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- Permanent Link:
- https://ufdc.ufl.edu/UFE0015606/00001
## Material Information- 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 )
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- 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 PAGE 5 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 PAGE 6 .................................... 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 PAGE 8 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 PAGE 10 .... 107 C-1Thevolumeargumentforavolumeofliquidwithagivenperturbation 119 D-1Sketchoftheproblemdepictingaliquidontopofair ........... 121 x PAGE 11 ThisdissertationadvancestheunderstandingoftheinstabilityofinterfacesthatoccurinRayleigh-Taylor(RT)andliquidbridgeproblemsandinvestigatestwomethodsfordelayingtheonsetofinstability,namely,changingthegeometryandjudiciouslyintroducinguidow.IntheRTinstability,itisshowntheoreticallythatanellipticalshapedinterfaceismorestablethanacircularoneofthesameareagiventhatonlyaxiymmetricdisturbancesareinictedonthelatter.Inacompanionstudyonbridges,itisexperimentallyshownthataliquidbridgewithellipticalendplatesismorestablethanacompanioncircularbridgewhoseendplatesareofthesameareaastheellipses.Usingtwodierentsizesofellipseswhosesemi-majoraxesweredeviatedfromtheradiiofthecompanioncirclesby20%,itwasfoundthattheellipticalbridge'sbreakupheightwasnearly3%longerthanthatofthecorrespondingcircularbridge. Anotherwaytostabilizeinterfacesistojudiciouslyuseuidow.Acom-prehensivetheoreticalstudyontheRTprobleminvolvingbothlinearandweaklynonlinearmethodsshowsthatmodeinteractionscandelaytheinstabilityofanerstwhileatinterfacebetweentwoviscousuidsdrivenbymovingwalls.Itis xi PAGE 12 xii PAGE 13 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 PAGE 14 Liquidbridgephotoa)Stableliquidbridgeb)Unstableliquidbridgeathigherheight. BothliquidbridgeandRayleigh-Taylorproblemshavenumeroustechnologicalapplications.Liquidbridgesoccur,forexample,intheproductionofsinglecrystalsbytheoatingzonemethod[ 1 2 ].Theyoccurintheformofowingjetsintheencapsulatedoilowinpipelines[ 3 ].Inthemeltspinningofbers,liquidjetsemittingfromnozzlesaccelerateandthinuntiltheyreachasteadystateand PAGE 15 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 ]. PAGE 16 Itisthecentralobjectiveofthisstudytoseehowtostabilizeliquidinterfacesbyapplyinganoutsideforceorbychangingthegeometryofthesystem.Forthatpurpose,understandingthephysicsofthesystem,includingthedissipationofdisturbancesandthenatureofthebreakupoftheinterfaceasafunctionofgeometryisveryimportant. Inapplicationsofliquidbridgessuchastheoatingzonetechnique,themoltencrystalissurroundedbyanotherliquidtoencapsulatethevolatilecomponentsandthepresenceoftemperaturegradientscausesow.Whethersuchowcancausestabilityornotisofinterest,sointhisstudyweshallconsidertheroleofshearinaliquidbridgeproblem.Anothereectthatisstudiedistheshapeofthesupportingsoliddisksonthestabilityofliquidbridges.Mostofthestudiesonliquidbridgespertaintobridgesofcircularendplates.Physicalargumentssuggestthatnoncircularbridgesoughttobemorestablesothisresearchalsodealswiththestabilityofnoncircularliquidbridges. Thecurrentresearchisbothexperimentalandtheoreticalincharacter.Thetheoreticalmethodsincludelinearstabilityanalysisviaperturbationcalculationsandweaklynonlinearanalysisviaadominantbalancemethod.Theexperimentalmethodsinvolvephotographyoftheinterfaceshapes.Theworkonliquidbridgeswillbeexperimentalinnatureonaccountofthedicultyinanalyzingtheproblemwithoutresorttocomputations.TheworkontheRayleigh-Taylorproblem,ontheotherhand,willbetheoreticalinnatureonaccountofdicultyinobtainingclearexperiments. Allinstabilityproblemsarecharacterizedbymodelsthatcontainnonlinearequations.Thismustbetruebecauseinstabilitybytheverynatureofitsdenitionmeansthatabasestatechangescharacterandevolvesintoanotherstate.Thefactthatwehaveatleasttwostatesisindicativethatwehavenonlinearityinthemodel.Ifthecompletenonlinearproblemcouldbesolved,thenallofthephysics PAGE 17 Shadowgraphimageshowingconvection. wouldbecomeevident.However,solvingnonlinearproblemsisbynomeansaneasytaskandoneendeavorstondthebehaviorbylinearizationofthemodelaboutaknownbasestatewhosestabilityisinquestion.Thislocallinearizationissucienttodeterminethenecessaryconditionsforinstabilityandintheabsenceofacompletesolutiontothemodelingequationsitwouldseembenecialtoobtaintheconditionsfortheonsetoftheinstability.Todeterminewhathappensbeyondthecriticalpointrequirestheuseofweaklynonlinearanalysis.Oncetheinstabilitysetsin,theinterfacecreatedintheordinaryliquidbridgeproblemandRayleigh-Taylorcongurationevolvestocompletebreakup.However,undersomeconditionseventhismaynotbetrueandwewillseelaterinthisdissertationthatasecondarystatemaybeobtainedifshearisapplied.Thereareinterfacialinstabilityproblemsthathavebeenstudiedwherepatternsmaybeobservedoncetheinstabilitysetsin.AnexampleofthisistheRayleigh-Benardproblemproblem,whichisaproblemofconvectiveonsetinauidthatisheatedfrombelow.Whenthetemperaturegradientacrossthelayerreachesacriticalvalue,patternsarepredictedandinfactarealsoobserved.Figure 1-3 isaphotographofsuchpatternsseeninanexperiment.Thefactthatsteadypatternsarepredictedandobservedimpliesasortof"saturation"ofsolutionsthatmightbeexpectedinaweakly PAGE 18 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. PAGE 19 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. PAGE 20 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 PAGE 21 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 PAGE 22 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 ]. PAGE 23 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 PAGE 24 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 PAGE 25 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 PAGE 26 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 PAGE 27 nonlinearregime[ 34 ].Inthisresearch,weareinterestedontheeectofgeometryandonshearonthestabilityoftheinterfaceinaRayleigh-Taylorconguration. Theequationsthatrepresentbothinstabilityproblemswithcorrespondingboundaryandinterfaceconditionsarepresentedinthenextsectionalongwiththemethodstosolvetheseequations. PAGE 28 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 PAGE 29 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. PAGE 30 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 PAGE 31 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 PAGE 32 orderisdenotedbyZ1,avariable,whichneedstobedeterminedduringthecourseofthecalculation.Notethatthesubscriptsrepresenttheorderoftheexpansion,e.g.thebasestatevariablesarerepresentedbyasubscriptzero.Wecanfurtherexpandv1andothersubscript'one'variablesusinganormalmodeexpansion.Consequently,thetimeandthespatialdependenciesoftheperturbedvariablesareseparatedas whereistheinversetimeconstantalsoknownasthegrowthordecayconstant.Thecriticalpointisattainedwhentherealpartofvanishes. WewilldiscussRayleigh-Taylorinstabilityinthenextchapterandapplythemodeldevelopedinthischaptertothisproblem. PAGE 33 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 PAGE 34 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) PAGE 35 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. PAGE 36 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 PAGE 37 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 PAGE 38 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. PAGE 39 ThephysicalproblemissketchedinFigure 4-2 .Observethattheradialpositiondependsontheazimuthalangle. SketchoftheRayleigh-Taylorproblemforanellipticalgeometry. ThemodelingequationsdeterminingthefateofadisturbanceareintroducedinChapter 3 .Inthisproblem,weareconsideringinviscidliquidsandthebasestateisaquiescentstatewheretheinterfaceisat.Thereforethenonlinearequationshaveatleastonesimplesolution.Itis andZ0=0.Weareinterestedinthestabilityofthisbasestatetosmalldistur-bances.Forthatpurposeweturntoperturbedequations.Theinterfaceposition PAGE 40 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 PAGE 41 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 PAGE 42 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 PAGE 43 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 PAGE 44 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 PAGE 45 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. PAGE 46 (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 PAGE 47 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. PAGE 48 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 PAGE 49 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 PAGE 50 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) PAGE 51 Aftertakingthecurloftheequationofmotionr4=0for0 PAGE 52 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 PAGE 53 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. PAGE 54 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) PAGE 55 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 PAGE 56 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 PAGE 57 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 PAGE 58 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 PAGE 59 (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 PAGE 60 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, PAGE 61 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. PAGE 62 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 PAGE 63 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 PAGE 64 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 PAGE 65 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 PAGE 66 (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 PAGE 67 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 PAGE 68 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 PAGE 69 (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 PAGE 70 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. PAGE 71 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 PAGE 72 lateralwalls,itispossibletoobtainananalyticalsolutionthough,itisnotpossibletouncouplethemodes.Infact,theworkinthissectionhasshowntheeectofmodeinteractionondelayingtheinstability. ThemainresultsofthischapterarethatanellipticalcrosssectionoersmorestabilitythanacompanioncircularcrosssectionsubjecttoaxisymmetricdisturbancesandthatsheardrivenowintheRTproblemcanstabilizetheclassicalinstabilityandleadtoalargercriticalwidth.TheseresultsmotivateustorunsomeexperimentsbutexperimentsontheRTproblemarenotsimpletoconstructandsoweconsiderbuildingliquidbridgeexperimentswithaviewofchangingthegeometryandintroducingowandseeingtheireectontheinstability. PAGE 73 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 PAGE 74 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) PAGE 75 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. PAGE 76 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 PAGE 77 and Thereisasimplesolutiontotheproblem.Itis~v=~0andP==R0whereR0istheradiusofthebridge. Theperturbedequationsatrstorderbecome and Likewisetheinterfaceconditionsatrstorderare and Thestabilityofthebasestatewillbedeterminedbysolvingtheperturbationequations.Toturntheproblemintoaneigenvalueproblem,substitute and intotherstorderequations.Intherstorderequationss,m,andkstandfortheinversetimeconstant,theazimuthalwavenumberandaxialwavenumberrespectively.Eliminatevelocitytoget drrd^P1 PAGE 78 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) PAGE 79 isobtained.Here,I0m(x)=d dxIm(x).Themostdangerousmodeiswhenmiszero.Then,theequationforthedispersioncurveis R301k2R20kR0I00(kR0) Tobegintheanalysisoftheproblem,wedrawtheattentionofthereadertoFigure 5-2 ,whichdepictsano-centeredbridgeinanouterencapsulant.Weareparticularlyinterestedinwhathappenstothedampingandgrowthratesofthe PAGE 80 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. PAGE 81 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. PAGE 82 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. PAGE 83 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() PAGE 84 Figure5-3: Thecross-sectionofano-centeredliquidbridge. Theouterliquid'sdomainequationcanbewrittensimilarly.Themassconservationandthenormalstressbalanceattheinterfacerequire and Inasimilarway,thedomainequationoforder11is Theconservationofmassequationattheinterfacebecomes whereR(1)0isthemappingfromthecurrentcongurationofano-centeredbridgetothereferencecongurationofthecenteredbridgeandwasshowntobecos().Asimilarsetofequationscanbewrittenfortheouterliquid.Thenormalstress PAGE 85 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 PAGE 86 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 PAGE 87 inequality[ 45 ],statesthat"R(0)1 5{52 ,weconcludethat2(1)iszero.Therefore,tondtheeectofo-centeringweneedtomoveontothenextorderinandget2(2). Theconservationofmassattheinterfacerequires1 where~n(2)0rP(0)1=sin2() PAGE 88 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 PAGE 89 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. PAGE 90 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 PAGE 91 lengthysowemoveontoagraphicaldepictionof2(2)andadiscussionofthephysicsoftheo-centering. 5-4 showstheeectofo-centeringonthegrowthrateconstant.Theneutralpointdidnotchange,whichisnotsurprisingbecauseattheneutralpointthepressureperturbationsareindeedzeroandsincethesystemisneutrallyatrest,itcannotdierentiatebetweencenteredando-centeredcongurations. Figure5-4. If`k'issmallerthanthecriticalwavenumber,kc,thebridgeisunstabletoinnitesimaldisturbances.AscanbeseenfromFigure 5-4 ,oncethebridgeis PAGE 92 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 PAGE 93 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. PAGE 94 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 PAGE 95 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 PAGE 96 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 PAGE 97 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 PAGE 98 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 PAGE 99 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. PAGE 100 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) PAGE 101 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 PAGE 102 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 PAGE 103 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 PAGE 104 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 PAGE 105 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 PAGE 106 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 PAGE 107 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. PAGE 108 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. PAGE 109 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. PAGE 110 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 PAGE 111 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. PAGE 112 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 PAGE 113 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 PAGE 114 Acartoonofabridgebulgingatthebottom.Thewallismoveddownwardwiththeobjectiveofobtainingasymmetricinterfacewithrespecttothemidplane. anon-shearingbridge,itwouldhavechangedtheowprolesandthereforethestabilitypointwhenthewallismoved. Theexpectationoftheexperimentswastoobtainmeasurablymorestablebridgeswithowthanwithoutow.TheheightofthebridgewasmeasuredwithaStarrettcaliperwitharesolutionof0.01mmandaccuracyof0:03mm.Thisistheonlyerrorthatmattersinourreportedresultsasonlypercentagechangesincriticalheightareofinterest. Theprocedureoftheexperimentwasasfollows.Thebridgewasrstcreatedintheabsenceofshear.Oncethedesiredvolumeofthebridgewasinjected,thevalveconnectedtotheinnerliquidinjectionportwasclosed.Thisisextremelyimportantasthepressuregradientcreatedinthechamberbymovingthewallortheupperdiskcanalterthebridgevolume.Thebreakuppointofthestaticbridgewasfoundatthisvolumebyslowlyincreasingtheheightoftheupperdiskinsmallincrementsandgivingampletimefordisturbancestosettledownorgrowbeforeeachincrement.Whenincreasingtheheightofthebridge,theencapsulantwasdrainedintotheoutsidecompartmentfromanexteriorliquidchamber.Oncethebreakuppointwasfound,thevolumeandthebreakupheightofthebridgesucedtocomputetheBondnumberusingtheSEsolver.Thewallwasmovedataconstantvelocityinthestabilizationdirectiononcethebridgestartedtobreak.Movingthewallchangedtheshapeofthebridgeimmediately.Whilethe PAGE 115 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. PAGE 116 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 PAGE 117 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 PAGE 118 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 PAGE 119 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 PAGE 120 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. PAGE 121 Inthischapter,themainresultsofthisdissertationarere-evaluatedandfutureworkisproposed.ThisdissertationhasinvolvedadvancingtheunderstandingoftheRayleigh-Taylor(RT)andliquidbridgeproblemsbycomparingthetwoproblemsandndingwaystodelaytheinstabilities.Itwasshownthatthestabilitypointhasbeenaectedbothbythegeometryofthesystemandtheow. InanattempttounderstandtheeectofgeometryandowonthestabilityofbothproblemsatheorywasadvancedfortheRTproblemwhileexperimentswereperformedforliquidbridges.TheRTproblemwasstudiedtheoreticallybecauseoftherelativesimplicityinusingthetwo-dimensionalrectangularCartesiancoordinatesystemtolearnaboutthephysicswhileexperiments,arecomplicatedbecauseoftheinabilitytoadheretothistwo-dimensionalassumption.Atheoryforliquidbridgesontheotherhandismorecomplicatedbecauseofthecylindricalcoordinatesystemwhile,theexperimentalcomplicationsseeninRTproblemareavoidedinexperimentsonthebridge. OnemajorconclusionofthetheoryintheRTproblemisthatinducingdiusionpathsforperturbationsenhancesthestability.Anothermajorconclusionofthetheoryisthatshear-drivenowenhancesthestabilityiftheoweldisclosedandtheinterfaceisallowedtobeatinthebasestate.Inaddition,anothermajorconclusionfromthetheoryisthattwowindowsofstabilitiesareobtainedforsomeparameters.Thismeansthattherearemultiplewidthrangeswhereowcanoerstability.However,iftheowwereopenregardlessofwhetheritisintheinertialorStokeslimittheinstabilitywouldeitherbeadvancedorremainunaected. 109 PAGE 122 Theconclusionsfromourtheoreticalstudyraisequestionsthatoughttobeaddressedinthefuture.Therstquestioniswhethertherearetheoremsonstabilitythatmaybeobtainedforgeometriesofarbitraryshapethatcouldgiveeitherupperorlowerboundsonthestabilityorboth.Thiswouldpossiblyinvolvetheuseofvariationalprinciples.Anotherquestionthatcouldbestudiedinthefutureiswhythereisasuddenchangefromadelayintheinstabilitywhenowispresentatanerstwhileatsurfacetothesuddenadvancementininstabilitywhentheinterfaceisnotatinthebasestate.Inotherwords,wemightwonderwhytheinstabilitydoesnotchangeslowlyandcontinuouslyastheinterfacegoesfrombeingatinthebasestatetonon-atinthebasestatewhenowispresent.Thiswouldinvolvetheoryofasymptoticsonimperfectionsandsuchatheorywouldalsohavetoaddressthesituationwheremultiplestabilitywindowsarepresent. ThemajorconclusionsoftheexperimentsonliquidbridgesarethatellipticalendplatesintheliquidbridgeenhancestabilityandowenhancesstabilityprovidedtheBondnumberisnon-zero.Theseconclusionsentertainseveralpossibilitiesforthefuture.Therstproblemforfutureresearchisconnectedtothemannerinwhichanellipticalbridgebreakso.Itdoessoinasymmetricmannerfromthemidpointi.e.thehalfwaypointbetweentheendplatespresumablybecausethemidpointisofcircularcrosssection.Iftheendplatesweretwistedwithrespecttoeachotherthebasestatetopologywouldchangeandthiswouldraisethequestiononwherecrosssectionswouldbecircularandhowthestabilitywouldbeaected.Ellipticalbridgesareopentomorequestions.Itwouldbeinterestingtoseewhatwouldhappentothestabilitypointifthedeviationoftheellipticalendplatefromthecirclewerenotsmall.Atheorysupportingtheseexperimentalresultsisalsoofinterest.Thetheorymaybedevelopedeitherbyusingaperturbationtheoryorbyusingellipticalcoordinates.Intherstcase,theellipseisdeviatedfromacirclebyasmallamount.Thelatteroersatheory, PAGE 123 whichisalsovalidforhighlyellipticalplatesthatwouldservefortwopurposes:obtainthestabilityofhighlyellipticalbridgesanddeterminethevalidityoftheperturbationcalculation. Anotherproblemforfuturestudyisconnectedtoowstabilizationofliquidbridges.Itwasobservedthatowstabilizesanon-zeroBondnumberbridgedependingonitsdirection.Abridgewithellipticalendplatescannotbevertical.Itwouldbeinterestingtobuildasetuptoinvestigatethestabilityofellipticalbridgessubjecttoow.Wouldtheyoergreaterstabilitycomparedtocircularbridges? PAGE 124 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 PAGE 125 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 PAGE 126 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. PAGE 127 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 PAGE 128 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. PAGE 129 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 PAGE 130 surfacedenedbyEquation B{1 ,themeancurvatureisgivenby 2H=Zxx ThesubscriptdenotesthederivativeofZwithrespecttothatvariable.ForthecylindricalsurfacedenedbyEquation B{2 ,thecurvatureis 2H=[1+R2z][R22R2+RR]2RRz[RRzRRz]+[R2+R2]RRzz Again,thesubscriptsdenotethederivatives. PAGE 131 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 PAGE 132 z+2Rsin2 zdz(C{2) Whentheintegralsareevaluated,thersttwotermsareequaltoeachotherforthevolumesgainedandlost.Theyare1 2R2and1 42.Ontheotherhandthelasttermforthegainedvolumeis2Rand2Rforthelostvolume.Hence,thevolumegainedismorethanthevolumelostunderthecurverotatedabouttheaxisofthejet. PAGE 133 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 PAGE 134 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 PAGE 135 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. PAGE 136 [1] D.Langbein,\Crystalgrowthfromliquidcolumns,"JournalofCrystalGrowth,vol.104,pp.47{59,1990. [2] R.A.Brown,\Theoryoftransportprocessesinsinglecrystalgrowthfromthemelt,"AIChEJournal,vol.34,pp.881{911,1988. [3] C.Hickox,\Instabilityduetoviscosityanddensitystraticationinaxisymmet-ricpipeow,"PhysicsofFluids,vol.14,pp.251{262,1971. [4] J.B.Grotberg,\Preface:Biouidmechanics,"PhysicsofFluids,vol.17,Art.No.031401,2005. 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PAGE 141 KeremUguzwasborninTurkey.HegraduatedfromBogaziciUniversityinIstanbul,Turkey,receivingaB.Sdegreein1999andaM.Sdegreein2001inchemicalengineering.Hismaster'sthesistitleis"SelectiveLowTemperatureCOOxidationinH2-richGasStreams".HethenattendedtheUniversityofFloridaforgraduatestudiesunderthesupervisionofProf.RangaNarayanan.In2006,hegraduatedfromtheUniversityofFloridawithaPh.Dinchemicalengineering. 129 |