
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00026579/00001
Material Information
 Title:
 Geometric effects on bilayer convection in cylindrical containers
 Creator:
 Johnson, Duane, 1970
 Publication Date:
 1996 [i.e. 1997]
 Language:
 English
 Physical Description:
 vii, 199 leaves : ill. ; 29 cm.
Subjects
 Subjects / Keywords:
 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 )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Summary:
 Liquid encapsulated crystal growth.
 Thesis:
 Thesis (Ph. D.)University of Florida, 1997.
 Bibliography:
 Includes bibliographical references (leaves 194198).
 Additional Physical Form:
 Also available online.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Duane Johnson.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright Duane Johnson. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 38855541 ( OCLC )
0028630001 ( ALEPH )

Downloads 
This item has the following downloads:

Full Text 
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
lightheartedness 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 352320 and NGT 51242 grants and from the
National Science Foundation, grant numbers CTS 9500393 and CTS 9307819.
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
GalerkinEckhaus 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
CodimensionTwo 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 ConvectionCoupling 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 codimensiontwo
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 gradientdriven flows.
It was also shown that changing the radius of the container can change the driving
force of convection from a surface tension gradientdriven to buoyancydriven and back
again. Finally, the weakly nonlinear analysis was able to give a qualitative description of
codimensiontwo 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. Liquidencapsulated 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 11. 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 buoyancydriven and interfacial tensiondriven 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 solidliquid 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 quasistatic. 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
Buoyancydriven 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
BuoyancyDriven Convection Interfacial TensionDriven Convection
Cold Cold
gas gas
" liquid 1 liquid c Iv ctio
Figure 12. 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 buoyancydriven convection.
The extent of buoyancydriven 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 buoyancydriven convection.
5
1.2.2. Marangoni Convection
Surface tension gradientdriven convection, unlike buoyancydriven 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 tensioninduced 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 buoyancydriven convection,
there exists a critical temperature difference where the surface tension gradientdriven
flow is not dampened by the thermal diffusivity or viscosity, and the fluid is set into
6
motion. Surface tension gradientdriven 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, buoyancydriven 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 13. 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 gradientdriven 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 gradientdriven
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, finitesized
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 13).
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 13,
there is an m = 1, n = 1 flow pattern, where m is the azimuthal mode and n is the radial
mode (see Figure 14). 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 crosssection. 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
codimensiontwo 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 14. 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 15. Schematic of the different types of convectioncoupling. 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 16. 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 15. 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 15a. This can also been
seen by plotting the velocity and temperature perturbations versus the fluid depth (Figure
16). Observe in Figure 16a 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 15b as counterrotating rolls
in the two fluids. This can also be denoted by the vertical component of velocity
switching sign at the interface (Figure 16b), 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 corotating. 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 16c). 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 noslip 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 15d). 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 16d) 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
fluidfluid 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 17. The four possible interfacial structures at a fluidfluid 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 17.
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 buoyancydriven
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 buoyancydriven 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 gradientdriven 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 buoyancydriven 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 buoyancydriven convection in the
upper phase, or by surface tension gradientdriven 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
RayleighMarangoni 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 RayleighMarangoni 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 nondeformable 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 gradientdriven
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
nondeformable. 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 gradientdriven 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 buoyancydrive convection is
dominant. He also notes, that in buoyancydriven convection, the fluid flows up into a
surface elevation and that in surface tension gradientdriven 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 gradientdriven
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
RayleighMarangoni 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 GalerkinEckhaus
expansion (Eckhaus, 1965; Manneville, 1990).
The most relevant result came from their analysis of codimensiontwo points, where
two different flow patterns coexist. For one of the codimensiontwo 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
codimensiontwo point. When the aspect ratio was decreased to the other side of the
codimensiontwo 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 vorticityfree boundary conditions with
realistic noslip 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 codimensiontwo 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 vorticityfree
sidewalls. In the paper by Zaman and Narayanan (1996), and later by Dauby et al.
(1997), a linear, three dimensional solution was found for RayleighMarangoni
convection in a cylinder. Both papers assumed that the interface was flat and that the
sidewalls of the cylinder were noslip. One of the most interesting observations in these
two papers, is that the progression of modes was not the same as the vorticityfree
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 noslip and vorticityfree sidewalls.
The vorticityfree calculations contradict the results of Koschmieder and Prahl (1990).
Additionally, the different progression of modes changes the codimensiontwo 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
codimensiontwo 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: buoyancydominated, 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 drivenconvection 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 RayleighTaylor instability.
The RayleighTaylor 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 GalerkinEckhaus 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 21. 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 21). 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 EI 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* Tb2, 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 nondimensional 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:
Vvi =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, noslip 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* =nk2VT2 (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 22 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)
I21 + 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 +LuLuN 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,...,N2
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 NavierStokes 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 )W2DI2 +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 =z1 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[ Nt 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 22. Plot of the Rayleigh number versus wave number.
A4 = RaB4 (2.82)
where (= (W,,i, 1,W2, 112, 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 22 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, noslip 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 nondeformable. 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 + 1W2 = 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
nondeformable 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,
Su = 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 21. 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 21 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 23 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 NavierStokes 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, onedimensional calculations from the linear
model section. These results will be used later to show the existence of codimensiontwo
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 23. Plot of the Rayleigh number versus aspect ratio. The plot is generated from
Figure 22 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 codimensiontwo 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 GalerkinEckhaus 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 nondeformable.
1 (va
Pr~ vi1 v VIV =V;I +V2vl +Rao1 z
Pr at
VV1 = 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 selfadjoint. Although it appears as if
it can be made selfadjoint 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
Vv* =0 (2.125)
V;2* +R w =0
iP P ip
1 1*
vV2V V2 + k2p Z
Vv* =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 nonadjoint 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 GalerkinEckhaus Expansion
The GalerkinEckhaus 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
SZA(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 LiapunozSchmidt 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 11 0
52
Substituting (2.118), (2.119), (2.129) and yp into (2.130) gives:
r{Kv ** / V2 /+ M {O* aO/+I P 2
Pr1 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:
MMp ( \ 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+np)(mn+p)(mnp) 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 =r1 {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 cutoff 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 nonzero. 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 pApl pqryAqAgr1 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 codimensiontwo points.
CHAPTER 3
EXPERIMENTAL APPARATUS AND PROCEDURE
The objectives of the experiments were to observe the behavior of the fluid
convection at codimensiontwo 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 buoyancydriven convection in one or both fluids, the bilayer
IR camera Electronic Hardware Unit
TV and VCR
/1\\\
Test Section
Computer J p~]
Figure 31. 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 oilair interface was detected using an infrared camera.
This chapter is split up into two parts, a description of the experimental apparatus
and a walkthrough 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 computercontrolled 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 wellestablished,
each had its drawback.
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 32. 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, platelike, 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 nonintrusive
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 MercuryCadmiumTelluride
(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" closeup 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 falsecolor 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 31. 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 cm3) 0.968 0.0012 5.27 5.96 1.19
Negative Thermal Expansion 9.6 33.3 0.078 0.501 7.3
(104 C1)
Thermal Conductivity 1.59 0.262 180 4010 1.7
(104erg 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)
(102 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 sidewalls 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 31.
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 oilair interface, the liquid insert contained a "pinning
edge" and a reservoir (Figure 33). 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 33. Crosssectional 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 oilair 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 overspill port. The over
spill port was simply a hole in the side of the bath where a tube was inserted. The bath
would be overfilled with water such that additional water would spill out through the
overspill 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 overspill 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 oilair interface. This problem was
resolved by coating the zinc selenide window with an antireflective polymer, which was
performed by IIVI 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 sidewalls. 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
computercontrolled 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 OL700 series,
thermistors. The thermistor located in the lower heating bath was a waterproof 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, waterproof 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 DAS1601 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 analogtodigital 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 48666 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
proportionalintegralderivative (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 34.
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. HeaterControl
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
XAxis YAxis
Data Data
Pause 
SiSend Digital
. . Update Graph Control Decision Output ToVCR
Shut Down
Experiment?
Yes
Send Digital Exit Program
Shutdown Signal Et Prog
Figure 34. 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
overnight. 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 builtin 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, pi 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 Fpr \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 E020.157 37 7 22
I :cm s)1 ,
erm 1.43E03 2 Lower Fluid pper Fluid
DiffusiviIty 0.12 Temp. Diff. Temp Diff.
I 13 57 iGraph Length
Thermal 2 08E04 3.33E03 ,Time sec
'Expansion E03
Gain T1.ul TauD
2.00 ASpec t 1 0. 0
rDensity 1 9120E03 5 T ei tlizte o r
Cmments Bang Bang 4 [.
o PI control [1i4 B1BE
PiD control
Figure 35. 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 closeup lens should be used such that the silicone oilair 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 overspill 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 .8Yaxis 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 36. 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 35. Once all of the necessary information was entered, the OK button
was clicked to go to the main program window (see Figure 36). 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 oilair 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 codimensiontwo 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 CODIMENSIONTWO 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 codimensiontwo
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 41 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 noslip 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 liquidgas 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 41. 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 41. 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 codimensiontwo 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 41, 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 41, 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 41. An infrared image of the toroidal flow pattern in
a cylindrical container. The picture is taken looking down
onto the oilair interface.
80
codimensiontwo 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 41. 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 42a).
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 42b). When this cell
reached some critical size, it split into two cells (Figure 42c). Here the flow pattern was
the same as the first bimodal flow pattern rotated be 900 (Figure 42d). This process then
repeated itself (Figure 42e 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 42. 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
codimensiontwo 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 RayleighMarangoni convection in square containers
(Ondarguhu et al., 1993). Although this work mentioned that the oscillating behavior was
a result of a TakensBogdanov (Golubitsky et al., 1988) bifurcation, which is associated
with codimensiontwo points, they did not prove that it was indeed a codimensiontwo
point. Secondly, the oscillations only occurred well into the supercritical region.
Codimensiontwo 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 codimensiontwo
points. Erneux and Reiss (1983) looked at supercritical bifurcations of two degenerate
eigenvalues (i.e. codimensiontwo 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 RayleighMarangoni 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 codimensiontwo point, it was possible for secondary Hopf bifurcations to
occur for aspect ratios slightly greater than the codimensiontwo point. However for the
m = 2, m = 0 codimensiontwo 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, vorticityfree sidewall
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 vorticityfree sidewall
condition generates m = 1, then m = 2, then m = 0 modes as the aspect ratio is increased,
whereas the noslip 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 noslip sidewalls will not have the m = 1, m = 2 codimensiontwo 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 noslip
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 codimensiontwo 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 RayleighMarangoni convection in cylindrical
containers at codimensiontwo 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
codimensiontwo 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 siliconeair 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 sidewalls
was obtained in bilayer convection at the expense of using unrealistic conditions. The
imposition of realistic noslip 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 noslip rigid sidewalls assumed a passive upper
85
phase and a nondeforming interface. All of the calculations used properties pertaining to
the silicone oilair 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 42
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 42 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 42. 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 42. 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*105
(o_ = 2.55) (co = 2.55) (to = 2.55) (co = 2.57)
1 237.6 237.8 241.7 243.4 5.14*102
(_ = 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 42 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 43. 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 43. Plot of the vertical component of velocity versus fluid depths, for 3 mm (a), 5
mm (b), and 9 mm (c) air heights. The liquidgas 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 (corotating), 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 (counterrotating), then the convection is considered to be mechanically coupled.
For the calculations given in Figure 43b, 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 43c)
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 vorticityfree sidewalls. 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 44. 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 44. Two observations can be made from Figure 44. 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 noslip sidewall
calculations.
A comparison can be made between the vorticityfree and the noslip sidewall
calculations. The vorticityfree sidewall calculations are shown in Figure 44 and the no
slip calculations are shown in Figure 45. It can be observed that the minimum
Marangoni number for each mode is the same for the vorticityfree sidewalls. For the no
slip sidewalls, 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 noslip calculations, the
modes quickly crowd together and become indistinguishable.
In Figure 45, 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 45, 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 noslip calculations as well as the vorticityfree 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
codimensiontwo 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 lightheartedness 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 352320 and NGT 51242 grants and from the National Science Foundation, grant numbers CTS 9500393 and CTS 9307819. ui
PAGE 4
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 GalerkinEckhaus 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
PAGE 5
4. RESULTS AND DISCUSSION 77 Introduction 77 CodimensionTwo 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 ConvectionCoupling 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
PAGE 6
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 codimensiontwo 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 gradientdriven flows. It was also shown that changing the radius of the container can change the driving force of convection from a surface tension gradientdriven to buoyancydriven and back again. Finally, the weakly nonlinear analysis was able to give a qualitative description of codimensiontwo 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. Liquidencapsulated 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 11. 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 buoyancydriven and interfacial tensiondriven 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 solidliquid 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 quasistatic. 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 Buoyancydriven 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 BuoyancyDriven Convection Cold gas Hot Interfacial TensionDriven Convection Cold gas liquid warmer >^ v' r^,,.. ^Â£% 1. Hot Figure 12. 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 buoyancydriven convection. The extent of buoyancydriven 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 buoyancydriven convection.
PAGE 12
1.2.2. Marangoni Convection Surface tension gradientdriven convection, unlike buoyancydriven 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 tensioninduced 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 buoyancydriven convection, there exists a critical temperature difference where the surface tension gradientdriven flow is not dampened by the thermal diffusivity or viscosity, and the fluid is set into
PAGE 13
motion. Surface tension gradientdriven 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, buoyancydriven convection is more prevalent, and 0.0 0.5 ao 90 ^ V 1Â— 1 \ =5 85 2 \ m=0 Â— m1 \ \ m=2 m=3 c 80 O D1 ^75nJ S 70 \ \ '^. \ . \ \ \ ^^^ Critical o en \ \ ^s Â— 55 iÂ— : ] _j 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 Aspect Ratio Figure 13. 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 gradientdriven 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 gradientdriven 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, finitesized 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 13). 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 13, there is an ni = 1, n = 1 flow pattern, where m is the azimuthal mode and n is the radial mode (see Figure 14). 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 crosssection. 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 codimensiontwo 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 14. Schematic of three different flow patterns. Circles represent fluid flowing up and X's represent fluid flowing down.
PAGE 16
Lower Dragging Mode Viscous Coupling TheiTnal Coupling Upper Dragging Mode Figure 15. Schematic of the different types of convectioncoupHng. 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
PAGE 17
10 a r\ a V / Lower layer b Upper layer b d d Figure 16. 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 15. Suppose that convection initiates in the lower layer. The upper layer responds by being dragged,
PAGE 18
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 l5a. This can also been seen by plotting the velocity and temperature perturbations versus the fluid depth (Figure 16). Observe in Figure l6a 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 l5b as counterrotating rolls in the two fluids. This can also be denoted by the vertical component of velocity switching sign at the interface (Figure l6b), 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 corotating. 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 l6c). Strictly speaking, the transverse components of velocity should be zero at the interface. However, thermal coupling is
PAGE 19
12 sometimes referred to the case when a small roll develops in one of the layers so as to satisfy the noslip 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 l5d). 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 l6d) 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 fluidfluid 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 17. The four possible interfacial structures at a fluidfluid interface. Each structure can give information about the driving force of the convection.
PAGE 20
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 17. 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 buoyancydriven 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 buoyancydriven 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 gradientdriven 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 buoyancydriven 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 buoyancydriven convection in the upper phase, or by surface tension gradientdriven convection where the surface tension
PAGE 21
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 RayleighMarangoni 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 RayleighMarangoni 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
PAGE 22
15 assumed that the interface between the two fluids was nondeformable 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 gradientdriven 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 nondeformable. 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: Â„. ^gas^ liquid '^ liquid^ gas Nield found that buoyancy and surface tension gradientdriven 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
PAGE 23
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 buoyancydrive convection is dominant. He also notes, that in buoyancydriven convection, the fluid flows up into a surface elevation and that in surface tension gradientdriven 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 gradientdriven convection leads to a long wavelength instability. He developed a nonlinear evolution of
PAGE 24
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 RayleighMarangoni 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 GalerkinEckhaus expansion (Eckhaus, 1965; Manneville, 1990). The most relevant result came from their analysis of codimensiontwo points, where two different flow patterns coexist. For one of the codimensiontwo 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 codimensiontwo point. When the aspect ratio was decreased to the other side of the codimensiontwo 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
PAGE 25
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 vorticityfree boundary conditions with realistic noslip 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 codimensiontwo 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 vorticityfree sidewalls. In the paper by Zaman and Narayanan (1996), and later by Dauby et al. (1997), a linear, three dimensional solution was found for RayleighMarangoni convection in a cylinder. Both papers assumed that the interface was flat and that the sidewalls of the cylinder were noslip. One of the most interesting observations in these two papers, is that the progression of modes was not the same as the vorticityfree calculations (Rosenblat et al., 1982a). That is, the flow pattern predicted at the onset of
PAGE 26
19 convection for a given aspect ratio is different for noslip and vorticityfree sidewalls. The vorticityfree calculations contradict the results of Koschmieder and Prahl (1990). Additionally, the different progression of modes changes the codimensiontwo 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 codimensiontwo 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: buoyancydominated, surface tension
PAGE 27
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 drivenconvection in
PAGE 28
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 RayleighTaylor instability. The RayleighTaylor 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.
PAGE 29
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: Â• A linear model of convection in two immiscible fluids, which are infinite in the horizontal direction. Â• A numerical calculation of the linear model equations using a Chebyshev spectral tau method. Â• A transformation and unfolding method used to map the results from the infinite, unbounded calculations into a bounded cylinder. Â• A weakly nonlinear analysis of convection of two immiscible fluids in a cylinder using a GalerkinEckhaus 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
PAGE 30
T =Tt T T. T T. raTTT' 23 iT in ^4 ""i >'>i1fl%f 'V Cold fluid #2 fluid #1 z = ri (x ,t ) z =0 z = d, Figure 21. 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 21). The equations which determine the velocity, pressure and temperature for each fluid are the familiar Boussinesq equations. Vv =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).
PAGE 31
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, aÂ„ 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,Â„. 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. VV2=0 (2.13) Pr Â— 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, noslip 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 nÂ„ 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) nyt,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. nT*iiiiX2n = a(v^n) (2.24) nTiiijnTjn, =nfV5a (2.25) where x is the stress tensor and V, is the surface gradient operator. Substituting the dimensionless variables, T, n, and nÂ„ 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)
PAGE 35
28 G2=X atz = / 5r 5ri Â—+ U, Â— Â— = w, for z = ri ot dx dr\ Â— (w2w,)=w, M2 for z = r dx (2.29) (2.30) (2.31) Â— (W]W2J=W2Wi /or z = ri 0, = 02 atz = T) ^502 aria02^ 501 5r50 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)
PAGE 36
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'Â™^ 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,\=Â—Y^ forz<0 (2.37) (2.38) 0,2 apRa J P0,2 =Â— ^ ^ 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)
PAGE 37
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(d22)82+W2 =0 K (2.45) (2.46) (2.47) where, D = Â— 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 AÂ„= A(s = 0) and in general, A = A(n(s),s). The term Â— 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
PAGE 38
31 8z An The derivative ^ = 1, as z = r\. We define rii and A, as follows, Â— = rii, and, dA zÂ— = A, Therefore we have: 5s a(ti(s),Â£)=AÂ„ + 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,(lX) atz = ^Z)0, =D0, atz = n2n,+ 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
PAGE 39
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
PAGE 40
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 ^Â— + 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,aÂ„{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.^Â„it)TÂ„{x) (2.59) The approximation error accrued by truncating the infinite series is given by Syy.
PAGE 41
34 du duj^ z,~zÂ— + '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)TÂ„{x) (2.61) =0 By adding x more variables, we need x more equations. These equations come from the boundary conditions. N+T I;Â„(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 (tÂ„ (x), T; (x)) = JT, (x%Â„ (x')(l x'^ )"^ dx' = ^ cÂ„5Â„Â„ (2.63) 1 ^ where 12 n = cÂ„ = [ n^ m 1 n = m \ n>Q and 5,Â„Â„ = S (2.64) n<0
PAGE 42
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={) Â„=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 LrÂ„(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 ^Â„ for n = 0, 1, ., N where: N N u^{xj)=YaÂ„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 aÂ„ 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 ^nbÂ„=2 Â£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.
PAGE 43
36 ^Â—T (+i)=(+irmx! IJ (2.69) By using equation (2.67) or (2.68), tlie relationship between Â„ 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 aÂ„<^^ 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,...,N2 ^~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
PAGE 44
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 NavierStokes 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 (Â£)^co^)lii?aD0i =0 (2.74) (d^co^)=), +Wi=0 (z)2co^)W2Dn2+ai?a02 =0 (D^G)^)T2ai?aÂ£)02 =0 (2.75) k(d^ (0^)02 +W2=0 The thirteen boundary conditions become: Wj = W; = at z = DW2 = DW, at z = n2n,+^Â— ^JtiÂ„+2(dW, ^i)W2)=0 atz = (d^+co^)w, 1^(0^+ CO ^)W2 = a ^ Ma (riÂ„01 ) atz = (2.76) A:Z)02=D0, atz = 0, =02+TiÂ„(l)^) atz = Z)Wj=Wi=0, =0 atz = l DWj = W2 = 02 = at z = /
PAGE 45
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 =Â— z1 for z> The change in independent variables requires the substitution. (2.78) d d dz dx dx ydz d 2 Â— for z < dx 2 d Â— Â— for z > ./ dx (2.79) Now expand each dependent variable in terms of Chebyshev polynomials. n,(z) .0.(4 W^iz) naCz) .02(4 NT a T; iz) (2.80) KCjj NT /, 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.
PAGE 46
39 1600 2.5 3,0 Wave Number 3.5 4.0 4.5 Figure 22. Plot of the Rayleigh number versus wave number. A^ = RaB^ (2.82) where ^ = (w,,ni,0, W2,n2,02,ry 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
PAGE 47
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 22 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, noslip 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 nondeformable. 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).
PAGE 48
41 ^Â•V=0 (2.83) V^,V.:+i?ae,z=^0 (2.84) V^Qi+Wi=0 (2.85) "^Â•^2=0 (2.86) ^V\,Vr,+aRaQ,z0 (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
PAGE 51
44 Table 21. 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 21 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 23 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.
PAGE 52
45 u^(r,(p,z)= y. cos(pj',Â„(^Â„Â„/)DW,.(z) / mn ^ v.(r,(?,z)=y2 sinm(pJÂ„(x,Â„Â„r)DW,(z) w, (r,(p,z)= cosm
PAGE 53
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 codimensiontwo 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 GalerkinEckhaus 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 nondeformable. = V:'i+v\i+i?fle,z f 7i ^ av, ^^1=0 (2.118)
PAGE 54
1 fdy, ^ 47 P P Pr VV2=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^ = Â—^ at z = oz oz ^li^+MÂ— ^ = 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\,pV,^+i?^0,^z "^^1^=0 (2.121) V20,^ + W,^=O
PAGE 55
48 vVSpV'2p+ai?^e2^z "^Â•^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:
PAGE 56
49 L, = V' v^ v^ R. 1 v^J and L2 = f V72 vV Upon inspection we can find the adjoint operator L' L = Ji 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 selfadjoint. Although it appears as if it can be made selfadjoint 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/; Vv =0 1/' (2.125) 2p p 2p ^ 2p VV* =0 KV^e* +aR* w* =0 ip p ip (2.126)
PAGE 57
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) arvp/ ar az =^ "' ''^^ ^""^ = ^'^ The solution to the adjoint problem is found similarly to the nonadjoint solution. "*, (^^'^)= ^m cosmcp j',Â„(x,Â„Â„r)z)W,.(z) v*^(r,(p,z)='%;sini.?9J^(:^^Â„r)z)W,(z) (2.128) \ (r,q>,^)= ^'Â„Â„ cosmcp J,Â„(:^Â™/)w,(z) for / = 1, e* (r,(p,z)= cosw(pJ,Â„(:V,Â„Â„r)0,.(z) 2.3.3 GalerkinEckhaus Expansion The GalerkinEckhaus 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.
PAGE 58
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 LiapunozSchmidt 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
PAGE 59
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;,(vVSiV,. (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:
PAGE 60
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) MMp ^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
PAGE 61
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 }Â• '^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) ^sf^p Â§J<;s,/'e{S5u}} (2.144)
PAGE 62
55 where ^^ and C,^ are arbitrary values (for example, Cu = 01 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 cutoff 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 nonzero. 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^Â—'^p^q yseS^ and p,q eS^ (2.145) Substituting equation (2.145) into (2.136) gives:
PAGE 63
a 56 P Â„ 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
PAGE 64
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 codimensiontwo points.
PAGE 65
CHAPTER 3 EXPERIMENTAL APPARATUS AND PROCEDURE The objectives of the experiments were to observe the behavior of the fluid convection at codimensiontwo 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 buoyancydriven convection in one or both fluids, the bilayer IR camera Electronic Hardware Unit Test Section Figure 31. Overall schematic of the experimental apparatus 58
PAGE 66
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 oilair interface was detected using an infrared camera. This chapter is split up into two parts, a description of the experimental apparatus and a walkthrough 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 computercontrolled 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 wellestablished, each had its drawback. 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
PAGE 67
60 Figure 32. 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, platelike, 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
PAGE 68
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 nonintrusive 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 MercuryCadmiumTelluride (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
PAGE 69
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" closeup 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 falsecolor image of the temperature field. Every 1/30"' of a second, a 640X480 frame of IR radiation is
PAGE 70
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 31. 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)
PAGE 71
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 sidewalls 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 31. 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 oilair interface, the liquid insert contained a "pinning edge" and a reservoir (Figure 33). 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
PAGE 72
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 33. Crosssectional 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 oilair 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
PAGE 73
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 overspill port. The overspill port was simply a hole in the side of the bath where a tube was inserted. The bath would be overfilled with water such that additional water would spill out through the overspill 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 overspill 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 0.7um and about 17^m. Zinc selenide is also slightly reflective to radiation in the 8 to 12 )am range. This reflection
PAGE 74
67 caused problems with the imaging of the siUcone oilair interface. This problem was resolved by coating the zinc selenide window with an antireflective polymer, which was performed by IIVI 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 sidewalls. 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.
PAGE 75
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 computercontrolled 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.
PAGE 76
69 Each of the temperatures were measured using a highly accurate thermistor. The types of thermistors used were Omega, Unear response, model OL700 series, thermistors. The thermistor located in the lower heating bath was a waterproof 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, waterproof 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 analogtodigital 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 48666 MHz with an ISA motherboard. Data was continually read from the data acquisition board by the control program.
PAGE 77
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 proportionalintegralderivative (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 34. 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
PAGE 78
( 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 Â— No Start Analog Data In Statistics (Data Averaging) Update Screen Variables Prepare XAxis \ Data Prepare YAxis 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 34. Flow chart of the programming logic
PAGE 79
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 overnight. 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 builtin 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
PAGE 80
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/cmC) 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 Â— 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) 'Â— Â— Â— J Comments: ? "^ Aspect Ratio &. Â— Control Parameters OBong Bang O PI control PID control Gain Taul TauD 1 1 1 4 1.4 A .1 .02 .1 Figure 35. The initialization program window.
PAGE 81
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 closeup lens should be used such that the silicone oilair 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 overspill tube.
PAGE 82
75 IR Program Version 3.0 1 Â— 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 Â— 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 Â— 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: Â— Segment Number: 1 of 7 VCR Status: Paused Control Type: PID / Â— (Si e. Pause VCR Exit Total Temperature Difference Temp Difference 720 1080 1440 Time 1800 Â•.I'lMMMWian.Figure 36. 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 35. Once all of the necessary information was entered, the OK button was clicked to go to the main program window (see Figure 36). 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.
PAGE 83
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 oilair interface changed into a particular pattern. At this point, the temperature difference and the flow pattern were recorded.
PAGE 84
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 codimensiontwo 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
PAGE 85
78 4.2 CODIMENSIONTWO 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 codimensiontwo 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 41 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 noslip 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 liquidgas 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 41. 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
PAGE 86
79 There are two items of information to be obtained from Table 41. 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 codimensiontwo 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 41, 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 41, 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 41. An infrared image of the toroidal flow pattern in a cylindrical container. The picture is taken looking down onto the oilair interface.
PAGE 87
80 codimensiontwo 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 41. 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 42a). 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 42b). When this cell reached some critical size, it split into two cells (Figure 42c). Here the flow pattern was the same as the first bimodal flow pattern rotated be 90 (Figure 42d). This process then repeated itself (Figure 42e 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 42. 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.
PAGE 88
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 codimensiontwo 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 RayleighMarangoni convection in square containers (Ondar9uhu et al, 1993). Although this work mentioned that the oscillating behavior was a result of a TakensBogdanov (Golubitsky et al., 1988) bifurcation, which is associated with codimensiontwo points, they did not prove that it was indeed a codimensiontwo point. Secondly, the oscillations only occurred well into the supercritical region. Codimensiontwo 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 codimensiontwo points. Erneux and Reiss (1983) looked at supercritical bifurcations of two degenerate eigenvalues (i.e. codimensiontwo points). They noted that when the supercritical
PAGE 89
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 RayleighMarangoni 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 codimensiontwo point, it was possible for secondary Hopf bifurcations to occur for aspect ratios slightly greater than the codimensiontwo point. However for the m = 2, m = codimensiontwo 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, vorticityfree sidewall 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 vorticityfree sidewall condition generates m = 1, then m = 2, then m = modes as the aspect ratio is increased, whereas the noslip 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 noslip sidewalls will not have the m = 1, m = 2 codimensiontwo 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 noslip boundary conditions. The eigenfunctions from these calculations could then be used in
PAGE 90
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 codimensiontwo 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 RayleighMarangoni convection in cylindrical containers at codimensiontwo 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 codimensiontwo 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 siliconeair 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
PAGE 91
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 sidewalls was obtained in bilayer convection at the expense of using unrealistic conditions. The imposition of realistic noslip 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 noslip rigid sidewalls assumed a passive upper
PAGE 92
85 phase and a nondeforming interface. All of the calculations used properties pertaining to the silicone oilair 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 42 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:
PAGE 93
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 42 gives a comparison of the critical Rayleigh number and the associated critical wave number, for various air depths.
PAGE 94
87 Two important points can be made from Table 42. 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 42. 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*102 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
PAGE 95
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.
PAGE 96
89 The critical Rayleigh numbers corresponding to Table 42 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.
PAGE 97
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 43. 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 43. Plot of the vertical component of velocity versus fluid depths, for 3 mm (a), 5 mm (b), and 9 mm (c) air heights. The liquidgas 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.
PAGE 98
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 (corotating), 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 (counterrotating), then the convection is considered to be mechanically coupled. For the calculations given in Figure 43b, 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 43 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 vorticityfree sidewalls. The formula is:
PAGE 99
92 73.00 65.00 Figure 44. 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. sÂ„Â„ 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 44. Two observations can be made from Figure 44. 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
PAGE 100
93 aspect ratio changes. Some of these observations carry over to the nosHp sidewall calculations. A comparison can be made between the vorticityfree and the noslip sidewall calculations. The vorticityfree sidewall calculations are shown in Figure 44 and the noslip calculations are shown in Figure 45. It can be observed that the minimum Marangoni number for each mode is the same for the vorticityfree sidewalls. For the noslip sidewalls, 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 noslip calculations, the modes quickly crowd together and become indistinguishable. In Figure 45, 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 45, 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 noslip calculations as well as the vorticityfree 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 codimensiontwo points and can be associated with oscillatory behavior in the immediate post onset regime of flow (Rosenblat et al., 1982a; Johnson and Narayanan, 1996).
PAGE 101
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 \ "*Â• "^ 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 45. Plot of the bounded, linear calculations using an insulating, noslip sidewalls. 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.
PAGE 102
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 codimensiontwo 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 polymethylsiloxanes 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.
PAGE 103
96 viscosity conveniently scales out of the problem and the bounded layer model results, given in Figure 45, 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 46 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 46. 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.
PAGE 104
97 (double toroid), pattern was observed (Figure 46a). The critical temperature difference across the liquid layer was 0.9C (8.4C overall). This flow pattern agrees with the noslip, threedimensional, 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 46b), the double toroidal pattern became more skewed, or lopsided, on one side. For the 20 mm air height (Figure 46c), 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 lopsided 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
PAGE 105
98 Table 43. Temperature differences across the silicone oil and overall temperature differences for the six experiments given in Figure 46 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 sidewalls, 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 43. 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
PAGE 106
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 46d 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 46e shows a flow pattern that looks like a combination of an m = and an m = 1 flow for the 14 mm air height. Figure 46f 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.
PAGE 107
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 sidewalls 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 44. 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 45 Air height Experiments (C) Hardin's Conducting sidewalls fC) Hardin's Insulating sidewalls (C) 14 mm 15.3 15.3 9.1 20 mm 8.4 8.8 4.2 In Figure 46e, 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 codimensiontwo point is predicted and both the m = 1 and m = modes are equally
PAGE 108
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 42. 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 45, one can conclude that the calculated Marangoni number will
PAGE 109
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 Codimensiontwo point section, a dynamic behavior termed "mode switching" was discovered for an aspect ratio of 2.5, which was near a codimensiontwo 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 codimensiontwo point as well. In fact, convecting upper air will shift the codimensiontwo point to the right. This can be seen by the decrease of the wave number in Table 42 for larger air heights. In Figure 47 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 46. When the air height decreases, the Biot number increases and the codimensiontwo point moves to the left. For the new Biot number, the flow pattern shifted from a codimensiontwo point to an m Figure 47. 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.
PAGE 110
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 SOT. 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 noslip calculations give a codimensiontwo 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 codimensiontwo point was found at an aspect ratio of 2.5. As mentioned above, an active air layer will move the codimensiontwo point closer to 2.5 By using both calculations and experiments, a new convectioncoupling 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 fluidfluid interface. These temperature gradients in turn cause surface tensiondriven 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.
PAGE 111
104 4.4 CHANGES IN CONVECTION COUPLING AND INTERFACIAL STRUCTURES In this section, the convectioncoupling 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 convectioncoupling 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 convectioncoupling 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 oilair system and the glycerolsilicone oil system. Their properties are shown in Table 45. Note the different signs of the interfacial tension gradient, a,, in each of the bilayer systems. Table 45. 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 GÂ„ (dyne/cm) 20.9 Â— 25 Â— Qi (dyne/cm C) 0.05 Â— 0.13 Â—
PAGE 112
105 Table 46. 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 46 gives the values of each ratio for each of the four possible interfacial structures (see Figure 17). As was described in the first chapter, the interfacial structures indicate the driving force for convection.
PAGE 113
106 4.4.1 Changes in ConvectionCoupling 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 49). 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 48. Schematic of the different types of convectioncoupling. 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 surfacedriven flow is caused by the upper fluid buoyantly convecting and simuhaneously inducing surface tensionor buoyancy driven convection in the lower layer, near the interface.
PAGE 114
107 TO ) Lower ^Â—^ 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 Â— ( r? N c/5 II 00 fi 6 E 3 en II id o rji 03 JO o 3 tio (U o 'o . 1) O .Â— 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( IÂ— 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 rl L_j f J r) V:; OB tH CN t/1 o 13
PAGE 115
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 48. As noted earlier, Figure 48a 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 oilair 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 48b 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 >Â• m Qi 40 20 \ \ \ 'Â• 3 mm \ \\ \ \ \ \ \ 5mm ^ \\ Â• \ \ \ \ \ 5.5 mm" """^ 6 mm Â— \ 7mm ~ \ 10mm _._Â—0.0 0.5 1.0 0.01 mm 1.5 2.0 \Na\B Number 2.5 3.0 3.5 Figure 49. 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.
PAGE 116
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 48c can be nearly depicted by Figure 410c. Observe that the velocities in both the upper and lower layer show comparable minima and a small counterroll has developed in the air layer to preserve the noslip condition between the fluids. It is possible to obtain a situation where no counterroll 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, corotating 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 48d can not be depicted in a silicone oilair system, because the air does not drag the silicone oil due to the very small ratio of dynamic viscosities. However, the calculations using glycerolsilicone oil system show this dragging effect well (Figure d). This situation is qualitatively the reverse of Figure 48a. The last convection mechanism. Figure 48e, is seen in calculations using silicone oilair (Figure e). Notice that within the scale of the plot, the lower velocity is nearly zero. A closer look at
PAGE 117
110 w a W b 2 mm 5 mm 2 mm 5 mm Figure 411. 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 41 1). At a wave number of co = 1.9, Figure 41 la shows mostly a thermal coupling of the silicone oil and air. For a larger wave number of CO = 2.7, Figure 41 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 49 using equation 2.116. The resuh is shown in Figure 412, 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
PAGE 118
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 412. Plot of the Rayleigh number of the silicone oil versus the cylindrical aspect ratio. This plot was generated from Figure 49 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 convectioncoupling 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 49, the height of the silicone oil is fixed at 2 mm. Therefore, Figure 412 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? ^
PAGE 119
112 The change in the convectioncoupling 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 ^''' Â— '3'0 mm ^.,''' Â— ^'2.5 mm Â— \ \ Case I Case II 0.5 1.0 1.5 2.0 2.5 V\/a\e Number 3.0 3.5 Figure 413. 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.
PAGE 120
113 because the critical Rayleigh number is a nonmono 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 48b for one aspect ratio and Figure 48c for another. Other mechanisms (Figure 48a, 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 oilair 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 tensiondriven convection and Case II indicates buoyancydriven 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 tensiondriven 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.
PAGE 121
114 When Figure 413 is unfolded, the dominating driving force for convection can change as the aspect ratio increases, and this is depicted in Figure 414. This is most pronounced at codimensiontwo 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 codimensiontwo points, though, the switch from one interfacial structure to the next can be abrupt. That is, on one side of the codimensiontwo point, the fluid is buoyancydriven with one spatial pattern, then switches to a surface tensiondriven flow on the other side of the codimensiontwo point, with a different spatial pattern. This can be seen at codimensiontwo 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 Â—J \ n = 2 \ Â— 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 414. 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.
PAGE 122
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 415. 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 ConvectionCoupling and Interfacial Structure In the last example the liquidliquid bilayer glycerolsilicone oil is examined. The thermophysical properties, which are listed in Table 45, 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 415 again gives a plot of the Rayleigh number versus wave number for a glycerolsilicone 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 nonzero, it was inferred that the onset of convection was oscillatory. While for real values of the Rayleigh number, the numbers are correct, for
PAGE 123
116 complex values, the real part of the Rayleigh number has no physical interpretation. The dotted lines in Figure 415 and Figure 417 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 416. 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 callouts in Figure 415 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 convectioncoupling 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 buoyancydriven 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
PAGE 124
117 4.15 cm 3 cm w /\ b vy 4.15 cm 3 cm Figure 416. 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 counterroll 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, buoyancydriven 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 gearlike 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.
PAGE 125
118 Unfolding Figure 415 gives an extremely interesting and complicated plot of the Rayleigh number versus the wave number (see Figure 417). 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 codimensiontwo points. For example, the codimensiontwo 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 codimensiontwo points, nonlinear interactions will be very dynamic. As was mentioned in the codimensiontwo 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 oilflourinert 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 codimensiontv/o points such as the 1.1 aspect ratio. Here a codimensiontwo point is close to an oscillating bimodal flow and an oscillating axisymmetric flow.
PAGE 126
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 417. Plot of the Rayleigh number of the glycerol versus the aspect ratio. The plot is generated from Figure 415 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 vorticityfree side walls and noslip sidewalls. 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 vorticityfree sidewalls and noslip sidewalls can not be ignored, the effects of aspect ratios on convection mechanisms and interfacial structures with realistic sidewall conditions are expected to be qualitatively similar to the vorticityfree 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
PAGE 127
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 oilair 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 sidewall conditions, it was assumed that the vertical and tangential component of vorticity vanished at the vertical sidewalls. 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 codimensiontwo 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 codimensiontwo points give interesting oscillating behavior. The effects of bounded geometry on bilayer convection should lead to many exciting experiments.
PAGE 128
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 codimensiontwo 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 codimensiontwo 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.
PAGE 129
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^pyp,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) MM^ RRÂ„ TÂ„, =Pr'(v;,v,Vv\ + (e;,v,.VeJ (4.13) and for two fluid layers:
PAGE 130
123 ap= / ir^r/ 7\ (414) ^lp^^\p) + \^2p'^2p MMp RRp "^ 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 44 (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)
PAGE 131
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) AÂ„=^^^^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, Â•^u j^ ~^\\^i\~^\\^\\ (421) where: \Y211111 / \Yoiiii c,, =2 / Y211111 / V \Yuii2! +Y 112111 /T +\Yiinoi +Y 110111/' Â£21 ^ ^ Â£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 (423)
PAGE 132
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 Â— 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 codimensiontwo 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: 5Â„= {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 /^2lA2 and the amplitude equation foxp = {21 } is:
PAGE 133
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 Â— 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
PAGE 134
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 Â— 2. (^ \7oi2121 / \7412121 J212IOI +7210121 J~~~ +Vy 212141 +7214121/7 Â•01 '41 4.5.4 Case 4 The last case looked at the codimensiontwo 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 Â— Â£)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:
PAGE 135
128 dh ^m J^ ~ ^01 ^01 "^YoiOlOl^OI ~^Yoi2121^21 "Â§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 2lyoio2oi + 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 2j212102 +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,Â„) for the single layer and bilayer systems at the various air heights (Table 47, Table 48, Table and Table 410). 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:
PAGE 136
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 48, Table 49, and Table 410. 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^Â„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
PAGE 137
[30 Table 47. 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 CodimensionTwo 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 CodimensionTwo 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
PAGE 138
131 Table 48. 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 1.48E05 8.82E03 1.42E02 1.90E05 8.32E03 1 .07E02 5.03E03 7.90E05 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 2.55E02 1.07E01 1.49E01 2.55E02 1.05E01 1.33E01 3.66E02 9.50E05 C01 3.21 E03 1.14E01 2.27E01 3.22E03 1.08E01 1.72E01 5.43E02 4.94E04 CodimensionTwo 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 8.44E03 2.70E02 3.66E02 8.42E03 2.67E02 3.34E02 1 .OOE02 2.27E04 gii2i 1.08E04 2.14E03 6.09E03 1.05E04 2.00E03 4.39E03 1 .93E03 7.09E05 C11 3.13E04 1 .34E02 3.46E02 3.15E04 1 .26E03 2.61 E02 5.39E03 1.21E04 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 6.54E03 3.07E02 5.98E02 6.46E03 3.17E02 5.08E02 1 .64E02 1.26E02 92111 3.51 E04 1.75E02 4.18E02 3.58E04 1 .63E02 3.13E02 7.78E03 1.57E05 C21 1 .60E04 7.81 E04 2.54E03 1.57E04 7.11 E04 1.69E03 8.30E04 3.94E05 CodimensionTwo 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 2.50E02 9.57E02 1.31E01 2.50E02 9.41 E02 1.20E01 3.52E02 1 .24E03 27012121 2.15E03 3.97E03 8.69E03 2.13E03 3.86E03 7.44E03 2.31 E03 1.91 E03 goi2i 6.32E04 1 .50E02 2.96E02 6.35E04 1.42E02 2.55E02 1.13E02 3.22E05 COI 2.40E03 6.45E02 1 .26E01 2.42E03 6.07E02 1.08E01 5.08E02 5.00E04 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 1.69E02 7.50E02 1.06E01 1.69E02 7.39E02 9.74E02 2.95E02 4.43E04 92101 1.38E03 4.38E02 8.86E02 1 .39E03 4.13E02 7.67E02 3.50E02 1.14E04 C21 8.17E05 4.16E03 9.32E03 7.93E05 3.91 E03 8.42E03 2.19E03 2.61 E05
PAGE 139
132 Table 49. 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 6.79E02 0.1303 0.1543 0.0680 0.1295 0.1440 0.0532 0.0321 C11 1 .85E05 8.88E03 1 .42E02 1 .94E05 8.32E03 1.07E02 4.76E03 1.51E04 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 2.53E02 1.06E01 1.48E01 2.55E02 1.05E01 1.33E01 3.12E02 1.72E02 C01 3.19E03 1.14E01 2.27E01 3.25E03 1.08E01 1.72E01 4.93E02 1.75E03 CodimensionTwo 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 8.55E03 2.72E02 3.67E02 8.41 E03 2.67E02 3.34E02 8.05E03 3.52E04 gii2i 1.11E04 2.17E03 6.15E03 1 .05E04 2.00E03 4.39E03 1 .06E03 5.94E05 C11 3.18E04 1.35E02 3.47E02 3.16E04 1.26E02 2.62E02 4.39E03 1 .24E04 ail 0.0519 0.0861 0.1098 0.0519 0.0854 0.1028 0.0421 0.0175 Y2nni 6.33E03 3.09E02 6.00E02 6.45E03 3.02E02 5.09E02 2.43E02 1.53E02 92111 3.63E04 1 .76E02 4.20E02 3.60E04 1.63E02 3.13E02 9.21 E03 2.04E04 C21 1 .59E04 8.11E04 2.60E03 1.56E04 7.13E04 1.70E03 6.16E04 5.48E05 CodimensionTwo 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 2.48E02 9.55E02 1.30E01 2.49E02 9.41 E02 1.20E01 2.96E02 1 .29E02 2yoi2i2i 2.07E03 4.07E03 8.80E03 2.12E03 3.88E03 7.46E03 1.72E03 3.93E03 goi2i 6.34E04 1.50E02 2.96E02 6.38E04 1 .42E02 2.55E02 1 .08E02 2.74E04 C01 2.38E03 6.44E02 1.26E01 2.44E03 6.09E02 1.08E01 4.82E02 1.50E03 a2i 0.075 0.135 0.173 0.075 0.134 0.167 0.064 0.036 f2101 1.73E02 7.53E02 1.06E01 1.69E02 7.39E02 9.74E02 2.99E02 7.36E04 92101 1.33E03 4.35E02 8.82E02 5.54E01 4.14E02 7.68E02 3.36E02 1.21 E03 C21 8.21 E05 4.23E03 9.44E03 7.86E05 3.92E03 8.43E03 2.14E03 9.67E05
PAGE 140
133 Table 410. 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 1.94E05 8.29E03 1.08E02 2.47E02 3.39E02 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 2.55E02 1.05E01 1.33E01 1.54E01 2.66E01 C01 3.25E03 1.07E01 1.73E01 3.00E01 2.77E01 CodimensionTwo 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 E03 2.67E02 3.35E02 3.46E02 5.25E02 gii2i 1.05E04 1.99E03 4.42E03 4.78E03 1.88E02 C11 3.16E04 1.25E02 2.63E02 2.20E02 2.25E02 a2i 0.0519 0.0853 0.1048 0.2495 1 .9394 7211111 6.45E03 3.01 E02 5.10E02 5.49E02 4.36E01 92111 3.59E04 1 .63E02 3.15E02 2.59E02 1.59E02 C21 1.56E04 7.10E04 1 .70E03 1.67E03 3.81 E03 CodimensionTwo 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 2.49E02 9.40E02 1.20E01 1.46E01 1.88E01 27012121 2.12E03 3.87E03 7.49E03 7.33E03 2.03E02 goi2i 6.38E04 1 .42E02 2.57E02 5.49E02 2.03E02 C01 2.43E03 6.06E02 1 .09E01 2.42E01 8.51 E02 a21 0.075 0.134 0.171 0.831 2.730 f2101 1 .69E02 7.38E02 9.79E02 1.18E01 2.03E01 92101 1.40E03 4.12E02 7.72E02 1.71E01 3.22E02 C21 7.86E05 3.90E03 8.48E03 1 .42E02 3.76E02
PAGE 141
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 JÂ„', M^ dr 0(z)0(z) Jz \Q(z)@*(z)dz mn' mn 2% 1 A.Â„Â„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.
PAGE 142
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 nondimensionalization 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 45). 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, = Â•"k,M "'^^ ''=ZAr "Â•=Z (4.49) TT e, = Â— ^ at; ^, = t' z' t/f/K, ^1 ~ ^ d, ^'~ ^T, h /' z' d;lK, ''
PAGE 143
136 where AT, and ATj are the temperature differences across the lower and upper fluid respectively, and TÂ„ is the temperature at the fluidfluid 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.
PAGE 144
137 dh. ."^ = SA,Y;Â„A,A, where: 1/. P = w. s = MMp 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 48 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 AÂ„ b ^ Figure 418. 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.
PAGE 145
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 48, Table 49, and Table 410 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 48 and Table 49, 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 48 and Table 49. 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.
PAGE 146
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 410. 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.
PAGE 147
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 codimensiontwo 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
PAGE 148
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 codimensiontwo 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 solidliquid 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
PAGE 149
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 codimensiontwo points. 5.3 NUMERICAL CALCULATIONS Finally, a series of numerical calculations could be very useful in explaining many questions. We have learned from the vorticityfree calculations and the noslip 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 liquidencapsulated 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.
PAGE 150
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
PAGE 151
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 A1 Table of different Matlab programs
PAGE 152
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 9.6e4; %negative thermal expansion coefficient of lower 3 3 3 e 3 ; 1.59e4 2 .62e3 l.le3 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) ;
PAGE 153
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;
PAGE 154
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 = 9.5e4; %negative thermal expansion coefficient thermcond = 1.59e4; %Thermal conductivity thermcondgas = 2.62e3; %Thermal conductivity of the gas thermdiff = l.le3; %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;
PAGE 155
148 omegaLast = 4.0; omegas tep = 0.1; omega = omegaFirst;
PAGE 156
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:N2,1:N) = 4 *E2 (1 :N2 : ) omega"2 *Iden (1 :N2 : ) ; %Row 1 Column 2 A{1:N2,N+1:2*N) = 2 *E1 {1 :N2 : ) ; %Row 2 Column 2 A(N1:2* (N2) ,N+1:2*N) = 4 *E2 (1 :N2 : ) omega'^2 *Iden (1 :N2 : ) ; %Row 3 Column 1 A(2*N3 :3* (N2) 1:N) = Iden (1 :N2 : ) ; %Row 3 Column 3 A(2*N3 :3* (N2) ,2*N+1:3*N) = 4 *E2 (1 :N2 : ) omega^2 *Iden (1 :N2 : ) ; %Row 4 Column 4 A(3*N5:4* (N2) ,3*N+1:4*N) = visc/density .* (4/depth"2 *E2 (1 :N2 : ) omega'^2 *Iden (l :N2 : ) ) ; %Row 4 Column 5 A(3*N5:4* (N2) ,4*N+1:5*N) = 2 / (density*depth) El(l:N2,:); %Row 5 Column 5 A(4*N7:5* (N2) ,4*N+1:5*N) = (l/density) (4/depth^2 *E2 (1 :N2 : ) omega''2 *Iden(l:N2, : ) ) ; %Row 6 Column 4 A(5*N9:6* (N2) ,3*N+1:4*N) = (l/thermcond) *Iden (1 :N2 : ) ; %Row 6 Column 5 A(5*N9:6* (N2) 5*N+1:6*N) = thermdif f (4/depth^2 *E2 (1 :N2 : ) omega*2.*Iden(l:N2, : ) ) ; % Begin Boundary Conditions *******************************************************
PAGE 157
150 %Row 7 Column 4 A{6*N11,3*N+1:4*N) = vecOn ( : ) ; %Row 8 Column 1 A(6*N10, 1:N) = vecO { : ) ; %Row 9 Column 1 A{6*N9,1:N) = depth *vecl {:)' ; %Row 9 Column 4 A(6*N9, 3*N+1:4*N) = vecln ( : ) ; %Row 10 Column 1 A{S*N8,1:N) = 4.*vecl( :) ; %Row 10 Column 2 A{6*N8,N+1:2*N) = vecO ( : ) ; %Row 10 Column 4 A(6*N8,3*N+1:4*N) = {4*visc/depth) *vecln ( : ) ; %Row 10 Column 5 A(6*N8,4*N+1:5*N) =vec0n(:)'; %Row 10 Column 7 A(6*N8,6*N+1) = (G + omega^2)/C; %Row 11 Column 1 A(6*N7,1:N) = 4 *vec2 { : ) + omega^2 *vec0 ( : ) ; %Row 11 Column 4 A(6*N7,3*N+1:4*N) = vise (4/depth''2 *vec2 ( : ) + omega^2 *vecO ( : ) ) ; %Row 12 Column 3 A(6*N6,2*N+1:3*N) = 2 *vecl { : ) ; %Row 12 Column 6 A(6*N6,5*N+1:6*N) = (2*thermcond/depth) *vecln ( : ) ; %Row 13 Column 3 A(6*N5,2*N+1:3*N) = vecO ( : ) ; %Row 13 Column 6 A(6*N5,5*N+1:6*N) = vecOn{:)'; %Row 13 Column 7 A{6*N5,6*N+1) = (1/thermcond 1) ; %Row 14 Column 4 A(6*N4, 3*N+1:4*N) = vecl ( : ) ; %Row 15 Column 4 A{6*N3,3*N+1:4*N) = vecO ( : ) ;
PAGE 158
151 %Row 16 Column 6 A(6*N2, 5*N+1:6*N) = vecO ( : ) ; %Row 17 Column 1 A{6*N1,1: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:N2,2*N+1:3*N) = Iden (1 :N2 : ) ; %Row 2 Column 3 B{N1:2* (N2) ,2*N+1:3*N) = 2 *El(l :N2 : ) ; %Row 4 Column 6 B(3*N5:4* (N2) 5*N+1:6*N) = thermexp *Iden (1 :N2 : ) ; %Row 5 Column 6 B{4*N7:5* (N2) ,5*N+1:6*N) = (2*thermexp/depth) *E1{1 :N2 : ) ; %Row 11 Column 3 B(6*N7,2*N+1:3*N) = (omega ^2 Gamma) *vecO (:)' ; %Row 11 Column 7 B(6*N7,6*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) ;
PAGE 159
152 if abs (imag (evals (indexl) ) ) < le10 % 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+n2l) = vel2(2:n2); temp(l:nl) = tempi (:) ; temp (nl+1 :nl+n2l) = temp2(2:n2); % Normalize the velocity and temperature maxvel = max (abs (real (vel) ) ) ;
PAGE 160
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) ;
PAGE 161
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:N2,1:N) = 4 *E2 (1 : N2 : ) omega'"2 *Iden (1 :N2 : ) ; %Row 1 Column 2 A(1:N2,N+1:2*N) = 2 *E1 (1 : N2 : ) ; %Row 2 Column 2 A(N1:2* {N2) ,N+1:2*N) = 4 *E2 {1 :N2 : ) omega'^2 *Iden (1 :N2 : ) ; %Row 3 Column 1 A(2*N3:3* (N2) ,1:N) = Iden (1 :N2 : ) ; %Row 3 Column 3 A(2*N3:3* {N2) ,2*N+1:3*N) = 4 *E2 (1 :N2 : ) omega^2 *Iden (1 :N2 : ) ; %Row 4 Column 4 A(3*N5:4* (N2) 3*N+1:4*N) = visc/density .* {4/depth^2 *E2 (1 :N2 : ) omega^2 *Iden (1 :N2 : ) ) ; %Row 4 Column 5 A(3*N5:4* (N2) ,4*N+1:5*N) = 2 / {density*depth) El(l:N2,:); %Row 5 Column 5 A(4*N7:5* {N2) ,4*N+1:5*N) = (l/density) {4/depth''2 *E2 (1 :N2 : ) omega*2 *Iden(l :N2, : ) ) ; %Row 6 Column 4 A(5*N9:6* (N2) ,3*N+1:4*N) = (l/thermcond) *Iden {1 :N2 : ) ; %Row 6 Column 6 A(5*N9:6* (N2) 5*N+1:6*N) = thermdif f {4/depth^2 *E2 (1 :N2 : ) omega^2.*Iden(l:N2, : ) ) ,% Begin Boundary Conditions for A matrix **************************************** %Row 7 Column 1 A(6*N11,1:N) = vecOn{:)';
PAGE 162
155 %Row 8 Column 1 A(6*N10, 1:N) = vecln{:)'; %Row 9 Column 3 A(6*N9,2*N+1:3*N) = vecOn ( : ) ; %Row 10 Column 4 A(6*N8,3*N+1:4*N) = vecO ( : ) ; %Row 11 Column 4 A(6*N7,3*N+1:4*N) = vecl ( : ) ; %Row 12 Column 6 A{6*N6, 5*N+1:6*N) = vecO ( : ) ; %Row 13 Column 1 A(6*N5,1:N) = vecO { : ) ; %Row 14 Column 4 A(6*N4,3*N+1:4*N) = vecOn{:)'; %Row 15 Column 1 A(6*N3,1:N) = depth. vecl (:)' ; %Row 15 Column 4 A(6*N3,3*N+1:4*N) = vecln ( : ) ; %Row 16 Column 3 A(6*N2,2*N+1:3*N) = vecO ( : ) ; %Row 16 Column 6 A(6*N2, 5*N+1:6*N) = vecOn ( : ) ; %Row 17 Column 3 A(6*N1,2*N+1:3*N) = vecl ( : ) ; %Row 17 Column 6 A(6*N1, 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:N2,2*N+1:3*N) = Iden (1 : N2 : ) ; %Row 2 Column 3 B(N1:2* (N2) ,2*N+1:3*N) = 2 *E1 (1 :N2 : ) ;
PAGE 163
156 %Row 4 Column 6 B(3*N5:4* (N2) ,5*N+1:6*N) = themexp *Iden (1 :N2 : ) ; %Row 5 Column 6 B(4*N7:5* (N2) ,5*N+1:G*N) = {2*thermexp/depth) *E1(1 :N2 : ) ; %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) ) ) < le10 % 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) ;
PAGE 164
157 Ra2 = Ral (depth^4 thermexp) / (thermdiff kinvisc thermcond) ;
PAGE 165
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 (N2,N) Q2 = zeros (N2,N) Ql = 4.*E2 {l:N2, Q2 = 4.*E2 (l:N2, %Row 1 Column 1 A(1:N2,1:N) = Ql ; ) omega^2 .* Iden (1 :N2 : ) ; ) (omega^2*depth^2) .* Iden (1 :N2 : ) ; %Row 1 Column 2 A(1:N2,N+1:2*N) = 2 *E1 (1 :N2 : ) ; %Row 2 Column 2 A{N1:2* (N2) ,N+1:2*N) = Ql ; %Row 3 Column 1 A{2*N3 :3* (N2) ,1:N) = Iden (1 :N2 : ) ; %Row 3 Column 3 A(2*N3:3* (N2) ,2*N+1:3*N) = Ql ; %Row 4 Column 4 A(3*N5:4* (N2) ,3*N+1:4*N) = Q2 ; %Row 4 Column 5 A(3*N5:4* (N2) ,4*N+1:5*N) = 2 *E1 (1 :N2 : ) ; %Row 5 Column 5 A(4*N7:5* (N2) ,4*N+1:5*N) = Q2 ; %Row 6 Column 4 A(5*N9:6* (N2) ,3*N+1:4*N) = Iden (1 :N2 : ) ; %Row 6 Column 6 A(5*N9:6* (N2) ,5*N+1:6*N) = Q2 ; % Begin Boundary Conditions
PAGE 166
159 %Row 7 Column 1 A(6*N11,1:N) =vecOn(:)'; %Row 8 Column 1 A{6*N10,1:N) = vecln(:)'; %Row 9 Column 3 A{S*N9,2*N+1:3*N) = vecOn { : ) ; %Row 10 Column 4 A(6*N8,3*N+1:4*N) = vecO ( : ) ; %Row 11 Column 4 A(6*N7,3*N+1:4*N) = vecl { : ) ; %Row 12 Column 6 A{6*N6,5*N+1:6*N) = vecO ( : ) ; %Row 13 Column 1 A(6*N5,1:N) = vecO { : ) ; %Row 14 Column 4 A(6*N4,3*N+1:4*N) =vecOn(:)'; %Row 15 Column 1 A(6*N3,1:N) = vecl( : ) ; %Row 15 Column 4 A(e*N3,3*N+l:4*N) = (thermdif f /depth^2 ) .* vecln ( : ) ; %Row 16 Column 3 A(6*N2,2*N+1:3*N) = vecO ( : ) ; %Row 16 Column 6 A(6*N2, 5*N+1:6*N) = depth/thermcond. *vecOn ( : ) ; %Row 17 Column 3 A(6*N1,2*N+1:3*N) = vecl ( : ) ; %Row 17 Column 6 A(6*N1, 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:N2,2*N+1:3*N) = Iden (1 :N2 : ) ;
PAGE 167
160 %Row 2 Column 3 B(N1:2* (N2) ,2*N+1:3*N) = 2 *E1 (1 :N2 : ) ; %Row 4 Column 6 B{3*N5:4* {N2) ,5*N+1:6*N) = RaRat *Iden (1 :N2 : ) ; %Row 5 Column 6 B(4*N7:5* (N2) ,5*N+1:6*N) = 2*RaRat *E1(1 :N2 : ) ; %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) ) ) < le10 % 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) ) ;
PAGE 168
161 Ma = Gamma Ral; Tempdiff = (thermdiffl kinviscl Ral) / (gravity thermexpl depthl*3) ; Ra2 = Ral (depth^4 therraexp) / (thermdiff kinvisc thermcond) ;
PAGE 169
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:N4,1:N) = 16 *E4 (1 :N4 : ) (8*omega^2) *E2 (1 :N4 : ) + omega'^4 .* Iden{l:N4, : ) ; %Row 2 Column 1 A(N3:2*N6,1:N) = Iden ( 1 :N2 : ) ; %Row 2 Column 2 A{N3:2*Ne,N+l:2*N) = 4 *E2 (1 :N2 : ) omega''2 .* Iden (1 :N2 : ) ; % Begin Boundary Conditions ***************************************************** %Row 4 Column 2 A{2*N5,N+1:2*N) = 2 *vecl ( : ) + Biot *vecO ( : ) ; %Row 5 Column 1 A(2*N4,1:N) = vecO ( : ) ; %Row 6 Column 1 A(2*N3,1:N) = 4 .* vec2 ( : ) ; %Row 7 Column 1 A(2*N2,1:N) = vecOn{:)'; %Row 8 Column 1 A{2*N1,1:N) =vecln(:)'; %Row 9 Column 2 A(2*N,N+1:2*N) = vecOn ( : ) ; % Define the B matrix *********************************************************** %Row 1 Column 2 B{1:N4,N+1:2*N) = omega"2 .* Iden (1 :N4 : ) ; %Row S Column 2
PAGE 170
163 B(2*N3,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) ) ) < le10 % 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) ;
PAGE 171
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:N4,1:N) = 16 *E4 (1 :N4 : ) 8*omega'^2 .* E2{l:N4,:) + omega''4 .* Iden(l:N4, : ) ; %Row 1 Column 2 A(1:N4,N+1:2*N) = Iden (1 :N4 : ) ; %Row 2 Column 2 A(N3:2*N6,N+1:2*N) = 4 *E2 (1 :N2 : ) omega^2 .* Iden (1 :N2 : ) ; % Begin Boundary Conditions ***************************************************** %Row 4 Column 2 A(2*N5,N+1:2*N) = 2 *vecl( : ) + Biot *vecO ( : ) ; %Row 5 Column 1 A{2*N4,1:N) = vecO { :) ; %Row 6 Column 1 A{2*N3,1:N) = vec2 ( : ) ; %Row 7 Column 1 A{2*N2,1:N) = vecOn ( : ) ; %Row 8 Column 1 A(2*N1,1:N) = vecln(:)'; %Row 9 Column 2 A(2*N,N+1:2*N) =vecOn{:)'; % Define the B matrix *********************************************************** %Row 2 Column 1 B(N3 :2*N6, 1:N) = omega'^2 .* Iden (1 :N2 : ) ; %Row 4 Column 1 B(2*N5,1:N) = (2 omega^2 Gamma) .* vecl( : ) ;
PAGE 172
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) ) ) < le10 % 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) ;
PAGE 173
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:N4,1:N) = 16 *E4 (1 :N4 : ) (8*omega^2 ) *E2 (1 :N4 : ) + omega'^4 .* Iden(l:N4, : ) ; %Row 2 Column 1 A(N3:2*N6,1:N) = Iden (1 :N2 : ) ; %Row 2 Column 2 A{N3:2*N6,N+1:2*N) = 4 *E2 (1 :N2 : ) omega"2 .* Iden (1 :N2 : ) ; %Row 3 Column 3 A(2*N5:3*N10,2*N+1:3*N) = kinvisc (16/depth^4 *E4 (1 :N4 : ) (8/depth*2*omega^2) .*E2 (l:N4, :) +omega^4 .* Iden (1 :N4 : ) ) ; %Row 4 Column 3 A{3*N9:4*N12,2*N+1:3*N) = (l/thermcond) .* Iden (1 :N2 : ) ; %Row 4 Column 4 A(3*N9:4*N12,3*N+1:4*N) =thermdiff .* (4/depth"2 *E2 (1 :N2 : ) omega^2 .* Iden (1 :N2, : ) ) ; % Begin Boundary Conditions %Row 5 Column 1 A(4*N11,1:N) = 4 .* vec2 { : ) ; %Row 5 Column 3 A(4*N11,2*N+1:3*N) = (4 vise / depth''2) .* vec2n ( : ) ; %Row 6 Column 1 A(4*N10, 1:N) = depth .* vecl( : ) ; %Row 6 Column 3 A(4*N10,2*N+1:3*N) = vecln ( : ) ; %Row 7 Column 2 A(4*N9,N+1:2*N) = depth .* vecl( : ) ;
PAGE 174
167 %Row 7 Column 4 A(4*N9,3*N+1:4*N) = thermcond .* vecln ( : ) ; %Row 8 Column 2 A(4*N8,N+1:2*N) = vecO ( : ) ; %Row 8 Column 4 A(4*N8, 3*N+1:4*N) = vecOn ( : ) ; %Row 9 Column 1 A(4*N7,1:N) = vecO ( : ) ; %Row 10 Column 3 A(4*N6,2*N+1:3*N) = vecOn{:)'; %Row 11 Column 1 A{4*N5,1:N) = vecOn ( : ) ; %Row 12 Column 1 A{4*N4,1:N) = vecln (:)' ; %Row 13 Column 2 A(4*N3,N+1:2*N) =vecOn{:)'; %Row 14 Column 3 A(4*N2,2*N+1:3*N) = vecO ( : ) ; %Row 15 Column 3 A(4*N1, 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:N4,N+1:2*N) = omega^2 .* Iden (1 :N4 : ) ; %Row 3 Column 4 B(2*N5:3*N10,3*N+1:4*N) = (thermexp omega''2) .* Iden (1 :N4, : ) ; %Row 5 Column 2 B(4*N11,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 = ;
PAGE 175
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) ) ) < le10 % 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) ;
PAGE 176
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 ***********Â•****************** + *************************Â•**** %Row 1 Column 1 A{1:N4,1:N) = 16 *E4 (1 : N4 : ) ( 8*omega"2 ) *E2 ( 1 :N4 : ) +omega"4 .* Iden(l:N4, :) ; %Row 1 Column 2 A(1:N4,N+1:2*N) = Iden (1 :N4 : ) ; %Row 2 Column 2 A(N3:2*N6,N+1:2*N) = 4 *E2 (1 :N2 : ) omega"2 .* Iden (1 :N2 : ) ; %Row 3 Column 3 A{2*N5:3*N10,2*N+1:3*N) = kinvisc (16/depth*4 *E4 (1 : N4 : ) {8/depth^2*omega^2) .*E2 (l:N4, :) + omega^4 .* Iden (1 :N4 : ) ) ; %Row 3 Column 4 A(2*N5:3*N10,3*N+1:4*N) = ( l/thermcond) .* Iden(l:N4, :) ; %Row 4 Column 4 A(3*N9:4*N12,3*N+1:4*N) =thermdiff .* (4/depth"2 *E2 (1 :N2 : ) omega*2 .* Iden (1 :N2 : ) ) ; % Begin Boundary Conditions %Row 5 Column 1 A(4*N11,1:N) = depth^2 .* vec2 ( : ) ; %Row 5 Column 3 A(4*N11,2*N+1:3*N) = kinvisc .* vec2n ( : ) ; %Row 6 Column 1 A(4*N10, 1:N) = visc*depth .* vecl ( : ) ; %Row 6 Column 3 A(4*N10,2*N+1:3*N) = kinvisc .* vecln(:)'; %Row 7 Column 2 A(4*N9,N+1:2*N) = vecl { : ) ;
PAGE 177
[70 %Row 7 Column 4 A{4*N9,3*N+1:4*N) = thermdiff /depth .* vecln ( : ) ; %Row 8 Column 2 A(4*N8,N+1:2*N) = thermcond .* vecO ( : ) ; %Row 8 Column 4 A(4*N8,3*N+1:4*N) = thermdiff .* vecOn(:)'; %Row 9 Column 1 A(4*N7,1:N) = vecO ( :) ; %Row 10 Column 3 A(4*N6,2*N+1:3*N) = vecOn ( : ) ; %Row 11 Column 1 A(4*N5,1:N) = vecOn ( : ) ; %Row 12 Column 1 A(4*N4,1:N) = vecln { : ) ; %Row 13 Column 2 A{4*N3,N+1:2*N) = vecOn ( : ) ; %Row 14 Column 3 A(4*N2,2*N+1:3*N) = vecO ( : ) ; %Row 15 Column 3 A(4*N1,2*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(N3 :2*Ne,l:N) = omega'^2 .* Iden (1 :N2 : ) ; %Row 4 Column 3 B(3*N9:4*N12,2*N+1:3*N) = (thermexp omega^2) .* Iden (1 :N2, : ) ; %Row 7 Column 1 B(4*N9,1: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
PAGE 178
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) ) ) < le10 % 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) ;
PAGE 179
172 Case 16 % caselG.m is an mfile 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:lastml for n = 1: lastn omega = lambda (m+l, n) / AspectRatio;
PAGE 180
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 = :lastml 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 = ;
PAGE 181
174 indexl = ; ind.ex2 = ; % Find the Ra s for the unstable and slave modes for m = 0:lastml 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) Â• 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 )
PAGE 182
175 Case 17 % caselV.m is an mfile 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;
PAGE 183
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:lastml for n = l:lastn omega = lambda {m+l,n) / AspectRatio; % Calculate the linear problem cd d: \users\johnd\projects\nonlinl\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 = :lastml 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 / ]);
PAGE 184
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:lastml 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 ) ;
PAGE 185
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 )
PAGE 186
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 ( (omegaLastomegaFirst) /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
PAGE 187
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 ( Â• ) ; disp(['Data written to file fileName]) disp ( ) ; end % End of the program disp (' Program casel9 completed')
PAGE 188
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 ( (omegaLastomegaFirst) /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
PAGE 189
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')
PAGE 190
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
PAGE 191
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 CrossSectional View Liquid Insert (Material: Lucite Drawn by: Duane Johnson Last Revised: August 11, 1997
PAGE 192
185 Air Insert Air Insert CrossSectional 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
PAGE 193
186 Ciamp Drawing Notes: 1) Drawing is not to scale Clamp Material: Lucite Drawn by: Duane Johnson Last Revised: August 11,1 997
PAGE 194
187 Rough Sketch of Program / Initialization \ Information ) Form !R Program Form
PAGE 195
[88 Initialization Flow Chart Start Program Initialization Information Form Add, Edit & Delete Fluid Parameters Initialize Data Acquisition Board To Data In
PAGE 196
189 Data In Flow Chart Pause Statistics (Data Averaging) To Display From Initialization Information Form IR Program Form From h* Â— Control Decision
PAGE 197
190 Display Flow Chart From Data In Update Screen Variables Prepare XAxis Data Prepare YAxis Data To Control Â— Decision
PAGE 198
191 Control Decision Flow Chart Change Segment Number
PAGE 199
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
PAGE 200
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 Â• 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 \, <: XA)(is ) \ Data / Prepare YAxis 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 )
PAGE 201
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 MultiFluid 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., PerezGarcia, C, Jamond, C, and Pantaloni, J., Wavelength Selection in BenardMarangoni 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 TwoLayer Mantle, Earth Plan. Sci. Lett. 76, 193, (1985) Dauby, P. and Lebon, G., BenardMarangoni 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
PAGE 202
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 RayleighBenardMarangoni 1 Flows in SmallAspectRatio Containers, J. Fluid Mech., 243, 73 1 (1 992) i Echebarria, B., Krmpotic, D., and PerezGarcia, 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 NonLinear Stability Theory, SpringerVerlag, New York, 1965 Ellsworth, K. & Schubert, G., Numerical Models of Thermally and Mechanically Coupled TwoLayer 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 RayleighBenardMarangoni Instability: Comparison between Theory and Experiment, J. NonEquil. 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., BuoyancyDriven 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)
PAGE 203
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 InterfacialTensionDriven Convection near a CodimensionTwo 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 LiquidAir 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 SurfaceTensionDriven Bernard Convection, J. Fluid Mech., 167, 49 (1986) Koschmieder, E.L., and Prahl, S.A., SurfaceTensiondriven 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)
PAGE 204
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. & PerezGarcia, Dynamical Patterns in BenardMarangoni Convection in a Square Container, Phys. Rev. Lett. 70, 3892(1993) Operator's Manual for the Inframetrics Model 760: IR Imaging Radiometer, document #07137000 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 RayleighBenard Convection in a TwoLayer System of Immiscible Liquids, Trans. ASME 118, 366 (1996) Rasenat, S., Busse, F. H. and Rehberg, I., A Theoretical and Experimental Study of DoubleLayer Convection, J. Fluid Mech. 199, 519 (1989). Renardy, Y.Y., Pattern Formation for Oscillatory BulkMode 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)
PAGE 205
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 SurfaceTension 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 NonLinear 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 BuoyancyDriven Convection Â— 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 RayleighMarangoni Convection: Transitions in Flow Structures at the Interface, Proc. R. Soc. Lond. A, 451, 487(1995)
PAGE 206
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
PAGE 207
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
PAGE 208
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
xml version 1.0 encoding UTF8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E4GHMO0U4_7GBQ5C INGEST_TIME 20150106T22:00:52Z PACKAGE AA00026579_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES

