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Stabilization of Liquid Interfaces

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Title:
Stabilization of Liquid Interfaces
Creator:
UGUZ, ABDULLAH KEREM ( Author, Primary )
Copyright Date:
2008

Subjects

Subjects / Keywords:
Bond number ( jstor )
Critical points ( jstor )
End plates ( jstor )
Fluids ( jstor )
Geometry ( jstor )
Liquid bridges ( jstor )
Liquids ( jstor )
Mathematical variables ( jstor )
Physics ( jstor )
Viscosity ( jstor )

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Source Institution:
University of Florida
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University of Florida
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Copyright Abdullah Kerem Uguz. Permission granted to University of Florida to digitize and display this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
8/31/2006
Resource Identifier:
485041430 ( OCLC )

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Full Text










STABILIZATION OF LIQUID INTERFACES


By
ABDULLAH K(EREM UGUZ
















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


2006
































Copyright 2006

by
Abdullah K~erem Uiguz
















To my Mom and my Dad















ACKNOWLEDGMENTS

First of all, I would like to thank Professor Ranga Narayanan for his support

and advice. He has been both a mentor and a friend. He ahr-l- .- emphasizes the

importance of enjoying your work. Dr. Narali- .Il .Il is enthusiastic about his work

and this is the best motivation for a student. His dedication to teaching and his

philosophy has inspired me to be in academia. I would like to thank Nick Alvarez.

He started as an undergraduate student helping me with my experiments. Then,

he became co-author of my papers. The members of my PhD committee, Prof.

Oscar D. Crisalle, Prof. Loc Vu-Quoc, and Prof. Dmitry K~opelevich also deserve

my gratitude. Also, I would like to thank Prof. Alex Oron for accepting to be in

my defense. I have really enjoi-x & taking classes from Prof. Vu-Quoc, Prof. Crisalle

and Prof. Narang. Their teaching philosophies of seeing the big picture have deeply

influenced me. Many thanks go to my friends Ozgur Ozen and Berk Usta for their

friendship. I am lucky to be their colleague.

Many thanks go to Sinem Ozyurt for her constant support throughout my

graduate education. I thank her for ahr-l- .- being there when I need her. She is

very special for me.

I would like to thank my brother, Erdem Uiguz, who has ahr-l- .- been with

me, and has motivated me for my work. I would like to express my highest

appreciation for my parents and my brother for their love and support throughout

my educational career. It has been difficult for them and for me because of the

large distance. Thank you for your patience, encouragement and your moral

support .

I would like to thank the University of Florida for an Alumni Fellowship.

















TABLE OF CONTENTS

page

ACK(NOWLEDGMENTS ......... .. iv

LIST OF TABLES ......... . vi

LIST OF FIGURES ......... . .. vii

ABSTRACT ......... .. .. viii

CHAPTER

1 INTRODUCTION . ...... ... .. 1

1.1 Why Were the Rayleigh-Taylor Instability and Liquid Bridges Stud-
ied? .... .. .. ............... 2
1.2 Organization of the Thesis . .... 6

2 THE PHYSICS OF THE PROBLEMS AND THE LITERATURE RE-
VIEW ........ .... .......... 8

3 A MATHEMATICAL MODEL . ..... 16

3.1 The Nonlinear Equations . ..... .. 16
3.2 The Linear Model ......... .. 18

4 THE RAYLEIGH-TAYLOR INSTABILITY ... .. .. 21

4. 1 Determining The Critical Width in Rayleigh-Taylor Instability by
Rayleigh's Work Principle .. .. .. .. . . 21
4.2 A Simple Derivation For The Critical Width For The Rayleigh-Taylor
Instability and The Weakly Nonlinear Analysis of the Rayleigh-
Taylor Problem ..... ... ...... .. ........ 23
4.3 The Effect of the Geometry on the Critical Point in Rayleigh-Taylor
Instability: Rayleigh-Taylor Instability with Elliptical Interface .. 27
4.4 Linear and Weakly Nonlinear All ll-k- of the Effect of Shear on
Rayleigh-Taylor Instability . . . .. 32
4.4. 1 Instability in Open C'I .Ill., I Couette Flow .. .. .. 36
4.4.2 Rayleigh-Taylor Instability in Closed Flow .. .. .. 38
4.5 Summary ......... ... 58

5 THE STABILITY OF LIQUID BRIDGES .... .. 61










5.1 The Breakup Point of a Liquid Bridge by Rayleigh's Work Princi-
ple ...... ...... .. .. ... ..... 61
5.2 A Simple Derivation To Obtain the Dispersion Curve for a Liquid
Bridge via a Perturbation Calculation ... .. .. .. .. 6:3
5.3 The Effect of Geometry on the Stability of Liquid Bridges .. .. 67
5.3.1 The Stability of an Encapsulated Cylindrical Liquid Bridge
Subject to Off-Centering ... .. 67
5.3.1.1 Perturbed equations: el problem .. .. .. 68
5.3.1.2 Mapping front the centered to the off-centered liq-
uid bridge . .... .. 70
5.3.1.3 Determining a2(1 ........ 7
5.3.1.4 Determining a2(2 .... ... .. 75
5.3.1.5 Results front the analysis and discussion .. .. 79
5.3.2 An Experimental Study on the Instability of Elliptical Liq-
uid Bridges ........... ....... 82
5.3.2.1 Results on experiments with circular end plates 86
5.3.2.2 Results on experiments with elliptical end plates 88
5.4 Shear-induced stabilization of liquid bridges ... .. .. 90
5.4. 1 A Model for Scoping Calculations .. .. .. .. 92
5.4.2 Determining the Bond Number ... .. .. .. 97
5.4.3 The Experiment . ..... .. .. 98
5.4.3.1 The experimental setup .. .. .. 98
5.4.3.2 The experimental procedure .. .. .. .. 100
5.4.4 The Results of the Experiments .. .. .. 10:3

6 CONCLUSIONS AND RECOMMENDATIONS .. .. .. 109

APPENDIX

A THE PERTURBATION EQUATIONS AND THE MAPPING .. .. 112

B SITRFACE VARIABLES ........ ... .. 115

B.1 The IUnit Nornial Vector . ..... .. .. 115
B.2 The IUnit Tangent Vector . .... .. 116
B.:3 The Surface Speed ......... ... .. 116
B.4 The Mean Curvature ....... ... .. 117

C THE VOLITME LOST AND GAINED FOR A LIQUID JET WITH A
GIVEN PERIODIC PERTURBATION .... .. .. 119

D THE EFFECT OF INERTIA IN THE R AYLEIGH-TAYLOR AND LIQ-
ITID JET PROBLEMS ......... .. .. 121

REFERENCES ......... . .. .. 124

BIOGRAPHICAL SK(ETCH ......... .. .. 129
















LIST OF TABLES
Table page

5-1 Physical properties of chemicals. . .... 84

5-2 Mean experimental break-up lengths for cylindrical liquid bridges. .. 87

5-3 Mean experimental break-up lengths for elliptical liquid bridges. .. .. 88

5-4 The effect of the viscosities on the maximum vertical velocity along the
liquid bridge interface. ......... ... 93

5-5 The effect of the liquid bridge radius on the maximum vertical velocity
along the liquid bridge interface. . .... 95

















LIST OF FIGURES

Figure page

1-1 Liquid bridge photo .. ... ... 2

1-2 Interface between heavier colored water on top of lighter transparent de-
cane in a conical tube .. ... ... :3

1-:3 Shadowgraph image showing convection .... .. 5

2-1 Photograph illustrating the jet instability .... .. .. 9

2-2 Liquid jet with a given perturbation .... ... 9

2-3 Dispersion curve for the jet . ... .. .. 11

2-4 Liquid bridge photograph front one of our experiments .. .. .. 11

2-5 Cartoon illustrating floating zone method ... .. .. 1:3

4-1 Sketch of the physical problem depicting two ininiscible liquids with the
heavy one on top of the light one . .... .. 22

4-2 Sketch of the Rayleigh-Taylor problem for an elliptical geometry .. .. 27

4-3 Two ininiscible liquids with density stratification ... .. .. :34

4-4 Base state stream function for closed flow Rayleigh-Taylor problem .. 41

4-5 Base state velocity field for closed flow Rayleigh-Taylor problem .. .. 42

4-6 Dispersion curves for the closed flow Rayleigh-Taylor problem for Ca=
10 and Bo =5 ...... ...... ......... 45

4-7 The dispersion curve for the closed flow Rayleigh-Taylor showing multi-
ple nmaxinia and nmininia for Ca=20 and Bo=500 ... .. .. 46

4-8 The effect of the wall speed on the stability of shear-induced Rayleigh-
Taylor for Bo=50 ......... .. 47

4-9 The effect of Bo on the stability of shear-induced Rayleigh-Taylor for
Ca=20 .... ........ .......... 48

4-10 The neutral stability curve for the shear-induced flow where Ca=20 .. 49

4-11 The neutral stability curve for the shear-induced flow where Ca=20 .. 50










4-12 Bifurcation diagrams ......... .. 57

5-1 Volume of liquid with a given periodic perturbation .. .. .. .. 62

5-2 Centered and off-centered liquid bridges .... .. 68

5-:3 The cross-section of an off-centered liquid bridge ... .. .. 72

5-4 a2(O) and a"2 ) (multiplied byi their scale factors) versus the wavienumber
for p*/p 1 and RIf /Rf ) 2 . ..... '79

5-5 C'!s lily,.- in a2 2) (multiplied byv its scale factor) for small to intermediate
density ratios for scaled wavenumber (kRF ) of 0.5 and Rf~ /Rf 2 830

5-6 C.!s lII,.- in? (Te2 (multiplied by its scale factor) large density ratios for
scaled wavenumber of 0.5 and Rf /Rf -) 2 ... .. .. .. 81

5-7 Os1 .II,,.- of a2 )' (multip~lied by its scale factor) versus outer to inner ra.-
dius ratio RF /RF1~ for scaled wavenlumber of 0.5 and pJ*/p 1 . 81

5-8 Sketch of the experimental set-up for elliptical bridge .. .. .. 8:3

5-9 Cylindrical liquid bridge ......... ... 87

5-10 Large elliptical liquid bridge . ..... . 88

5-11 Small elliptical liquid bridge . ..... . 89

5-12 The schematic of the returning flow created in the presence of an encap-
sulant in the floatingf zone technique ..... .. 92

5-13 The effect of the height of the bridge on the maximum axial velocity along
the liquid/liquid interface ......... .. 95

5-14 The effect of the encapsulant's viscosity on the ratio of maximum speed
observed at the interface to the wall speed .... .. 96

5-15 Photograph of the experimental set-up ..... .. 99

5-16 A cartoon of a bridge bulging at the bottom .. .. .. .. 102

5-17 The effect of wall speed on the percentage increase in the breakup height
of the bridge for various injected volumes ... .. .. .. 104

5-18 The effect of the volume on the percentage increase in the breakup height
of the bridge ......... .. .. 105

5-19 The effect of the wall speed on the percentage increase in the breakup
height of the bridge for various Bond numbers .. .. . .. 106










5-20 The effect of the wall speed on the percentage increase in the breakup
height of the bridge for various Bond numbers and larger volume .. 107

C-1 The volume argument for a volume of liquid with a given perturbation .119

D-1 Sketch of the problem depicting a liquid on top of air .. .. .. .. .. 121















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

STABILIZATION OF LIQUID INTERFACES

By

Abdullah K~erent I~guz

August 2006

C'I I!1-: Ranganatha Narayanan
1\ajor Department: C'! I. InuI Engineering

This dissertation advances the understanding of the instability of interfaces

that occur in Rayleigh-Taylor (RT) and liquid bridge problems and investigates two

methods for delaying the onset of instability, namely, changing the geometry and

judiciously introducing fluid flow. In the RT instability, it is shown theoretically

that an elliptical shaped interface is more stable than a circular one of the same

area given that only axiyninetric disturbances are inflicted on the latter. In a

companion study on bridges, it is experimentally shown that a liquid bridge with

elliptical end plates is more stable than a companion circular bridge whose end

plates are of the same area as the ellipses. Using two different sizes of ellipses

whose senli-1! I iB .r axes were deviated froni the radii of the companion circles by

211' it was found that the elliptical bridge's breakup height was nearly ;:' longer

than that of the corresponding circular bridge.

Another way to stabilize interfaces is to judiciously use fluid flow. A com-

prehensive theoretical study on the RT problem involving both linear and weakly

nonlinear methods shows that mode interactions can delay the instability of an

erstwhile flat interface between two viscous fluids driven by moving walls. It is










shown that when the flow is driven under Couette conditions the breakup point

remains unchanged compared to the classical RT instability. However, in a closed

two-dintensional container, shearing the fluids enhances the stability provided a

flat interface is an allowable base solution. In addition, for a selected choice of

parameters, three different critical points can he obtained. Therefore, there is a

second window of stability for the shear-induced RT problem. A weakly nonlinear

analysis using a dominant balance method showed the problem has either a back-

ward or forward pitchfork hifurcation depending on the critical point around which

the analysis is performed. In an experimental study investigating the effect of

shear-driven flow in a liquid bridge, it was shown that a returning flow in both the

encapsulating liquid and the bridge would increase the stability of a non-vertical

bridge depending on the direction of shear by as much as 1"'















CHAPTER 1
INTRODUCTION

This dissertation involves the study of two interfacial instability problems

with the objectives of understanding the underlying physics behind the instabilities

and finding v- .va~ to delay them. The two problems are the liquid bridge and the

Rayleigh-Taylor instabilities. A liquid bridge is a volume of liquid suspended

between two solid supports. It can be held together without breaking owing

to surface tension forces. However, at some critical height the surface tension

effects are not strong enough to maintain the integrity of the bridge between the

supporting disks and the bridge becomes unstable and collapses. A depiction of a

stable and an undulatingf bridge is given in Figure 1-1.

The instability occurs because there is a phI i-off between pressure gradients

that are generated due to transverse curvature and those caused by longitudinal

curvature. As the spacing between the end plates increases, the latter becomes

weak, an imbalance occurs and the necking becomes more pronounced leading to

ultimate breakup. The Rayleigh-Taylor instability, on the other hand, is observed

when a light fluid underlies a heavy one, and the common interface becomes

unstable at some width. For large enough widths, the stabilizing surface potential

energy is insufficient to withstand the destabilizing gravitational energy. Such

an instability is depicted in Figure 1-2. A basic understanding of the instability

is needed if there is any hope of altering the stability limit by, ;?-,, changing the

geometry or by applying an outside force to get more stability. A fair question to

ask is to why these two instability problems are chosen is addressed next.












.1


Allllh


Figure 1-1. Liquid bridge photo a) Stable liquid bridge b) Unstable liquid bridge
at higher height.


1.1 Why Were the Rayleigh-Taylor Instability and Liquid Bridges
Studied?

These two problems are similar in many v-wsi~. They exhibit instability when a

control parameter, which can be the height for a liquid bridge, or the width of the

container for the Rayleigh-Taylor instability, is exceeded beyond a critical value.

At the critical point, the interface deflects and proceeds to complete breakup. In

both problems the instability can be understood without taking viscosity into

account. We will also see that the physics of both problems can be explained by

the Rayleigh work principle. Also, in both problems shear can be induced in the

base state causing flow, which in turn may alter the stability limit. In addition,

understanding the Rayleigh-Taylor instability from a theoretical standpoint in the

much simpler Cartesian coordinates is instructive for studying liquid bridges whose

models are complicated because of the cylindrical coordinates.

Both liquid bridge and Rayleigh-Taylor problems have numerous technological

applications. Liquid bridges occur, for example, in the production of single crystals

by the floating zone method [1, 2]. They occur in the form of flowing jets in the

encapsulated oil flow in pipelines [3]. In the melt spinning of fibers, liquid jets

emitting from nozzles accelerate and thin until they reach a steady state and





















::k M -- WI~il

Figure 1-2. Interface between heavier colored water on top of lighter transparent
decane in a conical tube a) Stable interface b) Unstable interface at
higher diameter.


then they break on account of instability. Besides such technological applications

in materials science, liquid bridges have importance in biomedical science. For

example, Grotherg [4] shows the vast scope of biofluid mechanics ranging from

the importance of the cell topology in the reopening of the pulmonary airr- 1-~

[5] to the occluding of oxygen resulting from the capillary instabilities [6]. In all

these studies, the mucus that closes the airr- .1-4 is represented by a liquid bridge

configuration.

The Rayleigh-Taylor instability also pIIli-e a role in a number of situations,

some natural, others technological. For example, the inability to obtain any capil-

lary rise in large diameter tubes is a result of the Rayleigh-Taylor instability. When

a fluid '?il i-;-r is heated from below, it becomes top heavy and the interface can

become unstable even before convection sets in due to huois ma1y. In eI-IUphi.--les

the adverse stratification of densities in the star's gravitational field is responsible

for the overturn of the heavy elements in collapsing stars [7]. Rayleigh-Taylor in-

stability is also observed in inertial confinement fusion (ICF), where it is necessary

to compress the fuel to a density much higher than that of a solid. Rayleigh-Taylor

instability occurs in two different occasions during this process [8].










It is the central objective of this study to see how to stabilize liquid interfaces

by applying an outside force or by changing the geometry of the system. For

that purpose, understanding the physics of the system, including the dissipation

of disturbances and the nature of the breakup of the interface as a function of

geometry is very important.

In applications of liquid bridges such as the floatingf zone technique, the molten

( i--r I1 is surrounded by another liquid to encapsulate the volatile components

and the presence of temperature gradients causes flow. Whether such flow can

cause stability or not is of interest, so in this study we shall consider the role of

shear in a liquid bridge problem. Another effect that is studied is the shape of the

supporting solid disks on the stability of liquid bridges. Most of the studies on

liquid bridges pertain to bridges of circular end plates. Physical arguments -II----- -1

that noncircular bridges ought to be more stable so this research also deals with the

stability of noncircular liquid bridges.

The current research is both experimental and theoretical in character. The

theoretical methods include linear stability analysis via perturbation calculations

and weakly nonlinear analysis via a dominant balance method. The experimental

methods involve photography of the interface shapes. The work on liquid bridges

will be experimental in nature on account of the difficulty in analyzing the problem

without resort to computations. The work on the Rayleigh-Taylor problem, on the

other hand, will be theoretical in nature on account of difficulty in obtaining clear

experiments.

All instability problems are characterized by models that contain nonlinear

equations. This must be true because instability by the very nature of its definition

means that a base state changes character and evolves into another state. The

fact that we have at least two states is indicative that we have nonlinearity in the

model. If the complete nonlinear problem could be solved, then all of the physics

























Figure 1-3: Shadowgraph image showing convection.

would become evident. However, solving nonlinear problems is by no means an

easy task and one endeavors to find the behavior by linearization of the model

about a known base state whose stability is in question. This local linearization is

sufficient to determine the necessary conditions for instability and in the absence

of a complete solution to the modeling equations it would seem beneficial to

obtain the conditions for the onset of the instability. To determine what happens

beyond the critical point requires the use of weakly nonlinear analysis. Once the

instability sets in, the interface created in the ordinary liquid bridge problem and

Rayleigh-Taylor configuration evolves to complete breakup. However, under some

conditions even this may not be true and we will see later in this dissertation

that a secondary state may be obtained if shear is applied. There are interfacial

instability problems that have been studied where patterns may be observed once

the instability sets in. An example of this is the Rayleigh-Bi~nard problem problem,

which is a problem of convective onset in a fluid that is heated from below. When

the temperature gradient across the 1 e. -r reaches a critical value, patterns are

predicted and in fact are also observed. Figure 1-3 is a photograph of such patterns

seen in an experiment. The fact that steady patterns are predicted and observed

implies a sort of "saturation" of solutions that might be expected in a weakly










nonlinear analysis, weak in the sense that the >.1, ll-h- is confined to regions close

to the onset of the instability. Contrast this behavior with that expected of the

common Rayleigh-Taylor problem discussed earlier. In this problem the onset of

the instability leads to breakup and no saturation of solutions may be expected.

All this will become important in our discussion of this problem later on.

1.2 Organization of the Thesis

The rest of this thesis pertains to both experimental and theoretical aspects of

problems in Rayleigh-Taylor instability and liquid bridges. As stated, our goal is to

understand the reasons underlying these instabilities, to predict them and finally to

try to delay them.

C'!s Ilter 2 outlines the physics of the instability for both problems, namely

Rayleigh-Taylor and liquid bridges. This chapter includes a short discussion of

liquid jets because a preliminary study of liquid jets forms the basis for the study

of liquid bridges. In other words most of the physics pertaining to liquid bridges

can he understood more easily by studying liquid jets. A general literature review

and applications are also given in this chapter.

('!, Ilter 3 discusses the governing equations along with boundary and interface

equations in their general forms. The theoretical methods required to solve these

equations is also presented in this chapter.

C'!s Ilter 4 focuses on the Rayleigh-Taylor instability. In the first section, the

critical point is found using Rayleigh's work principle. Then, the same result is

obtained by a perturbation calculation. This is followed by a calculation that shows

the effect of changing the geometry on the stability by considering instability in

an elliptical interface via a perturbation calculation. The last section presents the

shear-introduced stabilization of the Rayleigh-Taylor problem where a theory is

advanced. The dispersion curves are plotted by using linear stability analysis while

the types of hifurcations are determined via a weakly nonlinear analysis.










C'!s Ilter 5, which deals with bridges, is organized in a manner similar to the

previous chapter. First, the critical point is determined using Rayleigh's work

principle. Then, a perturbation calculation is presented that obtains the same

result. This is followed by a calculation where the effect of off-centering a liquid

bridge with respect to its surrounding liquid on the stability of the liquid bridge

is studied. While the idea of off-centering seems peripheral to our objectives it

does introduce an imperfection and is important because we must make sure in

bridge experiments that this imperfection has little if any consequence. In addition

this configuration is an idealization of the fluid configuration that appears in the

floating zone crystal growth technique. The theoretical method to investigate

the off-centering problem involves the use of an energy method. The details of

the derivation, and the physical explanation of the results are emphasized in this

chapter. Thereafter this chapter contains the details and results of two series

of experiments. In the first series, we investigate the effect of the geometry via

the stability of elliptical liquid bridges. A physical explanation of the effect of

changing the end plates of a liquid bridge from circles to ellipses on the stability

of liquid bridges is given through the dissipation of disturbances. The breakup

point of elliptical liquid bridges is then determined by means of experiments. The

second series deals with the effect of shear on the stability of liquid bridges. The

experiments show the stabilizing effect of returning flow in a liquid bridge on its

stability and are assisted by rough scoping calculations on the base state.

('!s Ilter 6 is a general conclusion and presents a scope for a future study.















CHAPTER 2
THE PHYSICS OF THE PROBLEMS AND THE LITERATURE REVIEW

The purpose of this chapter is to familiarize the reader with the basic physics

and to provide a brief overview of the literature. We know from the previous

chapter that both liquid bridge and Rayleigh-Taylor problems may become

unstable. Here, we will give the details of the instability mechanisms. We start

with a discussion of liquid jets because it serves as a precursor to the study of

liquid bridges.

A liquid jet forms when it ejects from a nozzle as in ink-jet printing and

agricultural sprays. Such jets to some approximation are cylindrical in shape.

However, a cylindrical body of liquid in uniform motion or at rest does not

remain cylindrical for long and left to itself, spontaneously undulates and breaks

up. A picture of such a body of liquid is depicted in Figure 2-1. Given the fact

that a spherical body of liquid upon perturbation returns to its spherical shape

and a body of liquid in a rectangular trough also returns to its original planar

configuration we might wonder why a cylindrical volume of liquid behaves as

depicted in the picture leading to necking and breakup.

The physics of the instability can be explained by introducing Figure 2-2,

which depicts a volume of liquid with a perturbation imposed upon it. If viewed

from the ends as in Figure 2-2(a), the pressure in the neck exceeds the pressure in

the bulge and the thread gets thinner at the neck. This is the transverse curvature

effect. It reminds us of the fact that the pressure in small diameter bubbles is

greater than the pressure in large diameter bubbles. On the other hand if viewed

from the perspective of a front elevation as in Figure 2-2(b), the pressure under

a crest is larger than the pressure under the trough or neck and consequently,


























Figure 2-1. Photograph illustrating the jet instability. Reprinted from Journal of
Colloid Science, vol. 17, F. D. Rumscheidt and S. G. Mason, "Break-up
of stationary liquid threads," pp. 260-269, 1962, with permission from
Elsevier.












Figure 2-2. Liquid jet with a given perturbation a) Transverse curvature b) Longi-
tudinal curvature (Adapted from [10]).


the liquid moves towards the neck restoring the stability. This is the longitudinal

curvature effect. The longer the wavelength the weaker is this stabilizing effect.

The critical point is attained when there is a balance between these offsetting

curvatures.

The breakup of liquid jets has been extensively studied, both experimentally

and theoretically. Such studies can be tracked back to Savart's [11] experiments

and Plateau's observations [12], which led Plateau to study capillary instability.

Theoretical analysis had started with Rayleigh [13, 14] for an inviscid jet injected









into air. Neglecting the effects of the ambient air, Rayleigh showed through a linear

stability analysis that all wavelengths of disturbances exceeding the circumference

of the jet at rest would be unstable. He was also able to determine that one of

the modes had to grow faster. Rayleigh [15] conducted some experiments on the

breakup of jets and observed that the drops, which form after the breakup, were

not uniform. He attributed this nonuniformity to the presence of harmonics in

the tuning forks he used to sound the jet and create the disturbances. The effect

of viscosity was also considered by Rayleigh [16] for the viscosity dominant case.

The general case and the theory on liquid jets is summarized and extended in

several directions by C'I .!1. .l .ekhar [17]. The experimental work by Donnelly

and Glaberson [18] was in good agreement with ('I! .1..11~ I-ekhar's theory as

seen in Figure 2-3. Here, a dimensionless growth constant is plotted against a

dimensionless wave number, x. The critical point is reached when the dimensionless

wave number is equal to unity. In their experiments, Donnelly and Glaberson [18]

also saw the sort of nonuniformity of the drops that Rayleigh observed. Lafrance

[19] attributed this phenomenon to the nonlinearity. Through his calculation, he

was able to match the experimental data for early times. Alansour and Lundgren

[20] extended the calculation for large times.

In some applications, the jet is surrounded by another liquid as in the oil

flow in pipelines where an internal oil core is surrounded by an annular region of

water. In this regard, Tomotika [21] extended the Rayleigh stability to a viscous

cylindrical jet surrounded by another viscous liquid. A more general problem

was solved later using numerical methods hv Meister and Scheele [22] and the

reader is referred to the recent book by Lin [23] for an overview of the phenomena

of jet breakup. Although the study of liquid jets started more than a century

ago, this topic is still relevant due to applications in modern technology such as

nanotechnology [24].






















x = 27rB/A

Figure 2-3. Dispersion curve for the jet. The solid line represents C'I 1...4 I-lekhar's
theory [17]. Reprinted from Proceedings of the Royal Society of London
Series A mathematical and Physical Sciences, vol. 290, R. J. Donnelly
and W. Glaberson, "Experiments on capillary instability of a liquid
jet," pp. 547-556, 1966, with permission from the Royal Society.


When a liquid jet is confined between two solid supports a liquid bridge is

obtained as in Figure 2-4. This liquid bridge can attain a cvlindrical configuration

if it is surrounded hv another fluid of the same density.


Top disk

Liquid
Bri dge

Surrounding
Liquid



Bottorn disk

Figure 2-4: Liquid bridge photograph from one of our experiments.


Liquid bridges have been studied as far back as Plateau [12] who showed

theoretically that in a gravity-free environment, the length to radius ratio of a

cylindrical liquid bridge at breakup is 27r. This instability takes place because

of a competition between the stabilizing effect of longitudinal curvature and










destabilizing effect of transverse curvature as in the liquid jets. However, while

the physics of the instability of cylindrical jets and bridges are similar there are

subtle differences between these two configurations. First, there is no natural

control parameter when studying the instability of jets while the bridge does come

equipped with one; it is the length to radius ratio. Second, there is no mode with a

maximum growth rate in the liquid bridge problem.

To obtain a cylindrical configuration of a liquid bridge requires a gravity-free

environment. There are various v- .1-< to decrease the effect of the gravity during

an experiment. These include going to outer space, using density-matched liquids,

or using small liquid bridge radii. The effect of gravity is represented by the Bond

number, Bo, which is the ratio of gravitational effects to the effect of surface
gApR2
tension and is given by Bo = ;; where g is the constant of gravitational

acceleration, Ap is the absolute density difference between the inner and the outer

liquid, R is the radius and y the interfacial tension. Small radii can therefore cause

a decrease in the effect of gravity or the density mismatch. It might he noted that

while the Plateau limit was obtained for a gravity free case, instability limits for

non zero Bond numbers and for a variety of input liquid volumes have also been

calculated [25].

Liquid bridges have often been investigated for their importance in tech-

nological applications, such as in the floating zone method for
semi-conductors [1, 2], for their natural occurrence such as in lung airr- .1-< [4] and

for scientific curiosity [25, 26]. Liquid bridges, as they appear in
applications, are usually encapsulated by another liquid to control the escape of

volatile constituents. The floating zone method is used to produce high-resistivity

single-< t s--r I1 silicon and provides a crucible-free
technique, a molten zone, which is depicted in Figure 2-5, is created between a

pcl-l i --r I11;1,.- feed rod and a monol 1 ,--r I11;1,-- seed rod. The heaters are translated















H \\5
Heaters

SMolten zone




Monocrystalline
seed rod




Figure 2-5: Cartoon illustrating floating zone method.


uniformly thereby melting and 1. I i--r I11; .;~!_ a substance into a more desirable

state. The
stable molten zones or liquid bridges. Gravity is the 1!! li.r~ problem in the stability

of the melt. On earth, because of the hydrostatic pressure, the melt zone has to be

small, causing small crystals. In the case of GaSb for example, a material that is

used in electronic devices, the crystal that can be obtained is about 7.5 mm [28].

The maximum stable height of the molten zone is determined by gravity. However,

with the advent in microgravity research, it has been possible to obtain larger

liquid zones. It has been possible to grow GaAs crystals of 20 mm diameter by the

floating zone technique during the German Spacelab mission D2 in 1993 [29].

Apart from gravity, the temperature gradient strongly influences the shape

and stability of the crystal. The thermocapillary convection in the presence of an

encapsulant generates a shear flow and this shear flow has an effect on the float

zone or bridge stability. Our interest lies in the stability of the zone in the presence

of shear flow. A recirculating pattern appears upon shear-induced motion and the

effect of this type of shear flow on the bridge stability is a question of interest. The









focus of the research is on the enhancement of the stability of these bridges by

suitably changing the geometry of the end plates or by imposing shear.

Many satellite questions crop up in determining the stability of the liquid zone

in the presence of a closed encapsulant: What is the role of the viscosity on the

stability of the bridge? What is the role of the centering of the bridge? Do off-

center bridges help to stabilize the bridge itself? We will answer these questions in



The second problem of interest of this research is Rayleigh-Taylor instability.

It is well known that if a light fluid underlies a heavy one, the common interface

becomes unstable when the width of the interface increases beyond a critical

value. The instability is caused by an imbalance between the gravitational and the

surface potential energies. The latter ahr-l- .- increases upon perturbation and its

magnitude depends on the interfacial tension. This problem was first investigated

by Rayleigh [30] and then by Taylor [31]. If the fluids are incompressible and have

uniform densities, the thicknesses of the fluid 1.>. ris and the viscosities pti-li no

role in determining the critical width, we, which is given by we = x -
g[ p p*]'
Here, y is the surface tension, g is the gravitational constant, and p and p* are the

densities of the heavy and light fluids respectively. The nature of the bifurcation is

a backward pitchfork, i.e., when the instability initiates, it progresses to complete

breakup.

The interest in studying the stability of a dense liquid lying on top of a light

liquid continues because of its applications in other problems. For example, Voiltz

et al. [32] applied the idea of Rayleigh-Taylor instability to study the interface

between glycerin and glycerin-sand in a closed Hele-Shaw like cell. Another

different example of Rayleigh-Taylor instability is seen when miscible liquids have

been studied either to examine the stability of front moving problems in reaction

diffusion systems [33] or to understand the dynamics of the mixing zone in the










nonlinear regime [34]. In this research, we are interested on the effect of geometry

and on shear on the stability of the interface in a Rayleigh-Taylor configuration.

The equations that represent both instability problems with corresponding

boundary and interface conditions are presented in the next section along with the

methods to solve these equations.















CHAPTER 3
A MATHEMATICAL MODEL

This chapter includes the equations used to analyze both instability problems

and are given in vector form so that no special coordinate system need be chosen.

They can then he adapted to the specific problem of interest. The differences

between the problems and further assumptions, which will simplify the governing

equations, will be pointed out as each problem is studied.

In the first chapter, we pointed out that the instabilities are related to the

nonlinearities in the modeling equations. In this chapter we will observe that the

modeling equations are nonlinear because the interface position is coupled to the

fluid motion and the two depend upon each other.

3.1 The Nonlinear Equations

In both problems the physical system consists of two immiscible, non-reactive

liquids. The fluids are considered to have constant density and viscosity. Therefore,

the motion of each fluid is governed by the Navier-Stokes equation, which holds at

any point in the domain and boundary and is given by


p + V = -VP + pg + pVa- 3

Here v and P are the dimensionless velocity and pressure fields, g is the

gravitational constant, p and and p are the density and viscosity of the fluid

respectively. A similar equation for the second phase also holds. Alass conservation

in each phase is governed by the continuity equations. For each of the phases, it is


V F= (3-2)










Equations 3-1 and 3-2 represent a system of four equations in four unknowns,

these being the three components of the velocity and the pressure. We postpone

the scaling of the equations as the scales depend on the physical system of interest.

Depending on the dimensionless groups that arise, several simplifications can be

made all of which will be made later for each problem.

We continue with the modeling equations. All walls are considered to be

impermeable, therefore, v' 6= 0 holds. Here, n is the unit outward normal.

The no-slip condition applies along the walls, and gives rise to v'- t = 0 holds.

Here, t is the unit tangent vector.

At the interface, the mass balance equation is given by


p (v'- u) R=0= p* (v"- u)- 6 (3-3)


In the above equation u represents the surface speed. This equation yields two

interface conditions as there is no phase-change at the interface. Note that the

asterisk denotes the second phase.

At the interface, the tangential components of velocities of both fluids are

equal to each other, i.e.,

v t= v; t~ (3-4)

The interfacial tension at the interface comes into the picture through the force

balance, which satisfies


PF-6+ Vjf i+[Vi 6 B1]]n -6ir- P*i- p' [V17 +[Vvil -= ;172H 35

where y is the interfacial tension and 2H is the surface mean curvature. Observe

that as the direction of the normal determines the sign of the right hand side, we

don't want to specify its sign yet. The reader is referred to Appendix B for the

derivation of the surface variables in Cartesian and cylindrical coordinate systems.










The tangential and the normal stress balances are obtained by taking the dot

product of Equation 3-5 with the unit tangent and normal vectors respectively.

Finally, the volumes of both liquids must be fixed, i.e.,


d V = Vo(3-6)


where Vo is the original volume of one of the liquids. Equation 3-6 implies that

a given perturbation to the liquids does not change their volumes. This volume

constraint is the last condition needed to close the problem.

As we mentioned, the equations are nonlinear. The first nonlinearity is

observed in the domain equation because of the & Vv' term. However, in most

of the problems we study, as we will see in the following section, the base state

is quiescent and this term is usually not needed. The main nonlinearity comes

from the fact that the interface position depends on the fluid motion and the fluid

motion depends on the position of the interface. This nonlinearity is seen vividly

in the normal stress balance at the interface for it is an equation for the interface

position. To investigate the instability arising from small disturbances we move on

to the linearization of the equations.

3.2 The Linear Model

As our interest is primarily in the onset of instability, it is sufficient to

analyze a linearized model where the linearization is done about a base state. The

importance of linearization calls for an explanation.

The instability arises when a system, which was in equilibrium, is driven

away from the equilibrium state when small disturbances are imposed upon it

and when a control parameter exceeds a critical value. For example in the liquid

bridge problem, the control parameter may be the length of the bridge of a given

radius or it may be the width of the container in the Rayleigh-Taylor problem.

An equilibrium system is said to be stable if all disturbances imposed upon it










damp out over time and said to be unstable when they grow in time. Now if the

system becomes unstable to infinitesimal perturbations at some critical value of

the control parameter it is unconditionally unstable. It is crucial to note that

the disturbances are taken to be small for if a state is unstable to infinitesimal

disturbances it must be unstable to all disturbances. Also, this assumption

leads to the local linearization of the system. The theoretical approach that is

taken when studying the instability of the physical system is therefore to impose

infinitesimal disturbances on the base state and to linearize the nonlinear equations

describing the system around this base state. It should be pointed out that the

base state is .ll.-- li--< a solution to the nonlinear equations and often it might seem

defeating to look for a base state if it means solving these nonlinear equations.

However, in practice for a large class of problems the base state is seen almost

by inspection or by guessing it. For example, for a stationary cylindrical liquid

bridge in zero gravity, it is obvious that the base state is the quiescent state with a

vertical interface. On the other hand, for some other problems, one might need to

determine the flow profile in the base state as seen in the shear-induced Rayleigh-

Taylor problem. Often, we try to simplify the governing equations by making

assumptions such as creeping flow or an inviscid liquid. These assumptions are

emploi-e d if there is no loss of generality in the physics that we are interested. Most

of the time these simplifications can be introduced after the nonlinear equations are

made dimensionless.

Calling the base state variable for velocity, v'o, and indicating the amplitude of

the perturbation by e, the velocity and all dependent variables can be expanded as


v& Fo+e 01+zi +- (3-7)


Here zz is the mapping from the current state to the base state at first order. Its

meaning is explained in the Appendix A and, at the interface, the mapping at this










order is denoted by Z1, a variable, which needs to be determined during the course

of the calculation. Note that the subscripts represent the order of the expansion,

e.g. the base state variables are represented hv a subscript zero. We can further

expand vl and other subscript 'one' variables using a normal mode expansion.

Consequently, the time and the spatial dependencies of the perturbed variables are

separated as



where o- is the inverse time constant also known as the growth or decay constant.

The critical point is attained when the real part of o- vanishes.

We will discuss Rayleigh-Taylor instability in the next chapter and apply the

model developed in this chapter to this problem.















CHAPTER 4
THE R AYLEIGH-TAYLOR INSTABILITY

In this chapter, the instability of a flat interface between two immiscible fluids

where the light fluid underlies the heavy one is studied. The chapter is composed of

four sections. In the first section, we will employ Rayleigh's work principle to find

the critical width, introduced in ChI Ilpter 2, which is given by I, = Tr _.

In the second section, we obtain the same result by a perturbation calculation,

with a companion nonlinear analysis. The linear calculation is used in the third

section where a similar perturbation calculation in conjunction with another type of

perturbation is used to study the effect of a slightly deviated circular cross section

in the form of an elliptical cross section on the stability point. In the last section

we study the effect of shear on the Rayleigh-Taylor (RT) instability with a linear

and nonlinear analysis.

4.1 Determining The Critical Width in Rayleigh-Taylor Instability by
Rayleigh's Work Principle

The physical problem is sketched in Figure 4-1. A heavy fluid of density p lies

above a light fluid of density p* in a container of width w. We will make use of the

Rayleigh work principle as adapted from Johns and Narayanan [10] to determine

the critical width at which the common interface becomes unstable.

According to the Rayleigh work principle the stability of a system to a given

disturbance is related to the change of energy of the system where the total energy

of the system is the sum of gravitational and surface potential energies. The

change in the latter can he determined directly from the change in the surface area

multiplied by its surface tension [35]. Consequently, the critical or neutral point

is attained when there is no change in the total energy of the system for a given









w p


z=L




z = 0



z = -L


P





P Z(x)


1


Figure 4-1. Sketch of the physical problem depicting two immiscible liquids with
the heavy one on top of the light one.

disturbance. To set these thoughts to a calculation, let the displacement be


xz =Z(X) = e cs(kx)


(4-1)


where a represents the amplitude of the disturbance, assumed to be small, and k is

the wave number given by nx/lw, where n = 1, 2, The surface area is given by


/= drsd


(4-2)


(4-3)


Note that the system is in two-dimensions and the above equation is in fact the

energy per unit depth. Using Z, = -ek sin (kx), Equation 4-3 becomes


y4e2k2 (4


dz 1 dz
where ds is the arc length, given by de = 1 dx a 1 +' dx.

To order 62, the change in the potential energy can be written as


7 1 + Z dx 7 dx
0 0










The change in the gravitational potential energy per unit depth is given by
w L w Z w L 0

p~i.7 7 g dz x i .1 .;: -p* g dzd (4-5)
0 Z 0 -L* 0 0 0 -L*

Substituting the expression for Z, simplifies the above equation to

62~~~ _p CS k)d *cS k)d g [p p*] 62 _46)
0 0

The total energy change is therefore the sum of the energies given in Equations 4-4

and 4-6, i.e.



The critical point is attained when there is no change in the energy. Substitut-

ing k = x/lw into Equation 4-7, the critical width is obtained as


g[ p p*]

For all widths smaller than this, the system is stable. It is noteworthy that the

depths of the liquids phy? no role in determining the critical width.

In the next section, the same result is obtained by a perturbation calculation

and a weakly nonlinear analysis follows.

4.2 A Simple Derivation For The Critical Width For The
Rayleigh-Taylor Instability and The Weakly Nonlinear Analysis of
the Rayleigh-Taylor Problem

A simple perturbation calculation is used to determine the critical width

at which a heavy liquid on top of air becomes unstable and a weakly nonlinear

analysis is performed to determine the bifurcation type.

The physical problem is sketched in Figure 4-1. The bottom fluid in this

calculation is taken as air. The liquid is assumed to be inviscid.










The Euler and continuity equations are


pv' Vv'= -VP + py


(4-9)


and


(4-10)


V &= 0


These domain equations will be solved subject to the force balance and no

mass flow at the interface conditions given in ('! .pter 3, namely,


(4-11)


P = y2H


and


8 &= 8


(4-12)


The base state is assumed to be stationary. To investigate the stability of the

base state, linear stability analysis described in ('! .pter 3 is emploi- II For the

perturbed problem, the equation of motion and the continuity equation results in


V2 1 = 0


(4-13)


The walls are impermeable to flow, as a result the normal component of the

velocity is zero, or in terms of pressure we can write


n'o VPI = 0


(4-14)


Free end conditions are chosen for the contact of the liquid with the solid

sidewalls, i.e.,


dZ1
= 0


(4-15)


Therefore, each variable can be expanded as a cosine function in the horizontal

direction, e.g., Z1 = Z1 cos (kx) where k = ax/w. From the no-flow condition we get









P1 as a constant. Finally, the normal stress balance reads as

dPo d2Z1
Pi + Z1 Y (4-16)
dz dX2

Using the consta~nt-volume requirement, wYhich states f Zid = 0, the per-
turbedl pressuree, whichl was1 already found to) be ai constant, is determllined to be(

zero. Also, Z1 is found as A cos (kx). The critical point is determined by rewriting

Equation 4-16 as

[-pg + k2] Zl= 0 (4-17)

TIhe square of the critical wave number isiP = G'. Substitutinlg k = x/w,' thle

critical width is obtained as

we = xTI (4-18)

which is same as Equation 4-8. Now, our aim is to find what happens when the

critical point is advanced by a small amount as G = G, + 62. The responses of the

variables to this change in the critical point are given as


Z = E"Z3 (4-19)


Before moving to the weakly nonlinear analysis, let's rewrite the domain

equation as
p~ 1
-v Vv' --VP Gk (4-20)

When the expansions are substituted into the nonlinear equations, to the

lowest order in e, the base state problem, to the first order, the eigenvalue problem

where the critical point is determined, are recovered. The second order domain

equation becomes

0 = 1 VP2 2k (4-21)









Both the domain equation and the no-mass transfer condition at the interface

gives
1 dP2
0 =2 (4-22)

Hence, P2 is a conStant. The normal stress balance at this order is

do d2Z2
P2 + Z2 = ] (4-23)
dr dXI2

The pressure, which is a constant, turns out to be equal to zero by using the

constant volume requirement. Therefore Z2 is found as B cos (kxr). To determine

the value of A, hence the type of the hifurcation, the third order equations are

written. The domain equation is
dP3
= (4-24)

P3 turns out to be a constant as in the previous orders. The normal stress balance

at the third order is


d~o dP2 d2Z:3 d2Zi dZ1
P3 + Z3a + 3Zi ] 9 (-5


Observe that at this order there is a contribution to the pressure from the

second order and the denominator of the curvature also shows its signature at

this order. P3 turns out to be equal to zero as in the previous orders. Solvability

condition gives


t- 64 4 CO2 ..._9 4 COS2 (kir) sin2l (k-)i d O =I 0(4-2)
0 0

which can he simplified to

:3 4"~k4 = 0 (4-27)

As ,42 1S negative, G needs to be written as G = G, E2 which yields a positive

42. Therefore, the hifurcation type is a backward pitchfork.










4.3 The Effect of the Geometry on the Critical Point in
Rayleigh-Taylor Instability: Rayleigh-Taylor Instability with
Elliptical Interface

The breakup point of the RT instability with an elliptical interface is compared

to the RT instability with a circular interface. An enhancement in the stability is

obtained theoretically. It is assumed that the circular cross section will be subject

to only axisymmetric disturbances. The physical argument for the enhanced

stability is related to the dissipation of the disturbances. In a circular geometry,

this is achieved by radial dissipation. In an elliptical geometry dissipation can also

occur azimuthally.

The physical problem is sketched in Figure 4-2. Observe that the radial

position depends on the azimuthal angle.


z-L







z =-L*


Figure 4-2: Sketch of the Rayleigh-Taylor problem for an elliptical geometry.


The modeling equations determining the fate of a disturbance are introduced

in C'!s Ilter 3. In this problem, we are considering inviscid liquids and the base state

is a quiescent state where the interface is flat. Therefore the nonlinear equations

have at least one simple solution. It is


flo = O, Po -, ur: vS = and Po* = -p*gz (4-28)


and Zo = 0. We are interested in the stability of this base state to small distur-

bances. For that purpose we turn to perturbed equations. The interface position









can be expanded as


z = Z (r, 8, t, e) = Zo + eZI + e~22 + (4-29)


To first order upon perturbation, the equations of motion and continuity are


pf = VP~ and V i = (430J)

in the region Z (r, 8, t, e) < z < L. Combining the two equations we get


V2 1 = 0 (4-31)


with similar equation for the '*' fluid. The corresponding boundary conditions

are also written in the perturbed form. The no-flow condition at the sidewalls is

written as

1So vi = 0 = 1So v~ (4-32)

which is valid at r = R (0). Before introducing the remaining boundary conditions,

we want to draw the attention of the reader to this boundary condition. The

equation is written at the boundary, which depends on the azimuthal angle. This

is an inconvenient geometry. Therefore, to be able to carry out the calculation in

a more convenient geometry, we want to use perturbation theory and write the

equations at the reference state, which has a circular cross section.

The objective is to show that the RT problem with elliptical interface is more

stable than a companion RT problem where the interface is circular. The area

of the ellipse is assumed to be the same as that of the circle. Also, the ellipse

is assumed to deviate from the circle by a small amount so that a perturbation

calculation can be used. As the ellipse is considered as a perturbation of the ellipse,

first the mapping obtaining an ellipse from a circle needs to be determined.

Assume that the ellipse is deviated from the circle by a small amount 6 so that

the semi-major axis "a" of the ellipse is defined as a = R(O) [1 + 6] where is the









radius of the circle from which the ellipse is deviated. Then, the semi-minor axis

"b" of the ellipse is calculated by keeping the areas to be the same, i.e.,


xrR(of2 = wab


leading to

b=~ R(O) [1-6+62'

Observe that the surface position of the ellipse can be expanded in powers of



R = R(O) + 6R 1) + ~62 p(2) 43

The mappingfs R1 and R2 can be found using the equation for ellipse, which is

given by
X2 72
+= 1 (4-34)

Substituting the definitions for x, and y, which are R cos (0) and R sin (0),

respectively, also making use of the expansions for a, b, and R, one gets the

mappmngs as

R 1) = R(O) cos (20) a (4-35)

to first order in 6 and


R(2) __ p()I CS(20) + cos(4)436


to second order in 6.

The geometry of the physical system is determined through a perturbation

calculation. Now, we can return to our perturbation calculation.

The no-flow boundary conditions at the reference interface, i.e., z = 0, and at

the top wall, i.e., z = H, for the perturbed pressure can be written as

SPi
= 0 (4-37)
8z









Therefore P1 is a constant, which is found at each order in 6 using constant-volume

requirement. At the outer wall, the contact angle condition reads as

8Z1 1 8R8Z1
= (4-38)
Br R2 de d

The normal stress balance at the interface is


Pi + Zi z = [V2Zi] (-9

8p Po F
where = -pg and P1 is equal to a constant and = cl. Equation 4-39 can
8zx
be rewritten as

cl X2Z1 = V2ZI (4-40)

where X2 = p9. NOW, each variable is expanded in powers of 6 as





Similarly A, which determined the critical point is expanded as


cy A2 = V2IA2 (0 (1) 2(2)2 442)


Here, A(o)2 represents the critical point of the circle to axisymmetric disturbances.

Higher order terms in A are the corrections going from a circle to an ellipse.

To zeroth order in 5, the RT problem with a circular cross-section is recovered.

The normal stress balance at this order is


c o) X(o)2 Z O) = 2Z o) (4-43)

~(0)
From thle above equation, Z o) = AJo (X(O)R(O)) + hecnsatc )beoe

zero when the c~onsltant-volumne requirements is applied. The~refore Z o) turns out to

be


Z o) = AJo (X!o)R(o))


(4-44)









8~Z(O)
At the outer wall, = 0. Consequently, X(O) go) are found from
iir
JI (A(o)R(o)) = .
To first order in 5, the normal stress balance is given by


X(1)?Z o) (o")aZ ) -- VZ ) (4-45)

At the outer wall, R(1 1- = 0. Therefore, Z I) -- 4 ) (r) cos (20).

To find the constant AC ) the solvability condition is applied, i.e., Equation 4-43

is mnultipliedl w~ith Z I) and integrated ove~r Ithe surface, fromn which the integral of

the products of Equation 4-45 w~ith Z o) is subtractled. It turns out that AC )2 = 0 as
one would have expected. It means that the 1!! r ~ and minor axis of the ellipse can

be flipped and thle same result would be still valid. TIhe form of Z I) canl be found

from Equation 4-45 as

Z )! = B J (0() (0)) C~OS (20) (4-46)

The constant B is found from the outer wall condition as

A 2 0o (0()p(0))
B = (A(o)() ((O)o()) (4-47)

A similar approach is taken at second order in 6. The normal stress balance at

this order is

X(2) Z o) X(o)2Z 2) __ ~2Z2) 48

The solvability condition gives
R(o)
() Z o)= rdr =-R(o)Z o0) (r = Ro)) (4-49)


whler~e Z 2) is the H independent part of Z!" 2) Zo) is known,: and Z 2) can be found

from the outside wall condition given as

8Z 2) aZ 1) 2 3Z o) d2Z o) 2 8Z 1) 8R(1)
S+ 2R 1) + R() 1 + R(2) = 0 (4-50)
Br 872 dr3 d2 R(0)2 de d









After some algebraic manipulations, an equation for X(2)2 is Obtained as


X(2)? 382 (0)" _51)


As A(2)2 iS a positive number, the stability point is enhanced, which was expected

because of the dissipation of the disturbances argument.

4.4 Linear and Weakly Nonlinear Analysis of the Effect of Shear on
Rayleigh-Taylor Instability

In this section, the effect of shear on the RT instability is studied. Two

cases are considered: an open channel Couette flow and a closed two-dimensional

flow in a driven cavity. We will show that in the case of open channel flow, the

critical point remains unchanged compared to the classical Rayleigh-Taylor (RT)

instability, but it exhibits oscillations and the frequency of these oscillations

depends linearly on the wall speed. It is shown in Appendix D that such a result

also obtains if creeping flow is assumed while destabilization can be obtained if

only inertia is taken into account. The closed flow geometry is however different.

It is shown in this chapter that shearing the fluids by moving the walls stabilizes

the classical RT problem even in the creeping flow limit provided a flat interface

is an allowable base solution. This result would obtain only if both fluid 1.v. r~s

are taken as active. An interesting conclusion of the closed flow case is that for

a selected choice of parameters, three different critical points can be obtained.

Therefore, there is a second window of stability for the shear-induced RT problem.

To understand the nature of the bifurcation, a weakly nonlinear analysis is applied

via a dominant balance method by choosing the scaled wall speed (i.e., Capillary

number) as the control parameter. It will be shown that the problem has either a

backward or forward pitchfork bifurcation depending on the critical point.

The interest in the effect of shear on the interfacial instability is not new.

C'I. i. and Steen [36] showed that when constant shear is applied to a liquid that is










above an ambient gas, a return flow is created in the liquid deflectingf the interface.

Given that the symmetry is broken, the stability point is reduced, i.e., the critical

width at which the interface breaks up is lower than the classical RT limit given

earlier. However, if a flat interface is possible, the situation may be different. The

importance of a flat interface at the base state is seen in various other interfacial

instability problems; for example Hsieh [:37] studied the RT instability for inviscid

fluids with heat and mass transfer. He was able to show that evaporation or

condensation enhances the stability when the interface is taken to be flat in the

base state. Ho [:38] advanced this problem by adding viscosity to the model while

considering the lateral direction to be unbounded. With a flat base state, these

authors were able to obtain more stable configurations than the classical RT

problem. The reason for the stability of an interface of constant curvature during

evaporation is due to the fluid flow in the vapor, which tends to reduce interfacial

undulations and is even seen in problems of convection with phase change [:39].

There are other problems where the stability of a constant curvature base state

has been enhanced either by imposing potential that induce shear [40]. These

works motivate us to study the effect of shear on the RT problem with a constant

curvature base state and inquire whether the critical width of the interface changes

and if so, why and by how much. In many interfacial instability problems the

physics of the instability is studied by explaining the shape of the growth curves

where a growth constant, o-, is graphed against a disturbance wave number and in

most, but not all problems the curve shows a maximum growth rate at non-zero

values of the wave number. Here too, it is our aim to understand the physics of

shear effects by considering similar growth rate curves where the wave number is

replaced by scaled container width. Finally, it is of interest to see what the nature

of the hifurcation becomes when shear is imposed on the RT problem. To these

ends we move to a model.











Zw b





S= -L*
Ur U

(a) (b)

Figure 4-:3. Two immiscible liquids with density stratification a) Open channel flow
b) Closed flow.


The physical problem consists of two immiscible liquids where the heavy one

overlies the light one when shear is present. The shear is introduced by moving the

lower and bottom walls at constant speed. The parameters in the problem such as

the depths of the liquid compartments, the physical properties of the liquids and

the wall speeds are tuned to attain a flat interface between the two liquids. Two

problems are considered in this study. In the first, the horizontal extent is taken to

be infinity, while in the second, the fluids are enclosed by vertical sidewalls. The

purposes of considering the open channel flow problem are to introduce necessary

terminology and to understand some important characteristics, which will be

instructive when considering the closed flow problem. A sketch of the physical

problem can he seen in Figure 4-:3.

The two configurations seen in Figure 4-:3 are quite different from each

other. In both, a heavy liquid is on top of the light one and shear is created

by moving the walls. The waves travel in the open channel flow whereas in the

closed flow, the perturbations are impeded by the walls. In fact, the presence

of the sidewalls creates a return flow, which ought to affect the stability of the

interface. In the open channel flow, the speed of the lower and upper walls must he









different otherwise no effective motion will be observed. In both configurations, it is

assumed that the walls are moved slowly enough so that the inertia is ignored.

The scaled equation of motion and the continuity equation for a constant

density fluid with the creeping flow assumption are given by


VP = -B + V2v (4-52)


V i= (4-53)

Equations 4-52 and 4-53 are valid in Z(x) < z < 1. Similar equations for the lower

phase can be written as

VP* = -B* + V-~v- (4-54)

V il* = 0 (4-55)

The lower liquid is represented by *. The velocity scale is v and is chosen to be

the capillary velocity, i.e., y/p where p is the viscosity of the upper liquid. The

over-bars represent the scale factors. The pressure scale P is given by py/L. The

length scale is taken to be the upper compartment's depth, L. The dimensionless

variables B and B* are given by gp2and gpL2TSpectively. Now the domain

equations must be solved subject to boundary conditions. At the solid walls no-slip

and no-flow conditions hold. They are expressed as


v* = Ca and v* = 0 (4-56)


Note that, the no-slip condition at the bottom wall gives rise to the Capillary

number, i.e. vj = Ca, where vj is the x-component of the scaled velocity.

Similar equations can be written at the top wall. In addition to the conditions at

the top and bottom walls other conditions hold at the fluid-fluid interface. Here,

mass transfer is not permitted, the no-slip condition and the force balance hold.










Also, the volumes of both liquids must be fixed. These conditions are given in

C'!s Ilter 3 and will not be repeated here.

For the closed flow problem, the boundary conditions on the vertical walls,

which are located at x = 0 and w/L are also specified. These walls are imperme-

able and to get an analytic solution are assumed to be stress-free. These boundary

conditions translate into

8iv, 8iv*
v, = 0 = v* and O (4-57)


We are using linear stability analysis as described in OsI Ilpter 3. The role

of the wall speed on the critical point is questioned. The first problem, i.e., the

instability in open channel flow is presented in the next section.

4.4.1 Instability in Open Channel Couette Flow

In the open flow problem the bottom wall is moved with a constant speed

UJ while the top wall is kept stationary as only the relative motion of the walls is

important. Recall that the physical problem is sketched in Figure 4-3(a).

The conditions for a flat interface in the base state are determined by using

the normal stress balance at the interface. For a given viscosity ratio, a relation

between the wall speed and the ratio of the compartment lengths is established.

It turns out that if the viscosities of both liquids and the liquid depths are the

same, then the normal stress balance is automatically satisfied. The base state

velocity profile in the horizontal direction, i.e. v,,o, is linear whereas vz,o is equal to

zero. To determine the stability of this base state, the perturbed state is solved by

eliminating ve,i in favor of vz,i by using the continuity equation. Consequently, the

domain equation for the perturbed state becomes


V4Uz,1 = 0


(4-58)









84 4 d4
where the V4 o~pe~ratorl is de~fined as +; +i 2 j2i~ A similar Ir euation is,
valid for the phase. First, the time dependence of the velocity is separated by

using Equation 3 8. Then, vz,i is assumed to be vz,1 (z) eika Where k is the wave

number. From Equation 4-58, the form of the velocity can be expressed as


vz,i (z)= CzeIkz 2ZL;/ kz 36;;-kz 4 ;ZC-kz


Hereafter, the double hat symbol is dropped. To solve for the constants in the

above equation, the perturbed boundary conditions are imposed. The perturbed

no-penetration and no-slip conditions at the top wall are

dVz,1 49
vZ~i = 0 and = 0 4-9
dz

A similar equation is valid at the bottom wall. At the interface the perturbed

no-mass transfer condition becomes


vz,i = via, and vz,i = ikZlve,o + o-Z1 (4-60)

and the perturbed no-slip condition at the interface is


z~l (4-61)
dz dz

while the perturbed tangential stress balance is given by

d2Uz,1 d20 ,1
(4-62)
dz2 dX2

The perturbed velocities vz,i and v~,i are found in terms of o- and Z1 by using the

above equations. Then, these expressions for the velocities are substituted into the

normal stress balance, which is given by

iiPo i0i~v~ iiPo' d2 8 2Z]
P1 + Z1 2 P,* Z1 2 (4-63)
8z 8z 8z p- 8z dX2










The pressure terms from the normal stress balance are eliminated by using the

equations of motion. After these substitutions, Equation 4-63 becomes

3Uz1 -3k~~z, 30,1 3k2v z,1 + k2Z [Bo k] = (4-64)
dz3 drd3


where Bo is the Bond number defined as Bo=L2 From Equation

4-64, after some algebra it is found that the neutral point of the open channel

flow is the same as that of the classical RT problem but that the neutral point

is an oscillatory state, i.e. the imaginary part of o- is not zero. This result is in

agreement with physical intuition. One might expect that the real part of the

growth constants would be independent of Capillary number as they must be

independent of the direction of the wall movement. It must be noted that the

growth constant cannot depend on the square of Ca, as the base state problem

is homogeneous in the first power of Ca. The imaginary part of o-, on the other

hand, must appear in conjugate pairs and therefore must depend homogenously

on Ca. In general, the oscillation at the critical point is not surprising because the

perturbations are carried with the moving bottom wall and they are not impeded

in the horizontal direction. This will change in the second problem where the shear

induced RT instability in a closed container, is studied.

4.4.2 Rayleigh-Taylor Instability in Closed Flow

In this problem, the top and bottom walls are moved at constant speeds.

The wall speeds, the liquid depths and the viscosities are the parameters to be

determined to get a flat interface. The governing equations were presented earlier

along with the boundary and interface conditions. To simplify the calculation, a

stream function form is introduced. The stream function is defined via


ve and vz (4-65)
8z ~ 8ix









After taking the curl of the equation of motion


V41 = 0 for 0 < x < w/L and Z < z < 1
and (4-66)

V41~ *=0 for < X< w/L and -L*/L

are obtained. The solution to a similar fourth order equation can be found in [41].

For stress-free sidewalls, the solution can be written as


= i (x < z (4-67)


where k =with n = 1, 2, and I^ (z) = Aekz +Zk Ex" 6C-kz + Dze-kz

This stream function is expanded around a base state Iel, and the stability of this

base state is investigated.

The base state: The domain equations for the base state in terms of stream

functions are


V41',, = 0 for 0 < x < w/L and 0 < z < 1
and (4-68)

V41~ *=0 for < X< w/L and -L*/L
where





The z-dependent part of the stream function is given as


r;,, ,, (z) = Aoekoz B0Zekoz 06Co-koz + Doze-kol









noiT
where ko =with no = 1, 2, A similar result can be obtained for the *
WL
phase. At the top wall, no-penetration and no-slip imply


ve,o = aCau 4 sin (kox) = ar


and (4-69)

vz,o = 0 + ,


Similar equations can be written for the bottom wall. First, a flat interface for

the base state is assumed and then the conditions that allow it are found from the

normal component of the interfacial force balance. Now, at the interface, the mass

balance turns into
vUZ,o = 0 =~ v1, n,,=0 8 (4-70)


The no-slip condition becomes


vz,o = 4, u,no (4-71)


and the tangential stress balance can be written as

8ve~o 80zio #* 80 ,o Sti*
z,0 z,0(4-72)
8z~ 8ix p- 8zx 8x

which gives
+ll, k-1 u^,, ,, = ,, (4-73)


By using the eight conditions given above, I',, and ~~are determined in terms of

Ca. Then, the expressions are substituted into the normal stress balance, which is

given by
d'Uz,0 p~ i)U ,
Po 2 P,* +2 Z = 0 (4-74)
8z ~ p- 8z~




























Figure 4-4. Base state stream function for closed flow Rayleigh-Taylor problem for
Ca = 1, w/L = 1.


Replacing pressures with the stream functions, the new form of the normal

stress balance is given as

dir ,, 3k2" 36,o3kn2 rl u,1o = 0 (4-75)
dz3 0 r 3- 0x ~g

It turns out that the normal stress balance is satisfied if and only if the

viscosities of both liquids, the compartment depths, and upper and lower wall

speeds are the same, i.e., p = p*, L = L*, a = 1. With these conditions, the stream

functions for both fluids are the same, i.e., It'n = @*. The plots of the stream

functions and the velocity fields can be seen in Figures 4-4 and 4-5.

The stability of this base state is studied in the next section by introducing the

perturbed equations and solving the resulting eigfenvalue problem.

The perturbed state: The perturbed domain equations in terms of stream

functions are


V4161 = 0 for 0 < x < w/L and 0 < z < 1


(4-76)






42














Ca =r 1,wL=1

fo teuperpas. iilrlfo teloephs










aigre vali. Thyare solteved biy ail prodr coed that was ued frotaining thoe soluio






for the base state and require the use of the perturbed boundary conditions. At

the bottom wall, located at z = -1, the perturbed no-slip and the no-penetration

conditions give rise to
=~, 0 and T,n, = 0 (4-78)


A similar equation is valid at the top wall. Note that, the index that was no at the

base state is now changed to nl. These indices will pIIli a big role in the course of

solving the perturbed equations and so particular attention should be paid to them.

At the interface, mass balance is satisfied and thus


~lnl = ~*lni


(479)










and
8i~ 1 82,',, Z8',
=-Z1 + aZ1 (4-80)
8ix 8ixiiz dx 8z~

Observe that the x and z dependent parts of the variables in the above equation

were not separated, because there is coupling between the modes and each variable

needs to be written as a summation. Accordingly, Equation 4-80 becomes


m/L ~ ~ ~ ~ ~ ~ ~ T 1',, cos Lx =- Zncs L Ld os /L

mix mixI' ,. nox i

/LZim sin L zs /x+eZnm o L

(4-81)

The no-slip condition at the interface at this order becomes


&ln 1,n d 482)
dz dz

while the tangential stress balance is given by


dz2 d2

The viscosities do not appear in the tangential stress balance, because a flat base

state is satisfied only when the viscosities of both fluids are identical. By using

Equation 4-78 and its counterpart for the top fluid, and Equations 4-79, 4-82, and

4-83, seven of the constants of the stream functions are determined in terms of A .

Thus the stream functions can be written as


olni (z) = A1Yl,n (z) and T,n, (z) = A1Yl,n (z) (4-84)


where 1),,, and ~:,, are known. Thle last coefficient A; is detezrminedi by u~sing

Equation 4-81, which can then be written as














1~~~~8 ~ ~T '

ZI'm cos [mi


eCO mlim cos/Lx.


no] x/ +cos [m+ no] x/


+2 W/L Zimcs[i-n]/Lx o m o/Lx
(4-85)

To reducer Equatioin 4-85 into its momn ts,. it is mulrtip~lied by cos( x/ and
integrated over x. After some manipulations, Equation 4-85 becomes


m/LT~ ~l~ln '" '" "/LT ~[ 1,(nxo a 1,(ni-no)


(4-86)


In the above equation, Zl, _j) = Zl, y) where j is a positive integer. Note that j = 0

is ruled out by the constant-volume requirement given in Equation 3-6. The last

coefficient, AT, is found by substituting Equation 4-84 into Equation 4-86, i.e.,


Observe that Equation 4-87 is evaluated at z = 0. To close the problem
normal stress balance is used. It is written as


(4-87)


:m, the


-3k~ +; Zi,,, [-k1Bo +k ]


0 (4-88)


Whenthestram unctons1,, an I* are substituted into Equation 4-88,

an eigenvalue problem of the form MZIZ = aZ1 is obtained. Here, a are the

eigenvalues and MZ/ is a nondiagonal matrix that occurs as such because of the

coupling between the modes. As in the open channel flow, our aim is to see the

effect of the wall speed or the Capillary number on the RT instability. The input

variables are the physical properties of the liquids, the width of the box, the depth


-3~T, 3k










of the liquids, and the wall speed. In terms of dimensionless variables, they are Bo,

w/L, and Ca. The output variables are the growth constant a, or more precisely

the real and the imaginary parts of a and the eigenmodes.


02-

0 1-



S40 80 120 160
-0 05-
-0 1- / -20-w/

(a) (b)

Figure 4-6. Dispersion curves for the closed flow Rayleigh-Taylor problem for
Ca=10 and Bo=5. a) The ordinate is the leading eigenvalue, i.e., a35-
b) The ordinate of the upper curve is the leading a, and the ordinate of
the subsequent curves are 30th, 25th, and 20tha respectively.


There are infinite eigenvalues because of the summation of infinite terms in

Equation 4-87. The size of the matrix MZ/ depends on the number of terms taken

in the series, which is determined by the convergence of the leading eigenvalue. In

these calculations, 35 terms sufficed for all values of parameters. The eigfenvalues

are found using Maple 9TM. In Figure 4-6(a), the real part of the leading a, namely

a35, iS plotted against w/L. A variety of observations can be made from this

dispersion curve but first the reason for the instability is given. The stabilizing

mechanisms are due to the viscosities of the liquids and the surface tension. On

the other hand, transverse gradients of pressure between crests and troughs, which

depend on width, as well as gravity, which is width independent, destabilize the

system. When the width is extremely small, approaching zero, the system is stable

and the growth constant approaches negative infinity. This behavior is related

to the stabilizing effect of the surface tension, which acts more strongly on small

widths, in other words, on large curvature. When the width becomes larger, the






















5 10 15
w/L

Figure 4-7. The dispersion curve for the closed flow Rayleigh-Taylor showing nmul-
tiple nmaxinia and nmininia for Ca=20 and Bo=500.


surface tension can no longer provide as much stabilization and, as a result, the

curve rises to neutrality, where there is a balance between the opposing effects.

For larger width the surface tension effects get weaker and consequently, the

destabilizingf forces become dominant and the growth curve crosses the neutral

state and becomes positive. As the width increases even more, the curve continues

rising but at some point it passes through a nmaxiniun and starts decreasing as can

he seen in Figure 4-7. This calls for an explanation. This phenomenon, distinctive

of the closed flow problem, is attributed to the interaction of the modes. As the

width increases, higher modes must he acconinodated. This has a dual effect; when

a higher mode is introduced, the waves become choppier and surface tension acts

to stabilize the higher mode, while destabilizingf transverse pressure gradients also

act more strongly. Further increase in the width causes an increase in the distance

between crests and troughs and the stabilizing effect of surface tension becomes

weaker as also does the destabilizingf effect of transverse pressure gradients.

As the width increases, more and more modes now need to be acconinodated.

Consequently, the growth curve shows multiple nmaxinia and nxininia as can he seen

in Figure 4-7.











0 04-





-0 02-

-0 04-
w/L w/L
(a) (b)

Figure 4-8. The effect of the wall speed on the stability of shear-induced Rayleigh-
Taylor for Bo=50. a) The graphs correspond to Ca=1 (the most upper
curve), Ca=4, 10, 15, 20, 100, 500, and 5000. b) Close-up view near the
critical point for Ca=10 (the most left), Ca=15, 20, and 100.


In suninary, the inclusion of a higher mode as the width increases first makes

the waves choppier; but a further increase in the width makes the waves in the

new mode less choppy. Thus, stabilizing and destabilizing effects that are width

dependent get reversed in strength. In Figure 4-6(b), the real part of the leading a

and some of the lower growth constants are plotted for small widths. The pattern

of the other curves is similar to that of the leading one. However, more terms are

needed in the sunination in Equation 4-87 for the convergence of these curves in

Figure 4-6(b).

Our aim is to see the effect of the wall speed on the RT instability. For that

purpose, in Figure 4-8 the dispersion curves for the leading a are plotted against

w/L for several Capillary numbers at a fixed Bond number. Each curve shows a

similar behavior to the curves presented in Figure 4-6. As the width increases front

zero, the curves increase front negative infinity. They then exhibit several nmaxinia

and nxininia. For large Ca, the first nmaxiniun occurs when a is negative, i.e., the

system is stable. On the other hand, for small Ca, e.g. Ca = 1, the first nmaxiniun

is observed when the system is unstable. So, when the curve starts d.~ I 1. .0 II the

system becomes less unstable, but it remains unstable. A very interesting feature is












10 -








w/L

Figure 4-9. The effect of Bo on the stability of shear-induced Rayleigh-Taylor for
Ca=20. The curves correspond to Bo=200 (The most upper curve),
150, 110, 65, 50, and 5.


observed for the intermediate Capillary numbers. The first maximum is seen close

to the neutral point. Interestingly enough, the eigenvalue becomes negative one

more time. For those curves, like the second curve from the top in Figure 4-8(a),

it is possible to obtain a dispersion curve that has three critical points. In other

words, there are two regions for the width where the system is stable. The size of

this second stable window depends on Ca and Bo. This stability region builds a

basis for a very interesting experiment. The effect of the wall speed on the critical

point can be seen in Figure 4-8(b), which is a close-up view of Figure 4-8(a). The

system becomes more stable as the walls are moved faster. In Figure 4-8, the

dispersion curve is plotted at a fixed Bond number for different Capillary numbers

while in Figure 4-9, the Capillary number is kept fixed and the curves are similar.

The critical points are collected and the neutral curve is obtained in Figure 4-10.

The neutral curve depicted in Figure 4-10 is not a monotonically decreasing

curve. It is clear that for some Bo numbers there exist three critical points. A

neutral curve exhibiting three different critical points for a given wave number

is seen in the pure Marangoni problem [42]. However, it should be noted that

when gravity is added to the Marangoni problem, it does not exhibit the zero

wave number instability seen in the pure Marangoni problem and consequently,










200


150-


~j100-






0 2 4 6 8 10 12
w/L

Figure 4-10: The neutral stability curve for the shear-induced flow where Ca= 20.


does not have three critical points. The gravity is able to stabilize the small wave

number disturbances. A dispersion curve, and therefore a neutral curve similar to

those obtained in this study was observed by Agarwal et al. [43] in a solidification

problem. Besides these examples, such a dispersion curve is not common in most

interfacial instabilities. If one wants to compare the stability point of the shear-

induced RT problem to that of the classical RT problem, it would be more practical

to plot BoL VeTSUS w/L. If the depths are large enough, the classical RT stability
22
limit, which is Bo = r2, is TOCOVered because the effect of shear is lost.
L2
By using linear stability analysis, it was concluded that moving the walls and

creating a returning flow enhances the classical RT stability. The next question

to answer is what happens when the onset of instability is passed. In other

words, the type of bifurcation is of interest. The classical RT instability shows a

backward pitchfork subcriticall) bifurcation when the control parameter is the

width. Once the instability sets in, it goes to complete breakup. What would one

see in an experiment when the interface becomes unstable for the closed flow RT

configuration? To answer this question, a weakly nonlinear analysis is performed in

the next section.
















~400 -
o


0 2 4 6 8 10 12
w/L

Figure 4-11. The neutral stability curve for the shear-induced flow where Ca= 20.
The dashed line represents the critical value for the classical Rayleigh-
Taylor problem, which iS Xr2. Observe that the ordinate is independent
of L.


Weakly nonlinear analysis: In the weakly nonlinear analysis, the aim

of this study is to seek steady solutions, as one goes beyond a critical point by

increasing or decreasing a control parameter, X, from its critical value, Xc, by a

small amount. For that purpose, let each variable, "u", be expanded as follows


a = Uo + h Xl[ol X] + zzi)



11 Bh" ul 8 c??l, Buo 1
+ [ e]a 8 + 2zi + z, + z2 cX 3aX]
2 8z 8z2 8z 6;



3 3 1 i3x 2 1 2 1 33X + --(489
8z 8z z2 d23

In the above equation, zy, z2, and z3 are the mappingfs from the current state

to the reference or the base state [10]. The idea is to substitute the expansion into

the governing nonlinear equations and determine a~ from dominant balance as well

as the variable u, at various orders [44]. In this shear-induced RT problem, the

control parameter is chosen to be the scaled wall speed or the Capillary number,

Ca. Instead of determining a~, an alternative approach is to guess it, and the









correctness of this guess is checked throughout the calculation [44]. In anticipation

of a pitchfork bifurcation, a~ is set to 1/2 for this calculation. Thus, the expansion

can be written more conveniently as


~~Bu 1~ ~I 881 2 0r

a =~ no+ + zz1 + 62a, 82 + 2z + zz z

+du du3 c??3, a 1 3 2 1 2 1 1 3 + 4-0
6: 8z 8zx 8z2 2 = 3X

where e is such that Ca = Cac + 6~2. When the expansions are substituted into the

nonlinear equations, to the lowest order in e, the base state problem is recovered,

its solution is known. The first order problem in e is a homogenous problem and it

is identical to the eigenvalue problem provided o- is set to zero. It is important to

note that in this weakly nonlinear analysis we assume that both the real and the

imaginary parts of the largest growth constant is zero. Thus, if the neutral point is

purely imaginary, this method would not applicable. In this problem, some, but not

all, of the leading growth constants have imaginary parts. However, in what follows

we shall focus only on steady bifurcation points, as we are interested in steady

solutions .

The solution procedure is as follows. In the first order problem, the state

variables are solved in terms of Z1, which represents the surface deflection at first

order. This results in a homogenous problem being expressed as MZIZ = 0. Again,

MZ/ is a real non-symmetric matrix operator. At this order, the value of the critical

parameter, Cac, and the eigenvectors, up to an arbitrary constant, A, are found.

Then, the second order problem is obtained and is expected to be of the form

MZ2S%: = CWhere the constant c appears from the boundary condition at
the moving wall. A solvability condition has to be applied to this equation whence

A can be found. If it turns out that the solvability condition is automatically

satisfied, one needs to advance to the next order. At this order, the solvability










condition provides A2 Whose sign determines whether the pitchfork is forward or

backward. In the next section the second order equations are presented.

Second order problem: The perturbed domain equations at second order

are solved subject to the boundary conditions in a way similar to the previous

orders. At the bottom wall, the no-slip and the no-penetration conditions are given

by

=~n -1adO, (4 91)

A similar equation is valid at the top wall. At the interface, the second-order mass

balance equation satisfies

= 8,n,(4-92)

and
8 2 ',??',, Z'
= Z2 (4-93)
8ix 8ixiiz dx 8z~

Recall that at the base state I',, was found to be equal to I,, This leads to

several cancellations, for reasons of brevity the intermediate steps are omitted and

simplified versions of the equations are presented. As in previous order equations,

each variable is represented as a summation. As a result, (4-93) becomes





The no-slip condition is given by

'"~, (4-95)
dz dz

The tangential stress balance assumes the form

82,' 8 1 a 3~ a"
+2ZI 22 + Z" (4 96)
8z2 3 23









and the series expansion of the tangential stress balance yields

d 'd3 1(n i 3 1(nli


dz2 1,m13 d3



By using the above conditions, I'~ and I,,, are determined. To close the

problem, the normal stress balance is introduced in stream function form as


+ ,~"_ Z2,n? (-k2zBo + k 3) = 0 (4-98)


It turns out that after much algebraic manipulations, the normal stress balance

results in M Z2Z = 0. This means solvability is automatically satisfied; hence

Z2=BZ1 holds. Therefore, the third order problem needs to be introduced

with the hope of finding A2 and the nature of the pitchfork bifurcation. Before

introducing the third order equations, the meaning of the sign of A2 needs to be

given. Recall that an increase in Ca implies more stability; consequently, if A2

turns out to be positive at the next order, a curve of A versus 1/Ca represents a

backward subcriticall) pitchfork. However, if A2 Were determined to be negative,

this would be unallowable. Then, Ca must be decreased from Cac by an amount

1/2e2 leading to a positive A2, hence, a forward (supercritical) pitchfork in an A vs.

1/Ca graph.

Third order problem: The boundary conditions at the bottom wall are


'"34~ = 0 and ilj;.l = 0 (4-99)


At the interface, the mass balance equation satisfies












8,l' dZ1 2i~ 1 ?? 2"~ 1 2 3 dZ2 1i~
-3 +3ZI + 32 + 3Z +3j
8x d xdx 8x 8xx 8x] 8xdz2


dZ1 8 2~ dZ1 82 11
+3 +6ZI = (similar expression for phase) (4-100)
dr ix 8z dX 82

Note that in the above equation, the terms coming from the base state are not

shown because they canceled each other as I<',, = *~ holds. In addition, there

are some more cancellations that take place when the interface conditions of the

previous orders are introduced, eg., the second term in Equation 4-100 cancels

with the corresponding term of the phase by using Equation 4-79. Hereafter, as

the equations are very long, only the very simplified form of the interface conditions

will be provided without separating the x and z dependent parts. However, it

should be noted that as in the previous orders, each term has to be represented as

a summation because of the coupling of the modes. The no-mass transfer condition

at the interface gives rise to


=* (4-101)


and

Brl' 82', d3 ,',, dZI 2 1, dZ 2 iix2 321 :-2 3
+ 3 + 33Z+ Z 3
8ix 8ixiiz dr ix 8zdx 8x dx xr 8ixiz x~z



+ Z 3 + Z] 3Z +i~ 6Z1 +2~ 3Z 1 =? ~ 0 (4-102)
d xdz3 dr i~ ri~ rd2 1d i~3

The no-slip condition at the interface is

8,l' c?? '" _.~ 831 1d 2i I
+3ZI 3Z2 3 + Z 3Z2 rI103)
8z 8z2 1 3X ~Xd2 1 3X










The tangential stress balance assumes the form


: 3 Z + :3Z2 ~ + :32 i + :32~ '' (4-10 4)
~2 dX3 dX3 '~2 X3 dX3

Finally, the normal stress balance is given by

c? c? dZ1 c?' d2Zi C? ri
8:~x 80812 .r d2d 72 d2 1 54


d22 Zi3 1] dZ, c'l c 8 dZI 3'
+12Zi + 18Z1 + 9Z1 + 12 1 3~
dXI2 3X .r 3xd 2 .2 .r 3



+9Zf~ (similar expression for *)
8: 8 1.2


dZ, d Z, d2Z 2 dZ1 d" Z1 dZI 20(15
+ Bo + 8-9= (41)
d~r dxr dXI2 .r d.3 d.

The way to proceed from this point is very similar to the procedure applied at

the previous orders. First, the x-dependent part of the variables is separated and

the equations are written as a summation. Then, i' and gl; are solved in terms

of Za and the inhomogeneities. Finally, these expressions are substituted into the

normal stress balance and a problem of the form MZsZ = alZi + a2Z1Z2 a3Zi is

obtained. At the second order, M Z2Z was equal to zero. In fact, at the third order,

the constant a2 tuTI1S out to be zero for much the same reason. Now, the second

order correction to the interface deflection can he written as Z2 = BZ1 and the

constant B is not known but is not needed either. The unknown constant A or

more precisely, 242 determines the type of pitchfork hifurcation.

Using the equation from the first order, i.e., M ZiZ = 0, the solvability condition

can he applied as follows












tif, M$ Z:4= j ai +i~l i- (4106)


(MtS:~J 0, A) (4-107)

where the superscript t denotes the adjoint and (. .) stands for the inner product.

All the variables are solved in terms of the surface deflection. The last equation

to be used is the normal stress balance. In that equation, all parameters are

substituted and therefore MZ/ is a real matrix and its adjoint is therefore its

transpose. Then, by using Equation 4-106 and Equation 4-107, one can get


Z a~i +a:4Z = 0(4-108)

It is known that Z1 = A4Z1 where Zi was found at the first order. Equation 4-108

then can he expressed in terms of A as follows


n,44 P2 = 0 (4-109)


where n~ and 79 are constants which are determined at this third order. Let's

elaborate on how to obtain Equation 4-109. First, Ca and Bo are fixed. The

corresponding critical w/L is found front the first order calculation, which resulted

in Figure 4-10. When Bo is smaller than some value, which is approximately 70

for the choice of parameters in Figure 4-10, there is only one critical point and this

critical point has an imaginary part i.e., it is a Hopf hifurcation. As noted before,

this weakly nonlinear analysis traces only steady solutions and is therefore not

applicable to such critical points. However there is another region of Bo number

where there is only one critical point: Bo larger than approximately 110. In that

region, the critical point does not exhibit any imaginary part and this analysis

is applicable to such points, 242 18 ariT--.v positive arid the pitchfork is backward






57



Unstable AA stable


stable `\Iunstable stable I/unstable


1/Cac 1/Cac
unstable stable


(a) (b)

Figure 4-12. Bifurcation diagrams. a) Backward (Suberitical) pitchfork. b) For-
ward (Supercritical) pitchfork.



as depicted in Figure 4-12(a). When there are three critical points (For example,

Ca = 20, Bo = 70), the A2 COTTOSponding to the largest w/L is again positive

and the bifurcation is backward. If the bifurcation is backward, once the instability

sets in, it goes to complete breakup. In contrast with the largest critical w/L, the

smallest two critical points give rise to a negative A2. Then Ca must be decreased

from Cac in order to get a positive A2 and, for these cases, the nature of the

bifurcation is forward as depicted in Figure 4-12(b). Some more observations

can be made from the calculation. The inhomogeneities coming from the no-slip

condition, Equation 4-103, and the tangential stress balance, Equation 4-104, have

no effect on the constants a~ and p.

Once A is known, the variation of the actual magnitude of the disturbances

with respect to a parameter change can be calculated when Ca is advanced by

a small percentage beyond the critical point. For example, one can compare the

amplitude of the deflections of the first and second critical points for a fixed Ca

and Bo and something interesting but explicable turns up. It is found that A2

corresponding to the small w/L is one order of magnitude larger than A2 of the

larger w/L. This can be explained by looking at Figure 4-10 at the region where

three critical points occur. Focusing on the first two points, we observe that the










first critical point is where instability starts, while the second one is where stability

starts. This means that, any advancement into a nonlinear region from the first

critical point must produce a larger roughness, i.e., A2, COmpared to the second

critical point provided the nature of the pitchforks are the same, and indeed they

are.

4.5 Summary

The critical point of the RT instability is found using Rayleigh's work princi-

ple. The analysis requires determining the change in the total energy of the system,

which is composed of the gravitational and surface potential energies.

The theoretical study of the RT instability with elliptical interface turned

out to be more stable than its companion RT instability with circular interface.

This result is in agreement with our physical intuition based on the increased

possibilities of the dissipation of the disturbances switching from a circle to an

ellipse.

It is known in the RT problem that there is a decrease in stability when

the liquid is sheared with a constant stress. This decrease in the stability limit

is attributed to the symmetry breaking effect of the shear. In this study, we

show that the fluid mechanics of the light fluid is important and it changes the

characteristics of the problem. Under specific circumstances a flat interface is

permissible under shear. For the open channel flow, to get a flat interface in the

base state, the wall speed has to be adjusted according to the ratio of the liquid

heights and the viscosity ratios. If both ratios are unity then any wall speed is

allowed. On the other hand, for the closed flow problem, bias in the liquid heights,

the wall speeds or the viscosities is not permitted. If there is any difference between

the speeds of the upper and the lower walls or between the viscosity and depth of

the upper liquid and those of the lower liquid, then the system is less stable than

the classical RT problem.










In the open channel flow, the critical point remains unchanged compared

to the classical RT instability, but the critical point exhibits oscillations and the

frequency of the oscillations depends linearly on the wall speed. The perturbations

are carried in the horizontal direction by the moving wall resulting in an oscillatory

critical point. On the other hand, in a closed geometry, moving the wall stabilizes

the classical RT instability. The results show when, how and why shear can delay

the RT instability limit. Physical and mathematical reasons for the enhanced

stability are presented. In the closed flow problem, the lateral walls impede the

traveling waves and create a returning flow. The stability point increases with

increasing wall speed as expected. It is also concluded that the system is more

stable for shallow liquid depths. For large liquid depths, the shear has a long

distance to travel; consequently, it loses its effect. The classical RT instability is

recovered when the liquid depths are very large or the wall speed approaches zero.

The most interesting feature of this problem is the presence of the second window

of stability. For a given range of Ca and Bo, there exist three critical points, i.e.,

the system is stable for small widths, it is unstable at some width, but, it becomes

stable one more time for a larger width. We present a weakly nonlinear analysis via

a dominant balance method to study the nature of the bifurcation from the steady

bifurcation points. It is concluded that the problem shows a backward or forward

pitchfork bifurcation depending on the critical point.

Clearly, it would not be easy to conduct an experiment with the specifications

given in this section. The problem does not accommodate any bias in liquid depths

nor in viscosities of the liquids. Any small difference is going to cause a non-flat

interface and lead to an instability, which will occur even before the classical RT

instability. An ideal experiment might be carried out with porous sidewalls and

with two viscous liquids. However, from a mathematical point of view, the problem

shows interesting characteristics that have physical interpretations. For stress-free










lateral walls, it is possible to obtain an analytical solution though, it is not possible

to uncouple the modes. In fact, the work in this section has shown the effect of

mode interaction on delaying the instability.

The main results of this chapter are that an elliptical cross section offers

more stability than a companion circular cross section subject to axisyninetric

disturbances and that shear driven flow in the RT problem can stabilize the

classical instability and lead to a larger critical width. These results motivate

us to run some experiments but experiments on the RT problem are not simple

to construct and so we consider building liquid bridge experiments with a view

of changing the geometry and introducing flow and seeing their effect on the

instability.















CHAPTER 5
THE STABILITY OF LIQUID BRIDGES

This chapter deals with the stability of liquid bridges. The organization of

this chapter is the same as the previous chapter. We will start with Rayleigh's

work principle to investigate the critical point of a cylindrical liquid bridge in zero

gravity. Then, we will move on to the effect of geometry on the stability point.

This section contains two problems. The first one is the effect of off-centering

a liquid bridge with respect to its encapsulant. In the second part, elliptical

liquid bridges are studied. In fact, this section proves our intuition based on the

dissipation of the disturbances. Finally, the effect of shear is presented, which

helps us understand the effect of returning flow in the floating zone crystal growth

technique.

5.1 The Breakup Point of a Liquid Bridge by Rayleigh's Work
Principle

We know from Rayleigh's calculations that a liquid thread breaks up when the

wavelength of the disturbance exceeds its circumference. Let's begin by giving a

simple calculation to determine the critical length of a bridge. This calculation is

based on Rayleigh's work principle as adapted from Johns and Narali- Ilr Ilr [10]. We

will follow a procedure similar to the previous chapter.

According to the Rayleigh work principle the stability of a system to a given

disturbance is related to the change of energy of the system. In the liquid bridge

problem the surface energy is the surface area multiplied by its surface tension.

The critical or neutral point is attained when there is no change in the surface area

for a given disturbance. Consider a volume of liquid with a given perturbation on

it, as seen in Figure 5-1. The volume of the liquid under the crest is more than










Vlost
Vgamed







Figure 5-1: Volume of liquid with a given periodic perturbation.


the volume under the through (Appendix C); but the volume of the liquid needs to

be constant upon the given perturbation. Therefore, there is an imaginary volume

of liquid of smaller diameter whose volume upon perturbation is the same as the

actual volume. As a result, the surface area of the liquid is increased with the given

perturbation but it is also decreased because of the lower equivalent diameter. At

the critical point, there is a balance between the two effects and the surface area

remains constant.

To set these thoughts to a calculation consider the liquid having a radius Ro.

A one-dimensional disturbance changes the shape of the liquid to


r =R + cos(kx) (5-1)


where R is the equivalent radius, e represents the amplitude of the disturbance,

assumed to be small, and k is the wave number given by nx/lL with L heing the

length of the bridge. Using the above shape, the surface area is given by




A= j rddr 1 dr


where ds is the are length, given by ds do ia'' [~1 + do

So, the area per unit length turns out to be

4 1
S2xrR + xRE 2k2 (5-3)
L 2









Here R, the equivalent radius is found from the constant-volume requirement as

follows

V = xR,2 A = KT" (5-4)


1 e2
which implies R to be equal to Ro .Substituting this radius into the area
4 Ro
expression, the change in area is obtained as

1 e [(25;Ro)2 -L2] (5-5)
2 RoL2

The critical point is attained when the length of the bridge is equal to the

circumference of the bridge. There are two obvious questions that arise from this

calculation: what is the role of the disturbance type on the stability point and what

is the role of the liquid properties on the stability point? A particular disturbance

type, a cosine function is chosen for this calculation as every disturbance can be

broken into its Fourier components and the same calculation can be repeated. In

fact, the same calculation is performed by Johns and Nara i- Ilr Ilr [10] on page 10

for any function f(z) without decomposing into its Fourier components. Equation

5-5 tells us that the critical point does not depend on the properties of the liquid.

This can be understood from the pressure argument introduced in C'!s Ilter 2. At

the critical point, there is no flow. The viscosity and the surface tension pIIl i- a role

in determining the growth or decay rates of the disturbances. Such a curve can be

reproduced via a perturbation calculation and this is given next.

5.2 A Simple Derivation To Obtain the Dispersion Curve for a Liquid
Bridge via a Perturbation Calculation

A simple perturbation calculation is used to determine the critical length and

the dispersion curve of a liquid bridge. To make matters simple, the liquid bridge

is assumed to be composed of only one inviscid liquid, and the gravity is neglected.

This calculation will show the critical length as a function of its radius, the same

calculation methodology will also be applied in more complicated situations, such










as the case when a liquid encapsulates another liquid. The Euler and continuity

equations are:

p- + pv' Vv'= -VP (5-6)

and

V &= (5-7)

These domain equations will be solved subject to the force balance and no mass

flow at the interface i.e.,

P = -y2H (5-8)

and



Here 2H is the mean curvature, n the outward normal to the jet surface and u the

surface normal speed (Appendix B). To investigate the stability of the base state,

impose a perturbation upon it. Let e indicate the size of the perturbation and

expand and P in terms of 6, viz.

&=~~v Ce Ii -- and P = Po+ 17 +--- i~,] (5-10)


'rl' is the mapping from the current configuration of a perturbed jet to the

reference configuration of the cylindrical bridge. We presented the expansion of a

domain variable along the mapping Appendix A. More information can be found in

Johns and Narali- Ilr Ilr [10]. The radius of the bridge R in the current configuration

may also be expanded in terms of the reference configuration as


R (0, z, t, e) = Ro + eRI + (5-11)


Collecting terms to zeroth order in a we get

800
p~ +,~ -C -VC, = -VPo (5-12)









and

V = 0 (5-13)

There is a simple solution to the problem. It is & = 0 and P = y/Ro where Ro is

the radius of the bridge.

The perturbed equations at first order become


p =-P (5-14)

and

V 01 = 0 (5-15)

Likewise the interface conditions at first order are

Pi =1 -7O + 2R + 2 (5-16)
R~ R2 802 d2

and
881
iro v- 1= l (5-17)

The stability of the base state will be determined by solving the perturbation

equations. To turn the problem into an eigenvalue problem, substitute


P, = P, (r) eat ime cos (kz) (5-18)


and

R1 = Rze~e ime cos (kz) (5-19)

into the first order equations. In the first order equations s, m, and k stand for

the inverse time constant, the azimuthal wave number and axial wave number

respectively. Eliminate velocity to get


V2 1 = r'dr + k2r~ i P1 = 0 (5-20)









The corresponding boundary conditions for the perturbed pressure are


=P -e2 F1 (5-21)
dr

and

r, = [ -7 R R02 k2 1 (5-22)

The eigenvalues are the values of s at which this problem has a solution other

than the trivial solution. Let us first look at the neutral point, i.e., a2 = 0. The

solution to Equation 5-20 is of the form


P, = Alm (kr) (5-23)

where A must satisfy
dPi
(r = Ro) = 0 (5-24)
dr

From Equation 5-24, A vanishes. Using this in the only remaining equation,

i.e., Equation 5-22 gives

0 = [1 -m2 R k2 1 (5-25)


Now, for R1 to be other than zero [1 m2 -R k2] has to be equal to zero

which gives us the critical wave number of the bridge from k~dicAR~ = 1, hence the

critical length of the bridge is its circumference.

To obtain the dispersion curve, one needs to substitute Equation 5-22 into

Equation 5-21 to get


a2 pl [2 -n R k2] d1 (5-26)

Substituting:, the expressionc, fo from Equation 5-23 into the above equation


n2 __ 2 R k2] ':4:," (5-27)
pR~ Im (kRo)










is obtained. Here, I:,'(:r) = dIzl (:r). The~ most~ dangeroIus modeU is whenII mD is zero

Then, the equation for the dispersion curve is

o. __ [ k2R] /L1 n O (5-28)
pRo" lo (kRo)

5.3 The Effect of Geometry on the Stability of Liquid Bridges

In this section we will be concerned with two issues related to geometry. The

first has to do with the possible off-centering of a bridge. Recall that to obtain a

cylindrical bridge we have to encapsulate it hv another liquid of the same density.

This leads to the possibility that the bridge might he off centered and in turn this

raises questions on the stability of the bridge. The second problem has to do with

the end plates of the bridge. We ask whether the stability of the bridge can he

enhanced by making the end plates noncircular, specifically elliptic. The motivation

for this stems from our observations on the elliptic RT problem where azimuthal

pressure variations allowed us to obtain greater stability.

5.3.1 The Stability of an Encapsulated Cylindrical Liquid Bridge
Subject to Off-Centering

The liquid bridge is taken to be inviscid simply so as to simplify the calcula-

tions without much loss of essential physics. The perturbation theory explained in

the earlier chapters is used to study the stability of such a bridge subject to inertial

disturbances. At the end of the analysis we will learn that while the off-centered

nature does not change the neutral point it does affect the rate of growth and

decay of the disturbances causing the unstable regions to become less unstable and

stable regions to become less stable. Limiting conditions are considered in order to

provide a better understanding of the physics of off-centering.

To begin the analysis of the problem, we draw the attention of the reader to

Figure 5-2, which depicts an off-centered bridge in an outer encapsulant. We are

particularly interested in what happens to the damping and growth rates of the


















Figure 5-2: Centered and off-centered liquid bridges.

perturbations if the bridge is not centered. The stability is studied by imposing

small disturbances upon a quiescent cylindrical base state. Before this, we turn to

the governing nonlinear equations, which are given next.

The equation of motion and the continuity equation for an inviscid, constant

density fluid are given by

p- + pv' Vv'= -VP (5-29)

V &= (5-30)

Equations 5-29 and 5-30 are valid in a region 0 < r < R(0, z), where R(0, z)

is the position of the disturbed interface of the bridge. Here p is the density, and

& and P are the velocity and pressure fields. Similar equations for the outer fluid,

repre~sented by '*', c~an be written in the region R(0, z) < r < R ~o). The solution to
the base state problem is c'o = 0 = ,,u and Po P* = 7H ot htti
Ro
base state may be the centered or off-centered state. In the next sub section we will

present the higher order equations, which will then give us the dynamic behavior of

the disturbances.

5.3.1.1 Perturbed equations: a1 problem

To first order upon perturbation, the equations of motion and continuity are


p = -VPI and V -01= 0 (531)










in the region 0 < r < Ro(0). Combining the two equations we get


V2P1 = 0 0 < r < Roe (5-32)


with similar equation for the '*' fluid. The domain equations are second order

differential equations in both spatial directions. Consequently, eight constants

of integration must be determined along with R1, which is the surface mapping

evaluated at the base state. To find these unknown constants and R1, we write the

boundary conditions in perturbed form. At the interface, there is no-mass flow and

the normal component of the stress balance holds. Consequently


1S0 (1 U81) 80 -i (01~ U81) (0-00)


and

Pi P,* = -y2H, (5-34)

The walls are impermeable to flow, as a result the normal component of the

velocity is zero, or in terms of pressure we can write


n'o VPI = 0 (5-35)


A similar equation is valid for the '*' fluid. Free end conditions are chosen for

the contact of the bridge with the solid upper and lower walls, i.e.,

dR1
= at z = 0, Lo (5-36)
dz

The perturbed velocities, vi and vi can be eliminated from the boundary

equations by using Equation 5-31 and its counterpart for the '*' fluid. We separate

the time dependence from the spatial dependence by assuming that the pressure,

velocity and R1 can be expressed as K = Keat where K is the variable in question.










Equation 5-3:3 then becomes

1, -(T2R 1
-ifo VB' =' 2 0'Pl* (5-37)
P [' R,2 P*


Hereafter, the symbol, ^', will be removed from all variables. The problem

given by Equations 5-32 5-37 is an eigenvalue problem but the geometry is

inconvenient because Ro is a function of the azimuthal angle '8'. Therefore we use

perturbation theory and write the equations at the reference state i.e., the state

whe~n the shift dlistance 'b: is equal to zero and where Ro is equal to Roo") andc is

independent of '8'. All variables, at every order are expanded in a perturbation

series in 6, including the square of the inverse time constant a. Therefore a2 1S

2 __o 62'1' 2 ~ 2 2+ (5-38)


Our goal is to determine the variation of a2 at each order to find the effect of

the shift, 6, upon the stability of the bridge. The calculation of a2'0 1S well-known

and can he found in C'I 1...4 -lekhar's treatise [17]. Its value depends upon the

nature of the disturbances given to the reference bridge and can become positive

only for axisymmetric disturbances. Hence, the effect of 6 on the stability of the

bridge subjected to only axisymmetric disturbances in its reference on-centered

state is considered. To calculate the first non-vanishing correction to a2, we need

to determine the mapping from the displaced bridge configuration to the centered

configuration, and this is done next.

5.3.1.2 Mapping from the centered to the off-centered liquid bridge

In determining the mapping, we note that we have two different types of

perturbations: the physical disturbance represented hv e, and the displacement of

the liquid bridge represented by 6. Hence, we have an expansion in two variables.









To get this expansion, we observe that the surface of the disturbed liquid bridge is

denoted by


r = R(0, z, t, e, 5) (5-39)

Therefore R can be expanded as

Ro (o,S) a (o,z,t,S)

R = R o) +Itl -R+62 R2) ~~) 0) 61) 26 2~)

(5-40)
where R o) is the radius of the centered bridge and R' = dlo (6 = 0). Fig-
d6
ure 5-3 helps us to relate Ril and R ;2) tO 0i). By using the basic principles of

trigonometry, we can conclude that


R~ + 62 -26Rol cos(0) = R of2 (5-41)

Substituting the expansion of Ro from Equation 5-40 into Equation 5-41, we get


R ~) = cos(0) and R) 2) n(8


The mapping from centered to off-centered configuration having been found,

the effect of the displacement on the stability of the liquid bridge can be deter-

mined from the sign of 0.2'1', Which is given in the next section.

5.3.1.3 Determining o.241)

The perturbation expansions involve terms of mixed orders. The subscripts

represent the a disturbance while the superscripts in parentheses represent the 6

displacement. The domain equation of order e 50 is


V2 (0) = 0 0 < r < Ro)


(5-42)
























Figure 5-3: The cross-section of an off-centered liquid bridge.


The outer liquid's domain equation can be written similarly. The mass conservation

and the normal stress balance at the interface require

-1 o) p1(0) 2( o) 0i ) __0 10 ( -3


and

Pjo) p1(O)' = -:i2H o) (5-44)

In a similar way, the domain equation of order e161 is


V2 (1) =00

The conservation of mass equation at the interface becomes

-(1 V ~7(o) ) (0) + R (0) ] 2(1) (0) + 2(o)R 1) (5-46)


whler~e R~ is: the mrappinlg fromr th~e current configuration of an off-cenltered bridge

to the reference configuration of the centered bridge and was shown to be cos(0).

A similar set of equations can be written for the outer liquid. The normal stress









balance at the interface at this order is

p1(1) ~(1 + R P R -y2H () (5-47)


Wle use an energy method to get the sign of a2 1. By multiplying Equation

5-45 by Pj(o)/p, Equationl 542 by P /~p, integrating over thle volume V,' tak~inlg

their difference and adding to this a similar term arising from '*' fluid, we obtain

,(0 ) 2 l 1) P (1 ) 2 l 0
pp p



2~ 1() 1 OLP(1)* "() dV* = 0 (5-48)
P* pp*
The volume integrals can be transformed into surface integrals by using

Green's formula. The integral over the 'rz' surface vanishes because of symmetry,

i.e. because Pj(o) is the same at 'H' equal to zero anld 2xi. Thl~e inlteg-ral over thle

'rO8' surface vanishes because of the impermeable wall conditions. Equation 5-48

therefore becomes

R o)Lo 1() 1(1)i) 1(1~) 1(0)
oo pdr pd
(5-49)


-R o Lo x Pl0)*1 (1)* __)~ 0() ) 131:.1 = 0

Applying no-mass transfer equations at the interface i.e., Equations 5-43 and

5-46, Equation 5-49 becomes

rLo r2xr IPl [ (0) 2 (o1) + 2(1) 0t) g d ,0



p(0)L F2(o) (1) + 2(1) 0)



[P P~'C: ] [a2o) l 0) __ ) ~1,.1 := 0 (5-50)









Equation 5-50 is simplified by noting the fact that e'bo terms are '8' indepen-
detnt and that R l) is equal to c~os(0). Consequently, the integral of P,(o

and the corresponding term for the outer liquid over 'O' is zero. Substituting the

normal stress balances at each order, i.e. Equations 5-44 and 5-47, Equation 5-50
becomes


6'2H 0) 2(o) 1) R 1) )Tcl 0) 2 (o)' 0lP )] __ 0) I / =
(5-51)

To get the sign of a2 1' from Equation 5-51, we need to determine the form of
2H ) and therefore RI). But, the form of R(I can be guessed from Equation 5-46,

which h~as twvo types of inhomogeneities: RI anld a2(1) 0U). Thl~erefore, R l)
can be written as

R l) = A(z)a2'1' BZ) COS 8) + 0

where the constant C is zero because of the constant-volume requirement. Substi-

tuting thle formr of KRl into Equation 5-51, we obtainl

to~o 72xe2(o 2(1)1~o, A2A~z) R



+R 72x2(1 0))1 dz = 0 (5-52)

where we have used

(0)R ) d2 0) (1)1) d2 1) 2 R1)
2,H = +and 2H(1 = + +
1 R o)2 dz2 1 n0)2 n0)2 d2 d2

To determine the sign of a2 1' from Equation 5-52, the self-adjointness of the

d2 ,,,2 operator and thle correspondinlg boundary~ conditions onl Ro)(z) and A(z)

are used, rendering the term in Equation 5-52 in '{}' to zero. Also, the Rayleigh









inequality [45], states that


where X2 1S the lowest positive eigenvalue of the differential operator d2 ,,,2 and X2

is strictly positive. When we substitute this into Equation 5-52, we conclude that

a2'1' is ZeoO. Therefore, to find the effect of off-centeringf we need to move on to the

next order in 6 and get a2'2'

5.3.1.4 Determining a2(a)

The domain equation of the e'52 order is


2 1(2) = 0 (5-53)

The conservation of mass at the interface requires

-1 2) P(0) + 2 P Rr~1 3(0~))



(0)~ ~(2) :(1) V\ I1( 1 l0 2 l0
vlB r dr2 dr

2() () (1)2
2(3'() 0)+ 2a2'(1) ) 2(O) (2) 0( ) (5-54)

where
(2) (0) Si2 1(0U)
Ro) 7 21 R o 2

A similar mass balance equation for the outer fluid can be written. The normal
stress balance satisfies


1,(2) +t 2R +RR2


1(2) + 2R + R +1) R2~() 2~() = -y2H 2a) (5-55)









where the mean curvature is given by

(2) () hz 2 (2)iz 82 (2)
2H = + + R.T.
R o)% R o0)" iH2 d2

while R.T is given by

R.T'. = R o) [1- 3 cOS2() Siin2(H 0()2aNl +2silln(0)R o) -


(5-56)


4 cos(0)R no)


We proceed with an approach analogous to the previous section to predict the

sign of o.2' and we obtain the counterpart of Equation 5-48. We then use Green's
formula and introduce the no-mass transfer at the interface for the e 52 and the

elbo problems, viz. Equations 5-54 and 5-43 to obtain the analog of Equation 5-50,

which is

r Lo r 2 x 1(0 ) 2( ) ) 2( ) ) 02 ( o) )0S! i n 2(H


2i~DV cos(0)) 82l1 CS Pl(0) Sin2B 2~pl(0)
+ +
p 872 p 3r 0) 2

2 sin(0) B V1 sin2(B 1(0") ()
P~~~~~~ ~ PO1L d i) r (OLsimilar expression for liquid)


(5-57)


In order to simplify Equation 5-57 in a manner similar to the previous section,

we use the normal stress balance equations, i.e. Equations 5-44 and 5-55, the form

of R 2); Which is guessed from the no-mass transfer equation, i.e. Equation 5-54

and the self-adjointness of the d2 ,,,2 operator. We also use Equation 5-43, which

gives


sin2(H 1(0)
pR~j o)2SE
Then, Equation 5-57 becomes


2(o), 0) Sin2(H
R o)%


1t(2~) P~l"'(a2)* 2(0' ))] iJ





















8)


9)


p*k
2(o) (0) 2 1()* (1) 2 pl(0) d2 10)1() l0
o2k Br dr 872 d2 0() d

a2()0, 0i):i 0~ ) +k2 R0)2 R0) +2R~ oj)R (5-60)


NUote that PyU) and P ~) inl Equation 5-60 ar~e funlctionsu of only r and all of the
terms ar~e evaluated at the reference inlter~face, i.e. at r = R o(


ILo r2 x r (0) 2(2) (0) 2 cos(0) 82 l(1) COS 2H i3" l(0)
~o PO p d2 p p3

sin2(8 2 Pl(0) 2 sin(0) d P(1 (0) i .illr irr~inir rlqu
pR 0) i32 0Rj) 3

+1 2 o()+cs 1 1
Br dr 872 dr2

Si112(H)~2(o 0iYU" 3U ]1''1j)] 2(o) 0)7y(R.T) 1,31 = 0 (5-5


In principle, a2(2) can be found from the above equation. However, some
more work is needed as sterns such as R~! o)P1O) and P i) appe~ar. R o) c~an be

expressed as B cos(kz) for free end conditions, but the solution for the pres-
sur~e P,(i) is obtained fr~om thle domnain equation V2 Pf" = 0 anld uponl letting

P{ = P,"(r) cos(kz) cos(m0) the domain equation becomes

1 d dPfd mr2 B) (5
r dr dT T

where i and m are each zero for the e'bo order and equal to one for the e'61 order.

Using Equation 5-59, we evaluate the integrals in Equation 5-58 and obtain

^ 0) 2 ( 0) 2 l1) ll(0 2 V1l(0) (1)
Spk 872 28r3 2R o) a7.2 0j) 1









To find the sign of a2'2' from Equation 5-60, we need to solve for the per-

turbed pressures. Their forms are found from Equation 5-59 as

Pji) =~: A I(k ) +C K(kr) cos(kz) cos(m0)

and

Pi)* = As *Im(k~r)f +~ @Km (kr) cos(kz) cos(m0)

where C~2 is zero because thle pressure is bounded everyvwher~e.
To obtain the constants A, B, A* and C*, we substitute the form of the

pressures into the boundary equations at each order. To order elbo, from the no-
mass transfer, viz. Equation 5-43, the normal stress balance, viz. Equation 5-44
and the impermeable walls, we get


Ak0I((kRF)) = -pa2(O)8k~ (5-61)

Aloli~kRF )+CKAk )=-a2(o)k (5-62)

AkO0(kf )- Aolo(kRF ) C oKo(k:It ) = 1 -k p0 (-3

and

AloI:(kR o)*) + C oK:(kRF ') = 0 (5-64)

Whenl a2(o) is ZCoO, We See from1 Eq(uations 5-61, 5-62 and 5-64 that AlkO, AloT

andI CIo are all zero. From Equation 5-63, we recover the critical point, which is

k2Ri i ) When a2(O) is HOt zero, four equations must be solved simlultaneously
such that all of the constants not vanish at the same time.

Likiew~isel ,! and fl are solved by introdlucing the boundlary conditions
a t t e e 1 o r e r h e o l u t o n o t h e p e r u r b e p r e s u r e P a n d P a r e

substituted into Equation 5-60 to evaluate a2 2. The reader can see that an

analytical expression for a"20) 1S obtained. This expression, however, is extremely










lengthy so we move on to a graphical depiction of a"2? an~d a discussion of the

physics of the off-centering.

5.3.1.5 Results from the analysis and discussion

An immediate conclusion of the above derivations is that a2'1' is ZeoO. This

comes as no surprise because the deviation of the cylindrical bridge from the center

is symmetric. In other words, it does not matter whether the deviation is of an

amount equal to +6 or -6. In fact all odd order corrections to eigfenvalues will

therefore be equal to zero. Several figures are presented where the effect of off

centering is shown and the physics of off centering is discussed. The ordinates

and abscissas are given in terms of scaled quantities where the scale factors are

obvious from the labels. Figure 5-4 shows the effect of off-centering on the growth

rate constant a. The neutral point did not change, which is not surprising because

at the neutral point the pressure perturbations are indeed zero and since the

system is neutrally at rest, it cannot differentiate between centered and off-centered

configurations.



O06


O 02

Oi 02~ 06 0'8







Figure 5~-4. a2(o) anld e2'2' (multiplied by their scale factors) versus the wavenum?-
ber for p*/p 1 and R /R -~) 2.


If 'k' is smaller than the critical wavenumber, ke, the bridge is unstable to

infinitesimal disturbances. As can be seen from Figure 5-4, once the bridge is











10 0021

O 004









Figure 5~-5. C.!s !,- in (s2'a (multiplied byi its scale factor) for small to in~ter-
mediate density ratios for scaled wavenumb~er (kR )) of 0.5 and
R / IR ) 2.


unstable, the off-centering has a stabilizing effect. Although the neutral point

is unaffected, the rate of growth is reduced. The off-centering provides non-

axisymmetric disturbances, which in turn stabilize the bridge. However, lazy waves

amplify the effect of transverse curvature against the longitudinal curvature, con-

sequently, the bridge is ahr-l- .- unstable in this region. The longitudinal curvature

becomes more important for short wavelengths and in the stable region, each value

of a2 produces two values of a, which are purely imaginary and conjugate to each

other. The disturbances corresponding to the wavelengths in this region neither

settle nor grow. The bridge oscillates with small amplitude around its equilibrium

arrangement. The bridge cannot return to its equilibrium configuration without

viscosity, which is a damping factor. Once the bridge is stable, the off-centering

offers a destabilizing effect because the wall is close to one region of the bridge and

this d. 1 .1-< the settling effect of longitudinal curvature.

Limiting conditions, usually provide a better understanding of the physics.

In Figure 5-5, p*/p is allowed to vary and it approaches zero and its effect on

scaled a"24 is given. The figure shows that the outer fluid loses its role when p*/p

approaches zero because the fluids are inviscid. Therefore, the bridge is expected

to behave as if there were no encapsulant at all, thereby causing a22' tO Vanish. To




















2000 4000 moo En00 10000 p


Figure 5~-6. ('!, lII, in cs2 ) (multiplied byi its scale factor) large den~sity ratios for
scaled wavenumb~er of 0.5 anld R /R -~ 2.


see the behavior of the curve, the range of the plot is extended to p*/p = 14. When

p*/p is very large, as shown in Figure 5-6, the outer liquid serves as a rigid wall and

therefore a"2! approaches zero. In other words, a'2! approaches zero as p*/p goes

to either zero or infinity.

The ratio of the radii RF /RF~ is another paramelter that is examninedl and its

effect is shown in Figure 5-7. As the ratio approaches unity, the azimuthal effect

becomes more obvious. On the other hand, as the outer fluid occupies a very large

volume, the off-centering effect settles down. As a result, a2(2 approaches zero and

the bridge acts as if there was no outside fluid.

15 3











Figure 5-7. ('1! li!,- of a'2! multipliedd by' its scale factor) versus outer to inner
radius ratio R /)'RF for scaled wavenumber of 0.5 and p*/p 1 .









In summary, the physics of the problem indicate that the effect of off-centering

is such that it does not change the break-up point of the bridge but it does affect

the growth rate constant. The stable regions become less stable, meaning that

the perturbation settles over a longer period of time, whereas the unstable regions

become less unstable, therefore the disturbance grows slower. In addition, the

physics of the off-centered problem indicates that the effect of off-centering is seen

to even orders of 6 and this required an algebraically involved proof.

It is important to understand the effect of off-centering the bridge because it

can be technically difficult to center the bridge and this might have a technological

impact when a float zone is encapsulated by another liquid in the crystal growth

technique. Our next focus is to understand the complex interactions of geometry

on the stability of liquid bridges. We will present our physical explanation of why a

non-circular bridge can be more stable than its circular counterpart. We will prove

our reasoning with elliptical liquid bridge experiments.

5.3.2 An Experimental Study on the Instability of Elliptical Liquid
Bridges

In an earlier chapter we showed how an elliptic interface could help extend

the stability in the Rayleigh-Taylor problem In this chapter we will consider the

experimental extension of this idea to liquid bridges.

Liquid bridges have been studied experimentally as far back as Mason [46]

who used two density-matched liquids, namely water and isobutyl benzoate and

obtained a result for the ratio of the critical length to radius to within 0.05' of

the theoretical value [12]. While most of the theoretical and experimental papers

on liquid bridges pertain to bridges with circular cylindrical interfaces, there are

some, such as those by Meseguer et al. [47] and Laver6n-Simavilla et al. [48] who

have studied the stability of liquid bridges between almost circular disks. Using

perturbation theory for a problem where the upper disk is elliptical and the bottom




































Figure 5-8: Sketch of the experimental set-up for elliptical bridge.

disk is circular, they deduced that it is possible to stabilize an otherwise unstable

bridge for small but non-zero Bond number. Recall that the Bond number is given

by the ratio of gravitational forces to surface tension forces. The earlier work

of others and the earlier chapter on elliptical interfaces in the Rayleigh-Taylor

problem, therefore, has motivated us to conduct experiments on the stability of

liquid bridges between elliptical end plates and we now turn to the description

of these experiments. Figure 5-8 shows a diagram of the experimental set-up. It

depicts a transparent Plexiglas cylinder of diameter 18.50 cm, which can contain

the liquid bridge and the outer liquid. The bridge, in the experiments that were

performed, consisted of Dow Corning 710R, a phenylmethyl siloxane fluid that

has a density of 1.102 + 0.001 g/cm3 at 25 oC. The density was measured with a

pycnometer that was calibrated with ultra pure water at the same temperature.

The surrounding liquid was a mixture of ethylene glycol/water as -II__- -1h I1 by









Table 5-1: Physical properties of chemicals.


710R Mixture
Density (g/cm3) 1.102 + 0.001 1.102 + 0.001
Viscosity (cSt) [49] 500 7.94
Interfacial tension (N/m) [49] 0.012 + 0.002


Gallagher et al. [49]. The outer fluid is virtually insoluble in 710R. Table 5-1 gives

the physical properties of the chemicals used.

The bridge was formed between parallel, coaxial, equal diameter Teflon end

plates. The outer liquid was in contact with stainless steel disks. Furthermore, a

leveling device was used to make sure that the disks were parallel to each other.

To ensure the alignment of the top disk, the leveling device was kept on top of the

upper disk during the experiment. For the elliptical liquid bridge experiments, the

end plates were superimposed on each other. This was guaranteed by marking the

sides of the top and bottom disk, which were, in turn, tracked by a marked line

down the side of the Plexiglas outer chamber.

The key to creating a liquid bridge of known diameter, and making sure that

the disks are occupied completely by the proper fluids, is to control the wetting of

the inner and outer disks by the two fluids. If the 710R fluid contacts the stainless

steel surface, it will displace the outer fluid. Therefore, it was critical to keep the

steel disks free of 710R and this was assured by a retracting and protruding Teflon

disk mechanism. Prior to the experiment, the bottom Teflon disk was retracted

and the top Teflon disk protruded from the steel disks. This helped in starting

and creating the liquid bridge. Then, 710R fluid was injected from a syringe of

0.1 ml graduations through a hole of 20 thousandths of an inch (0.02 inches). A

liquid bridge of around 1 mm length was thus formed in the absence of the outer

liquid. Capillary forces kept this small-length bridge from collapsing. The outer

liquid was injected through two holes of 0.02 inches, 180 o from each other, so as










not to displace the 710R. The next step was to simultaneously increase the length

by raising the upper disk and adding the 710R and outer liquid.

A video camera was used to examine the bridge for small differences in density.

We were able to capture the image thanks to the difference in the refractive index

between the bridge and the outer liquid. The loss of symmetry in the liquid bridge

was an indication of the density mismatch. The elliptical liquid bridge is symmetric

around the mid plane of the bridge axis, while the circular bridge has a vertical

cylindrical interface, the shape of the bridge could then be checked via a digitized

image .

The density of the mixture was adjusted before the experiment to 0.001 g/cm3

by means of a pycnometer. However, during the experiment, finer density matching

was required, and either water or ethylene glycol was mixed accordingly to adjust

the density mismatch. The shape of the bridge was the best indicator to match the

densities. In addition, the accuracy of density matching was increased substantially

as the height of the bridge approached the stability limit. Extreme care was taken

to match the densities when the height was close to the break-up point due to

the fact that gravity decreases the stability point well below the Plateau limit for

circular liquid bridges [50]. For example, we were able to correct a slight density

mismatch, 4gof 10-s by adding 0.2 ml of water to 1 liter of surrounding liquid.

This density difference is observable by looking at the loss of symmetry in the

bridge. A similar argument also holds for elliptical liquid bridges. Depending on

the amount of liquid added, either water or ethylene glycol, mixing times ranged

from 10 to 30 minutes. In all experiments, sufficient time was allowed to elapse

after the mixing was achieved so that quiescence was reached.

The top disk was connected to a threaded rod, which was rotated to raise

it and increase the length of the bridge. The height of the bridge when critical

conditions were reached was ascertained at the end of the experiment by counter










rotating the rod downward until the end plates just touched. One full rotation

corresponded to 1.27 mm. The maximum possible error in height measurement

was determined to be 0.00:3 inches over a threaded length of 12 inches. Therefore,

the error in the total height measurement of the bridge was determined to be

less than 0.2 !' In addition to this, there was a backlash error that was no more

than 0.035 mm. It turns out that this error amounts to a maximum of 0.11

of the critical height. The total error in the height measurement technique was

therefore never more than 0.35' The volumes of fluid injected into the bridge for

the large and small bridges were 19.80 and 2.45 ml respectively. It may be noted

from Slobozhanin and Perales [51] as well as from Lowry [25] that a 1 decrease

or increase in the injected volume from the volume required for a cylindrical bridge

results in a decrease or increase by approximately 0.5' in the critical height,

respectively. Experiments with circular end plates were performed to ensure that

the maximum error was very small.

5.3.2.1 Results on experiments with circular end plates

The experiments with circular end plates were performed for two reasons.

First, the accuracy of the procedure and experimental set-up were verified by recov-

ering the Plateau limit. Second, the typical break-up time for the circular bridge

was measured to help estimate the waiting time for each increment when the ellip-

tical end plates were subsequently used. The diameters of the circular Teflon end

plates that were machined were measured by a Starrett 1\icrometer (T2:30XFL) to

an accuracy of +0.0025 mm as 20.02 mm and 10.01 mm respectively.

The lengths were increased in increments of 0.16 mm once the bridge height

was about ;:' lower than the critical height. Thereafter, for each increment the

waiting time was at least 45 minutes before advancing the height through the next

increment. When the critical height, as reported in Table 5-2, was reached the

necking was seen in about :30 minutes and total breakup occurred in around 15






















































I I --C-~L


Table 5-2. 1\ean experimental break-up lengths for cylindrical liquid bridges. Up-
per and lower deviations in experiments are given in brackets.



Break-up length (mm) ~ change in length of the mean

Large cylindrical bridge 62.84 (+0.02, -0.04) -0.08
Small cylindrical bridge :31.48 (+0.09, -0.05) +0.10


minutes after the initial necking could be discerned. Each experiment was repeated

at least :3 times and the results were quite reproducible. A typical stable bridge

at a height of 29.57 mm is depicted in Figure 5-9(a). The same bridge at breakup

is shown in Figure 5-9(b) at a height of :31.57 mm. The reported values in the

table do not account for the backlash and it should be noted that the increments

in height were done in steps of 0.16 mm. Taking this into account, it is evident

that the error in the experiment was very small, showing that the procedure and

the apparatus gave reliable results. This procedure was useful in the follow-up

experiments using elliptical end faces.


Figure 5-9. Cylindrical liquid bridge. Note that in this and all pictures the depicted
aspect ratio is not the true one due to distortions created by the refrac-
tive indices of the fluids residing in a circular container with obvious
curvature effects. (a) Stable bridge (b) Unstable bridge.


;1
~MI
























Figure 5-10. Large elliptical liquid bridge (a) Stable large elliptical liquid bridge.
(b) Unstable large elliptical liquid bridge, before break-up.


5.3.2.2 Results on experiments with elliptical end plates

The ill r ~ axes of the two elliptical Teflon end plates were measured to be

24.01 and 12.00 mm (+0.0025). The minor axes were measured to be 16.80 and

8.34 mm respectively. For the large disc, the radius of a hypothetical companion

circular end plate of the same area is 10.04 mm and for the small disc the compan-

ion radius is 5.00 mm, the deviation of the elliptical end plates from the companion

hypothetical circular plates of the same areas was therefore close to 211' .

Table 5-3. Mean experimental break-up lengths for elliptical liquid bridges. Upper
and lower deviations in experiments are given in brackets.



Break-up length (mm) ~ change in length from the
critical height of the hypotheti-
cal companion circular bridge
Large elliptical bridge 64.90 (+0.10, -0.05) 2.863
Small elliptical bridge 32.29 (+0.09, -0.09) 2.74


The procedure that was used for the bridge generated by elliptical end plates

was virtually identical to that used in the calibration experiments using circular

end plates, described earlier. Figures 5-10(a) and 5-10(b) show the large elliptical

liquid bridge at two different stages before and near break-up. Figures 5-11(a)




Full Text

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Firstofall,IwouldliketothankProfessorRangaNarayananforhissupportandadvice.Hehasbeenbothamentorandafriend.Healwaysemphasizestheimportanceofenjoyingyourwork.Dr.Narayananisenthusiasticabouthisworkandthisisthebestmotivationforastudent.Hisdedicationtoteachingandhisphilosophyhasinspiredmetobeinacademia.IwouldliketothankNickAlvarez.Hestartedasanundergraduatestudenthelpingmewithmyexperiments.Then,hebecameco-authorofmypapers.ThemembersofmyPhDcommittee,Prof.OscarD.Crisalle,Prof.LocVu-Quoc,andProf.DmitryKopelevichalsodeservemygratitude.Also,IwouldliketothankProf.AlexOronforacceptingtobeinmydefense.IhavereallyenjoyedtakingclassesfromProf.Vu-Quoc,Prof.CrisalleandProf.Narang.Theirteachingphilosophiesofseeingthebigpicturehavedeeplyinuencedme.ManythanksgotomyfriendsOzgurOzenandBerkUstafortheirfriendship.Iamluckytobetheircolleague.ManythanksgotoSinemOzyurtforherconstantsupportthroughoutmygraduateeducation.IthankherforalwaysbeingtherewhenIneedher.Sheisveryspecialforme.Iwouldliketothankmybrother,ErdemUguz,whohasalwaysbeenwithme,andhasmotivatedmeformywork.Iwouldliketoexpressmyhighestappreciationformyparentsandmybrotherfortheirloveandsupportthroughoutmyeducationalcareer.Ithasbeendicultforthemandformebecauseofthelargedistance.Thankyouforyourpatience,encouragementandyourmoralsupport.IwouldliketothanktheUniversityofFloridaforanAlumniFellowship. iv

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page ACKNOWLEDGMENTS ............................. iv LISTOFTABLES ................................. vi LISTOFFIGURES ................................ vii ABSTRACT .................................... viii CHAPTER 1INTRODUCTION .............................. 1 1.1WhyWeretheRayleigh-TaylorInstabilityandLiquidBridgesStud-ied? ................................... 2 1.2OrganizationoftheThesis ....................... 6 2THEPHYSICSOFTHEPROBLEMSANDTHELITERATURERE-VIEW ..................................... 8 3AMATHEMATICALMODEL ....................... 16 3.1TheNonlinearEquations ........................ 16 3.2TheLinearModel ............................ 18 4THERAYLEIGH-TAYLORINSTABILITY ................ 21 4.1DeterminingTheCriticalWidthinRayleigh-TaylorInstabilitybyRayleigh'sWorkPrinciple ....................... 21 4.2ASimpleDerivationForTheCriticalWidthForTheRayleigh-TaylorInstabilityandTheWeaklyNonlinearAnalysisoftheRayleigh-TaylorProblem ............................. 23 4.3TheEectoftheGeometryontheCriticalPointinRayleigh-TaylorInstability:Rayleigh-TaylorInstabilitywithEllipticalInterface ... 27 4.4LinearandWeaklyNonlinearAnalysisoftheEectofShearonRayleigh-TaylorInstability ....................... 32 4.4.1InstabilityinOpenChannelCouetteFlow .......... 36 4.4.2Rayleigh-TaylorInstabilityinClosedFlow .......... 38 4.5Summary ................................ 58 5THESTABILITYOFLIQUIDBRIDGES ................. 61 v

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.................................... 61 5.2ASimpleDerivationToObtaintheDispersionCurveforaLiquidBridgeviaaPerturbationCalculation ................. 63 5.3TheEectofGeometryontheStabilityofLiquidBridges ..... 67 5.3.1TheStabilityofanEncapsulatedCylindricalLiquidBridgeSubjecttoO-Centering .................... 67 5.3.1.1Perturbedequations:1problem .......... 68 5.3.1.2Mappingfromthecenteredtotheo-centeredliq-uidbridge ....................... 70 5.3.1.3Determining2(1) 71 5.3.1.4Determining2(2) 75 5.3.1.5Resultsfromtheanalysisanddiscussion ...... 79 5.3.2AnExperimentalStudyontheInstabilityofEllipticalLiq-uidBridges ............................ 82 5.3.2.1Resultsonexperimentswithcircularendplates .. 86 5.3.2.2Resultsonexperimentswithellipticalendplates .. 88 5.4Shear-inducedstabilizationofliquidbridges ............. 90 5.4.1AModelforScopingCalculations ............... 92 5.4.2DeterminingtheBondNumber ................. 97 5.4.3TheExperiment ......................... 98 5.4.3.1Theexperimentalsetup ............... 98 5.4.3.2Theexperimentalprocedure ............. 100 5.4.4TheResultsoftheExperiments ................ 103 6CONCLUSIONSANDRECOMMENDATIONS .............. 109 APPENDIX ATHEPERTURBATIONEQUATIONSANDTHEMAPPING ...... 112 BSURFACEVARIABLES ........................... 115 B.1TheUnitNormalVector ........................ 115 B.2TheUnitTangentVector ........................ 116 B.3TheSurfaceSpeed ........................... 116 B.4TheMeanCurvature .......................... 117 CTHEVOLUMELOSTANDGAINEDFORALIQUIDJETWITHAGIVENPERIODICPERTURBATION ................... 119 DTHEEFFECTOFINERTIAINTHERAYLEIGH-TAYLORANDLIQ-UIDJETPROBLEMS ............................ 121 REFERENCES ................................... 124 BIOGRAPHICALSKETCH ............................ 129 vi

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Table page 5-1Physicalpropertiesofchemicals. ....................... 84 5-2Meanexperimentalbreak-uplengthsforcylindricalliquidbridges. .... 87 5-3Meanexperimentalbreak-uplengthsforellipticalliquidbridges. ..... 88 5-4Theeectoftheviscositiesonthemaximumverticalvelocityalongtheliquidbridgeinterface. ............................ 93 5-5Theeectoftheliquidbridgeradiusonthemaximumverticalvelocityalongtheliquidbridgeinterface. ....................... 95 vii

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Figure page 1-1Liquidbridgephoto .............................. 2 1-2Interfacebetweenheaviercoloredwaterontopoflightertransparentde-caneinaconicaltube ............................ 3 1-3Shadowgraphimageshowingconvection .................. 5 2-1Photographillustratingthejetinstability .................. 9 2-2Liquidjetwithagivenperturbation .................... 9 2-3Dispersioncurveforthejet ......................... 11 2-4Liquidbridgephotographfromoneofourexperiments .......... 11 2-5Cartoonillustratingoatingzonemethod ................. 13 4-1Sketchofthephysicalproblemdepictingtwoimmiscibleliquidswiththeheavyoneontopofthelightone ...................... 22 4-2SketchoftheRayleigh-Taylorproblemforanellipticalgeometry ..... 27 4-3Twoimmiscibleliquidswithdensitystratication ............. 34 4-4BasestatestreamfunctionforclosedowRayleigh-Taylorproblem ... 41 4-5BasestatevelocityeldforclosedowRayleigh-Taylorproblem ..... 42 4-6DispersioncurvesfortheclosedowRayleigh-TaylorproblemforCa=10andBo=5 ................................ 45 4-7ThedispersioncurvefortheclosedowRayleigh-Taylorshowingmulti-plemaximaandminimaforCa=20andBo=500 .............. 46 4-8Theeectofthewallspeedonthestabilityofshear-inducedRayleigh-TaylorforBo=50 ............................... 47 4-9TheeectofBoonthestabilityofshear-inducedRayleigh-TaylorforCa=20 ..................................... 48 4-10Theneutralstabilitycurvefortheshear-inducedowwhereCa=20 ... 49 4-11Theneutralstabilitycurvefortheshear-inducedowwhereCa=20 ... 50 viii

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............................. 57 5-1Volumeofliquidwithagivenperiodicperturbation ............ 62 5-2Centeredando-centeredliquidbridges .................. 68 5-3Thecross-sectionofano-centeredliquidbridge .............. 72 5-42(0)and2(2)(multipliedbytheirscalefactors)versusthewavenumberfor==1andR(0)0=R(0)0=2 ....................... 79 5-5Changein2(2)(multipliedbyitsscalefactor)forsmalltointermediatedensityratiosforscaledwavenumber(kR(0)0)of0.5andR(0)0=R(0)0=2 .. 80 5-6Changein2(2)(multipliedbyitsscalefactor)largedensityratiosforscaledwavenumberof0.5andR(0)0=R(0)0=2 ................ 81 5-7Changeof2(2)(multipliedbyitsscalefactor)versusoutertoinnerra-diusratioR(0)0=R(0)0forscaledwavenumberof0.5and==1 ..... 81 5-8Sketchoftheexperimentalset-upforellipticalbridge ........... 83 5-9Cylindricalliquidbridge ........................... 87 5-10Largeellipticalliquidbridge ......................... 88 5-11Smallellipticalliquidbridge ......................... 89 5-12Theschematicofthereturningowcreatedinthepresenceofanencap-sulantintheoatingzonetechnique .................... 92 5-13Theeectoftheheightofthebridgeonthemaximumaxialvelocityalongtheliquid/liquidinterface .......................... 95 5-14Theeectoftheencapsulant'sviscosityontheratioofmaximumspeedobservedattheinterfacetothewallspeed ................. 96 5-15Photographoftheexperimentalset-up ................... 99 5-16Acartoonofabridgebulgingatthebottom ................ 102 5-17Theeectofwallspeedonthepercentageincreaseinthebreakupheightofthebridgeforvariousinjectedvolumes .................. 104 5-18Theeectofthevolumeonthepercentageincreaseinthebreakupheightofthebridge .................................. 105 5-19TheeectofthewallspeedonthepercentageincreaseinthebreakupheightofthebridgeforvariousBondnumbers ............... 106 ix

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.... 107 C-1Thevolumeargumentforavolumeofliquidwithagivenperturbation 119 D-1Sketchoftheproblemdepictingaliquidontopofair ........... 121 x

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ThisdissertationadvancestheunderstandingoftheinstabilityofinterfacesthatoccurinRayleigh-Taylor(RT)andliquidbridgeproblemsandinvestigatestwomethodsfordelayingtheonsetofinstability,namely,changingthegeometryandjudiciouslyintroducinguidow.IntheRTinstability,itisshowntheoreticallythatanellipticalshapedinterfaceismorestablethanacircularoneofthesameareagiventhatonlyaxiymmetricdisturbancesareinictedonthelatter.Inacompanionstudyonbridges,itisexperimentallyshownthataliquidbridgewithellipticalendplatesismorestablethanacompanioncircularbridgewhoseendplatesareofthesameareaastheellipses.Usingtwodierentsizesofellipseswhosesemi-majoraxesweredeviatedfromtheradiiofthecompanioncirclesby20%,itwasfoundthattheellipticalbridge'sbreakupheightwasnearly3%longerthanthatofthecorrespondingcircularbridge. Anotherwaytostabilizeinterfacesistojudiciouslyuseuidow.Acom-prehensivetheoreticalstudyontheRTprobleminvolvingbothlinearandweaklynonlinearmethodsshowsthatmodeinteractionscandelaytheinstabilityofanerstwhileatinterfacebetweentwoviscousuidsdrivenbymovingwalls.Itis xi

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xii

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Thisdissertationinvolvesthestudyoftwointerfacialinstabilityproblemswiththeobjectivesofunderstandingtheunderlyingphysicsbehindtheinstabilitiesandndingwaystodelaythem.ThetwoproblemsaretheliquidbridgeandtheRayleigh-Taylorinstabilities.Aliquidbridgeisavolumeofliquidsuspendedbetweentwosolidsupports.Itcanbeheldtogetherwithoutbreakingowingtosurfacetensionforces.However,atsomecriticalheightthesurfacetensioneectsarenotstrongenoughtomaintaintheintegrityofthebridgebetweenthesupportingdisksandthebridgebecomesunstableandcollapses.AdepictionofastableandanundulatingbridgeisgiveninFigure 1-1 Theinstabilityoccursbecausethereisaplayobetweenpressuregradientsthataregeneratedduetotransversecurvatureandthosecausedbylongitudinalcurvature.Asthespacingbetweentheendplatesincreases,thelatterbecomesweak,animbalanceoccursandtheneckingbecomesmorepronouncedleadingtoultimatebreakup.TheRayleigh-Taylorinstability,ontheotherhand,isobservedwhenalightuidunderliesaheavyone,andthecommoninterfacebecomesunstableatsomewidth.Forlargeenoughwidths,thestabilizingsurfacepotentialenergyisinsucienttowithstandthedestabilizinggravitationalenergy.SuchaninstabilityisdepictedinFigure 1-2 .Abasicunderstandingoftheinstabilityisneededifthereisanyhopeofalteringthestabilitylimitby,say,changingthegeometryorbyapplyinganoutsideforcetogetmorestability.Afairquestiontoaskistowhythesetwoinstabilityproblemsarechosenisaddressednext. 1

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Liquidbridgephotoa)Stableliquidbridgeb)Unstableliquidbridgeathigherheight. BothliquidbridgeandRayleigh-Taylorproblemshavenumeroustechnologicalapplications.Liquidbridgesoccur,forexample,intheproductionofsinglecrystalsbytheoatingzonemethod[ 1 2 ].Theyoccurintheformofowingjetsintheencapsulatedoilowinpipelines[ 3 ].Inthemeltspinningofbers,liquidjetsemittingfromnozzlesaccelerateandthinuntiltheyreachasteadystateand

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Interfacebetweenheaviercoloredwaterontopoflightertransparentdecaneinaconicaltubea)Stableinterfaceb)Unstableinterfaceathigherdiameter. thentheybreakonaccountofinstability.Besidessuchtechnologicalapplicationsinmaterialsscience,liquidbridgeshaveimportanceinbiomedicalscience.Forexample,Grotberg[ 4 ]showsthevastscopeofbiouidmechanicsrangingfromtheimportanceofthecelltopologyinthereopeningofthepulmonaryairways[ 5 ]totheoccludingofoxygenresultingfromthecapillaryinstabilities[ 6 ].Inallthesestudies,themucusthatclosestheairwaysisrepresentedbyaliquidbridgeconguration. TheRayleigh-Taylorinstabilityalsoplaysaroleinanumberofsituations,somenatural,otherstechnological.Forexample,theinabilitytoobtainanycapil-laryriseinlargediametertubesisaresultoftheRayleigh-Taylorinstability.Whenauidbilayerisheatedfrombelow,itbecomestopheavyandtheinterfacecanbecomeunstableevenbeforeconvectionsetsinduetobuoyancy.Inastrophysics,theadversestraticationofdensitiesinthestar'sgravitationaleldisresponsiblefortheoverturnoftheheavyelementsincollapsingstars[ 7 ].Rayleigh-Taylorin-stabilityisalsoobservedininertialconnementfusion(ICF),whereitisnecessarytocompressthefueltoadensitymuchhigherthanthatofasolid.Rayleigh-Taylorinstabilityoccursintwodierentoccasionsduringthisprocess[ 8 ].

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Itisthecentralobjectiveofthisstudytoseehowtostabilizeliquidinterfacesbyapplyinganoutsideforceorbychangingthegeometryofthesystem.Forthatpurpose,understandingthephysicsofthesystem,includingthedissipationofdisturbancesandthenatureofthebreakupoftheinterfaceasafunctionofgeometryisveryimportant. Inapplicationsofliquidbridgessuchastheoatingzonetechnique,themoltencrystalissurroundedbyanotherliquidtoencapsulatethevolatilecomponentsandthepresenceoftemperaturegradientscausesow.Whethersuchowcancausestabilityornotisofinterest,sointhisstudyweshallconsidertheroleofshearinaliquidbridgeproblem.Anothereectthatisstudiedistheshapeofthesupportingsoliddisksonthestabilityofliquidbridges.Mostofthestudiesonliquidbridgespertaintobridgesofcircularendplates.Physicalargumentssuggestthatnoncircularbridgesoughttobemorestablesothisresearchalsodealswiththestabilityofnoncircularliquidbridges. Thecurrentresearchisbothexperimentalandtheoreticalincharacter.Thetheoreticalmethodsincludelinearstabilityanalysisviaperturbationcalculationsandweaklynonlinearanalysisviaadominantbalancemethod.Theexperimentalmethodsinvolvephotographyoftheinterfaceshapes.Theworkonliquidbridgeswillbeexperimentalinnatureonaccountofthedicultyinanalyzingtheproblemwithoutresorttocomputations.TheworkontheRayleigh-Taylorproblem,ontheotherhand,willbetheoreticalinnatureonaccountofdicultyinobtainingclearexperiments. Allinstabilityproblemsarecharacterizedbymodelsthatcontainnonlinearequations.Thismustbetruebecauseinstabilitybytheverynatureofitsdenitionmeansthatabasestatechangescharacterandevolvesintoanotherstate.Thefactthatwehaveatleasttwostatesisindicativethatwehavenonlinearityinthemodel.Ifthecompletenonlinearproblemcouldbesolved,thenallofthephysics

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Shadowgraphimageshowingconvection. wouldbecomeevident.However,solvingnonlinearproblemsisbynomeansaneasytaskandoneendeavorstondthebehaviorbylinearizationofthemodelaboutaknownbasestatewhosestabilityisinquestion.Thislocallinearizationissucienttodeterminethenecessaryconditionsforinstabilityandintheabsenceofacompletesolutiontothemodelingequationsitwouldseembenecialtoobtaintheconditionsfortheonsetoftheinstability.Todeterminewhathappensbeyondthecriticalpointrequirestheuseofweaklynonlinearanalysis.Oncetheinstabilitysetsin,theinterfacecreatedintheordinaryliquidbridgeproblemandRayleigh-Taylorcongurationevolvestocompletebreakup.However,undersomeconditionseventhismaynotbetrueandwewillseelaterinthisdissertationthatasecondarystatemaybeobtainedifshearisapplied.Thereareinterfacialinstabilityproblemsthathavebeenstudiedwherepatternsmaybeobservedoncetheinstabilitysetsin.AnexampleofthisistheRayleigh-Benardproblemproblem,whichisaproblemofconvectiveonsetinauidthatisheatedfrombelow.Whenthetemperaturegradientacrossthelayerreachesacriticalvalue,patternsarepredictedandinfactarealsoobserved.Figure 1-3 isaphotographofsuchpatternsseeninanexperiment.Thefactthatsteadypatternsarepredictedandobservedimpliesasortof"saturation"ofsolutionsthatmightbeexpectedinaweakly

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nonlinearanalysis,weakinthesensethattheanalysisisconnedtoregionsclosetotheonsetoftheinstability.ContrastthisbehaviorwiththatexpectedofthecommonRayleigh-Taylorproblemdiscussedearlier.Inthisproblemtheonsetoftheinstabilityleadstobreakupandnosaturationofsolutionsmaybeexpected.Allthiswillbecomeimportantinourdiscussionofthisproblemlateron. Chapter 2 outlinesthephysicsoftheinstabilityforbothproblems,namelyRayleigh-Taylorandliquidbridges.Thischapterincludesashortdiscussionofliquidjetsbecauseapreliminarystudyofliquidjetsformsthebasisforthestudyofliquidbridges.Inotherwordsmostofthephysicspertainingtoliquidbridgescanbeunderstoodmoreeasilybystudyingliquidjets.Ageneralliteraturereviewandapplicationsarealsogiveninthischapter. Chapter 3 discussesthegoverningequationsalongwithboundaryandinterfaceequationsintheirgeneralforms.Thetheoreticalmethodsrequiredtosolvetheseequationsisalsopresentedinthischapter. Chapter 4 focusesontheRayleigh-Taylorinstability.Intherstsection,thecriticalpointisfoundusingRayleigh'sworkprinciple.Then,thesameresultisobtainedbyaperturbationcalculation.Thisisfollowedbyacalculationthatshowstheeectofchangingthegeometryonthestabilitybyconsideringinstabilityinanellipticalinterfaceviaaperturbationcalculation.Thelastsectionpresentstheshear-introducedstabilizationoftheRayleigh-Taylorproblemwhereatheoryisadvanced.Thedispersioncurvesareplottedbyusinglinearstabilityanalysiswhilethetypesofbifurcationsaredeterminedviaaweaklynonlinearanalysis.

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Chapter 5 ,whichdealswithbridges,isorganizedinamannersimilartothepreviouschapter.First,thecriticalpointisdeterminedusingRayleigh'sworkprinciple.Then,aperturbationcalculationispresentedthatobtainsthesameresult.Thisisfollowedbyacalculationwheretheeectofo-centeringaliquidbridgewithrespecttoitssurroundingliquidonthestabilityoftheliquidbridgeisstudied.Whiletheideaofo-centeringseemsperipheraltoourobjectivesitdoesintroduceanimperfectionandisimportantbecausewemustmakesureinbridgeexperimentsthatthisimperfectionhaslittleifanyconsequence.Inadditionthiscongurationisanidealizationoftheuidcongurationthatappearsintheoatingzonecrystalgrowthtechnique.Thetheoreticalmethodtoinvestigatetheo-centeringprobleminvolvestheuseofanenergymethod.Thedetailsofthederivation,andthephysicalexplanationoftheresultsareemphasizedinthischapter.Thereafterthischaptercontainsthedetailsandresultsoftwoseriesofexperiments.Intherstseries,weinvestigatetheeectofthegeometryviathestabilityofellipticalliquidbridges.Aphysicalexplanationoftheeectofchangingtheendplatesofaliquidbridgefromcirclestoellipsesonthestabilityofliquidbridgesisgiventhroughthedissipationofdisturbances.Thebreakuppointofellipticalliquidbridgesisthendeterminedbymeansofexperiments.Thesecondseriesdealswiththeeectofshearonthestabilityofliquidbridges.Theexperimentsshowthestabilizingeectofreturningowinaliquidbridgeonitsstabilityandareassistedbyroughscopingcalculationsonthebasestate. Chapter 6 isageneralconclusionandpresentsascopeforafuturestudy.

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Thepurposeofthischapteristofamiliarizethereaderwiththebasicphysicsandtoprovideabriefoverviewoftheliterature.WeknowfromthepreviouschapterthatbothliquidbridgeandRayleigh-Taylorproblemsmaybecomeunstable.Here,wewillgivethedetailsoftheinstabilitymechanisms.Westartwithadiscussionofliquidjetsbecauseitservesasaprecursortothestudyofliquidbridges. Aliquidjetformswhenitejectsfromanozzleasinink-jetprintingandagriculturalsprays.Suchjetstosomeapproximationarecylindricalinshape.However,acylindricalbodyofliquidinuniformmotionoratrestdoesnotremaincylindricalforlongandlefttoitself,spontaneouslyundulatesandbreaksup.ApictureofsuchabodyofliquidisdepictedinFigure 2-1 .Giventhefactthatasphericalbodyofliquiduponperturbationreturnstoitssphericalshapeandabodyofliquidinarectangulartroughalsoreturnstoitsoriginalplanarcongurationwemightwonderwhyacylindricalvolumeofliquidbehavesasdepictedinthepictureleadingtoneckingandbreakup. ThephysicsoftheinstabilitycanbeexplainedbyintroducingFigure 2-2 ,whichdepictsavolumeofliquidwithaperturbationimposeduponit.IfviewedfromtheendsasinFigure 2-2 (a),thepressureintheneckexceedsthepressureinthebulgeandthethreadgetsthinnerattheneck.Thisisthetransversecurvatureeect.Itremindsusofthefactthatthepressureinsmalldiameterbubblesisgreaterthanthepressureinlargediameterbubbles.OntheotherhandifviewedfromtheperspectiveofafrontelevationasinFigure 2-2 (b),thepressureunderacrestislargerthanthepressureunderthetroughorneckandconsequently, 8

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Photographillustratingthejetinstability.ReprintedfromJournalofColloidScience,vol.17,F.D.RumscheidtandS.G.Mason,"Break-upofstationaryliquidthreads,"pp.260-269,1962,withpermissionfromElsevier. Liquidjetwithagivenperturbationa)Transversecurvatureb)Longi-tudinalcurvature(Adaptedfrom[ 10 ]). theliquidmovestowardstheneckrestoringthestability.Thisisthelongitudinalcurvatureeect.Thelongerthewavelengththeweakeristhisstabilizingeect.Thecriticalpointisattainedwhenthereisabalancebetweentheseosettingcurvatures. Thebreakupofliquidjetshasbeenextensivelystudied,bothexperimentallyandtheoretically.SuchstudiescanbetrackedbacktoSavart's[ 11 ]experimentsandPlateau'sobservations[ 12 ],whichledPlateautostudycapillaryinstability.TheoreticalanalysishadstartedwithRayleigh[ 13 14 ]foraninviscidjetinjected

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intoair.Neglectingtheeectsoftheambientair,Rayleighshowedthroughalinearstabilityanalysisthatallwavelengthsofdisturbancesexceedingthecircumferenceofthejetatrestwouldbeunstable.Hewasalsoabletodeterminethatoneofthemodeshadtogrowfaster.Rayleigh[ 15 ]conductedsomeexperimentsonthebreakupofjetsandobservedthatthedrops,whichformafterthebreakup,werenotuniform.Heattributedthisnonuniformitytothepresenceofharmonicsinthetuningforksheusedtosoundthejetandcreatethedisturbances.TheeectofviscositywasalsoconsideredbyRayleigh[ 16 ]fortheviscositydominantcase.ThegeneralcaseandthetheoryonliquidjetsissummarizedandextendedinseveraldirectionsbyChandrasekhar[ 17 ].TheexperimentalworkbyDonnellyandGlaberson[ 18 ]wasingoodagreementwithChandrasekhar'stheoryasseeninFigure 2-3 .Here,adimensionlessgrowthconstantisplottedagainstadimensionlesswavenumber,x.Thecriticalpointisreachedwhenthedimensionlesswavenumberisequaltounity.Intheirexperiments,DonnellyandGlaberson[ 18 ]alsosawthesortofnonuniformityofthedropsthatRayleighobserved.Lafrance[ 19 ]attributedthisphenomenontothenonlinearity.Throughhiscalculation,hewasabletomatchtheexperimentaldataforearlytimes.MansourandLundgren[ 20 ]extendedthecalculationforlargetimes. Insomeapplications,thejetissurroundedbyanotherliquidasintheoilowinpipelineswhereaninternaloilcoreissurroundedbyanannularregionofwater.Inthisregard,Tomotika[ 21 ]extendedtheRayleighstabilitytoaviscouscylindricaljetsurroundedbyanotherviscousliquid.AmoregeneralproblemwassolvedlaterusingnumericalmethodsbyMeisterandScheele[ 22 ]andthereaderisreferredtotherecentbookbyLin[ 23 ]foranoverviewofthephenomenaofjetbreakup.Althoughthestudyofliquidjetsstartedmorethanacenturyago,thistopicisstillrelevantduetoapplicationsinmoderntechnologysuchasnanotechnology[ 24 ].

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Dispersioncurveforthejet.ThesolidlinerepresentsChandrasekhar'stheory[ 17 ].ReprintedfromProceedingsoftheRoyalSocietyofLondonSeriesA-MathematicalandPhysicalSciences,vol.290,R.J.DonnellyandW.Glaberson,"Experimentsoncapillaryinstabilityofaliquidjet,"pp.547-556,1966,withpermissionfromtheRoyalSociety. WhenaliquidjetisconnedbetweentwosolidsupportsaliquidbridgeisobtainedasinFigure 2-4 .Thisliquidbridgecanattainacylindricalcongurationifitissurroundedbyanotheruidofthesamedensity. Liquidbridgephotographfromoneofourexperiments. LiquidbridgeshavebeenstudiedasfarbackasPlateau[ 12 ]whoshowedtheoreticallythatinagravity-freeenvironment,thelengthtoradiusratioofacylindricalliquidbridgeatbreakupis2.Thisinstabilitytakesplacebecauseofacompetitionbetweenthestabilizingeectoflongitudinalcurvatureand

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destabilizingeectoftransversecurvatureasintheliquidjets.However,whilethephysicsoftheinstabilityofcylindricaljetsandbridgesaresimilartherearesubtledierencesbetweenthesetwocongurations.First,thereisnonaturalcontrolparameterwhenstudyingtheinstabilityofjetswhilethebridgedoescomeequippedwithone;itisthelengthtoradiusratio.Second,thereisnomodewithamaximumgrowthrateintheliquidbridgeproblem. Toobtainacylindricalcongurationofaliquidbridgerequiresagravity-freeenvironment.Therearevariouswaystodecreasetheeectofthegravityduringanexperiment.Theseincludegoingtoouterspace,usingdensity-matchedliquids,orusingsmallliquidbridgeradii.TheeectofgravityisrepresentedbytheBondnumber,Bo,whichistheratioofgravitationaleectstotheeectofsurfacetensionandisgivenbyBo=gR2 25 ]. Liquidbridgeshaveoftenbeeninvestigatedfortheirimportanceintech-nologicalapplications,suchasintheoatingzonemethodforcrystalgrowthofsemi-conductors[ 1 2 ],fortheirnaturaloccurrencesuchasinlungairways[ 4 ]andforscienticcuriosity[ 25 26 ].Liquidbridges,astheyappearincrystalgrowthapplications,areusuallyencapsulatedbyanotherliquidtocontroltheescapeofvolatileconstituents.Theoatingzonemethodisusedtoproducehigh-resistivitysingle-crystalsiliconandprovidesacrucible-freecrystallization[ 27 ].Inthistechnique,amoltenzone,whichisdepictedinFigure 2-5 ,iscreatedbetweenapolycrystallinefeedrodandamonocrystallineseedrod.Theheatersaretranslated

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Cartoonillustratingoatingzonemethod. uniformlytherebymeltingandrecrystallizingasubstanceintoamoredesirablestate.Thecrystalgrowsasthemeltsolidiesontheseed.Theaimistoobtainstablemoltenzonesorliquidbridges.Gravityisthemajorprobleminthestabilityofthemelt.Onearth,becauseofthehydrostaticpressure,themeltzonehastobesmall,causingsmallcrystals.InthecaseofGaSbforexample,amaterialthatisusedinelectronicdevices,thecrystalthatcanbeobtainedisabout7:5mm[ 28 ].Themaximumstableheightofthemoltenzoneisdeterminedbygravity.However,withtheadventinmicrogravityresearch,ithasbeenpossibletoobtainlargerliquidzones.IthasbeenpossibletogrowGaAscrystalsof20mmdiameterbytheoatingzonetechniqueduringtheGermanSpacelabmissionD2in1993[ 29 ]. Apartfromgravity,thetemperaturegradientstronglyinuencestheshapeandstabilityofthecrystal.Thethermocapillaryconvectioninthepresenceofanencapsulantgeneratesashearowandthisshearowhasaneectontheoatzoneorbridgestability.Ourinterestliesinthestabilityofthezoneinthepresenceofshearow.Arecirculatingpatternappearsuponshear-inducedmotionandtheeectofthistypeofshearowonthebridgestabilityisaquestionofinterest.The

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focusoftheresearchisontheenhancementofthestabilityofthesebridgesbysuitablychangingthegeometryoftheendplatesorbyimposingshear. Manysatellitequestionscropupindeterminingthestabilityoftheliquidzoneinthepresenceofaclosedencapsulant:Whatistheroleoftheviscosityonthestabilityofthebridge?Whatistheroleofthecenteringofthebridge?Doo-centerbridgeshelptostabilizethebridgeitself?WewillanswerthesequestionsinChapter 5 ThesecondproblemofinterestofthisresearchisRayleigh-Taylorinstability.Itiswellknownthatifalightuidunderliesaheavyone,thecommoninterfacebecomesunstablewhenthewidthoftheinterfaceincreasesbeyondacriticalvalue.Theinstabilityiscausedbyanimbalancebetweenthegravitationalandthesurfacepotentialenergies.Thelatteralwaysincreasesuponperturbationanditsmagnitudedependsontheinterfacialtension.ThisproblemwasrstinvestigatedbyRayleigh[ 30 ]andthenbyTaylor[ 31 ].Iftheuidsareincompressibleandhaveuniformdensities,thethicknessesoftheuidlayersandtheviscositiesplaynoroleindeterminingthecriticalwidth,wc,whichisgivenbywc=r g[].Here,isthesurfacetension,gisthegravitationalconstant,andandarethedensitiesoftheheavyandlightuidsrespectively.Thenatureofthebifurcationisabackwardpitchfork,i.e.,whentheinstabilityinitiates,itprogressestocompletebreakup. Theinterestinstudyingthestabilityofadenseliquidlyingontopofalightliquidcontinuesbecauseofitsapplicationsinotherproblems.Forexample,Voltzetal.[ 32 ]appliedtheideaofRayleigh-Taylorinstabilitytostudytheinterfacebetweenglycerinandglycerin-sandinaclosedHele-Shawlikecell.AnotherdierentexampleofRayleigh-Taylorinstabilityisseenwhenmiscibleliquidshavebeenstudiedeithertoexaminethestabilityoffrontmovingproblemsinreactiondiusionsystems[ 33 ]ortounderstandthedynamicsofthemixingzoneinthe

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nonlinearregime[ 34 ].Inthisresearch,weareinterestedontheeectofgeometryandonshearonthestabilityoftheinterfaceinaRayleigh-Taylorconguration. Theequationsthatrepresentbothinstabilityproblemswithcorrespondingboundaryandinterfaceconditionsarepresentedinthenextsectionalongwiththemethodstosolvetheseequations.

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Thischapterincludestheequationsusedtoanalyzebothinstabilityproblemsandaregiveninvectorformsothatnospecialcoordinatesystemneedbechosen.Theycanthenbeadaptedtothespecicproblemofinterest.Thedierencesbetweentheproblemsandfurtherassumptions,whichwillsimplifythegoverningequations,willbepointedoutaseachproblemisstudied. Intherstchapter,wepointedoutthattheinstabilitiesarerelatedtothenonlinearitiesinthemodelingequations.Inthischapterwewillobservethatthemodelingequationsarenonlinearbecausetheinterfacepositioniscoupledtotheuidmotionandthetwodependuponeachother. @t+~vr~v=rP+~g+r2~v(3{1) Here~vandParethedimensionlessvelocityandpressureelds,gisthegravitationalconstant,andandarethedensityandviscosityoftheuidrespectively.Asimilarequationforthesecondphasealsoholds.Massconservationineachphaseisgovernedbythecontinuityequations.Foreachofthephases,itis 16

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Equations 3{1 and 3{2 representasystemoffourequationsinfourunknowns,thesebeingthethreecomponentsofthevelocityandthepressure.Wepostponethescalingoftheequationsasthescalesdependonthephysicalsystemofinterest.Dependingonthedimensionlessgroupsthatarise,severalsimplicationscanbemadeallofwhichwillbemadelaterforeachproblem. Wecontinuewiththemodelingequations.Allwallsareconsideredtobeimpermeable,therefore,~v~n=0holds.Here,~nistheunitoutwardnormal. Theno-slipconditionappliesalongthewalls,andgivesriseto~v~t=0holds.Here,~tistheunittangentvector. Attheinterface,themassbalanceequationisgivenby Intheaboveequationurepresentsthesurfacespeed.Thisequationyieldstwointerfaceconditionsasthereisnophase-changeattheinterface.Notethattheasteriskdenotesthesecondphase. Attheinterface,thetangentialcomponentsofvelocitiesofbothuidsareequaltoeachother,i.e., Theinterfacialtensionattheinterfacecomesintothepicturethroughtheforcebalance,whichsatises whereistheinterfacialtensionand2Histhesurfacemeancurvature.Observethatasthedirectionofthenormaldeterminesthesignoftherighthandside,wedon'twanttospecifyitssignyet.ThereaderisreferredtoAppendix B forthederivationofthesurfacevariablesinCartesianandcylindricalcoordinatesystems.

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ThetangentialandthenormalstressbalancesareobtainedbytakingthedotproductofEquation 3{5 withtheunittangentandnormalvectorsrespectively. Finally,thevolumesofbothliquidsmustbexed,i.e., whereV0istheoriginalvolumeofoneoftheliquids.Equation 3{6 impliesthatagivenperturbationtotheliquidsdoesnotchangetheirvolumes.Thisvolumeconstraintisthelastconditionneededtoclosetheproblem. Aswementioned,theequationsarenonlinear.Therstnonlinearityisobservedinthedomainequationbecauseofthe~vr~vterm.However,inmostoftheproblemswestudy,aswewillseeinthefollowingsection,thebasestateisquiescentandthistermisusuallynotneeded.Themainnonlinearitycomesfromthefactthattheinterfacepositiondependsontheuidmotionandtheuidmotiondependsonthepositionoftheinterface.Thisnonlinearityisseenvividlyinthenormalstressbalanceattheinterfaceforitisanequationfortheinterfaceposition.Toinvestigatetheinstabilityarisingfromsmalldisturbanceswemoveontothelinearizationoftheequations. Theinstabilityariseswhenasystem,whichwasinequilibrium,isdrivenawayfromtheequilibriumstatewhensmalldisturbancesareimposeduponitandwhenacontrolparameterexceedsacriticalvalue.Forexampleintheliquidbridgeproblem,thecontrolparametermaybethelengthofthebridgeofagivenradiusoritmaybethewidthofthecontainerintheRayleigh-Taylorproblem.Anequilibriumsystemissaidtobestableifalldisturbancesimposeduponit

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dampoutovertimeandsaidtobeunstablewhentheygrowintime.Nowifthesystembecomesunstabletoinnitesimalperturbationsatsomecriticalvalueofthecontrolparameteritisunconditionallyunstable.Itiscrucialtonotethatthedisturbancesaretakentobesmallforifastateisunstabletoinnitesimaldisturbancesitmustbeunstabletoalldisturbances.Also,thisassumptionleadstothelocallinearizationofthesystem.Thetheoreticalapproachthatistakenwhenstudyingtheinstabilityofthephysicalsystemisthereforetoimposeinnitesimaldisturbancesonthebasestateandtolinearizethenonlinearequationsdescribingthesystemaroundthisbasestate.Itshouldbepointedoutthatthebasestateisalwaysasolutiontothenonlinearequationsandoftenitmightseemdefeatingtolookforabasestateifitmeanssolvingthesenonlinearequations.However,inpracticeforalargeclassofproblemsthebasestateisseenalmostbyinspectionorbyguessingit.Forexample,forastationarycylindricalliquidbridgeinzerogravity,itisobviousthatthebasestateisthequiescentstatewithaverticalinterface.Ontheotherhand,forsomeotherproblems,onemightneedtodeterminetheowproleinthebasestateasseenintheshear-inducedRayleigh-Taylorproblem.Often,wetrytosimplifythegoverningequationsbymakingassumptionssuchascreepingoworaninviscidliquid.Theseassumptionsareemployedifthereisnolossofgeneralityinthephysicsthatweareinterested.Mostofthetimethesesimplicationscanbeintroducedafterthenonlinearequationsaremadedimensionless. Callingthebasestatevariableforvelocity,~v0,andindicatingtheamplitudeoftheperturbationby,thevelocityandalldependentvariablescanbeexpandedas Herez1isthemappingfromthecurrentstatetothebasestateatrstorder.ItsmeaningisexplainedintheAppendix A and,attheinterface,themappingatthis

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orderisdenotedbyZ1,avariable,whichneedstobedeterminedduringthecourseofthecalculation.Notethatthesubscriptsrepresenttheorderoftheexpansion,e.g.thebasestatevariablesarerepresentedbyasubscriptzero.Wecanfurtherexpandv1andothersubscript'one'variablesusinganormalmodeexpansion.Consequently,thetimeandthespatialdependenciesoftheperturbedvariablesareseparatedas whereistheinversetimeconstantalsoknownasthegrowthordecayconstant.Thecriticalpointisattainedwhentherealpartofvanishes. WewilldiscussRayleigh-Taylorinstabilityinthenextchapterandapplythemodeldevelopedinthischaptertothisproblem.

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Inthischapter,theinstabilityofaatinterfacebetweentwoimmiscibleuidswherethelightuidunderliestheheavyoneisstudied.Thechapteriscomposedoffoursections.Intherstsection,wewillemployRayleigh'sworkprincipletondthecriticalwidth,introducedinChapter 2 ,whichisgivenbywc=r g[].Inthesecondsection,weobtainthesameresultbyaperturbationcalculation,withacompanionnonlinearanalysis.Thelinearcalculationisusedinthethirdsectionwhereasimilarperturbationcalculationinconjunctionwithanothertypeofperturbationisusedtostudytheeectofaslightlydeviatedcircularcrosssectionintheformofanellipticalcrosssectiononthestabilitypoint.InthelastsectionwestudytheeectofshearontheRayleigh-Taylor(RT)instabilitywithalinearandnonlinearanalysis. 4-1 .Aheavyuidofdensityliesabovealightuidofdensityinacontainerofwidthw.WewillmakeuseoftheRayleighworkprincipleasadaptedfromJohnsandNarayanan[ 10 ]todeterminethecriticalwidthatwhichthecommoninterfacebecomesunstable. AccordingtotheRayleighworkprinciplethestabilityofasystemtoagivendisturbanceisrelatedtothechangeofenergyofthesystemwherethetotalenergyofthesystemisthesumofgravitationalandsurfacepotentialenergies.Thechangeinthelattercanbedetermineddirectlyfromthechangeinthesurfaceareamultipliedbyitssurfacetension[ 35 ].Consequently,thecriticalorneutralpointisattainedwhenthereisnochangeinthetotalenergyofthesystemforagiven 21

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Sketchofthephysicalproblemdepictingtwoimmiscibleliquidswiththeheavyoneontopofthelightone. disturbance.Tosetthesethoughtstoacalculation,letthedisplacementbe whererepresentstheamplitudeofthedisturbance,assumedtobesmall,andkisthewavenumbergivenbyn/w,wheren=1;2;.Thesurfaceareaisgivenby dxdx(4{2) wheredsisthearclength,givenbyds="1+dz dx2#1=2dx"1+1 2dz dx2#dx.Toorder2,thechangeinthepotentialenergycanbewrittenas 2Z2xdxwZ0dx(4{3) Notethatthesystemisintwo-dimensionsandtheaboveequationisinfacttheenergyperunitdepth.UsingZx=ksin(kx),Equation 4{3 becomes 42k2w(4{4)

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Thechangeinthegravitationalpotentialenergyperunitdepthisgivenby SubstitutingtheexpressionforZ,simpliestheaboveequationto 2g24wZ0cos2(kx)dx+wZ0cos2(kx)dx35=1 4g[]2w(4{6) ThetotalenergychangeisthereforethesumoftheenergiesgiveninEquations 4{4 and 4{6 ,i.e. 1 42wk2g[](4{7) Thecriticalpointisattainedwhenthereisnochangeintheenergy.Substitut-ingk=/wintoEquation 4{7 ,thecriticalwidthisobtainedas g[](4{8) Forallwidthssmallerthanthis,thesystemisstable.Itisnoteworthythatthedepthsoftheliquidsplaynoroleindeterminingthecriticalwidth. Inthenextsection,thesameresultisobtainedbyaperturbationcalculationandaweaklynonlinearanalysisfollows. ThephysicalproblemissketchedinFigure 4-1 .Thebottomuidinthiscalculationistakenasair.Theliquidisassumedtobeinviscid.

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TheEulerandcontinuityequationsare and ThesedomainequationswillbesolvedsubjecttotheforcebalanceandnomassowattheinterfaceconditionsgiveninChapter 3 ,namely, and Thebasestateisassumedtobestationary.Toinvestigatethestabilityofthebasestate,linearstabilityanalysisdescribedinChapter 3 isemployed.Fortheperturbedproblem,theequationofmotionandthecontinuityequationresultsin Thewallsareimpermeabletoow;asaresultthenormalcomponentofthevelocityiszero,orintermsofpressurewecanwrite Freeendconditionsarechosenforthecontactoftheliquidwiththesolidsidewalls,i.e., Therefore,eachvariablecanbeexpandedasacosinefunctioninthehorizontaldirection,e.g.,Z1=^Z1cos(kx)wherek=n/w.Fromtheno-owconditionweget

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Usingtheconstant-volumerequirement,whichstateswR0Z1dx=0,theper-turbedpressure,whichwasalreadyfoundtobeaconstant,isdeterminedtobezero.Also,Z1isfoundasAcos(kx).ThecriticalpointisdeterminedbyrewritingEquation 4{16 as Thesquareofthecriticalwavenumberisg =G.Substitutingk=/w,thecriticalwidthisobtainedas g(4{18) whichissameasEquation 4{8 .Now,ouraimistondwhathappenswhenthecriticalpointisadvancedbyasmallamountasG=Gc+2.Theresponsesofthevariablestothischangeinthecriticalpointaregivenas Beforemovingtotheweaklynonlinearanalysis,let'srewritethedomainequationas ~vr~v=1 Whentheexpansionsaresubstitutedintothenonlinearequations,tothelowestorderin,thebasestateproblem,totherstorder,theeigenvalueproblemwherethecriticalpointisdetermined,arerecovered.Thesecondorderdomainequationbecomes 0=1

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Boththedomainequationandtheno-masstransferconditionattheinterfacegives 0=1 Hence,P2isaconstant.Thenormalstressbalanceatthisorderis Thepressure,whichisaconstant,turnsouttobeequaltozerobyusingtheconstantvolumerequirement.ThereforeZ2isfoundasBcos(kx).TodeterminethevalueofA,hencethetypeofthebifurcation,thethirdorderequationsarewritten.Thedomainequationis Observethatatthisorderthereisacontributiontothepressurefromthesecondorderandthedenominatorofthecurvaturealsoshowsitssignatureatthisorder.P3turnsouttobeequaltozeroasinthepreviousorders.Solvabilityconditiongives whichcanbesimpliedto 8A3k4=0(4{27) AsA2isnegative,GneedstobewrittenasG=Gc2whichyieldsapositiveA2.Therefore,thebifurcationtypeisabackwardpitchfork.

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ThephysicalproblemissketchedinFigure 4-2 .Observethattheradialpositiondependsontheazimuthalangle. SketchoftheRayleigh-Taylorproblemforanellipticalgeometry. ThemodelingequationsdeterminingthefateofadisturbanceareintroducedinChapter 3 .Inthisproblem,weareconsideringinviscidliquidsandthebasestateisaquiescentstatewheretheinterfaceisat.Thereforethenonlinearequationshaveatleastonesimplesolution.Itis andZ0=0.Weareinterestedinthestabilityofthisbasestatetosmalldistur-bances.Forthatpurposeweturntoperturbedequations.Theinterfaceposition

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canbeexpandedas 22Z2+(4{29) Torstorderuponperturbation,theequationsofmotionandcontinuityare intheregionZ(r;;t;)zL.Combiningthetwoequationsweget withsimilarequationforthe'*'uid.Thecorrespondingboundaryconditionsarealsowrittenintheperturbedform.Theno-owconditionatthesidewallsiswrittenas whichisvalidatr=R().Beforeintroducingtheremainingboundaryconditions,wewanttodrawtheattentionofthereadertothisboundarycondition.Theequationiswrittenattheboundary,whichdependsontheazimuthalangle.Thisisaninconvenientgeometry.Therefore,tobeabletocarryoutthecalculationinamoreconvenientgeometry,wewanttouseperturbationtheoryandwritetheequationsatthereferencestate,whichhasacircularcrosssection. TheobjectiveistoshowthattheRTproblemwithellipticalinterfaceismorestablethanacompanionRTproblemwheretheinterfaceiscircular.Theareaoftheellipseisassumedtobethesameasthatofthecircle.Also,theellipseisassumedtodeviatefromthecirclebyasmallamountsothataperturbationcalculationcanbeused.Astheellipseisconsideredasaperturbationoftheellipse,rstthemappingobtaininganellipsefromacircleneedstobedetermined. Assumethattheellipseisdeviatedfromthecirclebyasmallamountsothatthesemi-majoraxis"a"oftheellipseisdenedasa=R(0)[1+],whereisthe

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radiusofthecirclefromwhichtheellipseisdeviated.Then,thesemi-minoraxis"b"oftheellipseiscalculatedbykeepingtheareastobethesame,i.e.,R(0)2=ab 22R(2)+(4{33) ThemappingsR1andR2canbefoundusingtheequationforellipse,whichisgivenby Substitutingthedenitionsforx,andy,whichareRcos()andRsin(),respectively,alsomakinguseoftheexpansionsfora,b,andR,onegetsthemappingsas torstorderin,and 2cos(2)+3 2cos(4)(4{36) tosecondorderin. Thegeometryofthephysicalsystemisdeterminedthroughaperturbationcalculation.Now,wecanreturntoourperturbationcalculation. Theno-owboundaryconditionsatthereferenceinterface,i.e.,z=0,andatthetopwall,i.e.,z=H,fortheperturbedpressurecanbewrittenas

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ThereforeP1isaconstant,whichisfoundateachorderinusingconstant-volumerequirement.Attheouterwall,thecontactangleconditionreadsas @@Z1 Thenormalstressbalanceattheinterfaceis where@P0 4{39 canberewrittenas where2=g .Now,eachvariableisexpandedinpowersofas 22Z(2)1+(4{41) Similarly,whichdeterminedthecriticalpointisexpandedas 22(2)2+(4{42) Here,(0)2representsthecriticalpointofthecircletoaxisymmetricdisturbances.Higherordertermsinarethecorrectionsgoingfromacircletoanellipse. Tozerothorderin,theRTproblemwithacircularcross-sectionisrecovered.Thenormalstressbalanceatthisorderis Fromtheaboveequation,Z(0)1=AJ0(0)R(0)+c(0)1

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Attheouterwall,@Z(0)1 Torstorderin,thenormalstressbalanceisgivenby Attheouterwall,@Z(1)1 4{43 ismultipliedwithZ(1)1andintegratedoverthesurface,fromwhichtheintegraloftheproductofEquation 4{45 withZ(0)1issubtracted.Itturnsoutthat(1)2=0asonewouldhaveexpected.Itmeansthatthemajorandminoraxisoftheellipsecanbeippedandthesameresultwouldbestillvalid.TheformofZ(1)1canbefoundfromEquation 4{45 as TheconstantBisfoundfromtheouterwallconditionas Asimilarapproachistakenatsecondorderin.Thenormalstressbalanceatthisorderis Thesolvabilityconditiongives where^Z(2)1istheindependentpartofZ(2)1.Z(0)1isknown,and^Z(2)1canbefoundfromtheoutsidewallconditiongivenas

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Aftersomealgebraicmanipulations,anequationfor(2)2isobtainedas As(2)2isapositivenumber,thestabilitypointisenhanced,whichwasexpectedbecauseofthedissipationofthedisturbancesargument. D thatsucharesultalsoobtainsifcreepingowisassumedwhiledestabilizationcanbeobtainedifonlyinertiaistakenintoaccount.Theclosedowgeometryishoweverdierent.ItisshowninthischapterthatshearingtheuidsbymovingthewallsstabilizestheclassicalRTproblemeveninthecreepingowlimitprovidedaatinterfaceisanallowablebasesolution.Thisresultwouldobtainonlyifbothuidlayersaretakenasactive.Aninterestingconclusionoftheclosedowcaseisthatforaselectedchoiceofparameters,threedierentcriticalpointscanbeobtained.Therefore,thereisasecondwindowofstabilityfortheshear-inducedRTproblem.Tounderstandthenatureofthebifurcation,aweaklynonlinearanalysisisappliedviaadominantbalancemethodbychoosingthescaledwallspeed(i.e.,Capillarynumber)asthecontrolparameter.Itwillbeshownthattheproblemhaseitherabackwardorforwardpitchforkbifurcationdependingonthecriticalpoint. Theinterestintheeectofshearontheinterfacialinstabilityisnotnew.ChenandSteen[ 36 ]showedthatwhenconstantshearisappliedtoaliquidthatis

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aboveanambientgas,areturnowiscreatedintheliquiddeectingtheinterface.Giventhatthesymmetryisbroken,thestabilitypointisreduced,i.e.,thecriticalwidthatwhichtheinterfacebreaksupislowerthantheclassicalRTlimitgivenearlier.However,ifaatinterfaceispossible,thesituationmaybedierent.Theimportanceofaatinterfaceatthebasestateisseeninvariousotherinterfacialinstabilityproblems;forexampleHsieh[ 37 ]studiedtheRTinstabilityforinvisciduidswithheatandmasstransfer.Hewasabletoshowthatevaporationorcondensationenhancesthestabilitywhentheinterfaceistakentobeatinthebasestate.Ho[ 38 ]advancedthisproblembyaddingviscositytothemodelwhileconsideringthelateraldirectiontobeunbounded.Withaatbasestate,theseauthorswereabletoobtainmorestablecongurationsthantheclassicalRTproblem.Thereasonforthestabilityofaninterfaceofconstantcurvatureduringevaporationisduetotheuidowinthevapor,whichtendstoreduceinterfacialundulationsandisevenseeninproblemsofconvectionwithphasechange[ 39 ].Thereareotherproblemswherethestabilityofaconstantcurvaturebasestatehasbeenenhancedeitherbyimposingpotentialthatinduceshear[ 40 ].TheseworksmotivateustostudytheeectofshearontheRTproblemwithaconstantcurvaturebasestateandinquirewhetherthecriticalwidthoftheinterfacechangesandifso,whyandbyhowmuch.Inmanyinterfacialinstabilityproblemsthephysicsoftheinstabilityisstudiedbyexplainingtheshapeofthegrowthcurveswhereagrowthconstant,,isgraphedagainstadisturbancewavenumberandinmost,butnotallproblemsthecurveshowsamaximumgrowthrateatnon-zerovaluesofthewavenumber.Heretoo,itisouraimtounderstandthephysicsofsheareectsbyconsideringsimilargrowthratecurveswherethewavenumberisreplacedbyscaledcontainerwidth.Finally,itisofinteresttoseewhatthenatureofthebifurcationbecomeswhenshearisimposedontheRTproblem.Totheseendswemovetoamodel.

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(b) Twoimmiscibleliquidswithdensitystraticationa)Openchannelowb)Closedow. Thephysicalproblemconsistsoftwoimmiscibleliquidswheretheheavyoneoverliesthelightonewhenshearispresent.Theshearisintroducedbymovingthelowerandbottomwallsatconstantspeed.Theparametersintheproblemsuchasthedepthsoftheliquidcompartments,thephysicalpropertiesoftheliquidsandthewallspeedsaretunedtoattainaatinterfacebetweenthetwoliquids.Twoproblemsareconsideredinthisstudy.Intherst,thehorizontalextentistakentobeinnity,whileinthesecond,theuidsareenclosedbyverticalsidewalls.Thepurposesofconsideringtheopenchannelowproblemaretointroducenecessaryterminologyandtounderstandsomeimportantcharacteristics,whichwillbeinstructivewhenconsideringtheclosedowproblem.AsketchofthephysicalproblemcanbeseeninFigure 4-3 ThetwocongurationsseeninFigure 4-3 arequitedierentfromeachother.Inboth,aheavyliquidisontopofthelightoneandsheariscreatedbymovingthewalls.Thewavestravelintheopenchannelowwhereasintheclosedow,theperturbationsareimpededbythewalls.Infact,thepresenceofthesidewallscreatesareturnow,whichoughttoaectthestabilityoftheinterface.Intheopenchannelow,thespeedofthelowerandupperwallsmustbe

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dierentotherwisenoeectivemotionwillbeobserved.Inbothcongurations,itisassumedthatthewallsaremovedslowlyenoughsothattheinertiaisignored. Thescaledequationofmotionandthecontinuityequationforaconstantdensityuidwiththecreepingowassumptionaregivenby Equations 4{52 and 4{53 arevalidinZ(x)z1.Similarequationsforthelowerphasecanbewrittenas Thelowerliquidisrepresentedby*.Thevelocityscaleisvandischosentobethecapillaryvelocity,i.e.,=whereistheviscosityoftheupperliquid.Theover-barsrepresentthescalefactors.ThepressurescalePisgivenbyv=L.Thelengthscaleistakentobetheuppercompartment'sdepth,L.ThedimensionlessvariablesBandBaregivenbygL2 Notethat,theno-slipconditionatthebottomwallgivesrisetotheCapillarynumber,i.e.vx=U =Ca,wherevxisthex-componentofthescaledvelocity.Similarequationscanbewrittenatthetopwall.Inadditiontotheconditionsatthetopandbottomwallsotherconditionsholdattheuid-uidinterface.Here,masstransferisnotpermitted,theno-slipconditionandtheforcebalancehold.

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Also,thevolumesofbothliquidsmustbexed.TheseconditionsaregiveninChapter 3 andwillnotberepeatedhere. Fortheclosedowproblem,theboundaryconditionsontheverticalwalls,whicharelocatedatx=0andw=Larealsospecied.Thesewallsareimperme-ableandtogetananalyticsolutionareassumedtobestress-free.Theseboundaryconditionstranslateinto WeareusinglinearstabilityanalysisasdescribedinChapter 3 .Theroleofthewallspeedonthecriticalpointisquestioned.Therstproblem,i.e.,theinstabilityinopenchannelowispresentedinthenextsection. 4-3 (a). Theconditionsforaatinterfaceinthebasestatearedeterminedbyusingthenormalstressbalanceattheinterface.Foragivenviscosityratio,arelationbetweenthewallspeedandtheratioofthecompartmentlengthsisestablished.Itturnsoutthatiftheviscositiesofbothliquidsandtheliquiddepthsarethesame,thenthenormalstressbalanceisautomaticallysatised.Thebasestatevelocityproleinthehorizontaldirection,i.e.vx;0,islinearwhereasvz;0isequaltozero.Todeterminethestabilityofthisbasestate,theperturbedstateissolvedbyeliminatingvx;1infavorofvz;1byusingthecontinuityequation.Consequently,thedomainequationfortheperturbedstatebecomes

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wherether4operatorisdenedas@4 3{8 .Then,^vz;1isassumedtobe^^vz;1(z)eikxwherekisthewavenumber.FromEquation 4{58 ,theformofthevelocitycanbeexpressedas^^vz;1(z)=C1ekz+C2zekz+C3ekz+C4zekz Asimilarequationisvalidatthebottomwall.Attheinterfacetheperturbedno-masstransferconditionbecomes andtheperturbedno-slipconditionattheinterfaceis whiletheperturbedtangentialstressbalanceisgivenby Theperturbedvelocitiesvz;1andvz;1arefoundintermsofandZ1byusingtheaboveequations.Then,theseexpressionsforthevelocitiesaresubstitutedintothenormalstressbalance,whichisgivenby

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Thepressuretermsfromthenormalstressbalanceareeliminatedbyusingtheequationsofmotion.Afterthesesubstitutions,Equation 4{63 becomes whereBoistheBondnumberdenedasBo=gL2[] 4{64 ,aftersomealgebraitisfoundthattheneutralpointoftheopenchannelowisthesameasthatoftheclassicalRTproblembutthattheneutralpointisanoscillatorystate,i.e.theimaginarypartofisnotzero.Thisresultisinagreementwithphysicalintuition.OnemightexpectthattherealpartofthegrowthconstantswouldbeindependentofCapillarynumberastheymustbeindependentofthedirectionofthewallmovement.ItmustbenotedthatthegrowthconstantcannotdependonthesquareofCa,asthebasestateproblemishomogeneousintherstpowerofCa.Theimaginarypartof,ontheotherhand,mustappearinconjugatepairsandthereforemustdependhomogenouslyonCa.Ingeneral,theoscillationatthecriticalpointisnotsurprisingbecausetheperturbationsarecarriedwiththemovingbottomwallandtheyarenotimpededinthehorizontaldirection.ThiswillchangeinthesecondproblemwheretheshearinducedRTinstabilityinaclosedcontainer,isstudied. @zandvz=@ @x(4{65)

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Aftertakingthecurloftheequationofmotionr4=0for0
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wherek0=n0 w/Lwithn0=1;2;.Asimilarresultcanbeobtainedforthe*phase.Atthetopwall,no-penetrationandno-slipimplyvx;0=aCa)Xn0sin(k0x)d^0;n0 Similarequationscanbewrittenforthebottomwall.First,aatinterfaceforthebasestateisassumedandthentheconditionsthatallowitarefoundfromthenormalcomponentoftheinterfacialforcebalance.Now,attheinterface,themassbalanceturnsinto Theno-slipconditionbecomes andthetangentialstressbalancecanbewrittenas whichgives Byusingtheeightconditionsgivenabove,0and0aredeterminedintermsofCa.Then,theexpressionsaresubstitutedintothenormalstressbalance,whichisgivenby

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Figure4-4. BasestatestreamfunctionforclosedowRayleigh-TaylorproblemforCa=1,w/L=1. Replacingpressureswiththestreamfunctions,thenewformofthenormalstressbalanceisgivenas Itturnsoutthatthenormalstressbalanceissatisedifandonlyiftheviscositiesofbothliquids,thecompartmentdepths,andupperandlowerwallspeedsarethesame,i.e.,=;L=L;a=1.Withtheseconditions,thestreamfunctionsforbothuidsarethesame,i.e.,0=0.TheplotsofthestreamfunctionsandthevelocityeldscanbeseeninFigures 4-4 and 4-5 Thestabilityofthisbasestateisstudiedinthenextsectionbyintroducingtheperturbedequationsandsolvingtheresultingeigenvalueproblem.

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Figure4-5. BasestatevelocityeldforclosedowRayleigh-TaylorproblemforCa=1,w/L=1. fortheupperphase.Similarly,forthelowerphase arevalid.Theyaresolvedbyaprocedurethatwasusedforobtainingthesolutionforthebasestateandrequiretheuseoftheperturbedboundaryconditions.Atthebottomwall,locatedatz=1,theperturbedno-slipandtheno-penetrationconditionsgiveriseto Asimilarequationisvalidatthetopwall.Notethat,theindexthatwasn0atthebasestateisnowchangedton1.Theseindiceswillplayabigroleinthecourseofsolvingtheperturbedequationsandsoparticularattentionshouldbepaidtothem.Attheinterface,massbalanceissatisedandthus ^1;n1=^1;n1(4{79)

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and Observethatthexandzdependentpartsofthevariablesintheaboveequationwerenotseparated,becausethereiscouplingbetweenthemodesandeachvariableneedstobewrittenasasummation.Accordingly,Equation 4{80 becomesXn1n1 w/L^1;n1cosn1 w/Lx=Xm1^Z1;m1cosm1 w/LxXn0n0 w/Ld^0;n0 w/Lx w/L^Z1;m1sinm1 w/LxXn0d^0;n0 w/Lx+Xm1^Z1;m1cosm1 w/Lx(4{81) Theno-slipconditionattheinterfaceatthisorderbecomes whilethetangentialstressbalanceisgivenby Theviscositiesdonotappearinthetangentialstressbalance,becauseaatbasestateissatisedonlywhentheviscositiesofbothuidsareidentical.ByusingEquation 4{78 anditscounterpartforthetopuid,andEquations 4{79 4{82 ,and 4{83 ,sevenoftheconstantsofthestreamfunctionsaredeterminedintermsofA1.Thusthestreamfunctionscanbewrittenas ^1;n1(z)=A1^1;n1(z)and^1;n1(z)=A1^1;n1(z)(4{84) where^1;n1and^1;n1areknown.ThelastcoecientA1isdeterminedbyusingEquation 4{81 ,whichcanthenbewrittenas

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w/L^1;n1cosn1 w/Lx=Xm1^Z1;m1cosm1 w/Lx1 2Xm1Xn0n0 w/L^Z1;m1d^0;n0 w/Lx+cos[m1+n0] w/Lx 2Xm1Xn0m1 w/L^Z1;m1d^0;n0 w/Lxcos[m1+n0] w/Lx(4{85) ToreduceEquation 4{85 intoitsmoments,itismultipliedbycosj w=Lxandintegratedoverx.Aftersomemanipulations,Equation 4{85 becomes w/L^1;n1=^Z1;n1+1 2n1 w/LXn0d^0;n0 Intheaboveequation,Z1;(j)=Z1;(j)wherejisapositiveinteger.Notethatj=0isruledoutbytheconstant-volumerequirementgiveninEquation 3{6 .Thelastcoecient,A1,isfoundbysubstitutingEquation 4{84 intoEquation 4{86 ,i.e., w/LA1^1;n1=^Z1;n1+1 2n1 w/LXn0d^0;n0 ObservethatEquation 4{87 isevaluatedatz=0.Toclosetheproblem,thenormalstressbalanceisused.Itiswrittenas Whenthestreamfunctions^1;n1and^1;n1aresubstitutedintoEquation 4{88 ,aneigenvalueproblemoftheformM^Z1=^Z1isobtained.Here,aretheeigenvaluesandMisanondiagonalmatrixthatoccursassuchbecauseofthecouplingbetweenthemodes.Asintheopenchannelow,ouraimistoseetheeectofthewallspeedortheCapillarynumberontheRTinstability.Theinputvariablesarethephysicalpropertiesoftheliquids,thewidthofthebox,thedepth

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oftheliquids,andthewallspeed.Intermsofdimensionlessvariables,theyareBo,w/L,andCa.Theoutputvariablesarethegrowthconstant,ormorepreciselytherealandtheimaginarypartsofandtheeigenmodes. (b) DispersioncurvesfortheclosedowRayleigh-TaylorproblemforCa=10andBo=5.a)Theordinateistheleadingeigenvalue,i.e.,35.b)Theordinateoftheuppercurveistheleading,andtheordinateofthesubsequentcurvesare30th,25th,and20threspectively. ThereareinniteeigenvaluesbecauseofthesummationofinnitetermsinEquation 4{87 .ThesizeofthematrixMdependsonthenumberoftermstakenintheseries,whichisdeterminedbytheconvergenceoftheleadingeigenvalue.Inthesecalculations,35termssucedforallvaluesofparameters.TheeigenvaluesarefoundusingMaple9TM.InFigure 4-6 (a),therealpartoftheleading,namely35,isplottedagainstw=L.Avarietyofobservationscanbemadefromthisdispersioncurvebutrstthereasonfortheinstabilityisgiven.Thestabilizingmechanismsareduetotheviscositiesoftheliquidsandthesurfacetension.Ontheotherhand,transversegradientsofpressurebetweencrestsandtroughs,whichdependonwidth,aswellasgravity,whichiswidthindependent,destabilizethesystem.Whenthewidthisextremelysmall,approachingzero,thesystemisstableandthegrowthconstantapproachesnegativeinnity.Thisbehaviorisrelatedtothestabilizingeectofthesurfacetension,whichactsmorestronglyonsmallwidths,inotherwords,onlargecurvature.Whenthewidthbecomeslarger,the

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Figure4-7. ThedispersioncurvefortheclosedowRayleigh-Taylorshowingmul-tiplemaximaandminimaforCa=20andBo=500. surfacetensioncannolongerprovideasmuchstabilizationand,asaresult,thecurverisestoneutrality,wherethereisabalancebetweentheopposingeects.Forlargerwidththesurfacetensioneectsgetweakerandconsequently,thedestabilizingforcesbecomedominantandthegrowthcurvecrossestheneutralstateandbecomespositive.Asthewidthincreasesevenmore,thecurvecontinuesrisingbutatsomepointitpassesthroughamaximumandstartsdecreasingascanbeseeninFigure 4-7 .Thiscallsforanexplanation.Thisphenomenon,distinctiveoftheclosedowproblem,isattributedtotheinteractionofthemodes.Asthewidthincreases,highermodesmustbeaccommodated.Thishasadualeect;whenahighermodeisintroduced,thewavesbecomechoppierandsurfacetensionactstostabilizethehighermode,whiledestabilizingtransversepressuregradientsalsoactmorestrongly.Furtherincreaseinthewidthcausesanincreaseinthedistancebetweencrestsandtroughsandthestabilizingeectofsurfacetensionbecomesweakerasalsodoesthedestabilizingeectoftransversepressuregradients.Asthewidthincreases,moreandmoremodesnowneedtobeaccommodated.Consequently,thegrowthcurveshowsmultiplemaximaandminimaascanbeseeninFigure 4-7

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(b) Theeectofthewallspeedonthestabilityofshear-inducedRayleigh-TaylorforBo=50.a)ThegraphscorrespondtoCa=1(themostuppercurve),Ca=4,10,15,20,100,500,and5000.b)Close-upviewnearthecriticalpointforCa=10(themostleft),Ca=15,20,and100. Insummary,theinclusionofahighermodeasthewidthincreasesrstmakesthewaveschoppier;butafurtherincreaseinthewidthmakesthewavesinthenewmodelesschoppy.Thus,stabilizinganddestabilizingeectsthatarewidthdependentgetreversedinstrength.InFigure 4-6 (b),therealpartoftheleadingandsomeofthelowergrowthconstantsareplottedforsmallwidths.Thepatternoftheothercurvesissimilartothatoftheleadingone.However,moretermsareneededinthesummationinEquation 4{87 fortheconvergenceofthesecurvesinFigure 4-6 (b). OuraimistoseetheeectofthewallspeedontheRTinstability.Forthatpurpose,inFigure 4-8 thedispersioncurvesfortheleadingareplottedagainstw=LforseveralCapillarynumbersataxedBondnumber.EachcurveshowsasimilarbehaviortothecurvespresentedinFigure 4-6 .Asthewidthincreasesfromzero,thecurvesincreasefromnegativeinnity.Theythenexhibitseveralmaximaandminima.ForlargeCa,therstmaximumoccurswhenisnegative,i.e.,thesystemisstable.Ontheotherhand,forsmallCa,e.g.Ca=1,therstmaximumisobservedwhenthesystemisunstable.So,whenthecurvestartsdecreasing,thesystembecomeslessunstable,butitremainsunstable.Averyinterestingfeatureis

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Figure4-9. TheeectofBoonthestabilityofshear-inducedRayleigh-TaylorforCa=20.ThecurvescorrespondtoBo=200(Themostuppercurve),150,110,65,50,and5. observedfortheintermediateCapillarynumbers.Therstmaximumisseenclosetotheneutralpoint.Interestinglyenough,theeigenvaluebecomesnegativeonemoretime.Forthosecurves,likethesecondcurvefromthetopinFigure 4-8 (a),itispossibletoobtainadispersioncurvethathasthreecriticalpoints.Inotherwords,therearetworegionsforthewidthwherethesystemisstable.ThesizeofthissecondstablewindowdependsonCaandBo.Thisstabilityregionbuildsabasisforaveryinterestingexperiment.TheeectofthewallspeedonthecriticalpointcanbeseeninFigure 4-8 (b),whichisaclose-upviewofFigure 4-8 (a).Thesystembecomesmorestableasthewallsaremovedfaster.InFigure 4-8 ,thedispersioncurveisplottedataxedBondnumberfordierentCapillarynumberswhileinFigure 4-9 ,theCapillarynumberiskeptxedandthecurvesaresimilar.ThecriticalpointsarecollectedandtheneutralcurveisobtainedinFigure 4-10 TheneutralcurvedepictedinFigure 4-10 isnotamonotonicallydecreasingcurve.ItisclearthatforsomeBonumbersthereexistthreecriticalpoints.AneutralcurveexhibitingthreedierentcriticalpointsforagivenwavenumberisseeninthepureMarangoniproblem[ 42 ].However,itshouldbenotedthatwhengravityisaddedtotheMarangoniproblem,itdoesnotexhibitthezerowavenumberinstabilityseeninthepureMarangoniproblemandconsequently,

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Figure4-10: Theneutralstabilitycurvefortheshear-inducedowwhereCa=20. doesnothavethreecriticalpoints.Thegravityisabletostabilizethesmallwavenumberdisturbances.Adispersioncurve,andthereforeaneutralcurvesimilartothoseobtainedinthisstudywasobservedbyAgarwaletal.[ 43 ]inasolidicationproblem.Besidestheseexamples,suchadispersioncurveisnotcommoninmostinterfacialinstabilities.Ifonewantstocomparethestabilitypointoftheshear-inducedRTproblemtothatoftheclassicalRTproblem,itwouldbemorepracticaltoplotBow2 Byusinglinearstabilityanalysis,itwasconcludedthatmovingthewallsandcreatingareturningowenhancestheclassicalRTstability.Thenextquestiontoansweriswhathappenswhentheonsetofinstabilityispassed.Inotherwords,thetypeofbifurcationisofinterest.TheclassicalRTinstabilityshowsabackwardpitchfork(subcritical)bifurcationwhenthecontrolparameteristhewidth.Oncetheinstabilitysetsin,itgoestocompletebreakup.WhatwouldoneseeinanexperimentwhentheinterfacebecomesunstablefortheclosedowRTconguration?Toanswerthisquestion,aweaklynonlinearanalysisisperformedinthenextsection.

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Figure4-11. Theneutralstabilitycurvefortheshear-inducedowwhereCa=20.ThedashedlinerepresentsthecriticalvaluefortheclassicalRayleigh-Taylorproblem,whichis2.ObservethattheordinateisindependentofL. 2[c]2u2+2z1@u1 6[c]3 Intheaboveequation,z1,z2,andz3arethemappingsfromthecurrentstatetothereferenceorthebasestate[ 10 ].Theideaistosubstitutetheexpansionintothegoverningnonlinearequationsanddeterminefromdominantbalanceaswellasthevariableunatvariousorders[ 44 ].Inthisshear-inducedRTproblem,thecontrolparameterischosentobethescaledwallspeedortheCapillarynumber,Ca.Insteadofdetermining,analternativeapproachistoguessit,andthe

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correctnessofthisguessischeckedthroughoutthecalculation[ 44 ].Inanticipationofapitchforkbifurcation,issetto1=2forthiscalculation.Thus,theexpansioncanbewrittenmoreconvenientlyasu=u0+u1+z1@u0 22u2+2z1@u1 63u3+3z1@u2 whereissuchthatCa=Cac+1 22.Whentheexpansionsaresubstitutedintothenonlinearequations,tothelowestorderin,thebasestateproblemisrecovered;itssolutionisknown.Therstorderprobleminisahomogenousproblemanditisidenticaltotheeigenvalueproblemprovidedissettozero.Itisimportanttonotethatinthisweaklynonlinearanalysisweassumethatboththerealandtheimaginarypartsofthelargestgrowthconstantiszero.Thus,iftheneutralpointispurelyimaginary,thismethodwouldnotapplicable.Inthisproblem,some,butnotall,oftheleadinggrowthconstantshaveimaginaryparts.However,inwhatfollowsweshallfocusonlyonsteadybifurcationpoints,asweareinterestedinsteadysolutions. Thesolutionprocedureisasfollows.Intherstorderproblem,thestatevariablesaresolvedintermsofZ1,whichrepresentsthesurfacedeectionatrstorder.ThisresultsinahomogenousproblembeingexpressedasM^Z1=0.Again,Misarealnon-symmetricmatrixoperator.Atthisorder,thevalueofthecriticalparameter,Cac,andtheeigenvectors,uptoanarbitraryconstant,A,arefound.Then,thesecondorderproblemisobtainedandisexpectedtobeoftheformM^Z2=f^Z21+cwheretheconstantcappearsfromtheboundaryconditionatthemovingwall.AsolvabilityconditionhastobeappliedtothisequationwhenceAcanbefound.Ifitturnsoutthatthesolvabilityconditionisautomaticallysatised,oneneedstoadvancetothenextorder.Atthisorder,thesolvability

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conditionprovidesA2whosesigndetermineswhetherthepitchforkisforwardorbackward.Inthenextsectionthesecondorderequationsarepresented. Asimilarequationisvalidatthetopwall.Attheinterface,thesecond-ordermassbalanceequationsatises ^2;n2=^2;n2(4{92) and Recallthatatthebasestate0wasfoundtobeequalto0.Thisleadstoseveralcancellations;forreasonsofbrevitytheintermediatestepsareomittedandsimpliedversionsoftheequationsarepresented.Asinpreviousorderequations,eachvariableisrepresentedasasummation.Asaresult,( 4{93 )becomes ^2;n2=1 2Xn0d^0;n0 Theno-slipconditionisgivenby Thetangentialstressbalanceassumestheform

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andtheseriesexpansionofthetangentialstressbalanceyields Byusingtheaboveconditions,^2;n2and^2;n2aredetermined.Toclosetheproblem,thenormalstressbalanceisintroducedinstreamfunctionformas Itturnsoutthataftermuchalgebraicmanipulations,thenormalstressbalanceresultsinM^Z2=0.Thismeanssolvabilityisautomaticallysatised;hence^Z2=B^Z1holds.Therefore,thethirdorderproblemneedstobeintroducedwiththehopeofndingA2andthenatureofthepitchforkbifurcation.Beforeintroducingthethirdorderequations,themeaningofthesignofA2needstobegiven.RecallthatanincreaseinCaimpliesmorestability;consequently,ifA2turnsouttobepositiveatthenextorder,acurveofAversus1=Carepresentsabackward(subcritical)pitchfork.However,ifA2weredeterminedtobenegative,thiswouldbeunallowable.Then,CamustbedecreasedfromCacbyanamount1=22leadingtoapositiveA2,hence,aforward(supercritical)pitchforkinanAvs.1=Cagraph. Attheinterface,themassbalanceequationsatises

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(4{100) Notethatintheaboveequation,thetermscomingfromthebasestatearenotshownbecausetheycanceledeachotheras0=0holds.Inaddition,therearesomemorecancellationsthattakeplacewhentheinterfaceconditionsofthepreviousordersareintroduced,e.g.,thesecondterminEquation 4{100 cancelswiththecorrespondingtermofthe*phasebyusingEquation 4{79 .Hereafter,astheequationsareverylong,onlytheverysimpliedformoftheinterfaceconditionswillbeprovidedwithoutseparatingthexandzdependentparts.However,itshouldbenotedthatasinthepreviousorders,eachtermhastoberepresentedasasummationbecauseofthecouplingofthemodes.Theno-masstransferconditionattheinterfacegivesriseto and@3 (4{102) Theno-slipconditionattheinterfaceis

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Thetangentialstressbalanceassumestheform Finally,thenormalstressbalanceisgivenby@33 +dZ3 (4{105) Thewaytoproceedfromthispointisverysimilartotheprocedureappliedatthepreviousorders.First,thex-dependentpartofthevariablesisseparatedandtheequationsarewrittenasasummation.Then,^3and^3aresolvedintermsof^Z3andtheinhomogeneities.Finally,theseexpressionsaresubstitutedintothenormalstressbalanceandaproblemoftheformM^Z3=a1^Z31+a2^Z1^Z2+a3^Z1isobtained.Atthesecondorder,M^Z2wasequaltozero.Infact,atthethirdorder,theconstanta2turnsouttobezeroformuchthesamereason.Now,thesecondordercorrectiontotheinterfacedeectioncanbewrittenas^Z2=B^Z1andtheconstantBisnotknownbutisnotneededeither.TheunknownconstantAormoreprecisely,A2determinesthetypeofpitchforkbifurcation. Usingtheequationfromtherstorder,i.e.,M^Z1=0,thesolvabilityconditioncanbeappliedasfollows

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wherethesuperscriptydenotestheadjointandh:;:istandsfortheinnerproduct.Allthevariablesaresolvedintermsofthesurfacedeection.Thelastequationtobeusedisthenormalstressbalance.Inthatequation,allparametersaresubstitutedandthereforeMisarealmatrixanditsadjointisthereforeitstranspose.Then,byusingEquation 4{106 andEquation 4{107 ,onecanget (4{108) Itisknownthat^Z1=A^^Z1where^^Z1wasfoundattherstorder.Equation 4{108 thencanbeexpressedintermsofAasfollows (4{109) whereandareconstantswhicharedeterminedatthisthirdorder.Let'selaborateonhowtoobtainEquation 4{109 .First,CaandBoarexed.Thecorrespondingcriticalw=Lisfoundfromtherstordercalculation,whichresultedinFigure 4-10 .WhenBoissmallerthansomevalue,whichisapproximately70forthechoiceofparametersinFigure 4-10 ,thereisonlyonecriticalpointandthiscriticalpointhasanimaginaryparti.e.,itisaHopfbifurcation.Asnotedbefore,thisweaklynonlinearanalysistracesonlysteadysolutionsandisthereforenotapplicabletosuchcriticalpoints.HoweverthereisanotherregionofBonumberwherethereisonlyonecriticalpoint:Bolargerthanapproximately110.Inthatregion,thecriticalpointdoesnotexhibitanyimaginarypartandthisanalysisisapplicabletosuchpoints,A2isalwayspositiveandthepitchforkisbackward

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(b) Bifurcationdiagrams.a)Backward(Subcritical)pitchfork.b)For-ward(Supercritical)pitchfork. asdepictedinFigure 4-12 (a).Whentherearethreecriticalpoints(Forexample,Ca=20,Bo=70),theA2correspondingtothelargestw=Lisagainpositiveandthebifurcationisbackward.Ifthebifurcationisbackward,oncetheinstabilitysetsin,itgoestocompletebreakup.Incontrastwiththelargestcriticalw=L,thesmallesttwocriticalpointsgiverisetoanegativeA2.ThenCamustbedecreasedfromCacinordertogetapositiveA2and,forthesecases,thenatureofthebifurcationisforwardasdepictedinFigure 4-12 (b).Somemoreobservationscanbemadefromthecalculation.Theinhomogeneitiescomingfromtheno-slipcondition,Equation 4{103 ,andthetangentialstressbalance,Equation 4{104 ,havenoeectontheconstantsand. OnceAisknown,thevariationoftheactualmagnitudeofthedisturbanceswithrespecttoaparameterchangecanbecalculatedwhenCaisadvancedbyasmallpercentagebeyondthecriticalpoint.Forexample,onecancomparetheamplitudeofthedeectionsoftherstandsecondcriticalpointsforaxedCaandBoandsomethinginterestingbutexplicableturnsup.ItisfoundthatA2correspondingtothesmallw=LisoneorderofmagnitudelargerthanA2ofthelargerw=L.ThiscanbeexplainedbylookingatFigure 4-10 attheregionwherethreecriticalpointsoccur.Focusingonthersttwopoints,weobservethatthe

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rstcriticalpointiswhereinstabilitystarts,whilethesecondoneiswherestabilitystarts.Thismeansthat,anyadvancementintoanonlinearregionfromtherstcriticalpointmustproducealargerroughness,i.e.,A2,comparedtothesecondcriticalpointprovidedthenatureofthepitchforksarethesame;andindeedtheyare. ThetheoreticalstudyoftheRTinstabilitywithellipticalinterfaceturnedouttobemorestablethanitscompanionRTinstabilitywithcircularinterface.Thisresultisinagreementwithourphysicalintuitionbasedontheincreasedpossibilitiesofthedissipationofthedisturbancesswitchingfromacircletoanellipse. ItisknownintheRTproblemthatthereisadecreaseinstabilitywhentheliquidisshearedwithaconstantstress.Thisdecreaseinthestabilitylimitisattributedtothesymmetrybreakingeectoftheshear.Inthisstudy,weshowthattheuidmechanicsofthelightuidisimportantanditchangesthecharacteristicsoftheproblem.Underspeciccircumstancesaatinterfaceispermissibleundershear.Fortheopenchannelow,togetaatinterfaceinthebasestate,thewallspeedhastobeadjustedaccordingtotheratiooftheliquidheightsandtheviscosityratios.Ifbothratiosareunitythenanywallspeedisallowed.Ontheotherhand,fortheclosedowproblem,biasintheliquidheights,thewallspeedsortheviscositiesisnotpermitted.Ifthereisanydierencebetweenthespeedsoftheupperandthelowerwallsorbetweentheviscosityanddepthoftheupperliquidandthoseofthelowerliquid,thenthesystemislessstablethantheclassicalRTproblem.

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Intheopenchannelow,thecriticalpointremainsunchangedcomparedtotheclassicalRTinstability,butthecriticalpointexhibitsoscillationsandthefrequencyoftheoscillationsdependslinearlyonthewallspeed.Theperturbationsarecarriedinthehorizontaldirectionbythemovingwallresultinginanoscillatorycriticalpoint.Ontheotherhand,inaclosedgeometry,movingthewallstabilizestheclassicalRTinstability.Theresultsshowwhen,howandwhyshearcandelaytheRTinstabilitylimit.Physicalandmathematicalreasonsfortheenhancedstabilityarepresented.Intheclosedowproblem,thelateralwallsimpedethetravelingwavesandcreateareturningow.Thestabilitypointincreaseswithincreasingwallspeedasexpected.Itisalsoconcludedthatthesystemismorestableforshallowliquiddepths.Forlargeliquiddepths,theshearhasalongdistancetotravel;consequently,itlosesitseect.TheclassicalRTinstabilityisrecoveredwhentheliquiddepthsareverylargeorthewallspeedapproacheszero.Themostinterestingfeatureofthisproblemisthepresenceofthesecondwindowofstability.ForagivenrangeofCaandBo,thereexistthreecriticalpoints,i.e.,thesystemisstableforsmallwidths,itisunstableatsomewidth,but,itbecomesstableonemoretimeforalargerwidth.Wepresentaweaklynonlinearanalysisviaadominantbalancemethodtostudythenatureofthebifurcationfromthesteadybifurcationpoints.Itisconcludedthattheproblemshowsabackwardorforwardpitchforkbifurcationdependingonthecriticalpoint. Clearly,itwouldnotbeeasytoconductanexperimentwiththespecicationsgiveninthissection.Theproblemdoesnotaccommodateanybiasinliquiddepthsnorinviscositiesoftheliquids.Anysmalldierenceisgoingtocauseanon-atinterfaceandleadtoaninstability,whichwilloccurevenbeforetheclassicalRTinstability.Anidealexperimentmightbecarriedoutwithporoussidewallsandwithtwoviscousliquids.However,fromamathematicalpointofview,theproblemshowsinterestingcharacteristicsthathavephysicalinterpretations.Forstress-free

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lateralwalls,itispossibletoobtainananalyticalsolutionthough,itisnotpossibletouncouplethemodes.Infact,theworkinthissectionhasshowntheeectofmodeinteractionondelayingtheinstability. ThemainresultsofthischapterarethatanellipticalcrosssectionoersmorestabilitythanacompanioncircularcrosssectionsubjecttoaxisymmetricdisturbancesandthatsheardrivenowintheRTproblemcanstabilizetheclassicalinstabilityandleadtoalargercriticalwidth.TheseresultsmotivateustorunsomeexperimentsbutexperimentsontheRTproblemarenotsimpletoconstructandsoweconsiderbuildingliquidbridgeexperimentswithaviewofchangingthegeometryandintroducingowandseeingtheireectontheinstability.

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Thischapterdealswiththestabilityofliquidbridges.Theorganizationofthischapteristhesameasthepreviouschapter.WewillstartwithRayleigh'sworkprincipletoinvestigatethecriticalpointofacylindricalliquidbridgeinzerogravity.Then,wewillmoveontotheeectofgeometryonthestabilitypoint.Thissectioncontainstwoproblems.Therstoneistheeectofo-centeringaliquidbridgewithrespecttoitsencapsulant.Inthesecondpart,ellipticalliquidbridgesarestudied.Infact,thissectionprovesourintuitionbasedonthedissipationofthedisturbances.Finally,theeectofshearispresented,whichhelpsusunderstandtheeectofreturningowintheoatingzonecrystalgrowthtechnique. 10 ].Wewillfollowaproceduresimilartothepreviouschapter. AccordingtotheRayleighworkprinciplethestabilityofasystemtoagivendisturbanceisrelatedtothechangeofenergyofthesystem.Intheliquidbridgeproblemthesurfaceenergyisthesurfaceareamultipliedbyitssurfacetension.Thecriticalorneutralpointisattainedwhenthereisnochangeinthesurfaceareaforagivendisturbance.Consideravolumeofliquidwithagivenperturbationonit,asseeninFigure 5{1 .Thevolumeoftheliquidunderthecrestismorethan 61

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Volumeofliquidwithagivenperiodicperturbation. thevolumeunderthethrough(Appendix C );butthevolumeoftheliquidneedstobeconstantuponthegivenperturbation.Therefore,thereisanimaginaryvolumeofliquidofsmallerdiameterwhosevolumeuponperturbationisthesameastheactualvolume.Asaresult,thesurfaceareaoftheliquidisincreasedwiththegivenperturbationbutitisalsodecreasedbecauseofthelowerequivalentdiameter.Atthecriticalpoint,thereisabalancebetweenthetwoeectsandthesurfacearearemainsconstant. TosetthesethoughtstoacalculationconsidertheliquidhavingaradiusR0.Aone-dimensionaldisturbancechangestheshapeoftheliquidto whereRistheequivalentradius,representstheamplitudeofthedisturbance,assumedtobesmall,andkisthewavenumbergivenbyn/LwithLbeingthelengthofthebridge.Usingtheaboveshape,thesurfaceareaisgivenby dzdz(5{2) wheredsisthearclength,givenbyds="1+dr dz2#1=2dz"1+1 2dr dz2#dz.So,theareaperunitlengthturnsouttobe L=2R+1 2R2k2(5{3)

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HereR,theequivalentradiusisfoundfromtheconstant-volumerequirementasfollows whichimpliesRtobeequaltoR01 42 1 22 Thecriticalpointisattainedwhenthelengthofthebridgeisequaltothecircumferenceofthebridge.Therearetwoobviousquestionsthatarisefromthiscalculation:whatistheroleofthedisturbancetypeonthestabilitypointandwhatistheroleoftheliquidpropertiesonthestabilitypoint?Aparticulardisturbancetype,acosinefunctionischosenforthiscalculationaseverydisturbancecanbebrokenintoitsFouriercomponentsandthesamecalculationcanberepeated.Infact,thesamecalculationisperformedbyJohnsandNarayanan[ 10 ]onpage10foranyfunctionf(z)withoutdecomposingintoitsFouriercomponents.Equation 5{5 tellsusthatthecriticalpointdoesnotdependonthepropertiesoftheliquid.ThiscanbeunderstoodfromthepressureargumentintroducedinChapter 2 .Atthecriticalpoint,thereisnoow.Theviscosityandthesurfacetensionplayaroleindeterminingthegrowthordecayratesofthedisturbances.Suchacurvecanbereproducedviaaperturbationcalculationandthisisgivennext.

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asthecasewhenaliquidencapsulatesanotherliquid.TheEulerandcontinuityequationsare: @t+~vr~v=rP(5{6) and Thesedomainequationswillbesolvedsubjecttotheforcebalanceandnomassowattheinterfacei.e., and Here2Histhemeancurvature,~ntheoutwardnormaltothejetsurfaceanduthesurfacenormalspeed(Appendix B ).Toinvestigatethestabilityofthebasestate,imposeaperturbationuponit.LetindicatethesizeoftheperturbationandexpandandPintermsof,viz. 'r1'isthemappingfromthecurrentcongurationofaperturbedjettothereferencecongurationofthecylindricalbridge.WepresentedtheexpansionofadomainvariablealongthemappingAppendix A .MoreinformationcanbefoundinJohnsandNarayanan[ 10 ].TheradiusofthebridgeRinthecurrentcongurationmayalsobeexpandedintermsofthereferencecongurationas Collectingtermstozerothorderinweget

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and Thereisasimplesolutiontotheproblem.Itis~v=~0andP==R0whereR0istheradiusofthebridge. Theperturbedequationsatrstorderbecome and Likewisetheinterfaceconditionsatrstorderare and Thestabilityofthebasestatewillbedeterminedbysolvingtheperturbationequations.Toturntheproblemintoaneigenvalueproblem,substitute and intotherstorderequations.Intherstorderequationss,m,andkstandfortheinversetimeconstant,theazimuthalwavenumberandaxialwavenumberrespectively.Eliminatevelocitytoget drrd^P1

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Thecorrespondingboundaryconditionsfortheperturbedpressureare and ^P1=1 Theeigenvaluesarethevaluesofsatwhichthisproblemhasasolutionotherthanthetrivialsolution.Letusrstlookattheneutralpoint,i.e.,2=0.ThesolutiontoEquation 5{20 isoftheform ^P1=AIm(kr)(5{23) whereAmustsatisfy FromEquation 5{24 ,Avanishes.Usingthisintheonlyremainingequation,i.e.,Equation 5{22 gives 0= R201m2R20k2^R1(5{25) Now,for^R1tobeotherthanzero[1m2R20k2]hastobeequaltozerowhichgivesusthecriticalwavenumberofthebridgefromk2criticalR20=1,hencethecriticallengthofthebridgeisitscircumference. Toobtainthedispersioncurve,oneneedstosubstituteEquation 5{22 intoEquation 5{21 toget R201m2R20k2d^P1 Substitutingtheexpressionfor^P1fromEquation 5{23 intotheaboveequation R301m2R20k2kR0I0m(kR0)

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isobtained.Here,I0m(x)=d dxIm(x).Themostdangerousmodeiswhenmiszero.Then,theequationforthedispersioncurveis R301k2R20kR0I00(kR0) Tobegintheanalysisoftheproblem,wedrawtheattentionofthereadertoFigure 5-2 ,whichdepictsano-centeredbridgeinanouterencapsulant.Weareparticularlyinterestedinwhathappenstothedampingandgrowthratesofthe

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Figure5-2: Centeredando-centeredliquidbridges. perturbationsifthebridgeisnotcentered.Thestabilityisstudiedbyimposingsmalldisturbancesuponaquiescentcylindricalbasestate.Beforethis,weturntothegoverningnonlinearequations,whicharegivennext. Theequationofmotionandthecontinuityequationforaninviscid,constantdensityuidaregivenby @t+~vr~v=rP(5{29) Equations 5{29 and 5{30 arevalidinaregion0rR(;z),whereR(;z)isthepositionofthedisturbedinterfaceofthebridge.Hereisthedensity,and~vandParethevelocityandpressureelds.Similarequationsfortheouteruid,representedby'*',canbewrittenintheregionR(;z)rR(0)0.Thesolutiontothebasestateproblemis~v0=~0=~v0andP0P0=2H0= R0.Notethatthisbasestatemaybethecenteredoro-centeredstate.Inthenextsubsectionwewillpresentthehigherorderequations,whichwillthengiveusthedynamicbehaviorofthedisturbances.

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intheregion0rR0().Combiningthetwoequationsweget withsimilarequationforthe'*'uid.Thedomainequationsaresecondorderdierentialequationsinbothspatialdirections.Consequently,eightconstantsofintegrationmustbedeterminedalongwithR1,whichisthesurfacemappingevaluatedatthebasestate.TondtheseunknownconstantsandR1,wewritetheboundaryconditionsinperturbedform.Attheinterface,thereisno-massowandthenormalcomponentofthestressbalanceholds.Consequently and Thewallsareimpermeabletoow;asaresultthenormalcomponentofthevelocityiszero,orintermsofpressurewecanwrite Asimilarequationisvalidforthe'*'uid.Freeendconditionsarechosenforthecontactofthebridgewiththesolidupperandlowerwalls,i.e., Theperturbedvelocities,~v1and~v1canbeeliminatedfromtheboundaryequationsbyusingEquation 5{31 anditscounterpartforthe'*'uid.Weseparatethetimedependencefromthespatialdependencebyassumingthatthepressure,velocityandR1canbeexpressedasK=^KetwhereKisthevariableinquestion.

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Equation 5{33 thenbecomes 1 Hereafter,thesymbol,`^`,willberemovedfromallvariables.TheproblemgivenbyEquations 5{32 5{37 isaneigenvalueproblembutthegeometryisinconvenientbecauseR0isafunctionoftheazimuthalangle''.Thereforeweuseperturbationtheoryandwritetheequationsatthereferencestatei.e.,thestatewhentheshiftdistance''isequaltozeroandwhereR0isequaltoR(0)0andisindependentof''.Allvariables,ateveryorderareexpandedinaperturbationseriesin,includingthesquareoftheinversetimeconstant.Therefore2is 222(2)+(5{38) Ourgoalistodeterminethevariationof2ateachordertondtheeectoftheshift,,uponthestabilityofthebridge.Thecalculationof2(0)iswell-knownandcanbefoundinChandrasekhar'streatise[ 17 ].Itsvaluedependsuponthenatureofthedisturbancesgiventothereferencebridgeandcanbecomepositiveonlyforaxisymmetricdisturbances.Hence,theeectofonthestabilityofthebridgesubjectedtoonlyaxisymmetricdisturbancesinitsreferenceon-centeredstateisconsidered.Tocalculatetherstnon-vanishingcorrectionto2,weneedtodeterminethemappingfromthedisplacedbridgecongurationtothecenteredconguration,andthisisdonenext.

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Togetthisexpansion,weobservethatthesurfaceofthedisturbedliquidbridgeisdenotedby ThereforeRcanbeexpandedas }| {R(0)0+R(1)0+1 22R(2)0++R1(;z;t;)z }| {R(0)1+R(1)1+1 22R(2)1++(5{40) whereR(0)0istheradiusofthecenteredbridgeandR(1)0=dR0 5-3 helpsustorelateR(1)0andR(2)0toR(0)0.Byusingthebasicprinciplesoftrigonometry,wecanconcludethat SubstitutingtheexpansionofR0fromEquation 5{40 intoEquation 5{41 ,wegetR(1)0=cos()andR(2)0=sin2()

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Figure5-3: Thecross-sectionofano-centeredliquidbridge. Theouterliquid'sdomainequationcanbewrittensimilarly.Themassconservationandthenormalstressbalanceattheinterfacerequire and Inasimilarway,thedomainequationoforder11is Theconservationofmassequationattheinterfacebecomes whereR(1)0isthemappingfromthecurrentcongurationofano-centeredbridgetothereferencecongurationofthecenteredbridgeandwasshowntobecos().Asimilarsetofequationscanbewrittenfortheouterliquid.Thenormalstress

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balanceattheinterfaceatthisorderis Weuseanenergymethodtogetthesignof2(1).BymultiplyingEquation 5{45 byP(0)1=,Equation 5{42 byP(1)1=,integratingoverthevolume^V,takingtheirdierenceandaddingtothisasimilartermarisingfrom'*'uid,weobtainZ^V"P(0)1 ThevolumeintegralscanbetransformedintosurfaceintegralsbyusingGreen'sformula.Theintegraloverthe'rz'surfacevanishesbecauseofsymmetry,i.e.becauseP(0)1isthesameat''equaltozeroand2.Theintegraloverthe'r'surfacevanishesbecauseoftheimpermeablewallconditions.Equation 5{48 thereforebecomesR(0)0ZL00Z20"P(0)1@P(1)1 Applyingno-masstransferequationsattheinterfacei.e.,Equations 5{43 and 5{46 ,Equation 5{49 becomesZL00Z20P(0)1"2(0)R(1)1+2(1)R(0)1R(1)0

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Equation 5{50 issimpliedbynotingthefactthat10termsare''indepen-dentandthatR(1)0isequaltocos().Consequently,theintegralofP(0)1R(1)0 5{44 and 5{47 ,Equation 5{50 becomes Togetthesignof2(1)fromEquation 5{51 ,weneedtodeterminetheformof2H(1)1andthereforeR(1)1.But,theformofR(1)1canbeguessedfromEquation 5{46 ,whichhastwotypesofinhomogeneities:R(1)0@2P(0)1 5{51 ,weobtainZL0022(0)2(1)A(z)"R(0)1 wherewehaveused2H(0)1=R(0)1 5{52 ,theself-adjointnessofthed2=dz2operatorandthecorrespondingboundaryconditionsonR(0)1(z)andA(z)areused,renderingtheterminEquation 5{52 in'fg'tozero.Also,theRayleigh

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inequality[ 45 ],statesthat"R(0)1 5{52 ,weconcludethat2(1)iszero.Therefore,tondtheeectofo-centeringweneedtomoveontothenextorderinandget2(2). Theconservationofmassattheinterfacerequires1 where~n(2)0rP(0)1=sin2()

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wherethemeancurvatureisgivenby 2H(2)1=R(2)1 whileR.TisgivenbyR:T:=R(0)1[13cos2()]sin2()R(0)20@2R(1)1 5{48 .WethenuseGreen'sformulaandintroducetheno-masstransferattheinterfaceforthe12andthe10problems,viz.Equations 5{54 and 5{43 toobtaintheanalogofEquation 5{50 ,whichis +(P(2)1P(2)1)(2(0)R(0)1)#ddz=0(5{57) InordertosimplifyEquation 5{57 inamannersimilartotheprevioussection,weusethenormalstressbalanceequations,i.e.Equations 5{44 and 5{55 ,theformofR(2)1,whichisguessedfromtheno-masstransferequation,i.e.Equation 5{54 andtheself-adjointnessofthed2=dz2operator.WealsouseEquation 5{43 ,whichgivessin2() 5{57 becomes

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Inprinciple,2(2)canbefoundfromtheaboveequation.However,somemoreworkisneededastermssuchasR(0)1,P(0)1andP(1)1appear.R(0)1canbeexpressedasBcos(kz)forfreeendconditions,butthesolutionforthepres-sureP(i)1isobtainedfromthedomainequationr2Pi1=0anduponlettingPi1=^Pi1(r)cos(kz)cos(m)thedomainequationbecomes 1 drrd^Pi1 whereiandmareeachzeroforthe10orderandequaltooneforthe11order. UsingEquation 5{59 ,weevaluatetheintegralsinEquation 5{58 andobtain0=^P(0)12 2R(0)0@2^P(0)1 R(0)402 Notethat^P(0)1and^P(1)1inEquation 5{60 arefunctionsofonlyrandallofthetermsareevaluatedatthereferenceinterface,i.e.atr=R(0)0.

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Tondthesignof2(2)fromEquation 5{60 ,weneedtosolvefortheper-turbedpressures.TheirformsarefoundfromEquation 5{59 asP(i)1="A(i)kmIm(kr)+C(i)kmKm(kr)#cos(kz)cos(m) andP(i)1="A(i)kmIm(kr)+C(i)kmKm(kr)#cos(kz)cos(m) whereC(i)kmiszerobecausethepressureisboundedeverywhere. ToobtaintheconstantsA;B;AandC,wesubstitutetheformofthepressuresintotheboundaryequationsateachorder.Toorder10,fromtheno-masstransfer,viz.Equation 5{43 ,thenormalstressbalance,viz.Equation 5{44 andtheimpermeablewalls,weget and When2(0)iszero,weseefromEquations 5{61 5{62 and 5{64 thatAk0,Ak0andCk0areallzero.FromEquation 5{63 ,werecoverthecriticalpoint,whichisk2R(0)20=1.When2(0)isnotzero,fourequationsmustbesolvedsimultaneouslysuchthatalloftheconstantsnotvanishatthesametime. Likewise,^P(1)1and^P(1)1aresolvedbyintroducingtheboundaryconditionsatthe11order.Thesolutionoftheperturbedpressures,^P(i)1and^P(i)1aresubstitutedintoEquation 5{60 toevaluate2(2).Thereadercanseethatananalyticalexpressionfor2(2)isobtained.Thisexpression,however,isextremely

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lengthysowemoveontoagraphicaldepictionof2(2)andadiscussionofthephysicsoftheo-centering. 5-4 showstheeectofo-centeringonthegrowthrateconstant.Theneutralpointdidnotchange,whichisnotsurprisingbecauseattheneutralpointthepressureperturbationsareindeedzeroandsincethesystemisneutrallyatrest,itcannotdierentiatebetweencenteredando-centeredcongurations. Figure5-4. If`k'issmallerthanthecriticalwavenumber,kc,thebridgeisunstabletoinnitesimaldisturbances.AscanbeseenfromFigure 5-4 ,oncethebridgeis

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Figure5-5. Changein2(2)(multipliedbyitsscalefactor)forsmalltointer-mediatedensityratiosforscaledwavenumber(kR(0)0)of0.5andR(0)0=R(0)0=2. unstable,theo-centeringhasastabilizingeect.Althoughtheneutralpointisunaected,therateofgrowthisreduced.Theo-centeringprovidesnon-axisymmetricdisturbances,whichinturnstabilizethebridge.However,lazywavesamplifytheeectoftransversecurvatureagainstthelongitudinalcurvature;con-sequently,thebridgeisalwaysunstableinthisregion.Thelongitudinalcurvaturebecomesmoreimportantforshortwavelengthsandinthestableregion,eachvalueof2producestwovaluesof,whicharepurelyimaginaryandconjugatetoeachother.Thedisturbancescorrespondingtothewavelengthsinthisregionneithersettlenorgrow.Thebridgeoscillateswithsmallamplitudearounditsequilibriumarrangement.Thebridgecannotreturntoitsequilibriumcongurationwithoutviscosity,whichisadampingfactor.Oncethebridgeisstable,theo-centeringoersadestabilizingeectbecausethewallisclosetooneregionofthebridgeandthisdelaysthesettlingeectoflongitudinalcurvature. Limitingconditions,usuallyprovideabetterunderstandingofthephysics.InFigure 5-5 ,=isallowedtovaryanditapproacheszeroanditseectonscaled2(2)isgiven.Thegureshowsthattheouteruidlosesitsrolewhen=approacheszerobecausetheuidsareinviscid.Therefore,thebridgeisexpectedtobehaveasiftherewerenoencapsulantatall,therebycausing2(2)tovanish.To

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Figure5-6. Changein2(2)(multipliedbyitsscalefactor)largedensityratiosforscaledwavenumberof0.5andR(0)0=R(0)0=2. seethebehaviorofthecurve,therangeoftheplotisextendedto==14.When=isverylarge,asshowninFigure 5-6 ,theouterliquidservesasarigidwallandtherefore2(2)approacheszero.Inotherwords,2(2)approacheszeroas=goestoeitherzeroorinnity. TheratiooftheradiiR(0)0=R(0)0isanotherparameterthatisexaminedanditseectisshowninFigure 5-7 .Astheratioapproachesunity,theazimuthaleectbecomesmoreobvious.Ontheotherhand,astheouteruidoccupiesaverylargevolume,theo-centeringeectsettlesdown.Asaresult,2(2)approacheszeroandthebridgeactsasiftherewasnooutsideuid. Figure5-7. Changeof2(2)(multipliedbyitsscalefactor)versusoutertoinnerradiusratioR(0)0=R(0)0forscaledwavenumberof0.5and==1.

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Insummary,thephysicsoftheproblemindicatethattheeectofo-centeringissuchthatitdoesnotchangethebreak-uppointofthebridgebutitdoesaectthegrowthrateconstant.Thestableregionsbecomelessstable,meaningthattheperturbationsettlesoveralongerperiodoftime,whereastheunstableregionsbecomelessunstable,thereforethedisturbancegrowsslower.Inaddition,thephysicsoftheo-centeredproblemindicatesthattheeectofo-centeringisseentoevenordersofandthisrequiredanalgebraicallyinvolvedproof. Itisimportanttounderstandtheeectofo-centeringthebridgebecauseitcanbetechnicallydiculttocenterthebridgeandthismighthaveatechnologicalimpactwhenaoatzoneisencapsulatedbyanotherliquidinthecrystalgrowthtechnique.Ournextfocusistounderstandthecomplexinteractionsofgeometryonthestabilityofliquidbridges.Wewillpresentourphysicalexplanationofwhyanon-circularbridgecanbemorestablethanitscircularcounterpart.Wewillproveourreasoningwithellipticalliquidbridgeexperiments. LiquidbridgeshavebeenstudiedexperimentallyasfarbackasMason[ 46 ]whousedtwodensity-matchedliquids,namelywaterandisobutylbenzoateandobtainedaresultfortheratioofthecriticallengthtoradiustowithin0:05%ofthetheoreticalvalue[ 12 ].Whilemostofthetheoreticalandexperimentalpapersonliquidbridgespertaintobridgeswithcircularcylindricalinterfaces,therearesome,suchasthosebyMesegueretal.[ 47 ]andLaveron-Simavillaetal.[ 48 ]whohavestudiedthestabilityofliquidbridgesbetweenalmostcirculardisks.Usingperturbationtheoryforaproblemwheretheupperdiskisellipticalandthebottom

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Figure5-8: Sketchoftheexperimentalset-upforellipticalbridge. diskiscircular,theydeducedthatitispossibletostabilizeanotherwiseunstablebridgeforsmallbutnon-zeroBondnumber.RecallthattheBondnumberisgivenbytheratioofgravitationalforcestosurfacetensionforces.TheearlierworkofothersandtheearlierchapteronellipticalinterfacesintheRayleigh-Taylorproblem,therefore,hasmotivatedustoconductexperimentsonthestabilityofliquidbridgesbetweenellipticalendplatesandwenowturntothedescriptionoftheseexperiments.Figure 5-8 showsadiagramoftheexperimentalset-up.ItdepictsatransparentPlexiglascylinderofdiameter18.50cm,whichcancontaintheliquidbridgeandtheouterliquid.Thebridge,intheexperimentsthatwereperformed,consistedofDowCorning710R,aphenylmethylsiloxaneuidthathasadensityof1:1020:001g/cm3at25C.Thedensitywasmeasuredwithapycnometerthatwascalibratedwithultrapurewateratthesametemperature.Thesurroundingliquidwasamixtureofethyleneglycol/waterassuggestedby

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Table5-1: Physicalpropertiesofchemicals. 710R Density(g=cm3) 1:1020:001 1:1020:001Viscosity(cSt)[ 49 ] 500 7.94 Interfacialtension(N/m)[ 49 ] 0:0120:002 Gallagheretal.[ 49 ].Theouteruidisvirtuallyinsolublein710R.Table 5-1 givesthephysicalpropertiesofthechemicalsused. Thebridgewasformedbetweenparallel,coaxial,equaldiameterTeonendplates.Theouterliquidwasincontactwithstainlesssteeldisks.Furthermore,alevelingdevicewasusedtomakesurethatthediskswereparalleltoeachother.Toensurethealignmentofthetopdisk,thelevelingdevicewaskeptontopoftheupperdiskduringtheexperiment.Fortheellipticalliquidbridgeexperiments,theendplatesweresuperimposedoneachother.Thiswasguaranteedbymarkingthesidesofthetopandbottomdisk,whichwere,inturn,trackedbyamarkedlinedownthesideofthePlexiglasouterchamber. Thekeytocreatingaliquidbridgeofknowndiameter,andmakingsurethatthedisksareoccupiedcompletelybytheproperuids,istocontrolthewettingoftheinnerandouterdisksbythetwouids.Ifthe710Ruidcontactsthestainlesssteelsurface,itwilldisplacetheouteruid.Therefore,itwascriticaltokeepthesteeldisksfreeof710RandthiswasassuredbyaretractingandprotrudingTeondiskmechanism.Priortotheexperiment,thebottomTeondiskwasretractedandthetopTeondiskprotrudedfromthesteeldisks.Thishelpedinstartingandcreatingtheliquidbridge.Then,710Ruidwasinjectedfromasyringeof0.1mlgraduationsthroughaholeof20thousandthsofaninch(0.02inches).Aliquidbridgeofaround1mmlengthwasthusformedintheabsenceoftheouterliquid.Capillaryforceskeptthissmall-lengthbridgefromcollapsing.Theouterliquidwasinjectedthroughtwoholesof0.02inches,180fromeachother,soas

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nottodisplacethe710R.Thenextstepwastosimultaneouslyincreasethelengthbyraisingtheupperdiskandaddingthe710Randouterliquid. Avideocamerawasusedtoexaminethebridgeforsmalldierencesindensity.Wewereabletocapturetheimagethankstothedierenceintherefractiveindexbetweenthebridgeandtheouterliquid.Thelossofsymmetryintheliquidbridgewasanindicationofthedensitymismatch.Theellipticalliquidbridgeissymmetricaroundthemidplaneofthebridgeaxis,whilethecircularbridgehasaverticalcylindricalinterface;theshapeofthebridgecouldthenbecheckedviaadigitizedimage. Thedensityofthemixturewasadjustedbeforetheexperimentto0.001g/cm3bymeansofapycnometer.However,duringtheexperiment,nerdensitymatchingwasrequired,andeitherwaterorethyleneglycolwasmixedaccordinglytoadjustthedensitymismatch.Theshapeofthebridgewasthebestindicatortomatchthedensities.Inaddition,theaccuracyofdensitymatchingwasincreasedsubstantiallyastheheightofthebridgeapproachedthestabilitylimit.Extremecarewastakentomatchthedensitieswhentheheightwasclosetothebreak-uppointduetothefactthatgravitydecreasesthestabilitypointwellbelowthePlateaulimitforcircularliquidbridges[ 50 ].Forexample,wewereabletocorrectaslightdensitymismatch, of105byadding0.2mlofwaterto1literofsurroundingliquid.Thisdensitydierenceisobservablebylookingatthelossofsymmetryinthebridge.Asimilarargumentalsoholdsforellipticalliquidbridges.Dependingontheamountofliquidadded,eitherwaterorethyleneglycol,mixingtimesrangedfrom10to30minutes.Inallexperiments,sucienttimewasallowedtoelapseafterthemixingwasachievedsothatquiescencewasreached. Thetopdiskwasconnectedtoathreadedrod,whichwasrotatedtoraiseitandincreasethelengthofthebridge.Theheightofthebridgewhencriticalconditionswerereachedwasascertainedattheendoftheexperimentbycounter

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rotatingtheroddownwarduntiltheendplatesjusttouched.Onefullrotationcorrespondedto1.27mm.Themaximumpossibleerrorinheightmeasurementwasdeterminedtobe0.003inchesoverathreadedlengthof12inches.Therefore,theerrorinthetotalheightmeasurementofthebridgewasdeterminedtobelessthan0:24%.Inadditiontothis,therewasabacklasherrorthatwasnomorethan0.035mm.Itturnsoutthatthiserroramountstoamaximumof0:11%ofthecriticalheight.Thetotalerrorintheheightmeasurementtechniquewasthereforenevermorethan0:35%.Thevolumesofuidinjectedintothebridgeforthelargeandsmallbridgeswere19.80and2.45mlrespectively.ItmaybenotedfromSlobozhaninandPerales[ 51 ]aswellasfromLowry[ 25 ]thata1%decreaseorincreaseintheinjectedvolumefromthevolumerequiredforacylindricalbridgeresultsinadecreaseorincreasebyapproximately0:5%inthecriticalheight,respectively.Experimentswithcircularendplateswereperformedtoensurethatthemaximumerrorwasverysmall. Thelengthswereincreasedinincrementsof0.16mmoncethebridgeheightwasabout3%lowerthanthecriticalheight.Thereafter,foreachincrementthewaitingtimewasatleast45minutesbeforeadvancingtheheightthroughthenextincrement.Whenthecriticalheight,asreportedinTable 5-2 ,wasreachedtheneckingwasseeninabout30minutesandtotalbreakupoccurredinaround15

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Table5-2. Meanexperimentalbreak-uplengthsforcylindricalliquidbridges.Up-perandlowerdeviationsinexperimentsaregiveninbrackets. Break-uplength(mm) %changeinlengthofthemeanfromtheoreticalcritical Largecylindricalbridge 62.84(+0.02,-0.04) -0.08 Smallcylindricalbridge 31.48(+0.09,-0.05) +0.10 minutesaftertheinitialneckingcouldbediscerned.Eachexperimentwasrepeatedatleast3timesandtheresultswerequitereproducible.Atypicalstablebridgeataheightof29.57mmisdepictedinFigure 5-9 (a).ThesamebridgeatbreakupisshowninFigure 5-9 (b)ataheightof31.57mm.Thereportedvaluesinthetabledonotaccountforthebacklashanditshouldbenotedthattheincrementsinheightweredoneinstepsof0.16mm.Takingthisintoaccount,itisevidentthattheerrorintheexperimentwasverysmall,showingthattheprocedureandtheapparatusgavereliableresults.Thisprocedurewasusefulinthefollow-upexperimentsusingellipticalendfaces. Figure5-9. Cylindricalliquidbridge.Notethatinthisandallpicturesthedepictedaspectratioisnotthetrueoneduetodistortionscreatedbytherefrac-tiveindicesoftheuidsresidinginacircularcontainerwithobviouscurvatureeects.(a)Stablebridge(b)Unstablebridge.

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Figure5-10. Largeellipticalliquidbridge(a)Stablelargeellipticalliquidbridge.(b)Unstablelargeellipticalliquidbridge,beforebreak-up. Table5-3. Meanexperimentalbreak-uplengthsforellipticalliquidbridges.Upperandlowerdeviationsinexperimentsaregiveninbrackets. Break-uplength(mm) %changeinlengthfromthecriticalheightofthehypotheti-calcompanioncircularbridge Largeellipticalbridge 64.90(+0.10,-0.05) 2.86 Smallellipticalbridge 32.29(+0.09,-0.09) 2.74 Theprocedurethatwasusedforthebridgegeneratedbyellipticalendplateswasvirtuallyidenticaltothatusedinthecalibrationexperimentsusingcircularendplates,describedearlier.Figures 5-10 (a)and 5-10 (b)showthelargeellipticalliquidbridgeattwodierentstagesbeforeandnearbreak-up.Figures 5-11 (a)

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Figure5-11. Smallellipticalliquidbridge(a)Stablebridge.(b)Unstablebridge,beforebreak-up. and 5-11 (b)aretheanalogouspicturesforthesmallerareaellipticalbridge.Wefoundthattheincreaseinthebreak-uppointwasabout2:86%longerforthelargeendplateellipticalbridge,andnearly2:74%longerforthesmallellipticalbridgeshowingthatanellipticalbridgeisinfactmorestablethanthecompanioncircularbridge.ThebreakupheightsfortheellipticalliquidbridgeexperimentsaregiveninTable 5-3 Severalcommentsmaybemade.First,ascalinganalysisrevealsthattheratioofthecriticallengthofthedeviatedellipticalbridgetothecriticallengthofitscompanioncircularbridgecanonlydependonthepercentagedeviationoftheellipsefromthecircle,providedthattheBondnumberisnegligible.Thisiswhytheenhancementsinstabilityforthetwosetsofexperimentswithdierentellipsesareclosetoeachother.Second,fromageometricargument,onecanseethatthestabilitylimitcannotchangetorstorderwhentheellipticaldisksaredeviatedfromthecirculardisksbyasmallamount.Thisresultwasalsoobtained,albeitbycalculation,byMesegueretal.[ 47 ].Itwouldappearthatthechangeinstabilitycanbeseenonlyatsecondorder.Now,thedeviationintheendplatesusedareabout20%andtheobservedincreaseinstabilitycouldbeattributed

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tothemagnitudeofthisdeviationorsimplybecausethesecondordereectisstrongenoughtoshowthechange.Thethirdobservationisthatevenslightdensitymismatchesleadtoasymmetrywhichbecomesmostpronouncednearoratbreak-up.Thisisnotsurprisingasimperfectionsbecomedominantnearbifurcationpointsasseeninthetheoryofimperfections[ 52 ].Theimperfectionduetodensitymismatchcanonlyadvancethebreakupandsotheexperimentalresultsmustgivealowerboundtotheinstabilitylimitthatonewouldpredictfromtheory[ 52 ].Insummary,non-circularliquidbridgeswithgeometricallysimilarendplatescanbeexpectedtooergreaterstabilitythantheircircularcounterparts.Wehaveshownthistobetrueinthecaseofellipticalliquidbridgesbywayofexperiments. 53 { 55 ],Chenetal.[ 56 ]andAtreyaandSteen[ 57 ]toinvestigatehowbothdestabilizingeectscouldbejudiciouslycombinedtocanceloneanotherandactuallyenhancethestabilityofaliquidzone,evenenhancingthestabilitybeyondtheclassicalPlateaulimit.Itshouldbenotedhowever,thattosurpassthePlateaulimitisverydicultandwasneversuccessfullycompletedexperimentallywithaconstantowrate. Theyperformedaseriesofimpressiveexperimentswithbridgesofnon-zeroBondnumbers.Thesebridgeswereencapsulatedbyanouteruidthatwasallowedtoowthroughvertically.Theliquidbridgewasanchoredtotwoendplatesthatwereconnectedbyacenteringrod.Suchacenteringrodhasnoeectonthe

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stabilitywhenowisabsentbutitspresencedoesmodifytheowdynamicswhenashearingouteruidistakenintoaccount.Theexperimentsshowedstabilizationoftheinterfaceandthereasonsadvancedindicatedthatashearingowcould"straightenout"abulgingbridge,depending,ofcourse,onthedirectionofow.Inotherwords,theowcansuppressthedeviationsfromaverticalcylindricalinterface,balancegravity,andconsequently,stabilizeanotherwiseunstablestaticbridgebyasmuchas5%.ThestabilizationthattheyachievedevenreachedthePlateaulimit.Theexperiments,whichshowedstabilizationduetoshear,didnotproduceanystabilizationbeyondthePlateaulimitduetothenarrowrangeofsuchapossibilityandattendantexperimentaldiculties. Theworkinthisstudycontinuestheideaofstabilizationofnon-zeroBondnumberbridgesduetoshear.However,thequestionposedishowwouldowinducedinaclosedgeometrywhichisclosertothetechnologicalapplication,i.e.,oatingzonemethod,aectthestabilityofliquidbridges.Themajordierencebetweenthisworkandearliereortsisthattheuidowintheoutercompart-mentisinaconnedgeometry,notaowthroughconguration,noinnerrodwillbeused,andlargerdensitydierencesareexamined.Again,thereasonforconsideringthiscongurationismotivatedbythefactthattheliquidencapsulatedmeltzoneprocess,whichisaspecicFZtechnique,yieldsowprolesinclosedcompartments. Inshort,theoverallgoalisthereforetostudyshear-inducedrecirculatingow,asshowninFigure 5-12 ,anditseectonthestabilityofaliquidbridge.Therearemanyfactorsthatcomeintoplaywhenconsideringhowonemustdesignanapparatustoachieveourgoals.Forexamplechoosingtherightuidswithdesirableviscositiesandchoosingasensiblebridgeradiustoouterwallradiusratio.Webeginwithascopingnumericalcalculationthatwillassistestablishingthedimensionsoftheexperimentalsetupandthechemicalsthatconstitutethe

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Theschematicofthereturningowcreatedinthepresenceofanencapsulantintheoatingzonetechnique. bridgeandtheencapsulant.Oncethedimensionsandthechemicalsaredecided,amethodbywhichtocharacterizethestabilityofaliquidbridgeisdescribed.Then,theexperimentalsetup,thechemicals,andidentiestheaccuraciesinthemeasurementsarepresented.Lastly,theresultsoftheexperimentsarepresented.TheeectofeachparametersuchasthespeedofthemovingwallandtheeectoftheBoisstudiedalongwiththephysicalexplanationandcomparisonwiththenumericalresultswheneverpossible.Asanalpoint,asummaryoftheresultsandthecollectionofthemessagesaregiven. 5-12 .Bothinertialandviscoustermsaretakenintoaccountinthemodel.Theinputparameterstothemodelarethebridgeradius,theoutercompartmentradius,thelengthofthebridge,theviscositiesanddensitiesoftheuids,andthewallspeed.Thecalculatedinformationofinterestisthentheowprolesinbothregions,bridgeandencapsulant,whichalsodenestheconditionswhentheowsarenonaxisymmetric.Anonaxisymmetricowwouldcreateunwanteddisturbances

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Table5-4. Theeectoftheviscositiesonthemaximumverticalvelocityalongtheliquidbridgeinterface,vz;max.Thedensitiesoftheliquidsare1g/cceach,theheightofthebridgeis3cm,theouterradiusis2.5cm,andtheradiusofthebridgeis0.5cm.Theowsweredeterminedtobeaxisymmetric. 1 0.0015 50 50 0.0154 1 5 0.3455 1 1 0.1667 becausetheeectoftheseowsonbridgestabilityarenoteasilypredictable.Thenumericalmodelisapproximateinthataverticalshapeoftheinterfaceisassumed.ItisnoteworthythatamodelworkedintheStokeslimitwithstressfreehorizontalwalls,yetassumingadeformableinterfacewasproposedbyJohnson[ 58 ].Ournumericalsolutionwasobtainedwithnoslipconditionsandbyincludinginertialterms.FromJohnson'scalculations,wegatherthattheowintensityinthebridgeincreaseswithanincreaseinviscosityratiobetweentheouterandinneruids,whichwasalsoveriedbyourcomputations. Thefollowingcalculationsuseaniteelementmethodandweredonewithanaccuracyof=106(L2normofthecomputedresidual).Themodelismadeupof150x165quadrilateralniteelements(piecewiseQ2approximationforthevelocityeldandpiecewiseQ1forthepressure),builtupon301x331nodesintheradialandaxialdirections,respectively.Thisspatialdiscretizationleadstoanalgebraicsystemof199;262unknownstosolveforthevelocityeld.Thenumericalmodelwasextremelyhelpfulindeterminingappropriateviscositiesandradiusratios.Table 5-4 showssomeoftheresultsfortheinterfacevelocityscaledbythewallspeedforvariousviscosities.Threebridgeradiiof0.5,1,and1.5cmwerechosenforthecomputationswhiletheoutercompartmentradiuswasxedas2.5cm.Twooftheradiusvaluesrepresenttheactualdimensionsthatwereusedintheexperimentsandtheexperimentalchoicesreectedlogisticsaswellasmachining

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ease.Forcomputationalpurposes,theheightofthebridgewaschosentobe3cmforabridgeof0.5and1cmradiuswhileitwasvariedbetween4and6cmforabridgeof1.5cmradius.Theheightisaconvenientadjustablevariableinanexperiment.ThevaluesofheightchosenforthesmallerradiuscomputationswerebasedonbeinginthevicinityoftheRayleighPlateaulimiti.e.,height/radiusbeing2.Theheightschosenforthelargerradius,ontheotherhand,reectthefactthatowcouldonlyincreasebyincreasingtheheightfromthelowerradiusbridge,butnotsolargethatdicultiesduetoBowouldarise. Someofthefeaturesofthedetailednumericalmodelthatwasusedtodeter-minetheowprolesinthebridgecanbeguessedfromasimplescalinganalysis.Aroughscalingargumentfromaone-dimensionalmodelforatwouidsystemrevealsthedependenceofthevelocityalongtheinterfaceofthebridgeontheparametersintheproblematconstantouterwallspeed.Intheone-dimensionalmodel,amovingwallincontactwithanouteruidthatencapsulatesaninnercoreuidanchoredbyaverythinstationaryrodisassumed.Subscriptsoneandtworefertotheinnerandouteruidsrespectively,andthustheuidvelocityattheinterfacescaleswiththemovingwallspeedas [R2R1]=1vz(r=R1) whichyieldsto Thissimpleexpressionsuggestsincreasingtheoutsideliquid'sdynamicviscos-ity,decreasingthebridge'sviscosity,orincreasingthebridge'sradius,forxedwallradius,R2.ThisconclusionisalsojustiedbytheresultsofthedetailedcomputationsdisplayedinTable 5-4

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Table5-5. Theeectofviscosityonthemaximumverticalvelocityalongtheliquidbridgeinterface.Thedensitiesoftheliquidsare1g/cceach,theheightofthebridgeis3cm,theouterradiusis2.5cm,theviscositiesare1cPforeachliquid.Theowsweredeterminedtobeaxisymmetric. 0.1667 1.0 0.2500 1.5 0.3559 Theresultsinthetableportraymorethanthescalingargument.Forexampleobservethatwhilethescalingargumentrelatesthevelocitiestoviscosityratios,thecomputationsshowtheimportanceofindividualviscosities.Infactthecalculationsshowninthetabletellusthatiftheviscositiesarethesame,itisbettertohavelessviscousliquids.Asaninstancethevz;max=Uratiocanbeincreasedbyanorderofmagnitude,iftheviscositiesare1insteadof50cP. Theeectoftheheightofthebridgeonthemaximumaxialvelocityalongtheliquid/liquidinterface.Thedensitiesoftheliquidsare1g/cceach,theouterradiusis2.5cm,theradiusofthebridgeis1.5cm,andtheviscositiesare1.4and270cPforthebridgeandencapsulant,respectively.

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Itisnoteworthythatunliketheconclusionobtainedfromtheone-dimensionalmodeltheviscosityratiocannotalonedeterminetheowregimesinaclosedcompartmentmodel.Pressuregradientsareimportantandthusviscositycanneverbescaledasapureratioinclosedcompartmentmodels. Toseetheeectofradiusratio,morecalculationsweredoneassumingthattheviscositiesofbothuidsare1cP.TheresultsarepresentedinTable 5-5 .Asexpected,itisfoundthatastheradiusincreases,theratiovz;max=Uincreasesandthusmoremomentumistransferredtotheliquid/liquidinterface.Doublingtheradiusgaveanincreaseof1.5timesthevelocity. Toseetheeectofheightabridgeofradius1.5cmwaschoseninthecom-putations.ThespeedattheinterfaceforagivenUisexpectedtoincreaseby Theeectoftheencapsulant'sviscosityontheratioofmaximumspeedobservedattheinterfacetothewallspeed.Thedensitiesare1.616g/cc,theouterradiusis2.5cm,theradiusofthebridgeis1.5cm,andtheviscosityofthebridgeis1.2cP.Theheightofthebridgeis5.9cm.

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increasingtheheightofthebridgebecausethegapratio,L 5-13 .Forthesecomputations,theviscosi-tiesofthebridgeandthesurroundingliquidare1.4and270cP,respectively. Thesecalculationsledustochoosethechemicalsandtheradiusratios.Inourexperiments,wesettledupona3Mliquid,HFE7500,whichhasaviscosityofapproximately1.2cPfortheinneruidwhiletheoutsideliquidwasamixtureofsodiumpolytungstateandglycerineofviscosityaround250cP.Theradiusofthebridgewaschosentobeeither0.5cmor1.5cm. Onelastimportantparametertostudyistheeectoftheviscosityonthespeedattheinterface.Thisisimportantbecausetheviscositydependsontem-perature,whichcanchangeateachexperiment.AsseenfromFigure 5-14 ,eveniftheviscositychangesfromoneexperimenttoanother,themaximumspeedattheinterfacedoesnotchangeconsiderably.

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andusetheoreticalcalculationstoobtaintheBothatmustcorrespondtosuchabreakuplength.Here,weadoptedthelattermethodtodeterminetheBondnum-ber.InparticularweusedBrakke'sSurfaceEvolver(SE)programforthestabilitycalculation[ 59 60 ]. Theinputparametersforsuchacalculationarethevolumeoftheliquidinthebridge,thecontactangleattheendplates,thecriticalheightatbreakup,thebridgeradiusandaguessBondnumber.Theoutputofthecalculationisthetimeconstantforthedecayorgrowthofinnitesimaldisturbances.ForagivensetofparameterstheguessBondnumberischangeduntilneutralstabilityisobtainedi.e.,untilthetimeconstantisjustzero.TheguessBothatgivesneutralstabilityistheBonumberfortheexperimentalsystem.TheSEsoftwarewhoseaccuracydependsuponadjustablenumericaltuningparameterssuchasgridrenementwastestedinthezeroBondnumbercasebyrecoveringthePlateaulimitandalsobyverifyingtheresultsavailableingraphicalformbyLowry[ 25 ].OfcoursethismethodofdeterminingtheBondnumberofanexperimentassumesthatthecriticalheightcanbeaccuratelymeasured.AsexplainedlaterthiswasensuredbyrecoveringtheclassicalPlateaulimitforazeroBondnumbercongurationandinfactwashowwe'calibrated'thecorrectnessofourexperimentalprocedure.Wenowmovetoadiscussionoftheexperimentalsetup,thechemicalsusedandtheprocedureemployed. 5.4.3.1Theexperimentalsetup 5-15 .Theendplateswerecomposedoftwomaterials.TheinnerpartwasmadeofcircularTeondiskswithwhichthebridgewasincontact.Theencapsulatingliquidwasincontactwiththeouterpart

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Photographoftheexperimentalset-up. ofthedisks,whichwasmadeofstainlesssteel.Twosetsofliquidbridgeradiiwereusedintheseexperiments.ThediametersoftheseendplatesweremeasuredbyaStarrettMicrometer(T230XFL)as10and30mmwithanaccuracyof0:0025mm.Thetopdiskwasconnectedtoathreadedrod,whichwasrotatedtoraiseorloweritandtherebychangethelengthofthebridgewhilethebottomdiskwaskeptxed.ThePlexiglaswallcontainingtheliquidswasthreadedthroughtwolargerods,whichinturnwereconnectedtogearsattachedtoaservomotorBXM230-GFH2withagearreductionheadGFH2-G200.Themotor'sspeedwasadjustedbyitsowncontroller.Awiderangeofspeedswasaccessiblebyselectingdierentgearratios.Moreoverthedirectionofthemotioncouldbechanged.Teono-ringswereusedingroovesatbothtopandbottomdiskstoprovideaslipperysurfacebetweenthewallandthedisksandtoensurethattheencapsulantuiddidnotleakout.Figure 5-15 showsaphotographthatgivesaperspectiveoftheoperatingspanwithrespecttothetestsection.

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Thechemicalschosenwereasolutionofsodiumpolytungstateandglycerineastheencapsulantanda3MHFE-7500astheliquidbridge.ThedensityofHFE-7500is1.61g/ccanditsviscosityis1.2cP.Thedensityofsodiumpolytungstatesolutioncanbeeasilyadjustedfrom1.00g/ccto3.10g/cc.Intheseexperiments,westartedwithasolutionofdensity2.85g/ccandmixeditwithglycerinetoobtainnearlythesamedensityasthatofthebridge.Beforetheexperiment,thedensityoftheoutsidesolutionwasmeasuredwithahydrometertoanaccuracyof0.0001g/cc.Glycerineservedthedualpurposeofloweringthedensityofthesaltsolutionandincreasingitsviscosity.Thescopingcomputationsthatassistedinthedesignoftheexperimenttellusclearlythattheviscosityratiosoftheoutertoinneruidsmustbelargetoeectreasonableshear.Ourchoiceofuidsandtheneedtoadjusttheviscosityoftheouteruidreectedthemessagesconveyedbythesecalculations.Theviscosityoftheoutsidesolutionthereforewasvariedbetween200to250cPdependingonthesalt/glycerineratioforeachexperiment.Inthisregard,thereadermightobservefromFigure4thatthemaximumpossiblemomentumtransferisreachedevenwiththelowestviscosityof200cPfortheoutsideuid.Thisrangethereforeassuredthatviscositywouldnotplayafactorbetweendierentexperiments.Itisimportanttonotethatalthoughtheviscositieswerehigh,experimentswereconductedtoensurethatnoviscousheatingtookplace.Arotatingdiscviscometerinthenon-isothermalmodewasrepeatedlyrunforseveralminuteswiththeencapsulantuidtoseeifviscosityandtemperaturechangedovertime.Sinceviscosity,whichdependsontemperaturedidnotchangeovertime,therewasverylittleconcernthatviscousheatingwouldinturnaecttheBonumber. 61 ].Asobserved,theshapeofabridgedepends

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onthevolumeoftheliquidinjected,itsBondnumberandthespacingbetweentheendplates.TheaccuracyofspacingandradiusmeasurementsandthemethodfordeterminingtheBondnumberwerediscussedearlier.Thisleavesustospecifytheaccuracyofthevolumeofuidinjected,asthisisalsoimportant.Thevolumeswerecontrolledwithsyringes,whichhad0.1and0.2ccgraduationsforthe10and30mmdiameterbridgesrespectively. ThesetupandtheproceduretodeterminethebreakuppointwerecalibratedbyrecoveringthePlateaulimit.Thebreakuppointwasfoundtobe3.143cm0:010cmforthesmalldiameterbridge.ThecalibrationexperimentsweredonewiththesmalldiameterendplatestoensurethattheeectofslighttemperaturechangeswasminimalontheBondnumber.Itisimportanttoobservethatthechangeofdensityarisingfromtemperatureuctuationsisampliedbyninetimeswhenthelargedisksareused,asBoisproportionaltothesquareoftheradius.ThedetailsoftheprocedureandtheattendanterrorsinrecoveringthePlateaulimitarediscussedbyUguzetal.[ 61 ]. Guidedbythenumericalresults,keepingthevolumeoftheliquidinmind,thedensityandviscosityoftheoutsideliquidwasadjustedsothatshearcouldhaveaneectonthestabilityofthebridge.Theaimoftheexperimentswasthereforetocreateowintheoutsideliquidtominimizethedestabilizingeectofthedensityimbalanceandhelpstabilizethebridge.Shearingthewallcreatesareturningowintheoutsideliquid,whichinturncreatesareturningowintheliquidbridge.Notethattheowinthebridgeisinoppositedirectiontothedirectionofthewall(SeeFigure 5-16 ).Consequently,ifthebridgebulgesfromthebottom,thewallismoveddownwardtocreateaowsuchthattheinterfacebecomesmoresymmetric.Itisworthremindingthereaderthattherewasnocenteringrodusedintheseexperiments.Althoughsucharodwouldnothaveanyeectonthestabilityof

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Acartoonofabridgebulgingatthebottom.Thewallismoveddownwardwiththeobjectiveofobtainingasymmetricinterfacewithrespecttothemidplane. anon-shearingbridge,itwouldhavechangedtheowprolesandthereforethestabilitypointwhenthewallismoved. Theexpectationoftheexperimentswastoobtainmeasurablymorestablebridgeswithowthanwithoutow.TheheightofthebridgewasmeasuredwithaStarrettcaliperwitharesolutionof0.01mmandaccuracyof0:03mm.Thisistheonlyerrorthatmattersinourreportedresultsasonlypercentagechangesincriticalheightareofinterest. Theprocedureoftheexperimentwasasfollows.Thebridgewasrstcreatedintheabsenceofshear.Oncethedesiredvolumeofthebridgewasinjected,thevalveconnectedtotheinnerliquidinjectionportwasclosed.Thisisextremelyimportantasthepressuregradientcreatedinthechamberbymovingthewallortheupperdiskcanalterthebridgevolume.Thebreakuppointofthestaticbridgewasfoundatthisvolumebyslowlyincreasingtheheightoftheupperdiskinsmallincrementsandgivingampletimefordisturbancestosettledownorgrowbeforeeachincrement.Whenincreasingtheheightofthebridge,theencapsulantwasdrainedintotheoutsidecompartmentfromanexteriorliquidchamber.Oncethebreakuppointwasfound,thevolumeandthebreakupheightofthebridgesucedtocomputetheBondnumberusingtheSEsolver.Thewallwasmovedataconstantvelocityinthestabilizationdirectiononcethebridgestartedtobreak.Movingthewallchangedtheshapeofthebridgeimmediately.Whilethe

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wallwasmoving,thespacing,i.e.,theheightofthebridge,wasincreasedinsmallincrementsof0.008cm.Thebreakuppointofthebridgewasthenfoundinthepresenceofshearattheestablishedwallspeed.ThebreakupheightinthepresenceofshearandtheinjectedvolumewereusedtocomputeaBoasiftherewasnoow.ThisBoisreferredtoasthe"ApparentBo".Thuseachexperimentalsetforagiveninjectedvolumecomprisedofndingthebreakuppointforthestaticbridgeandthebreakuppointsfortheshearingfordierentwallspeedsrangingbetween42and168cm/hrwithamanufacturer'serrorof0:08cm/hr.Thenextstepwastoincreasethevolumeofthebridgeandrepeatthesetofexperiments.Sincetemperaturecouldchangeslightlyfromoneexperimentalsettoanother,thebreakuplengthsandtheBowerecalculatedforeachnewvolumeandwallspeeddata.Inthenextsectiontheresultsoftheexperimentsarepresented. 5-5 .Therefore,inthissectionweonlypresentthedatacorrespondingtothebridgeof1.5cmradius.Wewilldiscusstwomajoreectsonthepercentageincreaseofthebreakupheight:rst,theeectofthewallspeedandsecond,theeectoftheinjectedvolume.Thisclassicationallowsustoviewthedatawithdierentperspectives,asitishelpfulinidentifyingtheroleofeachparameterintheexperimentexplicitly.Asummarywillservetotietheresultstogether. WestartourdiscussionwithFigure 5-17 ,whichpresentsthepercentageincreaseinthebreakupheightinthepresenceofowforagivenBoandforvariousbridgevolumes.

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Theeectofwallspeedonthepercentageincreaseinthebreakupheightofthebridgeforvariousvolumes.Theradiusofthebridgeis1.5cm,theBois0:2180:006,andtoobtainthelinearwallspeedincm/hrmultiplytheabscissaby0.056. ThreeobservationscanbemadefromFigure 5-17 .First,introducingshearcertainlystabilizestheliquidbridge.Thereforeitcanbeconcludedthattheowactstoreducetheeectofgravity.Second,fortheexperimentsreportedinthegurethebreakupheightofthebridgeincreasesastheappliedspeedincreases.Visualobservationsofthebridgeshowedthatitdidnotachievenearsymmetryevenforthelargestwallspeedemployed.Thismeansthateventhelargestspeedwasnotenoughtoovercomethedestabilizingeectofgravityorinotherwords,correctthedensitydierence.Third,thegreatertheinjectedvolumethemorestabilizingtheowbecomes.ThisisseenexplicitlyinFigure 5-18 whichdisplaysthepercentagechangeversustheinjectedvolume.Tounderstandwhythisoccursobservethatasthevolumeinjectedincreases,thebreakupheightofthestaticbridgeincreases.Astheheighttoradiusratioincreases,theeectoftheow

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Theeectofthevolumeonthepercentageincreaseinthebreakupheightofthebridge.Theradiusofthebridgeis1.5cm,theBois0:1180:017,andthespeedis3000(168cm/hr). becomesmorepronounced.ThisalsoexpressesthenumericaltrendseenearlierinFigure 5-13 Itisnoteworthythatatsomeheight,thepercentageincreaseinthebreakupheightinthepresenceofowbeginstoplateauorbecomeconstant.Thisisbelievedtooccurbecausetheheighttoradiusratioisverylargeanddoesnotprovideanymoreincreaseinthemomentumtransferofowtothebridge.InfactinsomeexperimentsforagivenBo,weobservedadecreaseinthepercentagechangeforlargevolumes.Thisislogical,ifitisrememberedthatthebreakupheightofthebridgechangesveryslowlywithalargeincreaseinthevolume[ 25 ].Inaddition,thelargerthevolumethegreaterthe"weight"ofthebridgeandthemoredicultfortheowtohaveanimpact. Figure 5-19 whichdisplaysdataforaxedbridgevolumeshowsthatforagivenBo,increasingtheowrateenhancesthepercentagechangeinthebreakupheightofthebridge.Iftheowisstrongenough,theinterfacebecomessymmetric

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TheeectofthewallspeedonthepercentageincreaseinthebreakupheightofthebridgeforvariousBondnumbers.Theradiusofthebridgeis1.5cm,thevolumeis27.0cc.Toobtainthelinearwallspeedincm/hr,theabscissaismultipliedby0.056. atthebreakupoftheshear-inducedbridge.Theaimistogetazero"ApparentBo"withow.ThisstatementimpliesthatthegreatestpercentageincreasewouldoccurforthelargestBobridge.However,whenBoisverylarge,e.g.Bo=0:212itisseenfromthegurethattheavailableshearwasinsucienttoeectaconsiderablechangeinthestabilitypoint.ThisalsoimpliesthattheapparentBondnumberforallthreecaseswasnotthesameforagivenwallspeed.BycontrastwhentheBondnumberissmall,evenifazeroApparentBoisreachedbyintroducingow,thepercentagechangeintheheightislittlebecausethebridgeisalmostsymmetricwhenowisintroducedandlittlecorrectionoftheinterfaceshapeispermissible.Consequently,forthisexperimentalapparatus,themaximumpercentagestabilizationisobtainedforintermediateBobridgesandthisisthepointofFigure 5-19

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TheeectofthewallspeedonthepercentageincreaseinthebreakupheightofthebridgeforvariousBondnumbersandlargervolume.Theradiusofthebridgeis1.5cm,thevolumeis33.0cc.Thespeedisdi-mensionless.Toobtainthespeedincm/hr,thecurrentspeedneedstobemultipliedby0.056. Aswesawonewaytoenhancethemomentumtransferistoincreasethevolumeofthebridge.Figure 5-20 ,whichissimilartoFigure 5-19 ,isobtainedforahighervolumei.e.,for33.0ccofinjectedvolume.Asideofthefactthattheresultsaremoredramaticthereareotherfeaturesthatareinteresting.Forexampletheincreasingwallspeedinitiallycausesanincreaseinthestabilityuntilamaximumisreachedandthereafteradecreaseinthestabilityenhancement.Thiscallsforanexplanation.Whenthewallspeedissmall,thebridgewhichinitsstaticcongurationbulgesfromthebottom(say)becomesmoresymmetricandthestabilityisenhanced.AsalsoobservedbyLowry[ 25 ]asthewallspeedisincreasedandtheowgetsstronger,itactually"overcorrects"theshapeofthebridgeandipsthedirectionofthebulgei.e.,causingthebulgetoappearatthetop.ThisisparticularlytrueforthesmallBobridges,e.g.,Bo=0:04and0.08.Thus,therearetwopointsonthecurvewherethebreakupheightisthesamebutthebreakup

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occursfromthebottomfortherstpoint,andfromthetopforthesecondone.ThisisalsobelievedtobetrueforthelargerBondnumberbridgesbutthewallspeedneedstobecomelargeenoughtoseethemaximum,somethingthatwasnotpossiblewiththeavailableapparatus.ForexampleinthecaseofBo=0:13andBo=0:23,eventhelargestspeedpermittedbythecurrentapparatuswasnotenoughtoipthedirectionofthebulge.Consequently,nomaximumofpercentageincreasewasobservedfortheselargeBobridges. Asweconcludeourdiscussionoftheexperimentalresultswenotethatthe"ApparentBondnumber"whichisanotherwayofexpressingthecriticalheighttoradiusratiowhenshearisemployedcouldbecomeaslowas0.001.Thiswasobtainedataspeedof2000(112cm/hr)forabridgewhoseBondnumberwas0.124. Themainfeaturesoftheexperimentalresultsaresummarizedbythreestatements.First,foreverywallspeedthereexistsanoptimumBondnumberbridgewherethemaximumstabilityisobtained.Thevalueofthisoptimummustofcoursedependontheshearingapparatusemployedandtheuidschosen.LowBondnumberbridgeshaveanarrowwindowofstabilitywhilehighBondnumberbridgescannoteasilybestabilizedonaccountofshearinglimitations.Second,foreveryBondnumberthereisanoptimumwallspeedatwhichmaximumstabilityisobtainedforatlowwallspeedtheshearingisinsucientwhileatveryhighspeedstheshapeofthebridgeovercorrectsandbulgesfromtheoppositeend.Third,anincreaseinbridgevolumeleadstoanincreaseinmomentumtransfer.Thestabilizationchangeinthebridgethereforeincreasesuntilplateauisreached.

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Inthischapter,themainresultsofthisdissertationarere-evaluatedandfutureworkisproposed.ThisdissertationhasinvolvedadvancingtheunderstandingoftheRayleigh-Taylor(RT)andliquidbridgeproblemsbycomparingthetwoproblemsandndingwaystodelaytheinstabilities.Itwasshownthatthestabilitypointhasbeenaectedbothbythegeometryofthesystemandtheow. InanattempttounderstandtheeectofgeometryandowonthestabilityofbothproblemsatheorywasadvancedfortheRTproblemwhileexperimentswereperformedforliquidbridges.TheRTproblemwasstudiedtheoreticallybecauseoftherelativesimplicityinusingthetwo-dimensionalrectangularCartesiancoordinatesystemtolearnaboutthephysicswhileexperiments,arecomplicatedbecauseoftheinabilitytoadheretothistwo-dimensionalassumption.Atheoryforliquidbridgesontheotherhandismorecomplicatedbecauseofthecylindricalcoordinatesystemwhile,theexperimentalcomplicationsseeninRTproblemareavoidedinexperimentsonthebridge. OnemajorconclusionofthetheoryintheRTproblemisthatinducingdiusionpathsforperturbationsenhancesthestability.Anothermajorconclusionofthetheoryisthatshear-drivenowenhancesthestabilityiftheoweldisclosedandtheinterfaceisallowedtobeatinthebasestate.Inaddition,anothermajorconclusionfromthetheoryisthattwowindowsofstabilitiesareobtainedforsomeparameters.Thismeansthattherearemultiplewidthrangeswhereowcanoerstability.However,iftheowwereopenregardlessofwhetheritisintheinertialorStokeslimittheinstabilitywouldeitherbeadvancedorremainunaected. 109

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Theconclusionsfromourtheoreticalstudyraisequestionsthatoughttobeaddressedinthefuture.Therstquestioniswhethertherearetheoremsonstabilitythatmaybeobtainedforgeometriesofarbitraryshapethatcouldgiveeitherupperorlowerboundsonthestabilityorboth.Thiswouldpossiblyinvolvetheuseofvariationalprinciples.Anotherquestionthatcouldbestudiedinthefutureiswhythereisasuddenchangefromadelayintheinstabilitywhenowispresentatanerstwhileatsurfacetothesuddenadvancementininstabilitywhentheinterfaceisnotatinthebasestate.Inotherwords,wemightwonderwhytheinstabilitydoesnotchangeslowlyandcontinuouslyastheinterfacegoesfrombeingatinthebasestatetonon-atinthebasestatewhenowispresent.Thiswouldinvolvetheoryofasymptoticsonimperfectionsandsuchatheorywouldalsohavetoaddressthesituationwheremultiplestabilitywindowsarepresent. ThemajorconclusionsoftheexperimentsonliquidbridgesarethatellipticalendplatesintheliquidbridgeenhancestabilityandowenhancesstabilityprovidedtheBondnumberisnon-zero.Theseconclusionsentertainseveralpossibilitiesforthefuture.Therstproblemforfutureresearchisconnectedtothemannerinwhichanellipticalbridgebreakso.Itdoessoinasymmetricmannerfromthemidpointi.e.thehalfwaypointbetweentheendplatespresumablybecausethemidpointisofcircularcrosssection.Iftheendplatesweretwistedwithrespecttoeachotherthebasestatetopologywouldchangeandthiswouldraisethequestiononwherecrosssectionswouldbecircularandhowthestabilitywouldbeaected.Ellipticalbridgesareopentomorequestions.Itwouldbeinterestingtoseewhatwouldhappentothestabilitypointifthedeviationoftheellipticalendplatefromthecirclewerenotsmall.Atheorysupportingtheseexperimentalresultsisalsoofinterest.Thetheorymaybedevelopedeitherbyusingaperturbationtheoryorbyusingellipticalcoordinates.Intherstcase,theellipseisdeviatedfromacirclebyasmallamount.Thelatteroersatheory,

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whichisalsovalidforhighlyellipticalplatesthatwouldservefortwopurposes:obtainthestabilityofhighlyellipticalbridgesanddeterminethevalidityoftheperturbationcalculation. Anotherproblemforfuturestudyisconnectedtoowstabilizationofliquidbridges.Itwasobservedthatowstabilizesanon-zeroBondnumberbridgedependingonitsdirection.Abridgewithellipticalendplatescannotbevertical.Itwouldbeinterestingtobuildasetuptoinvestigatethestabilityofellipticalbridgessubjecttoow.Wouldtheyoergreaterstabilitycomparedtocircularbridges?

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Inthisappendix,theperturbationequationsandthemappingsusedinthetheoreticalworkareexplained.ThereaderisreferredtoJohnsandNarayanan[ 10 ]forthedetails. Let'u'denotethesolutionofaprobleminaninconvenientdomainDwhereDmaybenotspeciedandthenitmustbedeterminedaspartofthesolution.Themeaningoftheterminconvenientmaybeunderstoodwhenacalculationsimilartothatoftheo-centeredbridgepresentedinChapter 5 isstudied. ItwouldbepossibletoobtainthesolutionuandthedomainDifthesameproblemissolvedonaregulardomainD0,whichiscalledthereferencedomain.TheperturbationcalculationandthemappingrequiretheinconvenientdomainDneedstobeexpressedaroundD0inpowersofasmallparameter.ThereforethesolutionuandthedomainDaresolvedsimultaneouslyinaseriesofcompanionproblems.ThepointsofD0willbedenotedbythecoordinatey0andthoseofDbythecoordinatey.Thex-coordinateisassumedtoremainunchanged.Therefore,'u'mustbeafunctionofdirectlybecauseitliesonDandalsobecauseitisafunctionof'y'.ThepointyofthedomainDisthendeterminedintermsofthepointy0ofthereferencedomainD0bythemapping y=f(y0;)(A{1) Thefunctionf,canbeexpandedinpowersofas 22y2+(A{2) 112

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where Attheboundaryofthenewdomain,thefunctionyisreplacedbyYtopointoutthedierence.Itsexpansioninpowersofcanbewrittensimilarlyas 22Y2(Y0;=0)+(A{4) Lastly,thevariableu(y;)canbeexpandedinpowersofalongthemappingas u(y;)=u(y=y0;=0)+du(y=y0;=0) d+1 22d2u(y=y0;=0) d2+(A{5) Toobtainaformulafordu(y=y0;=0) d,dierentiateualongthemappingtakingytodependon,holdingy0xed.Usingthechainrule,thisgives du(y;) d=@u(y;) Whentheaboveequationisevaluatedat=0,weobtain du(y=y0;=0) d=u1(y0)+@u0 whereu1(y0)=@u(y0;=0) du(y=Y0;=0) d=u1(Y0)+@u0(Y0) Whenadditionalderivativesareobtainedandsubstitutedintotheexpansionofu,itbecomes u(y;)=u0+u1+y1@u0 22u2+2y1@u1

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Theaboveequationindicatesthatevenforthedomainequationsmappingneedstobeincludedinthegoverningequations.However,themappinginthedomaincannotbedetermined,infactitisnotneededneither.Wewillshowthisbymeansofanexampleandthenuseitasaruleofthumb.Let denedinourinconvenientdomain.Usingchainrule where@y0 A{2 .Holdingxed 22@y2 Thus,uptotherstorderin,thedomainequationbecomes Thedomainequationatthezerothorderinis Thedomainequationattherstorderinbecomes However,@2u0 A{15 becomes Themappingdoesnotappearinthedomainequations.However,themappingissavedforthesurfacevariablesascanbeseeninallproblemsstudiedinthisdissertation.

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InthisAppendix,weintroducethesurfacevariables,namelytheunitnormalvector,theunittangentvector,thesurfacespeedandthemeancurvature. inCartesiancoordinates,and incylindricalcoordinates.Thenormalpointsintotheregionwherefispositiveisgivenby ~n=rf Here, @xix+@f @ziz @rir+1 @i+@f @ziz @xix+iz @x2+1#1=2(B{4) 115

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inCartesiancoordinatesand @i@R @ziz @2+@R @z2#1=2(B{5) incylindricalcoordinates. ~t=ix+@Z @xiz @x2+1#1=2(B{6) inCartesiancoordinatesand @zir+iz @z2+1#1=2(B{7) or @~ir+"1+@R @z2#~i1 @@R @z~iz @2+"1+@R @z2#2+1 @@R @z2351=2(B{8) incylindricalcoordinates.

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Letthesurfacemoveasmalldistancesalongitsnormalintimet.Then,f(~rs~n;t+t)isgivenby whencef(~rs~n;t+t)=0=f(~r;t)requires Thenormalspeedofthesurface,u,isthengivenby u=s Now,usingthedenitionoftheunitnormalgivenearlierweget u=@f @t Inourproblems,thedenitionofubecomes u=@Z @t @x2+1#1=2(B{12) inCartesiancoordinatesand @t @2+@R @z2#1=2(B{13) incylindricalcoordinates. 10 ].Hereweprovidetheformulasforthesurfacesstudiedinthisdissertation.FortheCartesian

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surfacedenedbyEquation B{1 ,themeancurvatureisgivenby 2H=Zxx ThesubscriptdenotesthederivativeofZwithrespecttothatvariable.ForthecylindricalsurfacedenedbyEquation B{2 ,thecurvatureis 2H=[1+R2z][R22R2+RR]2RRz[RRzRRz]+[R2+R2]RRzz Again,thesubscriptsdenotethederivatives.

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ConsideravolumeofliquidwithagivenperiodicperturbationasseeninFigure 5{1 .Rotatedabouttheaxisofthejet,thevolumelostislessthanthevolumegained.Althoughthisstatementseemscounter-intuitive,yetitisnotdiculttoseethedierenceintheareas/volumeswhentwoslicesofsamethicknessofacylinderareconsidered.AscanbeseeninFigure C-1 ,theouterarea-similarlyvolume-isbiggerthantheinsidearea.TherotatedvolumeinFigure 5{1 issimilarinnature. InthisAppendixwewanttoprovemathematicallythatthegainedvolumeismorethanthelostvolume.IfwetakeFigure 5{1 asbasis,wecanrepresentthecurveasfollows z(C{1) ObservethatrisequaltoRwhenzis0,=2and.ThevolumegainedandlostcanbewrittenasVg=/2Z0R2+2sin22 z+2Rsin2 zdz Thevolumeargumentforavolumeofliquidwithagivenperturbation. 119

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z+2Rsin2 zdz(C{2) Whentheintegralsareevaluated,thersttwotermsareequaltoeachotherforthevolumesgainedandlost.Theyare1 2R2and1 42.Ontheotherhandthelasttermforthegainedvolumeis2Rand2Rforthelostvolume.Hence,thevolumegainedismorethanthevolumelostunderthecurverotatedabouttheaxisofthejet.

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TheaimofthiscalculationistoshowtheeectofowforaninviscidliquidintheRayleigh-Taylorandliquidjetproblems.TheproblemissketchedinFigure D-1 .Thefreesurfaceislocatedatz=1.Theliquidofdensityliesaboveapassivegas. Thegoverningnonlinearequationsarevx@vx @x vx@vz @z+g(D{1) and Thestabilityoftheproblemisdeterminedviaaperturbationanalysisde-scribedinChapter 3 .Thebasestatevelocityproleischosentobevx;0=f(z)whichsatisesthecontinuityequation.Thebasestateisgivenby Sketchoftheproblemdepictingaliquidontopofair. 121

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Theperturbedequationsaregivenasfollowsvx;0@vx;1 vx;0@vz;1 and Takingthecurloftheequationofmotion,oneobtains Lettingvx;0=CzwhereCisaconstant,eliminatingvx;1usingthecontinuityandnallyexpandingvz;1=^vz;1eikx,droppingthehat,onegets Thesolutiontotheaboveequationis Atz=0,theno-owcondition,vz;1=0,resultsinA=0.Attheinterface,z=1,theno-masstransferconditionisgivenby TheconstantBisfoundbysubstitutingtheexpressionforvz;1andvx;0asB=ikC

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Thepressureterminthenormalstressbalanceiseliminatedbyrsttakingthederivativeofitwithrespecttox,andusingtheequationsofmotion.Afterthesesubstitutions,Equation D{10 becomes isobtained.Observethatkcoth(k)islargerthanunity.AscanbeseenfromEquation D{11 ,theeectofthegravityisincreased,whichimpliesthatthecriticalwavelengthisdecreased.Therefore,theowmakesRayleigh-Taylorproblemlessstable. Thesecondproblemofinterestisowinajetwhereinertiaisdominant.Thegoverningnonlinearequationsareverysimilarbutwrittenincylindricalcoordinates.Thegoverningequation,counterpartofEquation D{7 is Aftersolvingforthedierentialequation,andapplyingboundaryconditions,theexpressionforthevelocityissubstitutedintothenormalstressbalance.Theresultingequationis R20+k2C2R20I1(kR0) Thetermcomingwiththeowisalwaysdestabilizing.Therefore,theowmakesliquidjetlessstable.

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[52] B.J.MatkowskyandE.Reiss,\Singularperturbationsofbifurcations,"SIAMJournalofAppliedMathematics,vol.33,pp.230{255,1977. [53] B.J.LowryandP.H.Steen,\Stabilizationofanaxisymmetricliquidbridgebyviscousow,"InternationalJournalofMultiphaseFlow,vol.20,pp.439{443,1994. [54] B.J.LowryandP.H.Steen,\Flowinuencedstabilizationofliquidcolumns,"JournalofColloidandInterfaceScience,vol.170,pp.38{43,1994. [55] B.J.LowryandP.H.Steen,\Stabilityofslenderliquidbridgessubjectedtoaxialows,"JournalofFluidMechanics,vol.330,pp.189{213,1997. [56] Y.J.Chen,N.D.Robinson,J.M.Herndon,andP.H.Steen,\Liquidbridgestabilization:theoryguidesacodimension-twoexperiment,"ComputationalMethodsinAppliedMechanicsandEngineering,vol.170,pp.209{221,1999. [57] S.AtreyaandP.H.Steen,\Stabilityanalysisoflongliquidbridgesinthepresenceofgravityandow,"ProceedingsoftheRoyalSocietyofLondonSeriesAMathematicalandPhysicalSciences,vol.458,pp.2645{2669,2002. [58] D.T.Johnson,\Viscouseectsinliquidencapsulatedliquidbridges,"InternationalJournalofHeatandFluidFlow,vol.23,pp.844{854,2002. [59] K.A.Brakke,\Thesurfaceevolver,"ExperimentalMathematics,vol.1,pp.141{165,1992. [60] K.A.Brakke,\Thesurfaceevolverandthestabilityofliquidsurfaces,"PhilosophicalTransactionsoftheRoyalSocietyofLondon,vol.354,pp.2143{2157,1996. [61] A.K.Uguz,N.J.Alvarez,andR.Narayanan,\Anexperimentalstudyontheinstabilityofellipticalliquidbridges,"PhysicsofFluids,vol.17,pp.0781061{0781064,2005.

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KeremUguzwasborninTurkey.HegraduatedfromBogaziciUniversityinIstanbul,Turkey,receivingaB.Sdegreein1999andaM.Sdegreein2001inchemicalengineering.Hismaster'sthesistitleis"SelectiveLowTemperatureCOOxidationinH2-richGasStreams".HethenattendedtheUniversityofFloridaforgraduatestudiesunderthesupervisionofProf.RangaNarayanan.In2006,hegraduatedfromtheUniversityofFloridawithaPh.Dinchemicalengineering. 129