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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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- 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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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)
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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
PAGE 7
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
PAGE 9
............................. 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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KeremUguzwasborninTurkey.HegraduatedfromBogaziciUniversityinIstanbul,Turkey,receivingaB.Sdegreein1999andaM.Sdegreein2001inchemicalengineering.Hismaster'sthesistitleis"SelectiveLowTemperatureCOOxidationinH2-richGasStreams".HethenattendedtheUniversityofFloridaforgraduatestudiesunderthesupervisionofProf.RangaNarayanan.In2006,hegraduatedfromtheUniversityofFloridawithaPh.Dinchemicalengineering. 129
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