marangoni Instability: Comparison between Theory and Experiment, J. Non-Equil. Thermodyn., 7, 169, (1982) Fujimura, K. and Renardy, Y.Y., The 2:1 Steady/Hopf Mode Interaction in the TwoLayer Benard Problem, Physica D 85, 25 (1995) Gershuni, G.Z. & Zhukhovitskii, E.M., Monotonic and Oscillatory Instabilities of a TwoLayer System of Immiscible Liquids Heated from Below, Sov. Phys. Dokl. 27, 531, (1982) Glicksman, M.E., Coriell, S.R., and McFadden, G.B., Interaction of Flows with the Crystal Meh Interface, Ann. Rev. Fluid Mech., 18, 307, (1986) Goldstein, R.J., Fluid Mechanics Measurements, Hemisphere press, Washington DC, p.377, 1983 Golubitsky, M., Stewart, I., and Schaeffer, D.G., SpringerVerlag, New York, p.450, 1988 Gottheb, D. and Orazag, S.A., Numerical Analysis of Spectral Methods, SIAM, Philadelphia, fourth edition, 1 986 Hardin, G.R., Sani, R.L., Henry, D. & Roux, B., Buoyancy-Driven Instability in a Vertical Cylinder: Binary Fluids with Soret Effect. Part I: General Theory and Stationary Stability Results, Int. J. Num Meth. Fluids 10, 79 (1990)

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196 Honda, S., Numerical Analysis of Layered Convection Marginal Stability and Finite Amplitude Analyses, Bull. Earthquake Res. Inst. 57, 273 (1982) Johnson, D. Chebyshev Polynomials in the Spectral Tau Method and Applications to Eigenvalue Problems, NASA Contractor Report 198451, (1996) Johnson, D. and Narayanan, R., Experimental Observation of Dynamic Mode Sw^itching in Interfacial-Tension-Driven Convection near a Codimension-Two Point, Phys. Rev. E 54, R3 102 (1996). Johnson, D. and Narayanan, R., Geometric Effects on Convective Coupling and Interfacial Structures in Bilayer Convection, to appear in Phys. Rev. E (1997) Johnson, D., Narayanan, R., and Dauby, P.C., The Effect of Air on the Pattern Formation in Liquid-Air Bilayer Convection How Passive is Air?, submitted to J. Fluid Mech., (1997). Koschmieder, E.L., Benard Cells and Taylor Vortices, Cambridge University Press, 1993 Koschmieder, E.L., and Biggerstaff, M.I., Onset of Surface-Tension-Driven Bernard Convection, J. Fluid Mech., 167, 49 (1986) Koschmieder, E.L., and Prahl, S.A., Surface-Tension-driven Benard Convection in Small Containers, J. Fluid Mech., 215, 571 (1990) Lin, C.C. and Segel, L.A., Mathematics Applied to Deterministic Problems in the Natural Sciences, SIAM, New York, 1994, 3'^ edition. Manneville, P., Dissipative Sructures and Weak Turbulence, Academic Press, San Diego, 1990 McFadden, G.B., Coriell, S.R., Boisvert, R.F., Glicksman, M.E., and Fang, Q.T., Morphological Stability in the Presence of Fluid Flow in the Meh, Metal. Trans. A, 15A, 2117(1984) Miiller, G., Crystal Growth from the Melt, SpringerVerlag, Berlin, 1988 Mullins, W.W. & Sekerka, R.F. Stability of a Planar Interface during Solidification of a Dilute Binary Alloy, J. Appl. Phys., 35, 444 (1964) Nataf, H.C., Moreno, S. & Cardin, P., What is Responsible for Thermal Coupling in Layered Convection, J. Phys. (Paris) 49, 1707 (1988) Nield, D.A., Surface Tension and Buoyancy Effects in Cellular Convection, J. Fluid Mech. 19, 1635 (1964)

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197 Normand, C, Pomeau, Y. and Velarde, M.G., Convective Instability: A Physicist's Approach, Rev. Mon. Phys., 199, 581 (1977) Ondar9uhu, T., Mindlin, G.B., Mancini, H.L., Garcimartin, A. & Perez-Garcia, Dynamical Patterns in Benard-Marangoni Convection in a Square Container, Phys. Rev. Lett. 70, 3892(1993) Operator's Manual for the Inframetrics Model 760: IR Imaging Radiometer, document #07137-000 rev. 2PA, Inframetrics Inc., Waltham, MA, 1991 Pearson, J.R.A., On Convection Cells Induced by Surface Tension, J. Fluid Mech., 4, 489, (1958) Pline, A., Wernet, M., Chung Hsieh, K.C., Ground Based PIV and Numerical Flow Visualization Results from the Surface Tension Drive Convection Experiment, proceedings from the Crystal Grow1;h in Space and Related Optical Diagnostics, 1557, 222(1991) Prakash, A. & Koster, J.N., Steady Rayleigh-Benard Convection in a Two-Layer System of Immiscible Liquids, Trans. ASME 118, 366 (1996) Rasenat, S., Busse, F. H. and Rehberg, I., A Theoretical and Experimental Study of Double-Layer Convection, J. Fluid Mech. 199, 519 (1989). Renardy, Y.Y., Pattern Formation for Oscillatory Bulk-Mode Competition in a TwoLayer Benard Problem, Z. Angew Math. Phys. 47, 567 (1996) Renardy, Y.Y. & Joseph, D.D., Oscillatory Instability in a Benard Problem of Two Fluids, Phys. Fluids 28, 788 (1985) Richter, F.M. & Johnson, C.E., Stability of a Chemically Layered Mantle, J. Geophys. Res. 79, 1635(1974) Rosenblat, S., Thermal Convection in a Vertical Circular Cylinder, J. Fluid Mech 122 395(1982) Rosenblat, S., Davis, S.H. and Homsy, G.M., Nonlinear Marangoni Convection in Bounded Layers. Part 1. Circular Cylindrical Containers, J. Fluid Mech. 120, 91 (1982a). Rosenblat, S., Davis, S.H. and Homsy, G.M., Nonlinear Marangoni Convection in Bounded Layers. Part 2. Rectangular Cylindrical Containers, J. Fluid Mech. 120, 123 (1982b) Schwabe, D., Marangoni Effects in Crystal Growth Melts, Physicochem. Hydrodyn., 2, 263(1981)

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198 Seborg, D.E., Edgar, T.F., Mellichamp, D.A., Process Dynamics and Control John Wiley and Sons, 1989 Smith, K.A., On Convective Instability Induced by Surface-Tension Gradients, 24, 401 (1966) Sparrow, E.M., Goldstein, R.J. & Jonsson, V.K., Thermal Instability in a Horizontal Fluid Layer: Effect of Boundary Conditions and Non-Linear Temperature Profile, J. Fluid Mech., 18, 33(1963) Stakgold, I., Green 's Functions and Boundary Value Problems, John Wiley & Sons, Inc., New York, 1979 Turner, J.S., Multicomponent Convection, Ann. Rev. Fluid Mech, 17, 1 1 (1985) Zaman, A. and Narayanan, R., Interfacial and Buoyancy-Driven Convection — The Effect of Geometry and Comparison with Experiments, J. Colloid Interface Sci. 179, 151 (1996) Zeren, R.W., and Reynolds, W.C, Thermal Instabilities in Two Fluid Horizontal Layers, J. Fluid Mech., 53, 305(1972) Zhao, A.X., Wagner, C, Narayanan, R., and Friedrich, R., Bilayer Rayleigh-Marangoni Convection: Transitions in Flow Structures at the Interface, Proc. R. Soc. Lond. A, 451, 487(1995)

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BIOGRAPHICAL SKETCH The author was born on June 1 1, 1970, in Van Nuys, California. He received an Associates in Science from Grand Rapids Community College, in 1991. In 1993, he graduated from Michigan State University with honors; receiving a B.S. in Chemical Engineering. He then attended the University of Florida, whereupon he received a NASA Graduate Student Research Program Fellowship. In 1997, he graduated from the University of Florida with a Ph.D. in Chemical Engineering. He is currently an NSFNATO postdoctoral fellow at the Universite Libre de Bruxelles in Brussels, Belgium. 199

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. rf ^ Ranganathan Narayanan, Chair Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Cj 'wUam d Lewis E. Johns Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in kope and quality, as a dissertation for the degree of Doctor of Philosophy. Ki \ Jorge Vifials Associate Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Dy KrishnarrvUrti Professor of Oceanography I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. \A\xk^/KK^KiiN' Ulrich H. Kurzweg Professor of Aerospace Engineering, Mechanics, and Engineering Science

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This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy December, 1997 /^U~-JWinfred M. Phillips Q Dean, College of Engineering Karen A. Holbrook Dean, Graduate School


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