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DATE DUE DATE DUE DATE DUE 1/98 animus-p.14 RAYIJEIGH-TAYLOR INSTABILITIES IN CONSTANT ACCELERATED FRAMES By Garlen Dale Wesson A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1997 ABSTRACT RAYLEIGH—TAYLOR IN STABILITIES IN CONSTANT ACCELERATED FRAMES By Garlen Dale Wesson An analogy between the velocity of progressive waves at the interface between two immiscible fluids and the mou'on of liquid droplets (or gas bubbles) moving through low- viscosity fluids was used to estimate the terminal velocity of drops and bubbles in accelerating fields. The work was conducted in three phases: first, the linear stability theory for accelerating flat interfaces was reviewed; second, the theory was tested experimentally for accelerating flat interfaces; and third, the extended wave analogy was tested experimentally for predicting the terminal velocity of bubbles and drops in accelerating environments. The linear stability analysis was developed and reviewed for the accelerating flat interface between two superposed fluids. The theoretical analysis predicted that the growth rate of unstable disturbances was stabilized by the presence of viscosity and interfacial tension. Interfacial tension was shown to completely stabilize an otherwise unstable arrangement for sufficiently large wavenumbers for the class of disturbances investigated. For an unstable arrangement, it was demonstrated that the depth of the viscous sublayer was largest on the side of the interface on which the fluid with the largest kinematic viscosity resided. The opposite was shown for the stable case. The hydrodynamic stability of an accelerating flat interface was investigated experimentally using three fluid/fluid systems: air/water, kerosene/water and Freon/water. The fluid/fluid pairs were accelerated up to 46 g while being videographed by a high speed video camera. The projected interface was analyzed by using a discrete Fourier transform. The growth rate of each individual harmonic component was measured and compared to linear stability theory. For the systems investigated, the growth rate followed the trends predicted by the theory, however these rates were significantly lower in the region of maximum growth rate and near the cut-off wave number. Stability was observed for the kerosene/water system in the region of the predicted cut-off wave number. Experiments were performed to determine the terminal velocity of bubbles and drops in accelerating frames and to test the extended wave analogy. The two systems studied were: kerosene droplets and air bubbles dispersed in a continuous water phase. The experimental results were grouped into two stable fluid particle shapes (spherical and spherical cap) and an unstable shape. The terminal velocity of the stable shapes predicted by the analogy fell within experimental error, however, the terminal velocity for the unstable shape was lower than that predicted by the wave analogy. “The allotments of Providence, when coupled with trouble and anxiety, often conceal from finite vision the wisdom and goodness on which they were sent; and frequently, what seemed a harsh and invidious dispensation, is converted by after experience into a happy and beneficial arrangement.” Frederick Douglass, My Bondage and My Freedom , 1855. This work is dedicated to the memory of my father, Garlen Wesson iv ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to Dr. Charles A. Petty, Advisory Committee Chairman, for his guidance and assistance throughout this study and in the preparation of this manuscript. I also wish to extend my appreciation to the members of my advisory committee: Dr. John F. Foss, Dr. K. Jayaraman, Dr. James P. Meyer and Dr. Susan E. M. Selke for their guidance and assistance. Other Michigan State University faculty members deserving of my thanks are Dr. Cordell Overby and Dr. Nicholas Altiero for their moral support during my graduate studies. To my mother, Rose M. Wesson; my brother, Dr. Donald E. Wesson; and my sister Dr. Rosemarie D. Wesson-Williams, I give sincere thanks for their constant love, understanding, and prayers without which this would not have been possible. Also I would like to extend my gratitude to all of my close friends; in particular, Dr. Valerie Adegbite, Bill and Demetra Allen, Vanessa Bradely-Hodge, Pree Buford, Dr. Janet Green, Maya Murshak and Dr. Steve Parks, Vanessa Otway, and Maria Victoria Tejada-Simon and Dr. Klaus Weispfennig who have made my graduate school experience a memorable one. Sincere and deepest love is reserved for my wife, Sandra Thompson-Wesson, whose love, encouragement, and support during times when frequent shifting deadlines were commonplace, made this journey worthwhile; and, my most perfect daughter, Galen S.T. Wesson, whose smile has made all the difference. TABLE OF CONTENTS Bass NOMENCLATURE .......................................................................... 1x LIST OF TABLES ............................................................................ xii LIST OF FIGURES .......................................................................... xvi CHAPTER 1 . INTRODUCTION .................................................................. 1 1.1 Introduction ......................................................... 1 1.2 Problem Description ............................................... 1 1.3 Background ......................................................... 2 1.4 Motivation For This Work ........................................ 4 1.5 Objectives ........................................................... 5 1.6 Methodology and Dissertation Outline .......................... 6 vi 2. LITERATUREREVIEW ........................................................... 8 2.1 2.2 2.3 Introduction ......................................................... 8 Accelerating Interfaces ............................................ 8 Bubble and Drop Terminal Velocities ........................... 12 3. STABILITY OF AN ACCELERATING FLUID/FLUID INTERFACE - THEORETICAL ............................... 15 3.1 Introduction ......................................................... 15 3 .2 Mathematical Formulation ........................................ 15 3 .3 Hydrostatics ........................................................ 20 3 .4 Linear Stability Analysis .......................................... 22 3.5 Inviscid Theory .................................................... 37 3.6 Viscous Theory .................................................... 45 3 .7 Viscous Theory: Marginal Stability ............................. 56 3 .8 Viscous Theory: Unstable Interfaces ........................... 57 3 .9 Stable Interfaces .................................................... 70 3.10 Conclusions ........................................................ 89 4. STABILTTY OF AN ACCELERATING FLUID/FLUID INTERFACE - EXPERIMENTAL ............................. 93 4.1 Introduction ......................................................... 93 4.2 Experimental Methodology ....................................... 93 4.3 Apparatus ........................................................... 99 4.4 Experimental Procedure ........................................... 102 4.5 Experimental Results .............................................. 104 4.6 Discussion of Results ............................................. 116 4.7 Conclusions ........................................................ 125 vii 5. TERMINAL VELOCTTY OF ACCELERATED FLUID PARTICLES ....................................................................... 127 5.1 Introduction ......................................................... 127 5.2 Background ......................................................... 128 5 .3 Apparatus ........................................................... 135 5.4 Experimental Procedure ........................................... 135 5.5 Results and Discussion ............................................ 137 5.6 Conclusions ........................................................ 155 6. CONCLUSIONS .................................................................... 157 7. RECOMMENDATIONS FOR FURTHER RESEARCH ...................... 164 APPENDICES A THE INTERFACIAL FORCE BALANCE .............................. 166 B ROTATIONAL WAVE NUMBERS ..................................... 171 C INTERFACIAL BOUNDARY CONDTTIONS ......................... 174 D COMPUTER PROGRAM AND RESULTS FOR THE EIGENVALUES OF THE RAYLEIGH—TAYLOR VISCOUS INTERFACES ................................................. 178 E DATA ERROR ANALYSIS ............................................... 197 F ACCELERATION DATA - FLAT SURFACE EXPERIMENTS . . . . 200 G SURFACE DATA - FLAT SURFACE EXPERIMENTS ............. 208 H ACCELERATION DATA - FLUID PARTICLE EXPERIMENTS . . 223 I VELOCTTY DATA - FLUID PARTICLE EXPERIMENTS .......... 229 LIST OF REFERENCES .................................................................... 235 viii NOMENCLATURE WW 0 waver "510 3300'?! "H frame acceleration; also see Eq.(3.87) see Eq.(3.58) velocity solenoidal vector field Atwood number See Table 3.1 Bond number, Bo“ particle Bond number wave velocity dL, dimensionless fluid particle diameter, see Eq.(S .4) D; , dimensionless fluid particle diameter, see Eq.(5.11) unit vector frequency , dimensionless functions fl and f2, see Eq.(5.5) planforrn function acceleration due to gravity complex viscous factor, see Eq.(3.112) identity tensor «Fr wave number fluid particle diameter; If, length scale, see Eq.(3.96) a“ 233 ”O H llt-l C: I: l€ viscous spacial growth factor, see Eq.(3.25) viscous factor, See Eqs.(3.66) and (3.91) Morton number unit normal vector viscosity parameter, See. Eq.(3.109) pressure fluid particle radius, see Eq.(5.15) Reynolds Number; Rep, particle Reynolds number growth rate material surface time total stress dyadic velocity; u, terminal velocity; a}, dimensionless terminal velocity U; dimensionless terminal velocity, see Eq.(5.10) vorticity Weber number, We*, particle Weber number; We“ modified particle Weber number x position dimensionless viscous sublayer depth y position 2 position relative to interface Greek Sam 13mm (1 viscosity ratio, see Eqs.(3.83) and (3.84) density ratio, see Eqs.(3.85) and (3.86) distance increment stability parameter, see Eq.(3.95) dimensionless growth parameters: it), growth rate, ‘1’; wave number velocity scalar potential relative acceleration parameter, see Eq.(3.79) wave length viscosity density kinematic viscosity interfacial tension shear component of stress tensor LIST OF TABLES Iable Base 1.1 Dimensional Analysis of Forces ........................................... 3 3.1 Components of the Dispersion Dyadic .................................... 34 3.2 Physical Property Parameters for Inviscid Theory ....................... 42 3.3 Physical Property Parameters for Viscous Theory ....................... 46 3.4 Dimensionless Physical Property Groups ................................ 53 3.5a Effect of Relative Acceleration on the Maximum Growth Rate of an Unstable Air/Water Interface (e=+ 1) ................................ 62 3 .Sb Effect of Relative Acceleration on the Maximum Growth Rate of an Unstable Kerosene/Water Interface (e=+ 1) ........................ 63 3 .5c Effect of Relative Acceleration on the Maximum Growth Rate of an Unstable Water/Freon Interface (e=+1) ............................ 64 3.6a Effect of Relative Acceleration on the Rotational and Irrotational Sublayers Near Unstable Air/Water Interface (e=+ 1) ................... 66 3 .6b Effect of Relative Acceleration on the Rotational and Irrotational Sublayers Near Unstable Kerosene/Water Interface (e=+1) ............ 67 3 .6c Effect of Relative Acceleration on the Rotational and Irrotational Sublayers Near Unstable Water/Freon Interface (e=+1) ................ 68 3 .7 Complex Eigenvalues for the Water/Freon Interface .................... 81 3.8a Effect of Relative Acceleration on the Complex Eigenvalues and the Rotational Sublayers of an Air/Water Interface Near the Minimum Wave Speed (e=-1) .......................................... 83 3.8a 3.8b 3.9 4.1 4.2 4.33 4.3b 4.4 5.1 5.2 D] D.2 D.3 D4 D5 D6 D7 D8 D9 F.1 F2 F3 F4 F.5 Effect of Relative Acceleration on the Complex Eigenvalues and the Rotational Sublayers of an Kerosene/Water Interface Near the Minimum Wave Speed (€=-l) .......................................... 84 Effect of Relative Acceleration on the Complex Eigenvalues and the Rotational Sublayers of an Water/Freon Interface Near the Minimum Wave Speed (e=-1) .......................................... 85 Solutions for Eq.(3.106) for Stable Interfaces (e=-l) ................... 88 Physical Property Data ...................................................... 106 Calculated Experimental Accelerations .................................... 108 Projected Interface Extracted Graphical Data Points ..................... 112 Results of FFT - Harmonic Amplitudes ................................... 112 Measured Growth Rates .................................................... 118 Experimental Results ........................................................ 146 Experimental Results - Wave Analogy .................................... 149 Program Notation ............................................................ 180 Program Listing: Eigenvalues for Rayleigh-Taylor Interfaces .......... 181 Examples Output of F indRoot Algorithm ................................. 183 Eigenvalues for Unstable Air/Water Interfaces ........................... 185 Eigenvalues for Unstable Kerosene/Water Interfaces ................... 187 Eigenvalues for Unstable Water/Freon Interfaces ........................ 189 Eigenvalues for Stable Air/Water Interfaces .............................. 191 Eigenvalues for Stable Kerosene/Water Interfaces ....................... 193 Eigenvalues for Stable Water/Freon Interfaces ........................... 195 Raw Acceleration Data - Experiment 0213-03 (Air/Water) ............. 201 Raw Acceleration Data - Experiment 0213-07 (Air/Water) ............. 202 Raw Acceleration Data - Experiment 0215-03 (Air/Water) ............. 203 Raw Acceleration Data - Experiment 0412-01 (Kerosene/Water) ...... 204 Raw Acceleration Data - Experiment 0412-04 (Kerosene/Water) ...... 205 F.6 F7 6.1 6.2 G3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 H] 11.2 11.3 11.4 11.5 1.1 1.2a I.2b 1.3 Raw Acceleration Data - Experiment 0412-05 (Kerosene/Water) ...... 206 Raw Acceleration Data - Experiment 0512-03 (W ater/Freon) .......... 207 Projected Surface Data - Experiment 0213-03 (Air/Water) ............. 209 Projected Surface Data - Experiment 0213-07 (Air/Water) ............. 210 Projected Surface Data - Experiment 0215-03 (Air/Water) ............. 211 Projected Surface Data - Experiment 0412-01 (Kerosene/Water) ...... 212 Projected Surface Data - Experiment 0412-04 (Kerosene/Water) ...... 213 Projected Surface Data - Experiment 0412-05 (Kerosene/Water) ...... 214 Projected Surface Data - Experiment 0512-03 (W ater/Freon) .......... 215 FFTCalculated Amplitudes - Experiment 021303 (Air/Water) ........ 216 FFTCalculated Amplitudes - Experiment 021307 (Air/Water) ........ 217’ FT Calculated Amplitudes - Experiment 0215-03 (Air/Water) ........ 218 FFT Calculated Amplitudes - Experiment 0412-01 (Kerosene/Water) 219 FFT Calculated Amplitudes - Experiment 0412-04 (Kerosene/Water) 220 FFT Calculated Amplitudes - Experiment 0412-05 (Kerosene/Water) 221 FFT Calculated Amplitudes - Experiment 0512—03 (W ater/Freon) .... 222 Raw Acceleration Data - Experiment 0815-02 (Air/Water) ............. 224 Raw Acceleration Data - Experiment 0815-02 (Air/Water) ............. 225 Raw Acceleration Data - Experiment 0815-03 (Air/Water) ............. 226 Raw Acceleration Data - Experiment 0815-05 (Kerosene/Water) ...... 227 Raw Acceleration Data - Experiment 0815-05 (Kerosene/Water) ...... 228 Raw Particle Position Data - Experiment 0815-01 (Air/Water) ......... 230 Raw Particle Position Data - Experiment 0815-02 (Air/Water) Upper Bubble ................................................................ 231 Raw Particle Position Data - Experiment 0815-02 (Air/Water) Lower Bubble ................................................................ 231 Raw Particle Position Data - Experiment 0815-03 (Air/Water) ......... 232 xiv 1.4a I.4b 1.5a I.5b Raw Particle Position Data - Experiment 0815-05 (Kerosene/Water) Upper Drop ................................................................... 233 Raw Particle Position Data - Experiment 0815-05 (Kerosene/Water) Lower Drop ................................................................... 233 Raw Particle Position Data - Experiment 0816—01 (Kerosene/Water) Upper Drop ................................................................... 234 Raw Particle Position Data - Experiment 0816-01 (Kerosene/Water) Lower Drop ................................................................... 234 XV LIST OF FIGURES Him Base 3.1 Planar View of an Accelerating Interface ...................................... 17 3.2 Hydrostatic Pressure Distribution for p2>p,,aF>g. .......................... 21 3.3 Growth Rate of Unstable Inviscid Interfaces (aF=0) ......................... 41 3 .4 The Effect of Frame Acceleration on the Maximum Growth Rate for Unstable Inviscid Fluids .................................................... 44 3.5 Solution Flow Chart for Solving Eq.(3.90) .................................. 50 3 .6 Dimensionless Growth Rates for Unstable Viscous Interfaces ............. 58 3.7 The Effect of 'y on the Maximum Growth Rate ............................... 60 3.8 The Effect of Wave Length on the Wave Speed for Marginally Stable Interfaces ..................................................... 72 3 .9 The Effect of Frame Acceleration on the Minimum Wave Speed for Marginally Stable Interfaces ................................................ 74 3.10 Relationship Between the Weber Number and the Bond Number for Progressive Waves Supported by an Inviscid Interface ................. 76 3.11 The Effect of Wavelength on the Wave Speed for Stable Viscous Interfaces ........................................................ 80 4.1 Elevator Dimensions and Viewing Prospective ............................... 95 4.2 Frame Acceleration vs. Critical Wave Number ............................... 98 4.3 Experimental Apparatus ......................................................... 100 4.4 Square Root Elevator Position vs. Time Experiment 0213-03 (Typical) .................................................. 107 xvi 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 Video Image(l93x239 Pixels) - Experiment 0213-03, t=10ms ............. 110 Projected Interface Graph for Experiment 0213-03, t=10ms ............... 111 Experiment 0213-03, t=lms; a) actual video frame; b) graphical representation; c)Discrete Fourier Transform .................. 113 Experiment 0213-03, t=10ms; a) actual video frame; b) graphical representation; c)Discrete Fourier Transform .................. 1 l4 Experiment 0213-03, t=16ms; a) actual video frame; b) graphical representation; c)Discrete Fourier Transform .................. 1 15 Amplitude of k=1.35 cm’l Wave vs. Time for Experiment 0213-03 (typical) ................................................................. 117 Dimensionless Growth Rate vs. Dimensionless Wavenumber Air-Water .......................................................................... 119 Dimensionless Growth Rate vs. Dimensionless Wavenumber Kerosene-Water .................................................................. 120 Dimensionless Growth Rate vs. Dimensionless Wavenumber Water-Freon ....................................................................... 121 Experimental Results Comparison ............................................. 122 Dimensionless Growth Rate vs. Dimensionless Wave Number Experimental Results Comparison with Other Researchers ................. 124 Generalized Correlation of Mersmann, 1983 ................................. 131 Weber number vs. Bond Number - wave Analogy .......................... 134 Experimental Apparatus ......................................................... 136 Video Frames for Experiment 0814—01 Kerosene Drop in Water, aF= 0 g .............................................. 138 Video Frames for Experiment 0815-05 Kerosene Drops in Water, aF= 12.6 g ......................................... 139 Video Frames for Experiment 0814-01 Air Bubbles in Water, aF= 46.3 g .............................................. 141 Observed Fluid Particle Shapes ................................................. 142 Square Root Elevator Position vs. Time Experiment 0815-03 ( Typical) ................................................. 143 Fluid Particle Displacement vs. Time Experiment 08150—03 (Typical) ................................................ 144 xvii 5.10 5.11 5.12 5.13 5.14 5.15 A.1 Generalized Correlation of Mersmann, 1983 (Air/Water", Kerosene/Water; Kerosene/Water, 7:1) ......................... 147 Generalized Correlation of Mersmann, 1983 (Air/Water; Kerosene/Water; Kerosene/Water, 7:1 Peebles and Garber, 1953) ...................................................... 148 Weber Number vs. Bond Number (Air/Water; Kerosene/Water; Kerosene/Water, 7:1) ......................... 150 Weber Number vs. Bond Number (Air/Water; Kerosene/Water; Kerosene/Water, F1) ......................... 152 Weber Number vs. Bond Number Generalized Correlation of Mersmann, 1983 (Air/Water; Kerosene/Water; Kerosene/Water, 7:1 Peebles and Garber,l953) ....................................................... 153 Modified Weber Number vs. Bond Number (Air/Water; Kerosene/Water; Kerosene/Water, 7:1) ......................... 154 Arbituary Surface Element ...................................................... 168 xviii CHAPTER 1: INTRODUCTION 1.1 Introduction This chapter introduces the reader to the problem studied in this work. This is accomplished by defining the problem in Section 1.2 and subsequently developing the background for the forces involved and the dimensionless groups anticipated. Next, the motivation for this work is presented followed by the objectives to be accomplished. The research methodology and a dissertation summary conclude this chapter to familiarize the reader with the structure of this work. 1.2 Problem Description The investigation of the instability of disturbances at the interface between two immiscible fluids is investigated in this work This is first done theoretically by applying a 2 linear stability analysis. The theory is then tested experimentally by observing accelerating fluid/fluid interfaces. Linear stability theory is also used to predict the terminal velocity of liquid droplets and gas bubbles (hereinafter referred to as fluid particles) in linearly accelerating fields by extending an earlier observation by Mendelson (1967). The terminal velocity prediction based on this wave analogy is tested experimentally by observing the terminal velocity of fluid particles in linearly accelerated environments. 1.3 Background When investigating the forces involved in the destruction of an interface in the presence of no other extemal body forces other that gravity, four forces can be envisioned. The forces include: 1 An inertial force, resulting from velocity of the fluids at the interface; 2. A viscous force, resulting from the presence of viscosity in the fluids; 3. An accelerating force, rising from a density difference between the fluids in the presence of an acceleration; and, 4. An interfacial tension force, arising from an imbalance of molecular forces caused by the presence of two different fluids at the interface. These four forces, when represented as ratios, can be expressed by three independent dimensionless groups. The matrix shown in Figure 1.1. represents all possible force pair combinations. The vast majority of theoretical and experimental investigations of the break up of an interface have focused on environments in which the viscous and inertial destructive forces dominate with the surface force serving as the restoration force. The Table 1.1 Dimensional Analysis of Forces Inertial Viscous Surface Acceleration Inertia 1 i L a Re We We Viscous Re 1 g BoRe We We Surface We & 1 Bo Re Acceleration E We L 1 Bo BoRe Bo Reynolds Number, Re = Inertial Vrscous Weber Number, We = hem“ Surface Bond Number, B0 = Acceleration Surface 2 Morton Number Mo = We Bo Re4 acceleration force is considered, by omission, to be negligible. From Table 1.1 we see that the dimensionless groups controlling such processes are the Weber and Reynolds numbers. Examples of such engineering applications where these forces dominate include: turbulent pipe flow, tank agitation, and shearing valves. Little attention has been given to engineering applications in which the acceleration force dominates the drop breakup process. Again, from Table 1.1, it is shown that the groups controlling such a process are the Bond and Weber numbers. This may occur in such common engineering applications as centrifugal separators, flow contractions, and Milk transfers of low viscosity fluids. For applications in which all four forces are important, many combinations of the four forces can be envisioned. One such group is the Morton number, which may be formed as a combination of all four forces, see Table 1.1. The magnitude of Mo indicates the mlative importance of the combination of the acceleration and viscous forces to the combination of the surface and inertia forces. Systems with extremely small Mo may be enViSi0r1ed to be approximated by neglecting either the viscous or acceleration forces and thus l‘e-ducing to the first and second cases mentioned above. 1'4 Motivation for this Work Contamination of fluid layers is a frequent and often undesirable occurrence in Various industries. Examples include ships’ oily bilge waters (Kumar and Kannan, 1994) and tank transfer of two phase systems. This phenomena can be caused by Rayleigh- TaYlor instabilities growing unbounded at the interface to ultimately cause turbulent mixing of the phases (Read, 1984)- The essential problems of hydrodynamic stability were recognized and formulated in the nineteen century, most notably by Helmhotz, Kelvin, Rayleigh, and Reynolds (Drazin and Reid,1984). Investigators have since continued to develop experimental and theoretical analysis of this phenomena in an effort to predict the onset of these instabilities. A fundamental understanding of the onset of these instabilities is desired since, in the case of spherical interfaces, may lead to drop or bubble breakup. Experimental validation of the linearized theory is also desired. When designing inertial separation processes where the separation efficiency may be a function of the drop size, such as hydrocyclones and centrifuges, knowledge of the break up process is essential. Also germane is the ability to predict the terminal velocity of the fluid particles in an accelerating environment. In the case of hydrocyclone design, the terminal velocity is often approximated using Stokes’ law. This concept is only valid for a particle Reynolds number of less than 0.1. Other models, developed in inertial frames are available but are not based on fundamental concepts and therefore are not easily converted to the accelerating environment. A model valid for the accelerating environment is therefore P’Pferred. Experimental validation of the initial growth rates predicted by linearized stability theory and the development of a model which predicts the terminal velocity of fluid pal“licles in accelerating environments provide the motivation for this work. 1'5 Objectives The principal goal of this research is to determine how well the linear hydrodynamic s‘i‘bility theory predicts the onset and growth of instabilities at an accelerating interface. The objective will be accomplished by performing the following tasks: 6 y... . Develop in detail the linear hydrodynamic stability theory for an accelerating flat interface between two immiscible fluid layers including the effect of interfacial tension; 2. Build an apparatus to measure the growth rate of disturbances introduced into the interface. These growth rates will be compared to the linear stability theory developed in part 1; 3. Expand the observation of Mendelson (1967) to include accelerating environments to develop a model that can be used to predict the terminal velocity of fluid particles in an accelerating field; and, 4. Measure experimentally the terminal velocity of bubbles and drops and compare with the prediction of the new model. 1 - 6 Methodology and Dissertation Outline The instability of an accelerating interface was investigated experimentally and theot‘etically. This instability is hypothesized to be the cause of the initial growth of disturbances which lead to the eventual break up of the interface exposed to conditions reIDl‘esented by high Bond numbers. Chapter 1 formally introduces the reader to the stability problem and outlines the 0blectives and methodology of the research. Chapter 2 gives a historical review of the theOretical and experimental studies previously conducted on the Rayleigh-Taylor iIIStability. Chapter 2 also outlines experimental and theoretical work done in the study of leminal velocity of bubbles and drops. Chapter 3 and 4 are focused on the stability of an accelerating flat interface. Chapter 3 develops the linear stability theory for two immiscible, viscous fluids including 7 the effects of interfacial tension. The stabilizing effects of interfacial tension and viscosity are also investigated. Chapter 4 presents the results of an experimental study of flat interfaces. In this chapter, the reader is introduced to the novel concept of decomposing the projection of the interface into its fundamental waves by way of Fourier decomposition. The growth rates of the individual wave lengths are measured and compared with those predicted by the linearized stability theory. Chapter 5 introduces the analogy first developed by Mendelson(l967) of relating the terminal velocity of drops and bubbles to the progressive wave velocity at the interface between two immiscible fluids in a stable configuration. The analogy is expanded to include a constant accelerating environment. The model is formulated by the application 'of linear stability theory presented in Chapter 3 for the propagation of progressive waves at the interface of viscous fluids. Chapter 5 also presents the experimental analysis of the study of drops and bubbles mOving through a fluid while in an accelerating environment. The experimental results are first compared with the empirical analysis of Mersmann( 1967), and then expanded to include the effects of an accelerating frame. The terminal velocity prediction of the expatrded Mendelson analogy is compared with the experimental results of this work and ‘hOSe of Peebles and Garber(1953). Chapter 6 gives the conclusion of this research and finally, Chapter 7 gives the recommendations for further research. CHAPTER 2: LITERATURE REVIEW 2-1 Introduction This chapter summarizes the current available literature in the area of linear hydrodynamic stability of accelerating interfaces. The review is sectioned further into the theonetical developments and experimental analysis. The review of theoretical and experimental analysis of drop terminal velocity is also reviewed. The current literature is lit"lited to those studies in the inertial domain. Apparently, no studies have been conducted m cOnstant accelerated frames. 2'2 Accelerating Interfaces TheOrerical Developments More than 100 years ago, Lord Rayleigh (1883) published the classical treatment entitled “luvestigation of the Character of the Equilibrium of an Incompressible Fluid of Variable Density”. It was motivated as an illustration of the theory of cirrus clouds but it was not Specific for this case. The work describes the linear stability eigenvalue problem for incompressible fluids under the influence of gravity. The general stability criterion for 9 incompressible inviscid fluids, without the effects of interfacial tension, is discussed and particular perturbation solutions for exponential density variations are derived. Rayleigh is given credit to being the first to pose the stability problem in a general manner and to recognize its principle significance for atmospheric stratifications. During his time, stable surface waves had already been well understood theoretically (Kull,1991). Some early observations of the dependence of oscillation period on the densities of the two fluids were apparently made by Benjamin Franklin in a letter dated 1762 (Lamb,l932). In that letter, Franklin noted “the natural periods of oscillation of the common surface of two fluids of very nearly equal density are very long compared with those of a free surface of similar extent” (Lamb, 1932). This observation is indicative of the influence of the Atwood number on the oscillation frequency of a stable interfacial arrangement. Harrison( 1908) next included the effect of viscosity in the treatment of superimposed fluid layers. In this paper, a complete analytical discussion of two problems is offered: 1) the case of two superposed fluids of infinite extent and, 2) the case of a fluid of finite depth superposed on a fluid of infinite depth. The solution offered an analytical explanation of the phenomena observed by Ekrnan (1904) in a Norwegian north polar expedition. He remarked that a ship moving in the Norwegian Foirds experiences great resistance owing to considerable waves being set up at the common interface of a layer of fresh water and sea water. Such waves are quickly damped, and would therefore drain a great amount of energy from the ship. Several topics of interfacial instabilities of superposed fluids are treated in the hydrodynamic text of Sir Horace Lamb(1932) and serves as an excellent treatise on the subject through 1932. Other related topics covered are gravity waves (Art. 449; pp. 625- 8), the oscillations of liquid drops (Art 2755; p. 473-5 and Art. 355; p. 639-41) and the effects of surface tension on the oscillation of a surface (Arts. 266 and 267; pp. 456-461). Taylor(1950) later proposed that the stability of superposed fluids accelerating with a constant acceleration in a direction perpendicular to the plane of stratification can be 10 treated by the same formalism originally pr0posed by Rayleigh(1833). By the application of the principle of equivalence in dynamics, Taylor replaced the acceleration due to gravity in Rayleigh’s formulation with an apparent or net acceleration of the system. In Taylor’s formulation, however, he followed the lead of Rayleigh and developed the analysis for an inviscid fluid without the effect of interfacial tension. After the publication of this paper, the phenomena became known as the Rayleigh-Taylor (R-T) instability. -The effects of interfacial tension and viscosity were next theoretically added to the Taylor analysis by Bellman and Pennington(1954). Two very interesting conclusions of the work were that with the introduction of interfacial tension the tendency for disturbances of large wave numbers (small wave lengths) no longer increased without bound. The theory showed that a critical wave number exists above which all wave numbers were stable. Also, because of the quadratic nature of the resulting growth rate prediction, there exists a wave number that exhibits a maximum growth rate. It was also shown in this work, that although the presence of viscosity lowered the growth rate of higher wave number disturbances, its effectalone was not sufficient to completely stabilize any wave numbers. In succession to the work of Harrison and Pennington, the instability theory was rapidly developed in various theoretical directions. These now include the effects on the R-T instability due to magnetic fields (Kruskal, et al., 1954), spherical geometries (Chang, 1959), compressibility effects (Mikalean, 1993) and non-linear modeling (Kull, 1983; Read, 1984; Cafero and Cima, 1993). There also exist excellent reviews on the R-T stability. Most notably those dedicated specifically to the topic by Sharpe(1984) and Kull(1991). Also reviews of the Rayleigh-Taylor instability is reviewed can be found in the texts of Lamb(1932), Chandrasekhar(l96l), Rosensweig(1985), and Drazin and Reid(l981). 11 Experimental Developments The first reported experimental demonstration of the R-T phenomena within a reference frame of constant acceleration was conducted by Lewis(1950). The experimental apparatus accelerated the two immiscible fluids through a containment shaft using air pressure and a specially designed breakaway diaphragm. Pictures of the interface were captured by high speed still photograph lighted with strobe light energized by a calibrated timing mechanism. In this work several experiments were performed with accelerations as high as 130 g. Although the vast majority of the experiments were conducted with an air- water system, Lewis also included an air-glycerin system and the liquid-liquid system of benzene and water. In these experiments, an initial disturbance was placed in the interface and the “peak-trough” distance was measured as twice the amplitude of the disturbance within the interface. This amplitude was monitored as a function of time and found to initially grow exponentially as predicted by Taylor (1950). It was also noted that after a period of time the growth rate was no longer well predicted by the Taylor theory. This was attributed by Lewis to the development of non-linearities not taken into account by the theory. Another series of experiments were conducted by Emmons et al (1962). The objective of this work was to test the linear stability theory including interfacial tension. The experimental apparatus used in these experiments had similar dimensions as that of Lewis, however the container in which the fluid resided was accelerated using elastic bands. The pictures of the accelerating interface were taken using a similar strobe timing mechanism employed by Lewis (1950). Lewis examined two air-liquid systems: air/methanol and air/carbon tetrachloride. In these experiments accelerations up to 7 g were obtained. An initial disturbance was placed in the interface similarly to that done by Lewis (1950) with a range of wave numbers 12 spanning the predicted cut-off wavenumber. The technique for determining the disturbance amplitude was identical to that used by Lewis. The growth rates agreed well with the linear stability theory showing a lowering of growth rates in the vicinity of the predicted cutoff wave number. However, complete stability was not observed. This lack of agreement with the predictions of the linearized theory was explained in part within accuracies in the determination of the initial amplitude of the disturbance and in part due to neglected role in non-linearities in the theory. The R-T instability was next reconsidered by Cole and Tankin (1973). The experimental apparatus was identical with that of Emmons et a1. (1962). The objective was to determine if better agreement with the linearized theory would be obtained for the individual growth rates by approximating the disturbance using a cosine wave. The reported results were similar to those of Emmons et a1 (1962) with no complete stabilization observed for wave numbers greater than the critical wave number. Cole and Tankin ( 1973) also attributed this to nonlinearities ignored in the development of the linearized theory. The instability evolution between two fluids of nearly the same density (octyl alcohol and water) was investigated by Ratafia(1973). In this work the density dependence was shown to be expressed by the Atwood number (See Chapter 3) leading to appreciably lower growth rates for fluid with nearly equal densities. Because of the lower growth rates observed, these experiments showed the development of the Kelvin-Hemholtz instability during the latter stages of growth. 2.3 Bubble and Drop Terminal Velocities The terminal velocity of individual bubbles in various media has been the subject of many experimental (Emmons et al., 1962; Haberman and Morton, 1953; Harmathy, 1960; Hu and Kintner, 1955; Peebles and Garber, 1953; Collins, 1965; Maneri and Zuber, 1974; 13 Maxworthy, et al, 1996; Uno and Kitner, 1956) and analytical (Birkoff and Carter, 1957; Davies and Taylor“, 1950; Garabedian, 1957; Grace et al, 1976; Lehrer, 1976; Maneri, 1995; Wairegi and Grace, 1976) investigations and at least one textbook (Clift, et aL,1974) In addition, a bubble rise model was developed by Mendelson (1967) for distorted (oblate spheroid) and spherical cap bubbles rising in low viscosity fluids, based on the similarity he observed between the measured terminal velocities and waves on the surface of a fluid. Maneri and Mendelson (1968) extended this analogy to include both three- dimensional and plane bubbles rising in infinite and finite media. Lehrer (1976) later reformulated this analogy as a balance of the gravity force with the buoyant and resulting drag forces. Plane bubbles are formed between infinite parallel plates and in rectangular ducts when the bubble volume is large enough that the bubble is in contact with both plates of the larger face of the rectangular duct. In all cases, the bubble geometry investigated is compatible with the medium, namely a 3-D bubble (spherical, spherical cap, cylindrical, etc.) rising in infinite media or circular tubes and plane bubbles rising between infinite parallel plates or in rectangular ducts. Maneri (1995) , using the analogy developed by Mendelson, developed an expression for the terminal velocity of a bubble of any size rising in a rectangular duct. During the development and subsequent theory, Maneri found that it was necessary to ignore the inertial effects of the dispersed phase in order for the analogy to fit the experimental results. This was justified by using the results of Marrucci et a1. (1970) who studied the motion of liquid drops in non-Newtonian fluids. Mersmann (see Churchill, 1988;p. 470) developed a generalized correlation for bubbles, droplets in gas, and drops rising and falling in immiscible fluids based on the Bond (Ebtv'os) and Weber number relations. Later Mersmann (1983) subsequently proposed a graphical conelation which illustrates the various stability of drops and bubbles rising or falling as various shapes. This graphical representation gives the locus of stability 14 for bubble or droplet breakup. It is the intent of this work to adapt this figure to a linearly accelerating field to predict the stability of drops and bubbles. No studies have been reported for the more general case of a fluid particle of any size or geometry rising or falling in a linearly accelerating environment. This research extends the analogy of Mendelson (1967), and Maneri and Mendelson ( 1968) to include accelerating environments. CHAPTER 3: STABILITY OF AN ACCELERATING FLUID/FLUID INTERFACE - THEORETICAL 3.1 Introduction The objective of this chapter is to review the linear stability theory for an accelerating interface between two irnnriscible fluids. The presentation uses the Helmhotz’s theorem to decompose the disturbance velocity into rotational and irrotational contributions. The analysis, which extends the earlier work of Reid (1961), yields stability criteria in terms of the disturbance wavenumber, the relative acceleration, and the physical properties of the fluid pair. The wave velocity of a progressive wave at the interface of a stable fluid- fluid anangement is also estimated in this chapter and will be further utilized in Chapter 5. This chapter provides a theoretical framework for the experimental design and testing conducted in Chapter 4. 3.2 Mathematical Formulation A force balance at an interface between two immiscible fluids relates the forces of each fluid acting on the interface to an opposing interfacial tension. Appendix A gives a 15 16 detailed derivation of this balance equation with the result that (see p. 202, Lea], 1992) (In) _I(2)).Eu) +Vn0—O'QU’V-Q”) = Q . (3.1) In Eq.(3.1), g‘” is a unit normal vector defined over the interface and is directed into Fluid 1. The interfacial tension is represented byo. The local curvature of the interface is given by V-g‘”. For a flat interface, V-g‘” = 0. 1‘” denotes the total stress dyadic of an incompressible, Newtonian fluid (i): I=—Pl+#(VE+V!T) . (3.2) Figure 3.1 illustrates the physical situation for which both fluids are subjected to a constant linear acceleration in addition to the constant acceleration due to gravity. At point ‘a’ on the surface the curvature is negative and, in the absence of viscous normal stresses, the interfacial tension supports a larger pressure for Fluid 1 (i.e., p‘” > pm). At point ‘b’ . l ‘2’ > p") inasmuch as V '11“ > O. on the surface, p The interface is a material surface. Its shape is defined by a scalar valued function S(x, t) for which S(xr(t),t) = 0 . (3.3) where x,(t) is the motion of the interface. The interface can be neither a source nor a sink of matter, therefore, continuity at the interface requires (see p 69, Leal,l988) l7 Figure 3.1 Planar View of an Accelerating Interface 18 as DxI __ _-.V = + Dr S O (3.4) and LE"? =u“’(x.(t).t)=u‘2’(xr(t).t) . (3.5) u‘” and 2(2) represent, respectively, the velocity of Fluid 1 and the velocity of Fluid 2 relative to the * -frame (see Figure 3.1). If the coordinates of the interface are denoted by (x,,y,,z,), then the interfacial shape function can be represented as S=§'§;-z](xpyht)a (36) where x is a position vector relative to the * - frame. The gradient of S evaluated on the surface is orthogonal to a local tangent plane of S and points in the direction of increasing values of 8. Therefore, the unit normal vector Q“) can be written as VS £1-(l) ___ _ ’ (37) "VS" where ||VS||2=VS-VS=1+ Q 2+ 25— 2. (3.8) 8x, 8y, 19 The local curvature of the surface is given by (3.9) V (l) = [E] = _ V12121 _ VS ' V“VS" llVSll llVSll "V5“2 For small deflections in the shape of the interface, |VS|| 5 l and Eq.(3.9) reduces to V-g‘” E—Vflz, , (3.10) where V3, represents the two dimensional Laplacian operator Via-7+— . (3.11) The continuity equation and the equation of motion for an incompressible, Newtonian fluid govern the velocity and pressure distributions for Fluid 1 (z > 2,) and Fluid 2 (z < 21 ). Relative to the * - frame, these equations are and p[%—%+§F)=-Vp+p\72g+pg . (3-13) to \T' .I..\ % :71 4 I 20 As indicated in Figure 3.1, the density and viscosity of Fluid 1 are p1 and 11,, respectively. p2 and #2 represent the density and viscosity of Fluid 2. In Eqs. (3.12) and (3.13), the velocity of the fluid is relative to the * - frame, which moves at a constant acceleration 3F- The substantial (or material) derivative in Eq.(3.13) represents the time rate of change of the velocity of a fluid element relative to the * - frame. For notational simplicity hereinafter, the coordinates x,y, and z are understood to be relative to the *- frame. 3.3 Hydrostatics For g") = Q and 2(2) = Q, Eq.(3.13) governs the hydrostatic pressure distribution near a planar interface with a semi-infinite fluid layer of density p2 above a semi-infinite fluid layer of density p, (see Figure 3.1). The hydrostatic pressure distribution, p55, is given by: pf—p,(aF-g)z . 220 p55: (3.14) pf-p2(aF—g)z , 230. The steady state pressure distribution satisfies Eq. (3.13) with n = Q for each phase. For aF < g, the hydrostatic pressure increases as 2 -> +00 and decreases as 2 —-) —oo; however, for aF > g , the hydrostatic pressure decreases as 2 —> +00 and increases as 2 —> —o<> . The following heuristic argument anticipates that the interface is unstable forp2 g, 0:0, [1, =0, and 112 =0. Figure 3.2 compares the steady state hydrostatic pressure distribution (solid line) and a hydrostatic response (dotted line) to a 21 PB' < pA' ; unstable Interface >P p3 > p A ; unstable '21 Figure 3.2 Hydrostatic Pressure Distribution for p2 < p1, at: > g. 22 small surface deflection. Note that for z > 0, the pressure at Point B is larger than the pressure at Point A. Because Eq.(3.1) requires the normal forces to balance at the interface (i.e., pA = pa), 2] moves to larger values of z (i.e., 21 —> oo) in response to p8 > pA. Likewise, for z < 0 the pressure at Point B’ is smaller than the pressure at Point A’. Therefore, in order to balance the normal forces at the interface, zl moves to smaller values of z (i.e., zI -—> —oo) in response to ply > p8, . This heuristic argument for instability also supports the stability of the interface for p2 < p1 and aF < g. The above argument also anticipates an unstable interface for p2 > p1 and aF = 0 for all wave lengths, 0 < ,1 < oo. However, for sufficiently small values of A, it is a well known observation that capillary forces can balance the adverse pressure distribution across the interface. Thus, Eq.(3.1) can be satisfied for finite curvature (i.e., lV-g"’| O and )1 is not too large. 3.4 Linear Stability Analysis The linear stability of the hydrostatic pressure distribution and the complementary quiescent velocity field can be determined mathematically by introducing an arbitrary, infinitesimal deflection of the interface as an initial condition. If the amplitude of the surface disturbance increases as time increases, then the hydrostatic state is unstable. For small amplitude disturbances, the hydrodynamic equations can be linearized about the hydrostatic state with a pressure gradient given by Vpss=-p(aF_g)§.z ' (315) 23 A disturbance pressure gradient can be defined as Vpd EVp—Vpss . (3.16) Because 95, = Q, the linear form of Eq.(3.13) is 8 2 1 ——VV u=-—V , 3.17 (at )— p Pd ( ) Eq. (3.17) applies for z < z, and for z > 21 with the kinematic viscosity v and the density p corresponding to Fluid 2 and Fluid 1, respectively (see Figure 3.1). Because V p = 0, it follows directly from Eq.(3.17) that the disturbance pressure pd satisfies Laplace’s equation: Vzpd =0. (3.18) It also follows directly from Eq.(3.17) that the disturbance vorticity (EV. a V x Q) satisfies the unsteady state, vector-valued, difi‘usion equation: [58?- vzjflzg , (3.19) Helrnholtz’s decomposition theorem (see p. 70 of Aris, 1962) gives a unique representation of the velocity field in terms of a scalar potential <1) and a solenoidal vector fieldA (i.e., V - A = 0): 24 y.=-V+V>oo . The planform function Fk(x,y) in Eq.(3.24) satisfies Helmholtz’s equation ViFr = -k2Fr . (3.27) and periodic boundary conditions in the xy-plane. Therefore, Ft = exp(ikxx + ikyy) . (3.28) The parameter ‘s’ in Eq.(3.24) represents a growth rate associated with a disturbance having a wave number vector g=k e' +ke‘ (3.29) A _=.-— and A E—. (3.30) 26 If the real part of s is positive, then the linear mode is unstable. A similar analysis of Eqs.(3.18) and (3.21) shows that the linear modes associated with the scalar potential (I) and the disturbance pressure pd are proportional to exp(ikz+st) Fk(x,y) , (3.31) where the positive sign is used for z < O, and the negative sign is used for z > 0. The vector field A has three components A = A,(X.y.2.t) 9; + A,(x,y,2.t) e; + A,(x,y.z,t) e; . (3.32) Because V - A = 0, the z-component of A can be related to Ax and Ay with the result that ikxAx + ilrylo.y (im) Az(x9 errt) = — (3.33) In the above equation, (+m2) applies for z < 0 and (~m,) for z > 0. Once again, the real part of ml and m2 must be positive. Appendix B shows how the real part of m is related to the real and imaginary parts of the growth rate 5. It follows directly from Eqs.(3.20) and (3.33) that the three scalar components of u(x,t) are determined by the three scalar functions (I) , Ax and Ay: u: ._ 3.34 , 8x ( ) do 3A, 8A, + —-— , 8y 82 27 are 3A,, 3A, uy ——-§;+[ az ax J , (3.35) and u, "5; (3.36) 84) 3A, 3A,, + — — . 8x 8y The linear disturbance modes for the scalar potential <1) and the components of the vector potential A satisfy periodic boundary conditions in the xy-plane. For |2| —-> co, the disturbance must be bounded, so lim (<1),A,,,Ay) = (0,0,0) . (3.37) z—itw It follows directly from Eqs.(3.24) and (3.31), with the (+) sign for Fluid 2 (z < 2,) and the (-) sign for Fluid 1 (2 > 2,), that the k-th mode of and A can be represented as: cf” exp(+kz + st) F, (x, y), z < 2, 11)“) = , (3.38) cl,” exp(-kz + st) F,,(x, y), 2 > 2, cl,” exp(+m22 + st) F,‘ (x, y), z < 2, Am = i , (3.39) cf,” exp(-m,z + st)F,l (x, y), z > 2I 28 and C‘s“ eXp(+m22 + st) F,(x, y), 2 < 2, A‘y'” = . (3.40) c2” exp(-—m,2 + st) Fk(x, y), 2 > 2, The A?) component of A can be calculated by using Eq.(3.33): ' (k) (k) .c, kx +c5 k —1 ’ exp(+m22 + st) F,(x, y), z < 2, “‘2 A1“ = 4 . (3.41 ) +, cf,"’kx + 0‘6“’ky l exp(—m,z + st) F,,(x, y), 2 > z, t m, The disturbance modes associated with the normal component of the velocity follow by combining Eq.(3.36) with Eqs.(3.38) - (3 .40). The analysis gives [alk’ exp(+kz) + a?) exp(+m22)]exp(st) Fk(x, y), z < 2, u‘“ = , (3.42) Z [a‘,"’ exp(-k2) + a?) exp(—m,z)]exp(st)F,, (x, y), 2 > z, (k) .. m (k) _ - (k) (k) m _ (k) 00 _ - (k) (k) where a, =—kc, , a2 =r(k,,c5 --kyc3 ), a3 =+kc2, and a4 =r(k,,c6 -kyc,, ). The modal coefficients a“) and a“) are associated with irrotational disturbances whereas 1 3 a‘," and a?) are associated with rotational disturbances. The variation of the disturbance pressure normal to the interface follows directly from Eq.(3.17): 29 -_—:(§—t-— V2)“; , (3.43) Therefore, Eqs.(3.42) and (3.43) imply that i (1:) —p2 a‘k S exp(+kz + st)F,, (x, y), z < 2, (k) pd = i - (3.44) (k) +p, Ekfiiexpkkz + st)F,,(x, y), 2 > z, \ Eq.(3.44) shows that the rotational disturbance does not influence the pressure modes (k) (k) (k) explicitly. The four modal coefficients a, , a2 , a, (k) and a4 are determined by continuity conditions at the interface. The other two coefficients introduced with the set of equations for (I) and A (see Eqs.(3.38) - (3.40)) can be related to the above four coefficients by satisfying the x - and y - component equations for the velocity field (see Eq.(3. 17)). The continuity equation (see Eq.(3.12)) implies that each velocity mode must satisfy the following condition 00 an, 22 = ikxu‘f’ + ikyu‘,” . (3.45) Therefore, Eq.(3 .45) and continuity of each velocity component across the interface implies that (u;k))nuidl@lt = (utzo)md2@z| (3.46) ..u 1!; ~10 30 and a1") a1“) [ a; J =( a; j . (3.47) Fluidl@z, Fiuid2@z, Eqs.(3 .46) and (3.47) are satisfied provided a“) + a?’- - 3‘,” + a?) (3.48) and k3?” + mza a‘z")- =—-ka‘3‘° m ,af,“ . (3.49) For a Newtonian fluid, each linear mode for the shear components of the stress (see Eq.(3.2)) can be written in terms of the velocity components as follows (see p 432, Chandrasekhar, 1961) (k) (k) (k) r“’- =,u[auax +822]: u[ik, u‘k’+a;;] (3.50) and allk) alik) . ally“ 1;? =fl[— ; +7]: ”(118,090 '1’?) . (351) 31 Multiplying Eqs.(3.50) and (3.51) by ik, and iky, respectively, and adding the two resulting equations gives the following result ik 1“" +ik 1“" -— [a—z—l—um +k2u m] 3 52 x n y y: — fl &2 ' ( t ) Eqs.(3.26) and (3.45) were used to obtain Eq.(3.52). This result shows that continuity of the shear components across the interface implies that (L211 u(It) 2 (1:) 82,100 2 (k) -},r, z +ku ] = - ,u,[ ——;—“ +ku J , (3.53) 312 Fluidl@ z, 32 Fluid 2 @ 2. which requires tr,(2k2a‘,"’ + (m,2 + k2)a‘4“’) = 11, (218“) + (m; + k2)a”") . (3.54) The normal component of the interfacial force balance (see Eq.(3.1)) reduces to the following result Bu [pl(aF - g)zr ’ Pd '1' 2’11 .372) Fluidl® II (3.55) 8a 2 ”avilzl + (P2 (3r " g)zl ' Pd + 2”: j) . Fluld2@z, where the hydrostatic pressure distribution defined by Eq.(3.14) has been used together with the curvature approximation for infinitesimal disturbances (see Eq.(3.10)). The kinematic condition that this interface is a material surface (see Eq.(3.5)) requires that the 32 infinitesimal disturbance modes associated with the interfacial position satisfy 82:“ = (“(210) , Fluldl@2. : (u:k))Fluid2@zl ' (3'56) Eqs.(3.42), (3.44), and (3.56) imply that the interfacial condition given by Eq.(3.55) is satisfied provided as. - g) + + + s 2 (k) (k) (k) (k) 2 —%§a§"’ +2u,[ (3.57) 0']:2 af” + 21‘," + a‘,“ + af," 2 + s 2 (k) (k) (it) it) +p2(aF—g) 3] +32 +33 +34 s 2 (k) (k) (k) (k) +73, '1' Zflz . 2 (k) 2 The above expression explicitly exploits the condition that u and its normal derivative are continuous across the interface (see Eqs.(3.48) and (3.49)). An explicit derivation of Eq.(3.57) is presented in Appendix C. The above four homogenous linear equations for the modal coefficients (i.e., Eqs(3.48), (3.49), (3.54), and (3.57)) can be written as 33 llII> In: ll 10 (3.58) where the components of the vector 3 are given by QH m . (3.59) The sixteen coefficients of the dyadic valued operator A are listed in Table 3.1. This operator was first developed for two dimensional disturbances by Hanison (1908) and later studied extensively by Bellman and Pennington (1954) and by Reid (1961). The derivation developed here was partially motivated by the presentation given by Chandrasekhar (see pp. 441-443, 1961). Unlike the earlier developments, the theoretical derivation here uses the Helmholtz decomposition theorem (see Eq.(3.20)) to explicitly represent the velocity perturbations in terms of irrotational and rotational contributions. The necessary and sufficient condition (see, for example, p. 427, Kreyszig,1993) for the existence of a non-trivial vector 3 which satisfies Eq.(3.58) is that det(A)=O . (3.60) Eq.(3.60), which is often referred to as a dispersion relation, is the key result of this section and provides a means to determine the growth rate parameter s for the tlrree- component, three-dimensional disturbances to the hydrostatic pressure distribution given by Eq.(3.14). As noted by Chandrasekar (p.439, 1961), both 5 and its conjugate satisfy Eq.(3.60). For Re(s) = 5,, > 0, infinitesimal disturbances increase exponentially in time 34 Table 3.1 Components of the Dispersion Dyadic A4, =B+([12 —p,)k+£li-§ A42 =B+(u’z “#1)“!2 A43 =B’U‘2 ‘f‘i)k+!—;fi A... = B- (#2 -#r)mt B a (P2 " pl)(aF -g)+Ok2 28 1.11.; 35 and the interface is considered unstable. However, if 5,, < 0, then initial disturbances will decay exponentially in time. The growth or decay process will have a periodic behavior if Im(s) E s, at 0. The marginal state occurs for sR = O. The dynamic behavior of the marginally stable mode depends on the value of 5.. The determinant of A can be expanded to yield a nonlinear equation for the growth rate 5. .The following identity follows by (l) subtracting the first column of A (see Table 3.1) from the second; (2) subtracting the third column from the fourth; and, adding the first column to the third (see p. 443, Chandrasekhar, 1961): 1 0 0 0 k m2 — k 2k m, - k 2k2flz p23 218012 - #1) "pls A41 A42 " A41 A43 + A41 A44 " A43 det( ||> ):-: det (3.61) tn,z — k 2k m, - k = det p25 215012 " [11) -p18 A42 " A41 A43 + A41 A44 "' A43 The development of Eq.(3.61) about the last row yields: det(A) = —(A,, - A,,)[2ltp,s + 2k2 (11, - 11,)(m, — k)] + (A43 'l' A4l)[(m2 " k)pls + p25(ml ' 10] (3-62) + (A4,, — A,,)[(m2 — M21801, - 14)— p,s2k]. 36 It now follows from Table 3.1 that Eq.(3.62) can be written as det(A) = [2B + W]{(mz -— k)p,s + (m, —— k)pzs] (3.63) -4k=[(a - #1th- k) + flu. — axm. - k) — Bf], where B is defined by (see Table 3.1) __ _ 2 B 5 (p2 pl)(aF 8) + 01‘ (3.64) 23 Eq.(3.63) is the major result of this section and is equivalent to the result derived earlier by Chandrasekhar (p. 343, 1961). For detLA) = 0, Eq.(3.63) implies that nip—”IEQE = M' , (3.65) where the dimensional viscous factor M’ is defined by male. - #1.er - k) +1?le - A)(mr - k)- %j M’ E [(312 - k)p,8+(m1 - k)p2$] 37 3.5 Inviscid Theory The dispersion relation (i.e., Eq.(3.60)) simplifies significantly for inviscid fluids. For 11, —> 0 and IL, —> 0, it follows from Eq.(3.25) that m, -> co and m2 —) oo; therefore, the components of the velocity vector potential A (see Eqs.(3.39) - (3.40)) are zero and the disturbance velocity is strictly irrotational (see Eq.(3.20)). Although the normal component of the velocity is still continuous across the interface, the physical conditions expressed by Eqs.(3.47) and (3.53) no longer constrain the interfacial disturbance. Therefore, for an inviscid fluid, the dispersion relation reduces to a condition that the normal stress and the normal component of the velocity are continuous across the interface. Mathematically, these two boundary conditions are satisfied for nontrivial infinitesimal disturbances provided 1 —1 det[ ]= O , (3.67) A41 A43 where A,, and A,3 are defined by Table 3.1 with u, = 1.12 = 0 . Eq.(3.67) implies that A,, + A43=0 or, equivalently, ZB+(p,+p2)E=0 . (3.68) With B defined by Table 3.1, the above result gives 2 __ (pz -pl)(aF—g)k+ 01‘3 ' - _ _ (3.69) s (n+m) (n+m) 38 This equation was first derived by Harrison(1908) for aF = 0 and later extended by Taylor(1950) for aF 3* 0 (also see p. 435, Chandrasekhar,l961). Eq.(3.68) also follows directly from Eq.(3.65) by noting that lim M’ —) O , fill—’0 h inasmuchas m, =1/ 11, and m, ==1/ ,u, for (u,,u,)-—>(0,0) . Stable Inviscid Interfaces Eq.(3.69) shows that s is a pure imaginary number if (p. - pr)(ar - g) > 0 . (3.70) Thus, Eq.(3.70) is a sufficient condition for the stability of the interface between two immiscible, inviscid fluids. Physically, if the lighter fluid is above the heavier fluid (i.e., p2 < p, ), then the interface is stable to infinitesimal irrotational disturbances provided a, < g. Moreover, if the heavier fluid is above the lighter fluid (i.e., p2 > p, ), then Eq.(3.69) also predicts that the interface is stable provided a, > g. The normal component of the velocity mode at the interface is given by (see Eq.(3.42) for a?) = 0 and a?) = 0 ) u?” = exp(s,,t)exp[i(_k - x_ :t s,t)] . (3.71) 39 As previously indicated, these disturbance modes decay if SR < 0, and grow if 5,, > 0. Thus, if Ineq.(3.70) holds, sR = 0 and infinitesimal disturbances for inviscid fluids neither grow nor decay. The i sign in Eq.(3.71) explicitly acknowledges that the complex conjugate of s is also a solution to the dispersion relation det(A) = 0. Unstable Inviscid Interfaces The interface is unstable to infinitesimal disturbances if the real part of s is positive (SR > 0). Because the physical parameters on the right-hand side of Eq.(3.69) are real numbers, 3 can be either real (i.e., s, = 0) or pure imaginary (i.e., SR =0). With 3 = sR, Eq.(3.69) reduces to S2 = “(p2 - p1)(aF - g)k _ 0'18 “ (p.+p2) (p.+p2) ' (3.72) Thus, Eq.(3.72) shows that sR > 0 for all wave numbers in the range 0 < k < k”, where k” = \l-(pz ' pl)(aF '8) . (3.73) The above conclusion assumes that the interface has an unstable configuration which requires (cf. Ineq.(3.70)) (p2 -pr)(ar-g)<0 . (3.74) Clearly, the above inequality is a necessary condition for an unstable interface between two 4O immiscible, inviscid fluids. Physically, Eq.(3.72) implies that for a lighter fluid above a heavier fluid (p2 < p,), the interface is unstable to long wavelength disturbances (i.e., k < k“) if a, > g. Eq.(3.72) also predicts that all wavelengths are unstable for o = 0, provided Ineq.(3.74) holds. Figure 3.3 shows the inviscid growth rates for three different fluid/fluid interfaces for a,, = 0: air/water; kerosene/water; and water/freon. Table 3.2 gives the physical property parameters used in the calculations. A maximum growth rate occurs at an intermediate wave number for each system. It follows directly from Eq.(3.72) that 2 _ _2_ ”(p2 ' Pr)(ar " g) (SR) 3 (p, + p2) km , (3.75) where (3.76) tial? max Figure 3.3 is for the special case a, = 0. Note that the air/water interface (i.e., water over air) is unstable to infinitesimal disturbances with wave lengths larger than 27t/ k,1 .=_1.73cm. The air/water and water/freon interfaces have comparable cut-off wave numbers because (pH - p,_)/ a is about the same for these two systems (see Eq.(3.73) and Table 3.2). Note also that the Atwood group, At =- (p,, — pL)/(pH + p,) , for the air/water interface (At a 0.997) is much larger than the Atwood group for the water/freon interface (At 5 0.222) . This causes the maximum inviscid growth rate (due primarily to buoyancy) for the air/water interface to be about twice the maximum growth rate for the water/freon interface. However, the kerosene/water interface has an even smaller Atwood number 41 Inviscid System: k° cm' s°r,mn, 1113'1 k“ cm'l A. Air/Water 3.64 48.8 2.10 B. Kerosene/Water 2.05 10.4 1.18 C. Water/Freon 3.74 23.3 2.16 50 45 Figure 3.3 Growth Rate for Unstable Inviscid Interfaces (a, =0) 42 Table 3.2 Physical Property Parameters for Inviscid Theory System p L p" O Light/Heavy [g/cm’] [g/cm3] [gls’] Air/Water 0.0013“) 1.0“) 74(3) Kerosene IWater Q8501) 1.00) 35(4) Water/Freon 1.0“) 1.570) 40“) (1) See p. 3-210, Peny’s Chemical Engineering Handbook, 5111 ed. (1973) (2’ See p. 3-196, Perry’s Chemical Engineering Handbook, 5th ed. (1973) ‘3’ See p. 40, Adamson (1982) ‘" This research, See Chapter 4 43 (At _=_ 0.081) as well as a smaller value for (pH — p,)/ 0'. Thus, the growth rate of the most unstable inviscid mode for a kerosene/water interface is smaller than the other two examples. Furthermore, the critical wave length at which interfacial tension can stabilize the growth rate for a kerosene/water interface occurs at larger wave lengths, i.e., 27r/kM s 3.00cm. Eqs.(3.75) and (3.76) can also be written as 2 58.1w = km Elf—g. (3.77) 8,2,,“ kg,“ 8 and 1 kanzfiazflzfiiz (3.78) k2... 4.... kit 8 ’ where s3,“ and kfw represent, respectively, the maximum growth rate and corresponding wave number for aF=0 and (p2 - p, ) = (pH - p,) > 0 (i.e., the heavy fluid over the light fluid). Eqs.(3.77 and (3.78) show that the relative acceleration parameter 7, defined by y = a, ‘3' , (3.79) determines SEW/s1“, and kw /kfm. Figure 3.4 illustrates the effect of you the growth rate of the most unstable inviscid mode. An unstable heavy fluid over a light fluid (a,_. < g) Inviscid S ater ater ater reon 2 heavy fluid light fluid M M light fluid heavy fluid 1.5 Sim/Sow sr ma/s°. m... 1 . . 0.5 Aim/790.... Mom L L L 1 19 u 44 l -2 -1 O 1 2 (ar' g) / E Figure 3.4 The Effect of Frame Acceleration on the Maximum Growth Rate for Unstable Inviscid Fluids 45 has the same growth rate as an unstable light fluid over a heavy fluid (aF > g). For aF = g, all infinitesimal disturbances are stable, which is equivalent to the neutrally buoyant case pH = p,. However, as 7 increases (i.e., either aF << g or aF >> g), the wave length of the most unstable disturbance decreases to zero or, equivalently, km —9 oo. Thus, only the smallest wave length disturbances can be stabilized by interfacial tension (AM=27r/kM—)0 as 7—)»). 3.6 Viscous Theory Reid (1961) and Chandrasekhar (1961) studied the behavior of Eq.(3.65) for the special case vH = vL . Here the growth rate parameter will be determined for a wider range of conditions in order to quantify the effect of viscosity on the stability of an accelerated interface near the inviscid limit as well as to quantify the growth rates for fluid/fluid interfaces for which vH 9* vL . Table 3.3 summarizes some of the important physical properties of the three systems treated in the previous section. Clearly, Eq.(3.65) still holds for these examples, but here the viscous factor M’ will no longer be neglected. Note that the water/freon system satisfies the special condition imposed by Reid, vi2., v,, = vL . In Section 3.7, the mathematical nature of the marginal stability state under the hypothesis that Re(s) = 0 and Im(s) = 0 will be examined; the initial growth rates of unstable infinitesimal disturbances for Re(s) > 0 and Im(s) = 0 will be explored in Section 3.8. In Section 3.9, the decay process associated with stable progressive waves is examined for Re(s) < 0 and Im(s) ¢ 0. The results complement the comprehensive study by Reid(l96l), who examined a class of problems for which v,, = v,_. 46 Table 3.3 Physical Property Parameters for Viscous Theory System ”L “H PL Pa Vt Light / Heavy [g/cm -s] [g/cm -s] [g/cm3] [g/cm3] [cmzls] Ain’Water 0.00018“) 0.010“) 0.0013“) 1.0“) 0.010 Kerosene lWater 0.014(4) 0.010“) 0.85“) 1.0“) 0.013 Water/Freon 0.01“) 0.016(4) 1.0“) 1.570) 0.010 (1) See p. 3-210, Perry’s Chemical Engineering Handbook, 5th ed. (1973) ‘2’ See p. 3-196, Perry’s Chemical Engineering Handbook, 5th ed. (1973) ‘3’ See p. 40, Adamson (1982) ‘" This research (See Chapter 4) Two phase kinematic viscosity: v,E——”"+fl" . Pa +91. 47 Eq.(3.65) can be rewritten in terms of the following dimensionless quantities: S XE , 3.80 v,k2 < > m a Ya—i= 1+—l-X , 3.81 ' k a ( ) m a Y 54: l+-lx , 3.82 2 k [32 ( ) a,e__p.'_ , (3.83) pl+p2 a,= ”2 =1—a,, (3.84) Pl'l'Pz lie ”1 . (3.85) ' raw. #2 5 =1- , 3.86 132 ”mu. [3. ( ) 2 a 5 "(pz ’ pl)(aF -g)—O’k , (3.87) 48 bis 3 . (3.88) In Eqs.(3.80) and (3.88) the two-phase kinematic viscosity v, is defined by “‘ “‘2 . (3.89) pl +p2 v, 5 It follows directly from Eq.(3.65) and the above definitions that the dimensionless eigenvalues satisfy the following nonlinear algebraic equation F(X)=-ab2 +x2 —M=O , (3.90) where the dimensionless viscous factor M is defined by (P2 ’ P1 )(Y, " 1) + aIX][(fi2 " B1 )(Yz '1)- ale . M(X) E4 . (Y2 -l)al+(Y1 ’1)a2 (3.91) The factor M(X) does not depend on the orientation of the two phases. For instance, if the phases are reversed (i.e., a, a: 0:2 and [3, 4: [32 ), M(X) is unaffected inasmuch as M(Xiavaztfivfiz) = M(xiazravfizrfil) ' However, the sign of the dimensionless group ab2 in Eq.(3.90) does depend on the sign of 49 (p2 - p, )(aF - g)as well as on the magnitude of the wave number. The unstable growth rates (or eigenvalues) are determined by solving Eq.(3.90) for Re(X) > 0 and Im(X) = 0. For this case Y, and Y2 are both real and larger than unity, Y, > 1 and Y2 > 1 (see Eqs.(3.81) and (3.82) with Im(X) = 0 and Re(X) > 0). Eq.(3.90) can be solved to yield values of X for a given physical system. A direct search method was conducted by Mathematica‘” to find the complex eigenvalues X. The solution methodology is outlined in Figure 3.5. Appendix D gives the program listing and illustrates the method. The dimensionless group ab2 in Eq.(3.90) can also be rewritten in terms of a dimensionless wave number 412 defined by k — 3.92 45: kc . ( ) where the wave number kc is given by kc 5 JW ’ pl)“ . (3.93) 0' Note that kc coincidently corresponds to the cut-off wave number k" for an unstable inviscid interface (see Eq.(3.73)). The relative acceleration group 7 is defined by Eq.(3.79). It follows directly from Eqs.(3.87), (3.88), (3.92) and (3.93) that 2 ab2 = Mo'”2 [29—] . (3.94) 2 50 Input Physical Properties PH, PL, 11H- llL, 0 V Calculate two-phase kinematic viscosity, v, l Calculate Density and Viscosity Ratios O‘H- “L: PH, PL 1 Specify Acceleration Parameters 31:, 'Y l Dimensionless Complex Growth Rate X l Solve Eq.(3.90) using Mathematica® See Appendix D l Calculate ab2 see Eqs.(3.87) and (3.88) l Specify Wave Number k A Select Wave number 412 Figure 3.5 Solution Flow Chart for Solving Eq.(3.90) 51 where the parameter a is either +1 for an unstable interface or -1 for a stable interface: ”(p2 ’p,)(a,, —g) — +1 ’(pz " pl)(aF ‘8) < 0 8 E (pH—pL)laF—gl — (3.95) "1 ,(Pz _ pl)(aF "8) > O ‘The dimensionless group M0 is the Morton number (see p. 417, Churchill, 1988). This group only depends on the intrinsic properties of the fluid/fluid interface and the effective acceleration parameter g7. Mo can be expressed as a measure of the relative importance of buoyancy to the interfacial tension at an intrinsic length scale [c defined by (pa + pL )V12 = (“a + ”J"! (”a + ”my 3 . 0' 0' (pa + pL )0 C (3.96) Thus, MO?= (pa “1313(ng = At (”a +“L)487 . 7 0' (pH +pl.)03 (3 9 ) where At represents the Atwood group, (p,, - p,)/(p,, + p,). The two-phase viscous length scale A is extremely small. For the water/freon system, [c = 600A . Because Mo/ y 510'”, buoyancy effects relative to capillary effects at scales comparable to (c are unimportant. The Morton number for y = 1 (i.e., aF = 0 or 2g) associated with the three physical examples selected for study range from about 26 x 10'12 for the air/water interface to 600 x 10'12 for the water/freon interface. For an inviscid theory (i.e., 52 11,, = 11, = 0), the Morton group is zero. Table 3.4 lists the physical property groups 01,, and [3,, for the three systems defined by Tables 3.2 and 3.3. The density ratio 01,, and the viscosity ratio 13,, cover a wide range of possibilities. The water/freon interface approximates the assumption that 0,, / 01,, = ,6, / 01L, assumed in earlier theoretical work by Reid( 1961 ). 'With ab2 given by Eq.(3.96), it follows that Eq.(3.90) can be rewritten as Mo”2¢;X2 = up, -¢,3 +Mo“2 63M . (3.98) The left-hand side of Eq.(3.98) can be rewritten as follows 2 MOI/2 (pgx2 = [i] , (3.99) S C where the characteristic growth rate sc is defined by _ 1/2 sf = At gy 1rc -_- At(£fl-E-’3L(gy)3) . (3.100) Note that sc does not depend on the viscous properties of either fluid. Furthermore, sc physically represents the growth rate at the cut-off wavenumber kc of an unstable interface for which 6 = 0 (see Eq.(3.72) and Taylor, 1950). Therefore, with 415-8- . (3.101) 53 Table 3.4 Dimensionless Physical Property Groups System 01,, [3" MO” Light/Heavy Eq.(3.83) Eq.(3.85) [x 1010] Air/Water 0.9987 0.9823 0.2624 Kerosene /Water 0.5405 0.4167 3.355 Water/Freon 0.6109 0.6154 6.005 54 Eq.(3.98) can be rewritten as 9.2 =89. -¢§+Mo“2¢§M . (3.102) The factor M, defined by Eq.(3.91), can be rewritten in terms of Y, and Y2 alone by eliminating X inasmuch as (see Eqs.(3.81) and (3.82)) %x = Y3 —1=(Y. +001 —1) 0103) l and %£X=Y; —1=(Y2 +1)(Y, —1) . (3.104) 2 With the above results, Eq.(3.91) can be rewritten as Y1 ’1)(Y2 ’ ”(P2 + B1Y1)(Bl+ fizYz) [(Yz - Dal + (Y1 "1)a2] M=—4( __ as, , ([32+AY1)(A+I32Y2) - 433x (Yl+1)(Y2+l)[(Y2—l)al+(Yl-l)a2] . (3.105) As noted earlier, M does not depend on the relative position of the heavy and light phases and is a symmetric function of the phase configuration (i.e., 1 4: 2 ). Eq.(3.102) can be simplified further by using Eqs.(3.99) and (3.105) with the result that 55 2 _ 84,2 ’d’; where N =+4 alaZ ([32 + filYlXBl + fizYz) 7 B102 (Y1 +1)(Y2 +1)l(Y2 ’ 1)a1 + (Y1 ’ 1)o‘zl (32 + B1Y1)(Bl+ fizYz) :(1+Y,),B, +(1+Y2)B,: 4 =+— 3.107 X ( ) Clearly N depends on the dimensionless grth rate it), because Y, and Y2 depend on X and, according to Eq.(3.99), 1/4 2 X 491 Therefore, with Eq.(3.108), N can be rewritten as N :4MOl/4 fl (fiZ + BlYl)(fil + fiZYZ) (3109) ¢, [(1+Y,)/3,+(1+ Y,)[3,: Eq.(3.106) is equivalent to Eq.(3.67) as well as Eq.(3.90). The dimensionless growth rate (1), for unstable (8 = +1) and for stable (8 = -1) interfaces must satisfy this nonlinear equation. The theory shows that the dimensionless growth rate 0, depends on 56 the dimensionless wave number (I), and three physical property groups: on“, B“, and Mo. Eq.(3.106) reduces to the well-known inviscid theory by setting Mo = 0 (see Section 3.3 and esp., Eq.(3.69). With (1), 5 (121° for Mo = O, Eq.(3.106) can be expressed as ¢1=H¢10 (3.110) where ¢.° =«/e¢2 -¢§ (3.111) and the dimensionless complex viscous factor H is defined by H=+ /_1_, (3.112) 1+N It follows from Eq.(3.109) that H depends on on“ (= a1), [3,, (=B1), Mo, oz, and $1- 3.7 Viscous Theory: Marginal Stabilty At the marginal state, Re(s) = O. For viscous fluids, Chandrasekhar (see p.449, 1961) has noted that the marginal state is also a stationary state. Thus, the surface displacement modes satisfy the following static equation subject to spatially periodic boundary conditions 57 OVIZI 2:“ = (p2 - p1)(ar= - all“ - (3-113) Because 2:” == Fk (x, y), Eqs.(3.27) and (3.113) imply that the marginal wave numbers are given by k: = ”(92 -pO1-)(aF —g) ___ (pH —:L)g7 (3.114) for (p2 — p,)(aF — g) < 0. Thus, as expected for a hydrostatic result, viscosity plays no role in determining the marginal wavenumber. 3.8 Viscous Theory: Unstable Interfaces The effect of viscosity on the growth rate of an unstable interface (a = +1) can be evaluated by examining the real part of 4:, for disturbance wave numbers ranging from zero to unity: 0 < ‘1’: < 1. Figure 3.6 shows the dependence of o, (= s/sc) on ¢2(= k/kc) for y=l (i.e., aF =00r 2g). The physical property groups on", [3", and Mo/y corresponding to the three fluid/fluid interfaces are defined by Tables 3.2 - 3.4. Figure 3.6 also shows that the inviscid theory (i.e., Eq.(3.111)) gives an upper bound on the viscous growth rates, as anticipated by Eq.(3.112) inasmuch as N>0 for this case. The dimensionless growth rates for the air/water system are close to the inviscid limit; however the kerosene/water and the water/freon systems show lower values for (1),. This occurs 0.63 r 0.62 0.61 ‘91 0.60 0.59 Inviscid Theory Eq.(3.113) 58 0.7 - 0.5 Figure 3.6 Dimensionless Growth Rates for Unstable Viscous Interfaces (AzAir/Water; BzKerosenelWater; C:Water/Freon) 59 because the Morton number for systems B and C are an order of magnitude larger than M0 for system A (see Table 3.3). The insert in Figure 3.6 shows that the maximum deviation due to viscous effects occurs near the maximum growth rate, (02 ~ 0.6. The viscous dimensionless growth rate approaches the inviscid limit at the end points of the unstable curve, i.e., 4), —> 0 and (1), —>1 . The observation is expected since the viscous effects are small for large wavelengths (i.e., (p2 -) 0) when the velocity gradients are small; and, are dominated by interfacial tension as the wave number approaches kM (i.e., 422 -—> 1 ). The effect of the relative acceleration group on the maximum dimensionless growth rate is summarized by Figure 3.7. The three curves are parameterized by the group N, which increases significantly as 7 increases over several orders of magnitude. Increasing the relative acceleration parameter clearly influences the Morton group defined by Eq.(3.99). Thus, the retardation of the dimensionless growth rate by viscous effects is enhanced as 7 increases. The same effect could also be achieved by decreasing the interfacial tension. The impact of viscosity on the grth rate, albeit small for the three systems examined here, will be much larger for systems with larger values of the two phase viscous length scale (c. The magnitude of [c for each of the three systems examined in this dissertation follow directly from Eq.(3.96) and the physical properties: air/water ............... e, =1403. kerosene/water ....... [c = 890 1;; water/freon ............ (c z 6501:. 60 0.63 : Inviscid Theory 0.62 ' 0.61 $1.1m: : 0.6 0.59 0 200 400 600 800 1000 Figure 3.7 The Effect of y on the Maximum Growth Rate (AzAir/Water; BzKerosene/Water; C:Water/Freon) 61 The previous development shows that the dimensional growth rate for unstable viscous interfaces can be expressed as (see, esp., Eq.(3. 100) and (101)) 3 “4 where the Atwood group At is defined as Ate-M . (3.116) pH+pL Figure 3.6 shows how (p1 changes with $2 (a k/ kc) and with the physical property groups 0t", BB and Mo. Tables 3.5a - 3.5c compares the maximum growth rates for all three interfaces with the inviscid estimate obtained by setting Mo = 0. The wavelength of the maximum growth rate disturbance is also indicated in the tables. The calculations clearly show that the viscosity has a relatively small effect on the maximum growth rate for 0.1 S y S 1000. This somewhat surprising conclusion will be discussed further momentarily. The spatial decay of the velocity disturbance normal to the interface is strongly influenced by the viscosity. Because the derivative of the normal component of the velocity is continuous across the interface (see Eq.(3.47)), the jump in the viscous normal stress appearing in the boundary condition given by Eq.(3.55) can be written as 2(u1-u2)(%)fl m . (3.117) .-¢< II>.n--tvx ,hnu .-u-u...~,..~ 5V7». 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I“. m, I. 11., “in 65 Eq.(3.42) indicates that the normal component of the velocity gradient has two distinct physical contributions: an irrotational term which stems directly from the velocity scalar potential (I); and, a rotational term which stems directly from the velocity vector potential A The exponential spatial dampening of the irrotational component of the disturbance velocity is governed by the wave number R (see Eq.(3.26)); the exponential spatial dampening of the rotational component depends on the wave number of the disturbance, the growth rate, and the kinematic viscosity of the fluid. For Fluid 1, the rotational component of the disturbance mode decreases over a length scale comparable to (c.f. Eq.(3.25)) z~__= . (3.118) A similar expression holds for Fluid 2 with vI replaced by v2 . Tables 3.6a - 3.6c give the relative decay factor m,/k and mzlk at the maximum growth rate for 0.1 S y .<_ 1000. It follows directly from Eq.(3.92) that the wave number at the maximum growth rate is given by (pH-’pn)g7 . (3.119) k = ¢2.mukc = ¢2.max 0 As expected, the viscous (or rotational) contribution to the unstable disturbance diminishes much faster than the irrotational contribution as |z| >> 0 . The calculations show that for an air/water interface with y = 0.1 and 4b = (bum , the viscous length scale, llmH , for the water side of the interface (Fluid 2) is about 1/38 times smaller than the inviscid length scale, 1km . On the other hand,the viscous length scales for the air side is only 1/10 times Ill Fifi! Ill VIIOIHYHIU. hUvUvJ/Vt \U) I ..h 1.1 '1‘. My. yhfiv ~;Jltv\\...w.l 1 . Ficrmll “V‘Fhlfllht 66 £5.30 N 3.0. .780 2.x owed n J. 39.3” C 386 m6 «Sod m.~_ $23 0.3 who 2:: SSd m6 good We— vaod EON Rd A... — 386 C6 0086 QNN NE .o one 3.9 c — wooed as $36 w.w~ 2.06 :.~ and a. — mm: .o ed— 386 Yam 3: cued and n . c Eu. ...—.2 ..> Eu. :5: => 50......32 ...—8......— rniae > 8.2... 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Thus, although the kinematic viscosity and the relative acceleration have a significant quantitative impact on the actual size of the viscous sublayer region near an unstable interface, the relative ratio YH/YL remains approximately constant for 0.1 S y S 1000: air/water ..................... YH lYL z 3.60 kerosene/water ............. YH IYL = 1.28 water/freon .................. YH / YL = 0.99 . Clearly, the extent of the vorticity sublayer depends on the kinematic viscosity of the fluid. The above results show that the heavy and the light sublayers for the water/freon system are comparable because VH = VL . However, the vorticity sublayer on the air side of an air/water interface is much larger than its counterpart on the water side inasmuch as vH >> vL . Table 3.3 shows that for the kerosene/water interface, vH > vL and, consequently, YH/YL > 1. This result implies that the vorticity sublayer in the kerosene phase is larger than the vorticity sublayer on the water side. The spatially nonuniform vorticity distribution near the interface may provide an initial state which triggers nonlinear phenomena associated with the entrainment of one fluid by another. For YH/YL > 1, perhaps the heavier fluid, which has a smaller vorticity sublayer, entrains the lighter fluid. This hypothesis would suggest that as a consequence of Rayleigh-Taylor instability, small air bubbles or small drops of kerosene may be engulfed by the heavier water phase. Further discussion of this speculative hypothesis is beyond the scope of this dissertation. 70 3.9 Stable Interfaces Eq.(3.71) introduces velocity disturbances as linear combinations of two dimensional standing waves with wave numbers k" and ky (k2 2 k3 +ki) and with temporal periods given by 27r/s,. Alternatively, arbitrary disturbances can also be interpreted as linear combinations of two-dimensional progressive waves with wave numbers k, and ky and with a wave speed given by =i— . (3.120) Eq.(3. 120) explicitly acknowledges that the set of eigenvalues include complex conjugate pairs. Therefore, the wave speed may be either positive or negative. The wave length A is related to 11, and A, (see by Eq(3.30)) by the following relationship: ——=—+— . (3.121) ’12 1i ii Stable Inviscid Interfaces With 3 = isl , Eq.(3.69) for inviscid fluids is equivalent to a prediction of the wave speed for the propagation of an interfacial disturbance: :;# £1 E?’ 71 =+Atgy—+——— . (3.122) The above expression gives the wave speed for O < y < oo regardless of the orientation of the two phases inasmuch as (p2 — p,)(aF — g) > 0 for a stable interface. Eq.(3.122) is equivalent to Eq.(3.106) with 8 = —1 and N = 0. Figure 3.8 shows the dependence of the wave speed on the wave length of the disturbance for the three physical examples defined by Table 3.2. In the absence of viscosity, the buoyant contribution controls the wave speed for large wave length disturbances whereas the interfacial tension governs the wave speed for short wave length disturbances. It follows directly from Eq.(3.122) that a minimum wave speed occurs at an intermediate wave length given by (see Figure 3.8) 1/2 0' 11m = = 27: . (3.123) km... [(105 - mgr] =—-9i—"— . (3.124) 72 PM, cm c°,.,,.,, cm/s 0.01 0.1 1 10 7», cm Figure 3.8 The Effect of Wave Length on the Wave Speed for Marginally Stable Interfaces (7:1; A: Air/Water ; B: Kerosene/Water; C: Water/Freon) 73 With k a 27r/11 and with the Bond number defined by Bo 5 (P11 ’ PL )8?’ [=1 buoyancy force k20’ interfacial tension ’ (3.125) the inviscid theory predicts that a minimum wave speed associated with marginally stable inviscid progressive waves occurs for a Bond number equal to unity, Bolmm 5 (“12‘3” =1 . (3.126) min The above result follows by combining Eqs(3.123) for kmin and the definition of the Bond number. Eqs(3.123) and (3.124) can also be written as 2‘0 k ' 1/2 __mrm: 3.127 A... k3... y ‘ ’ and :3“ =7“ , (3.128) where 1:,“ and c° represent, respectively, the minimum wave length and corresponding wave speed for the special reference case of ap = 0( i.e., y = 1). Clearly, for this special case, the light fluid must be above the heavy fluid for a marginally stable interface. Figure 3.9 shows the modes for the minimum wave speed. This result is similar to the theory governing unstable inviscid interfaces (see Figure 3.4). 74 Inviscid S ater ater ater reon 2 light flUid heavy fluid M M heavy fluid light fluid 1.5 1 Cm/Cmo C .../Cm" 0. ‘. MM MM 1 1 A 19 j -2 -1 0 1 2 (ar' g) I E Figure 3.9 The Effect of Frame Acceleration on the Minimum Wave Speed for Marginally Stable Interfaces 75 Eq.(3.122) stems from the interfacial force balance given by Eq.(3.1) when restricted to inviscid fluids. Lamb(see p. 434, 1931) and Mendelson (1967) have noted that Eq.(3.122) expresses a balance between three physical forces acting at the interface: (pH + pL) k c2 ....... an effective inertial force per unit volume, ‘(pH — pL )gy ......... an effective buoyancy force per unit volume, and kzo’ .................... an effective interfacial tension force per unit volume. Eq.(3.122) generalizes the earlier results of Lamb and Mendelson to a simple class ‘of noninertial frames for which aF = constant ¢ 0. As noted by Taylor (1950), as well as by Chandrasekhar (1961), the generalization simply requires the replacement of g by g'y. Eq.(3.122) can also be rewritten as We=Bo+l (3.129) where the Bond number is defined by Eq.(3.125) and the Weber number,We, is defined by Wes( pH + pL)kc2 [z] inertial force kza interfacial tension ' (3.130) Eq.(3.129) is presented in Figure 3.10. The Weber number at which the wave speed is a minimum occurs for a bond number of unity (i.e., B0 = 1, We = 2). For very large Bond numbers (i.e., Bo >> 1), Eq.(3.129) implies that We = Bo. Thus the interfacial forces and the buoyancy forces effectively balance one another for large Bo; on the other 76 100 : 10 : We = Bo + 1 : Inertial Forces : Bouyancy forces We . 2 1— ——————————————— l I 1 _ : ; Inertial Forces : Minimum Wave ’ Interfacial Forces : Speed I i I l t I l I O 1 n 1 1 n I 1 1 4 1 1 1. . 1 1 0 O 1 10 100 Bo Figure 3.10 Relationship Between the Weber Number and the Bond Number For Progressive Waves Supported by an Inviscid Interface 77 hand, for Bo << 1, the inertial forces are effectively balanced by the interfacial tension (i.e., We ~ 1). Stable Viscous Interfaces For stable viscous interfaces, the following inequality must hold (cf. Ineq.(3.74)) (pz—p,)(aF—g)>0 . (3.131) Thus, if aF < g, then p2 = pL and p1 = pH. That is the heavier fluid is below the lighter fluid (see Figure 3.1). On the other hand, if aF > g, then p2 = pH and p1 = pL . For this case, Eq.(3.106) with e = -1 determines the complex growth rate s( = ¢1Sc)3 2:12.313 .3, ”N (3.132) For an inviscid interface (i.e., Mo=0), N=O and Eq.(3.132) is equivalent to Eq.(3.122): (¢,°)2=-(¢2+¢§) . Mo=o . (3133) Because $2 is real and positive, solutions to Eq.(3.133) are pure imaginary numbers with an imaginary part given by 78 1m(¢f)=: ¢2+¢§, Mo=0 . (3.134) The wave speed for a stable viscous interface (i.e., Mo > O) can be related to the imaginary part of a), as follows (see Eq.(3.120)): c ___ Im(s) = Im(¢,) _s_c_ k (112 k , (3.135) where kc and 5c are defined by Eqs.(3.93) and (3.100), respectively. The complex growth rate (it, (= Re(¢,) + i Im(¢,)) is related to the dimensionless complex growth rate X by Eq.(3.110): Re(¢,) + 11m(¢,) = Mo‘”¢f[Re(x) + iIm(X)] . (3.136) Appendix D discusses the use of Mathematica® to find the eigenvalues X for stable interfaces, (i.e., Eq.(3.90) with ab2 < 0). Figure 3.11 shows the wave speed predicted by Eq.(3.132) for 'y = 1. The wave speed and wave length are made dimensionless by using the minimum wave speed Cm and its associated wave length Km predicted by the inviscid theory (see Eq.(3.123 and (3. 124)): (3.137 Cmin E $2 ) 79 and A. E =— . (3.138) The inviscid theory, defined by Eq.(3.133), is also shown on Figure 3.11 for comparison. It is noteworthy that for the three fluid pairs studied, viscous effects are only apparent for very small wavelength disturbances. Table 3.7 gives the complex eigenvalues for the water/freon system for 0.05 .<. 412 S 270. XI and XR satisfy Eq.(3.90). The real and imaginary parts of q), are calculated by using Eq.(3.108) with Mo = 600x10“. The dimensionless wave speed c* and the corresponding dimensionless wave length N are also listed in Table 3.7. c' and 1', which are graphed in Figure 3.11, were calculated using Eqs.(3.133) and (3.134), respectively. Tabulated results for the air/water and kerosene/water interfaces are given in Appendix D. Eq.(3.110) implies that the real and imaginary parts of 1), can be written as ¢1n = -(iH,)(i¢fi) (3.139) and «A. = +(iHR )(iai) (3.140) where HR and H, represent the real and imaginary parts of the viscous factor defined by 80 2.5 - 1 Capillary Waves Gravitational Waves 2 . c. 1.5 - Inviscid Theory A ‘\ \ / \\ / . ‘ B l r- ————————————— , I C l r I l l 1 i 0.5 #_. 1 -4444 1 . . ....-. 0.1 l 10 Figure 3.11 The Effect of Wave Length on the Wave Speed for Stable Viscous Interfaces (A:Air/Water; BzKersoenelWater", C:WaterlFreon) 81 Table 3.7 Complex Eigenvalues for the Water/Freon Interface £=-1; 7:1; Mo=600x10'”; 1min=l.68 cm; cm=10.79 cm/s, kc = 3.37 cm" k, cm" (1,2 XR XI ¢m 4),, 7V c’ 0.2 0.053 43.20 16,273 —6.13x10“ 0.231 18.7 3.05 0.4 0.107 -25.88 5,769 -l.47x10'3 0.367 9.34 2.16 2.0 0.535 -8.564 576.26 -1.2)I£10'2 0.818 1.87 1.08 4.0 1.07 ~5.824 261.35 --3.3lx10'2 1.48 9.34x10'I 0.981 20 5.35 -3.499 85.535 -4.97x10'l 12.1 1.87x10’I 1.60 40 10.7 -2.976 59.272 1.69 33.7 9.34x10’2 2.22 200 53.5 -2.090 25.794 29.7 366 1.87x10'2 4.84 400 107 -1.805 17.990 102 1020 9.34x10'3 6.75 1000 268 -1.495 11.12 531 3950 3.7x10'3 10.43 82 Eq.(3.112). Eqs.(3.139) and (3.140) give the components of (b, ( = (0”, + 1(1),, ) and its complex conjugate 42," (= on, — i¢,, ) . Figure 3.12 shows how the viscous moduli HR and H, depend on o, for the water/freon interface. Table 3.83 summarizes the eigenvalues for an air/water interface at the minimum wave speed for several different values of the acceleration group. Tables 3% and 3.9c give similar results for the kerosene/water and the water/freon interface. The temporal relaxation of the initial disturbance is determined by the real part of the eigenvalue inasmuch as exp[st]=exp[s,,t] exp[is,t] , (3.141) where sR = XRv,k2 (3.142) and s, = x,v,1c2 . (3.143) The two-phase kinematic viscosity coefficient v, is defined by Eq.(3.89). It is noteworthy that X, for an air/water interface is essentially constant ( XR = -2) as 7 changes whereas XR for kerosene/water and water/freon interfaces show a significant dependence on 'y. The imaginary part of the eigenvalue decreases as 7 increases for all three interfaces (see Tables 3.8a - 3.8c). Eqs.(3.90), (3.91) and (3.94) show that the complex eigenvalue X 83 Table 3.83 Effect of Relative Acceleration on the Complex Eigenvalues and the Rotational Sublayers of an Air/Water Interface Near the Minimum Wave Speed (e=-l) Heavy Phase LightPhase (112 'y X 11 X1 Yin: Yin YLR YLI 0.8695 0.1 -2.151 1288 6.91 6.85 25.58 25.60 0.8695 10 -2.015 407.3 3.92 3.81 14.37 14.41 1.100 1 -2.051 574.2 4.63 4.53 17.02 17.05 1.100 100 -1.937 180.5 2.66 2.49 9.55 9.60 34 Table 3.8b Effect of Relative Acceleration on the Complex Eigenvalues and the Rotational Sublayers of an Kerosene/Water Interface Near the Minimum Wave Speed (e=-1) Heavy Phase LightPhase (112 y X,, X1 YER YHI YLR YLI 1.080 0.1 -8.445 537.9 14.58 14.65 18.55 18.81 1.080 10 -4.862 168.9 8.05 8.22 10.32 10.57 0.976 1 -6.713 332.9 11.36 11.55 14.57 14.83 0.976 100 -3.888 103.7 6.31 6.47 8.08 8.33 85 Table 3.8c Effect of Relative Acceleration on the Complex Eigenvalues and the Rotational Sublayers of an Water/Freon Interface Near the Minimum Wave Speed (e=-l) Heavy Phase LightPhase q)2 7 X1: X1 Yuk Yrrr YLR Yu 0.846 0.1 -8.59 595.4 17.25 17.47 17.30 17.30 0.846 10 -4.98 186.3 9.60 9.81 9.51 9.72 1.071 1 -5.82 261.4 11.39 11.60 11.28 11.50 1 .071 100 -3.42 81.3 6.32 6.51 6.26 6.45 86 depends explicitly on five dimensionless groups: a, on", BH, M0 and ¢2- The forgoing observations on the influence of v on XR and X, for fixed values of ‘1’; stem from the changes in Mo cc 7 (see Eq.(3.97)). Apparently for low values of the Morton number (i.e., air/water), XR is a relatively weak function of Mo whereas X, is influenced significantly by Mo. However, as indicated by Tables 3.8b and 3.8c this observation depends on (an, B"). Table 3.8a-3.8c also show the interesting result that Y,,,,/Y,,, ~ 1 and YLR/YL, ~ 1 for each of the calculations. Also, in contrast to the unstable interfaces (see Tables 3.6a - 3.6c) for which Y,, > YL if v,, vL (water/freon), stable interfaces (see Tables 3.8a - 3.8c) have just the opposite structure, viz., YHR ~ Ylll < YLR 7' Yu if VH < V1. (3-145) and Y,,,, ~ Y,,, > Y,_,, ~ Y,, if v,, > vL . (3.146) The above inequalities imwy that the rotational sublayer for stable interfaces penetrates the heavy phase more than the light phase if v,, v,_, the rotational sublayer penetrates the lighter phase more. It is noteworthy that for the unstable interfaces the relative spatial extent of the rotational sublayers in the heavy and light phases reverses. The physical significance of this interesting discovery of Rayleigh-Taylor interfaces requires additional study. 87 The complex growth rate 3 can also be sealed with sc (see Eq.(3.101)), s=¢lsc where 1/4 5, = At (fig-fimf] . (3.147) Eq.(3.106) implies that the dimensionless complex eigenvalue 4% depends on the same five dimensionless groups as X, viz., e, a", 0", Mo, and ()2. As previously shown, the use of 5c as a scale factor isolates the viscous effects in the dimensionless growth rate (I), or, equivalently, the complex modulus H (see Eq.(3.112)). Thus, in , 4. =(¢. +¢§) [-H. +1H..] . (3.148) The norm of the complex eigenvalue 4), is given by ||¢.|l= W = 74>: +43. = (o, +¢j)”2,/Hf, +11,2 . (3.149) Table 3.9 gives the real and imaginary components of the eigenvalue (1), for stable interfaces. The dimensionless groups or“, B“, Mo, and (1), were varied over a wide range in 88 Table 3.9 Solutions for Eq.(3.106) for Stable Interfaces (£=-1) a. B. ”0‘10" «1. 4... 4.. "a "a 4. 0.9987 0.9823 2.62 0.869 -0.0021 1.235 1.000 0.002 1.235 26.2 1.100 -0.0056 1.558 1.000 0.004 1.558 262 0.870 -0.0061 1.235 1.000 0.005 1.235 2,620 1.100 -0.019 1.557 0.999 0.017 1.557 0.5405 0.4167 33.6 1.080 -0.0237 1.507 0.985 0.015 1.507 336 0.976 -0.0273 1.354 0.981 0.020 1.354 3,360 1.080 -0.0431 1.489 0.975 0.028 1.490 33,600 0.976 -0.0500 1.334 0.966 0.082 1.335 0.6 l 09 0.6154 60.0 0.846 —0.017 1.189 0.986 0.014 1.189 600 1.071 -0.061 1.484 0.982 0.022 1.485 6,000 0.846 -0.017 1.176 0.976 0.026 1 . 176 60,000 1.071 -0.061 1.458 0.967 0.040 1.459 89 order to examine the stability properties of the air/water, kerosene/water and water/freon interfaces. 3.10 Conclusions .Classical linear hydrodynamic stability theory shows that the interface between two immiscible, inviscid fluids subjected to a uniform acceleration is unstable to arbitrary infinitesimal disturbances if (p2 - p,)(aF - g) < 0, where p2 is the density of the fluid layer situated above another fluid of density p, (see Figure 3.1). A stable interface obtains for (p2 - p,)(aF - g) > 0. Physically, a light fluid over a heavy fluid (i.e., p2=pL < p,, = p ,) is unstable if the acceleration of the interface aF > g and stable if a, < g. The familiar example of a heavy fluid over a light fluid (i.e., p,=p,, > pL p ,) is unstable for a, < g, but stable for aF > g. These interesting results have been understood for more than a century and represent the most elemental predictions of a general class of problems known as Rayleigh-Taylor instability. In this chapter, the earlier results of Reid (1961), which examined the influence of viscosity on the classical Rayleigh-Taylor problem for a special class of fluids, has been extended to include commonly encountered fluids for which v,, at v,. The earlier theoretical work of Reid (1961) and of Chandrasekhar (1961) was restricted to v,, = VL. Apparently more recent generalizations of the classical Rayleigh—Taylor problem have continued with this earlier simplification (see, esp., Kull, 1991). However, here the influence of v,, / vL $1 on the dynamic behavior of stable and unstable interfaces was quantified by calculating the eigenvalues associated with the temporal behavior of the interface. 90 The property ratio v,, Iv, has a significant quantitative and qualitative effect on the wave numbers associated with the rotational component of the eigenfunctions normal to the fluid/fluid interface. This discovery associated with the classical Rayleigh-Taylor problem is new with this research. The mathematical representation of the problem as a superposition of rotational and irrotational disturbances (see Eq.(3.20)) allowed the nonlinear dispersion equation to be rewritten as a multiplicative product between a complex viscous modulus and the inviscid spectrum (see Eq.(3.110)). This technique permitted a convenient evaluation of the viscous influence on the eigenvalues for stable and unstable interfaces. The complex growth rates were examined relative to the viscous scaling, X = sl(v,k2), and relative to an inviscid scaling, 4), s s/sc with sc defined by Eq.(3.100). The dimensionless growth rate, either X or 4),, depends on five dimensionless groups: 8, the stability group; a", the fractional density ratio of the heavy phase; B“, the fractional viscosity ratio of the heavy phase; M0, the Morton number; and, the dimensionless wave number 4),. The dimensionless rotational complex wave numbers Y, (E m,/ k) and Y2 (2 m2 / k) also depend on these five groups. Noteworthy conclusions related to each type of stability problem examined in this chapter are summarized below. Inviscid Theory: Stable Interfaces The condition (p2 - p l)(aF - g) > O is sufficient for the stability of the interface between two inviscid fluids. For (p,-p,)(aF-g)<0, there exists a marginal wavenumber kM (Eq.(3.73)) for which interfacial tension stabilizes the interface for disturbances with k 2 kw. This marginal wavenumber is independent of viscosity and is 91 a direct result of the increasing normal force due to interfacial tension with increasing curvature of the interface. For stable inviscid interfaces, the eigenvalues are pure imaginary numbers. The stable disturbance at an inviscid interface is therefore represented by a standing wave. In the absence of viscosity, the velocity field is irrotational. Inviscid Theory: Unstable Interfaces The condition (p2 - p,)(aF-g) < 0 is necessary for instability of an inviscid fluid. Disturbances with wavenumbers less than kw, are unstable and are associated with eigenvalues that are real numbers and represent an exponentially growing disturbance with no oscillatory behavior. Moreover, unlike the case when interfacial tension is absent, there exists a mode of maximum instability for which the amplitude of the disturbance grows fastest. Viscous Theory: Marginal Stability The marginal state for viscous fluids represents a stationary state, i.e. the real part of the growth rate equals zero. The marginal wave number that determines this state is identical to the marginal wave number for the inviscid result. This shows that viscosity plays no role in determining the marginal wavenumber. Viscous Theory: Unstable Interfaces Viscosity retards the grow rate of an otherwise unstable interface with its 92 effectiveness being greatest in the region of maximum growth rate. The effect is diminished for small wave numbers where the curvature is low and the velocity gradients are small, and the interfacial tension dominates (i.e., (p, -> 1). This effect also depends on the acceleration parameter 7, increasing significantly as 7 increases several orders of magnitude. The impact of viscosity is small for the three systems investigated but is anticipated to be much larger for systems in which the two phase viscosity length scale is much larger (see Eq.(3.96)). The spatial decay of the velocity disturbance normal to the interface is strongly influenced by viscosity. For systems in which the two fluids had different kinematic viscosities, the depth of the vorticity sublayer was greatest in those fluids with the higher kinematic viscosity. This spatial nonuniforrnity of vorticity near the interface may provide an initial state which triggers nonlinear phenomena associated with the entrainment of one fluid by another. Viscous Theory: Stable Interfaces Unlike the case with the unstable interface, the dimensionless spatial decay of the vorticity is nearly equal on either side of the interface. For systems of fluids with different kinematic viscosities, it was observed that the viscous sublayer was smaller for the fluid with the higher kinematic viscosity. This observation was just the opposite of that found for the unstable case. For the three systems studied, the viscous effects are only apparent for very small wave lengths. Moreover, the progressive wave velocity is well approximated by the inviscid theory. Apparently the cause for this is the relatively low value of M0 for each of the systems studied. It is anticipated that for fluids with higher values of Mo, viscosity can, in principle, completely suppress the progressive wave velocity for small wavelength disturbances by completely dampening the oscillatory behavior of the interface. CHAPTER 4: STABILITY OF AN ACCELERATING FLUID/FLUID INTERFACE - EXPERIMENTAL 4.1 Introduction The objective of this chapter is to present new experimental results on disturbances located at the interface between two immiscible accelerating fluids. An apparatus was designed and built to videograph the interface. A two-dimensional planar image of the interface was decomposed into harmonic components using a discrete Fourier transform. This technique enabled multiple wave numbers to be identified in a single experiment The resulting growth rates were measured and compared to those predicted by the linearized theory. The results, which extend the work of Lewis(1950) and Emmons(l960), yield growth rates for an extended range of physical property parameters. 4.2 Experimental Methodology The experimental apparatus was designed to accelerate a container containing a fluid/fluid pair while the interface was videographed. All of the kinematic quantities were 93 94 calculated from the videographed images. The projected interfacial surface was decomposed into harmonic components using a discrete Fourier transform. The growth rate of individual harmonic components was determined by monitoring the amplitudes as a function of time. The fluid container, referred to hereinafter as an elevator, was 150 mm tall by 50 mm wide, and with a depth of 25 mm. The fluid interface was viewed through the 150 mm by 50 mm window. Figure 4.1 is an illustration of the elevator including the dimensions and the viewing perspective. The elevator dimensions were chosen to approximate those of Lewis (1950) and Emmons (1961), each of whom conducted similar accelerating interface experiments. The materials used in the construction of the elevator and acceleration shaft are covered in Section 4.3 below. The three fluid pairs studied were air/water, kerosene/water and freon/water. The air/water system was previously investigated by Lewis (1950). The kerosene/water fluid pair was chosen since its physical properties closely approximate produced water on offshore platforms and bilge water contained in sea going vessels. Experiments conducted using this pair relate to the creation of oil/water emulsions by accelerated interfaces. The freon/water fluid pair was chosen since these fluids have approximately equal kinematic viscosities. Experiments from this system can be compared directly to the theoretical results of Reid(l961) and Chandrasekhar (1961). The interface was videographed using a high speed video camera. The camera is capable of a full frame recording rate of 1000 frames per second and up to 12,000 frames per second with 1/ 12 of the full frame view. The recording rate of 2000 frames per second (full frame view) was chosen as the best compromise of image size and recording rate. The upper limit on the observable acceleration is determined by the viewing area, video recording rate, and the minimum number of observed video frames. Using a minimum of 95 150mm Figure 4.1 Elevator Dimensions and Viewing Prospective 96 50 frames for observation, a recording rate of 2000 frames per second, and a viewing area of 1l2 of the elevator height (75 mm), the maximum observable acceleration is 48.9 g. The elevator is accelerated using air pressure supplied to the top of the container. The maximum pressure difference required is that which will produce the maximum observable acceleration. This is determined by assuming that the pressure difference is only 75% efficient in producing the acceleration. This inefficiency accounts for energy losses (i.e., flow of air around the sides of the elevator, frictional losses, etc.). For a total mass of 750 gm (500 gm elevator and 250 gm fluid), a pressure difference of 57 psi is required to attain the maximum acceleration of approximately 49 g. At 2000 frames per second, 50 video frames gives an observation time of approximately 25 ms. Therefore, the maximum observable growth rate of an initial disturbance having an amplitude of 2 mm is 17 x 103 s’1 for a wave which grows to 1/2 of the elevator height in 25 ms. This is well above the maximum growth rate of approximately 680 s", predicted by the inviscid theory for the air/water system for aF= 50 g (see Eqs.(3.73), (3.75) and (3.76) of Chapter 3). To determine the growth rate of disturbances at the interface, the interface is decomposed into its harmonic components using a discrete Fourier transform. This is done by dividing the viewing area into 16 evenly spaced intervals and measuring the position of the interface at each of the 16 intervals. For a sine wave to be completely defined, it is required that it be sampled at least two times per cycle. The interval spacing will therefore define the limitations on the smallest wave length or largest wave number disturbance observed. This limitation within the Fourier transform technique is commonly referred to as the Nyquist critical frequency (see p. 494, Press, 1992). The Nyquist critical frequency is defined as 97 (4.1) _1_. 9 f N2A where A is the sampling interval spacing. Equivalently, the Nyquist wave number may be defined as k, (4.2) .75. A . T'he Nyquist wave number limitation is useful here because it can be viewed as the largest wavenumber resolution based on the sampling interval chosen for the experiments. Using a viewing area of 95% of the total window width of 50 mm and 16 equal division gives a limiting Nyquist wave number of 10.6 cm". Consequently, disturbances with wavenumbers larger than 10.6 cm‘1 can not be observed. It is predicted in Chapter 3 that all wavenumber disturbances above a marginal value k... (see Eq.(3.73) of Chapter 3) will be stabilized by interfacial tension for an otherwise unStable arrangement. In order to experimentally observe this phenomena using the I=0ul‘ier transform technique described above, k,1 ( or kc ) must be less than kN . Figure 4.2 shows the behavior of kc predicted by Eq.(3.73) for the three fluid systems investigated. It is clear from Figure 4.2 that complete stabilization due to interfacial tension should be ObSetVed using the designed apparatus at only moderate accelerations (~10 g) for the air/Water and the freon/water systems. Slightly higher accelerations are permitted for the ket'osenelwater system (~27 g). 98 3O Observable , B k—— Wavenumbers L 1:51;" | 25 . l » I - l ._ ' t 1 2° C I A . I 31:38 1 i 15 - : C . l ' 1 L I 10 - i : i : 5 i I I : I 1 :k.=k.=1o.6cm-' 0 ............................... o 5 10 15 kc. cm‘ Figure 4.2 Frame Acceleration vs. Critical wave number (A: Air/Water”, B:KerosenefWater; C:WatedFreon) 99 4.3 Apparatus An accelerometer was built to observe the growth rates of disturbances on a flat interface between two immiscible fluids. A general schematic is shown in Figure 4.3. The experimental apparatus consisted of four parts: 1. A container or elevator where the two fluids were contained and through which they were viewed; 2. an “elevator shaft” through which the elevator was accelerated; 3. a pressure reservoir from which the force causing the acceleration was supplied; and, 4. a high-speed digital video camera by which the events were recorded and analyzed. The air-water experiments were conducted in an elevator in which all six sides were made of acrylic sheets. The elevator was filled by removing the acrylic top lid. The sides Were assembled and sealed with screws and a 1/32 inch flexible gasket material with a thin lay er of silicone sealant. The combination of flexible gaskets and silicone made the elevator leak proof. The acrylic elevator designs did not survive high acceleration (and subsequent deceleration) and therefore a new improved design was used for the kerosene-water and water—Freon experiments. The new design, which had the same outside dimensions as the previous elevator, Was made of four sides of machined aluminum with two acrylic viewing windows. A 1/2 inch threaded hole was drilled through the top of the elevator to allow for filling. After the e1e‘hfitor was filled, the hole was plugged with a rubber stopper. The windows and 100 Enclosed Plexiglass Container Plexiglass Guide Chamber 3/ [j— Video Process! Deceleration Container Figure 4.3 Experimental Apparatus 101 aluminum walls of the elevator were sealed with a combination of flexible gaskets and silicone similar to the acrylic prototype. The shaft was made in two sections: a top section where the elevator was placed at the start of the experiment and a lower section, through which it was accelerated and videographed during the experiment. Each section of the shaft was made of 1/4 inch acrylic sheets. These sheets were assembled and sealed using a cork gasket material brushed with a silicone sealant. The inside dimensions of the shaft were identical to that of the elevator with an addition of 1 mm to allow for slight variations in acrylic sheet thi(Zlcrress. This allowed the elevator to slide freely within the shaft. The flanges were made of acrylic and secured to the shaft sections using screws and Sealed with silicone brushed cork gaskets. To ensure alignment of the two shafts thI‘Oughout the experiment, alignment screws were placed in the connecting flanges and the t“"0 sections were held together using quick release welder’s clamps. These clamps maintained a secure connection while allowing for quick disconnect and assembly of the Shaft. The lower flange on the bottom section was used to secure the shaft to a table with t""0 ‘C’ clamps. A 1/2 inch diameter hole was drilled into the top section where a 1/2 inch high pressure hose was attached and connected from the pressure reservoir to the shaft. The pressure reservoir was a 20 liter pressure vessel connected to a pressurized air source capable of supplying up to 90 psig air. The reservoir was connected to the shaft by a two foot 1/2 inch high pressure hose. The pressure was monitored using a gauge mounted directly on the vessel. To ensure a constant pressure in the reservoir during the acceleration of the elevator, the pressure reservoir vessel volume was chosen to be approximately 100 times that of the shaft. The large volume of the reservoir relative to the vol‘ll'ne of the shaft assured that the elevator experienced a constant acceleration while being ViE:“‘Ved by the video camera. 102 The interface was videographed using a Kodak” Ektapro” high speed digital video camera. The camera was fitted with a Vivitar" 28 - 105 mm zoom lens. The camera was positioned approximately 1.5 m from the center of the apparatus. The apparatus was illurtiinated by three 600 W tungsten-halogen projector lamps. Two of the lamps illuminated the front of the apparatus, directed at approximately 45 degrees to the right and left. The third lamp was positioned about 0.5 m directly behind the apparatus focused on a White sheet of paper which served to diffuse the light. The camera was capable of a full frame recording rate of 1000 frames per second and up to 12,000 frames per second with 1/ 12 of the full frame view. The recording rate of 2000 frames per second was chosen as the best compromise of image size and recording I‘Elte. The video images were down loaded to a computer and stored on magnetic tape for later analysis. The video images were viewed and enhanced using personal computer software Adobe Photoshope 4.0. on an Apple® Macintosh” IIci personal computer equipped with a 2 1 inch video monitor. The combination of software and large monitor allowed the image t0 be enlarged and enhanced utilizing all information encoded within the digital image. The digital image resolution was 94 x 239 pixels. 4.4 Experimental Procedure The experimental protocol for measuring the disturbance growth rate consisted of the following steps: 1) The video camera was positioned and focused so that all sides of the elevator were in view. A still frame was videographed to calculate the pixel to millimeter scale factor. 2) 3) 4) 5) 6) 7) 103 The elevator was filled with the fluids to be tested. The interface was positioned at approximately half of the height of the elevator resulting in equal amounts of the top and bottom fluids. Each approximately 7.5 cm in depth. A single layer of paper, the diaphragm, was positioned on the top of the lower section of the elevator and the elevator placed on top of it. The top section of the shaft was placed over the elevator and secured with the positioning screws and welders clamps. The pressure reservoir was charged to the desired pressure (30 - 90 psig) using pressurized air and then isolated from the source by closing the valve. The video camera was readied by setting the recording rate and auto shut-off command. The auto shut—off command places the digital image processor in a mode that will automatically shut-off when its maximum storage capacity was reached (1,600 frames). At 2,000 frames per second this gives a maximum recording time of 0.8 seconds. The video camera was turned on and immediately the pressure released from the pressure reservoir to the elevator shaft by manually opening the valve. The captured video sequence was played back to visually determine the fiarnes in which the acceleration took place. The selected frames (typically 100-150 frames) were then downloaded to an IBM” 486-DX personal computer via a GPIB (General Purpose Interface Board) connection to the digital image processor. 104 8) The images were compressed and stored onto two magnetic tapes (one for archival storage) for subsequent analysis. 9) For analysis, the individual digital images were uploaded from the IBM“D 486-DX computer to the Apple” Macintosh® IIci computer and the analysis conducted. 4.5 Experimental Results Fluid Physical Property Measurements The density of the kerosene was measured using a picnometer at room temperature. Since the actual temperature was not measured, it is assumed that it was 20 'C +/- 3 'C. “re error attributed to the kerosene density measurement was determined from the eStirnated variation of the specific gravity of organic fluids near room temperature (see Perry, 1973, fifth edition) of 0.001 'C“. The densities of water, air and freon listed are those reported by Peny and Chilton (see pg. 3-210, fifth edition, 1973). The viscosity of kerosene and Freon was measured using an Ostwald-Cannon- Fenske viscometer. The reported values represent the average of six measurements. The experimental error reported is two times the sample standard deviation. The values are Consistent with those reported by Klee and Treybal (1956) for kerosene at 18 'C. The visCosity of air and water are values reported by Perry and Chilton (see pg. 3-211, fifth edition, 1973). The interfacial tension for the kerosene/water and water/Freon fluid pairs was measured using a pendant drop method. The values reported are the average of six samples 105 of each fluid taken. The error reported is twice the sample standard deviation. These results agree well with those reported by Murshak ( 1995) and Klee and Treybal (1956) for kerosene and water at 18 'C. The interfacial tension for the air/water system used is that reported by Perry and Chilton (see pg. 3-211, fifth edition, 1973). Table 4.1 summarizes the physical properties measured. The information contained in Table 4.1 is identical to Table 3.3, shown previously in Chapter 3, with the addition of the experimental error attributed to the individual measuring processes. In cases where the data were retrieved from a reported source, the error represents i one unit of the last significant digit reported. Elevator Accelerations The frame acceleration for each experiment was calculated by monitoring the change in position of the top of the elevator as a function of time. This procedure is illustrated by the graphical insert in Figure 4.4. The distance 2,. is measured relative to an arbitrarily fixed position in the video image, held constant for each experiment. Shown in Figure 4.4 is a plot of the square root of the relative position of the elevator vs. time for experiment 0213-07 (see Table 4.2). The scatter in the data is typical of the experiments reviewed. Also plotted in Figure 4.4 is the linear regression fit of the data. The slope of the regression line was used to determine the acceleration of the elevator using the equation displayed. The experimental start time is determined as the intersection of the regression line with the time axis. Table 4.2 lists the calculated frame, accelerations for each experiment. The error associated with the acceleration was estimated as twice the standard deviation of the data 106 Table 4.1 Physical Property Data p, gm/cc rt, cp o, dynes/cm Air 0.0013 ‘” 0.018 "’ n/a Water 1.0 m 1.0 m rrla Kerosene 0.85 :1: 0.003 1.8 :1: 0.1 n/a Freon 1.57 ‘2’ 1.6 i 0.1 n/a Air-Water n/a n/a 74 ‘3’ Kerosene-Water n/a n/a 35 :1: 12 Water-Freon n/a nla 40 :1: 15 "’ See p. 3-210, Perry’s Chemical Engineering Handbook, 5th ed. (1973) ‘2’ See p. 3-196, Perry’s Chemical Engineering Handbook, 5th ed. (1973) ‘3’ See p. 40, Adamson (1982) n/a = Not Applicable 12 107 Figure 4.4 Square Root Elevator Position vs Time Experiment 0213-03 (typical) 108 Table 4.2 Calculated Experimental Accelerations Experiment System a, , g 0213-03 Air-Water 21.5 :1: 1.5 0213-07 Air-Water 12.7 :1: 1.3 0215-03 Air-Water 16.5 :t 1.5 0412-01 Kerosene - Water 41.3 :1: 2.8 0412-04 Kerosene - Water 22.5 :1: 2.0 0412-05 Kerosene - Water 16.0 :1: 1.8 0512-03 Water- Freon 48.7 :1: 3.1 109 relative to the least square approximation. The displacement data used for the calculation of acceleration for each experiment is tabulated in Appendix F. Disturbance Growth Rates Figure 4.5 shows the video image of the accelerating interface for experiment 0213-03 for 10 ms into the acceleration. To the left and top are the frame pixel reference marks. These give the actual pixel locations on the video frame image. The white line was drawn by hand onto the image to highlight the projected interfacial interface. Although the image was satisfactory for viewing on the 21 inch video monitor, it is far from satisfactory viewing here as a printed image. The image pixel dimensions are converted to dimensions of length using the scale taken from a still image. For this experiment, it is shown as 0.558 pixels/mm. The result of this conversion is displayed in Figure 4.6. The scale imaged is divided into 16 equally spaced intervals so that the discrete Fourier transform may be performed. The tabulated values are shown in Table 4.3a. with the results of the Fourier transform are displayed in Table 4.3b. The tabulated Fourier transform results for each experiment are listed in Appendix 1. Figures 4.7a, 4.8a, 4.9a illustrate the evolution of the projected interfacial surface at different times. The curves were produced using the data used for the discrete Fourier transform described in Section 4.2. Figures 4.7c - 4.9c show the evolution of the discrete Fourier transforms of the surface projection at each time displayed. The transforms represent an arbitrary disturbance that entered the accelerometer and is growing based upon the physical parameters of the system. The amplitude of the harmonic is displayed on the y-axis while the corresponding wave number is shown on the x-axis. The amplitude of 110 Scale = 0.558 mm/pixel Figure 4.5 Video Image (193 x 239 Pixels) - Experiment 0213-03, t = 10 ms 111 X, mm 30 40 50 60 N O 70 80 10 20 30 40 50 y. mm ' I I I I I I I I I I I I I I I I I ' E d 60 70 80 100 Figure 4.6 Projected Interface Graph for Experiment 0213-03, t=10ms 112 as _a: 7:: _8._ :2 _2s :2: _85 738: _ sea. 8.... 7.33 :3 Tee _e..m _3... _ 2." _m2 _ 8... _ ...—e... reggae. ices: - E a £33. 3 new .....m 9% _. _o e. _ c 5.8 9% him ...nn Nfin efim ado fine fine v.3 new Ana-5» ES. 22% code 2.3 3.3 amen «in 5.9. «New 9.: 3.3 3.: 8.3 Mmdn ”can ocdn ABE: en— mn— eu- :— 9: men 2: ma ea me an as. as no 00 mm Agouivu 358 5.5 .8295 325...". 8a.»...— 3835 a £15. Eustace ....s 3 u e 83:..5 38:5: - E... a £33. 3 as. sea 55 8.0me uses... 38.3: a 3. use. 113 Surface evolution X, mm 30 50 70 0 .L i 20 - W E 40 ‘ E >'. 60 - 80 - 100 (b) DFI‘ Sine Harmonic Amplitude vs Wavenumber 40 530- E 0' 320— E. <10- 0 Drnrgr r 1:11—1— 2»: 2 s 8. e 2 a 8. '-' N V W \D co 0" S k,cm' (C) Figure 4.7 Experiment 0213-03, t= 1 ms; a) actual video frame, b) graphical representation, c) Discrete Fourier Transform 114 Surface evolution x, mm 30 50 70 O : c 20 4 e 40 ‘ E >':. 60 -W 80 - 100 (a) (b) DFI‘ Sine Harmonic Amplitude vs Wavenumber 40 w 0 Amplitude, mm 8 Figure 4.8 Experiment 0213-03, t= 10 ms; a) actual video frame, b) graphical representation, c) Discrete Fourier Transform 115 Surface evolution 71, mm 30 40 50 60 70 1 1 1 1 r I v y,mm 8883383888: (a) (b) DFI' Sine Harmonic Amplitude vs Wavenumber 40 E 30 » E o“ g 20 » i. E < 10 H 0 ” ‘ . : . ‘3. ‘5? 8. no as 2 Figure 4.9 Experiment 0213—03, t= 16ms; a) actual video frame, b) graphical representation, c) Discrete Fourier Transform 116 each wave number is shown to grow with time. From these figures, the amplitude of each harmonic is monitored and the exponential growth rate calculated. The amplitude of the harmonic with wave number equal to 1.35 cm'1 associated with experiment 0213-03 is plotted in Figure 4.10. The saw-tooth nature of the data is attributed to a random distribution of error in measurement of the projected interfacial surface. Also plotted is the regression of the exponential function from which the growth rate can be estimated. The figure shows a growth parameter, s, of 80 ms". The initial wave amplitude, A0, is estimated to be 5.6 mm. The results indicate that the initial disturbance contained only very small amplitudes. A complete listing of the amplitudes for each experiment at various times is presented in Appendix G. Table 4.4 lists the calculated growth rates for each experiment and fundamental harmonic wave number. The errors reported were calculated using the procedure recommended by Holman and Gajda (1989) and outlined in Appendix E. No error is associated in the determination of the harmonic wavenumber. The experimental results for the dimensionless growth rates, (1),, plotted versus the dimensionless wavenumber, (1),, are displayed in Figures 4.11, 4.12, and 4.13 for the experiments using air/water, kerosene/water and water/Freon, respectively. In each figure the error bars represent an approximation of the actual experimental value residing within the area between the bars. The growth rate prediction given by Eq.(3.111) for inviscid fluids including the effects of interfacial tension is also plotted on the figures. Amplitude, mm — O t I 117 20 H U! I v 0 5 10 15 20 Time, ms Figure 4.10 Amplitude of k=1.35 cm-l wave vs time Experiment 0213-03 (typical) Table 4.4 Measured Grth Rates 118 Wavenumber k , [cm'l]; Growth Rate 3, [ms'l] Experiment (Table 4.2) 1.35 2.7 4.05 5.4 6.75 8.1 9.45 10.8 0213-03 80:4 135 :4 220:4 300:4 280:4 200:4 250:4 300:4 0213-07 100:4 160:4 180:4 200:4 250:4 240:4 250:4 250:4 0215-03 90 : 4 160 : 4 190 : 4 200 : 4 240 : 4 300 : 4 240 : 4 320 : 4 0412-01 30:4 60:4 80:4 90:4 130:4 70:4 120:4 120:4 0412-04 30:4 40:4 65:4 70:4 50:4 30:4 0.05:4 1:4 0412-05 30:4 45:4 60:4 15:4 0.1 :4 0.1 :4 0.05:4 0.05:4 0509-05 50 : 8 120 : 8 130 :t 8 140 : 8 200 : 8 280 : 8 300 : 8 325 : 8 119 833}? he 39:35.63 82:32085 .m> Sum 5380 mmoEomEoEB :4. 0.5!..— Aboofi. Emma: - ”5am 35810an 3 Ne 2 _ we .5 been... 283:— 120 Abcofi. 2835”. .338a gunfitoaxm u 5 hoaéucomobv— he Eggnog? awe—communes .m> 83— 5326 mun—commcofia S .v 0.59... : _ 3.5 bong. 28$:— 121 Aboofi. 2835”- “8:53— anfituaxmn+v cooumtofikr Bu “385533 mac—commcofimn— .m> 83— 5380 mmoEommcoEE m _ .v 932"— N6 v._ m._ N._ 2 _ ad ad 56 ed 2. v6 m6 Nd _d : _ 3.3m boo: 283c— 4L1] 122 Acoemxuo§$¥ whoaugcomEov—nx coaguanov 333800 3:53— .ScoEton—xm E .v Baum"— Ne v; m._ NA _._ : _ 3.5 bog—H 2835 123 4.5 Discussion of Results In Figure 4.14, all of the experimental results are plotted together for comparison with Eq.(3.111). In general the resulting experimental growth rates are lower than those predicted by the inviscid theory. Recall, from Chapter 3, for the systems investigated, the viscous effects are very small, owing to the very small M0 for the system. Therefore Eq.(3.111) should represent a reasonable approximation. The dampening effects of viscosity alone do not explain the lowering of the growth rates. These results, however, may be attributed to aliasing of larger wavenumber growth rates within the discete Fourier transformation. The effects of wave numbers larger than the Nyquist wave number are superimposed onto the effects within the known range. If the initial disturbance contained wave numbers larger than the Nyquist wave number and also was higher than the critical wave number, its amplitude would be superimposed onto the measuring range. For all wave number greater than (mm the grth rate is smaller and would in turn lower the grth rate within the observable range. This may be attributed to the lower than predicted observed growth rates. This would be true for not only stable disturbances (¢1>1) but for any disturbance for which (i), is greater than 4),”, or (p, equal to approximately 0.62. In Figure 4.15, the results of Emmons et al. (1962) are ploued with the results of this work. Note that Emmons also observed lower than predicted results of the inviscid case for ¢, < 0.7. Emmons employed the experimental technique of imparting a particular wave length disturbance into the interface and measured the growth rate during the acceleration. Since Emmons et al. (1962) did not use a Fourier transformation, the above explanation would not hold. The results did not exhibit a maximum and return to zero as is 124 Acofifieo~a3u+ voyagcomeuovwx neSaEeauo - x83 we “cg—£364” ”med—.EoEEmnov comm—3:80 838d .8585qu ..353263 mac—comedofia m> 39m 5380 mmoioficoea m _ .v oSmE «e v; N— d u 1 IxP ‘1'”. 22.8.5 boo—E. 28$:— \ ed aeevmaoaemu n m a nu W 83353 . m - 3 a 4 ad .4 ad L _ 126 analyzed by decomposing the disturbance into harmonic components by taking the Fourier transform and then monitoring the growth rates of individual Fourier components. The experimental apparatus was built to videograph the interface between the two immiscible fluids while the pair was being accelerated. The dimensions of the container were chosen to agree with the experimental apparatus of Lewis(1950) and Emmons(l960). The combination of the physical dimensions of the container and the discrete Fourier transform Operation limited the investigation of wavenumbers greater than the critical wave number to the kerosene/water experiments. The experimental results were compared with the growth rates predicted by the linear stability theory presented in Chapter 3. It was shown in Chapter 3, for the three fluid/fluid pairs investigated in the range of tested accelerations, that the inviscid analysis serves as a good approximation. The dimensionless growth rates were consistently lower than predicted values. This may be accounted for by the manner in which the data were analyzed, namely by the inherent limitations of the discrete Fourier transformation. The aliasing of wavenumbers outside the N yquist frequency may account for the lowering of the observed growth rates. The lowering of the observed growth rate may also be due to the non-linear effects ignored by the linearized theory. The experimental results were compared to those of Lewis(1950) and Emmons(l962) and found to have the same general behavior. The experiments did show wavenumber disturbances, with wavenumbers approaching that of the critical wavenumber, exhibiting stability or zero growth. This stabilization, which is predicted by the linearized stability theory, is due to the effects of interfacial tension. CHAPTER 5: TERMINAL VELOCITY OF ACCELERATED FLUID PARTICLES 5.1 Introduction The objective of this chapter is twofold: first to measure the terminal velocity of drops and bubbles in accelerated frames; and, second to compare the measured values with those predicted by a progressive wave model first proposed by Mendelson (1967). The experimental results are also compared with those of Pebbles and Garber (1953) for drops and bubbles rising through a stagnant fluid. The experimental results are correlated with an empirical relationship developed by Mersmann (1983), which includes the effect of an accelerating frame. The experiments were performed in an accelerating frame of reference using air and kerosene as the dispersed phases and water as the continuous phase. Accelerations ranging from 0 - 46 g (gravitational units) were tested with fluid particles having diameters between 6-14 mm. The resulting ranges in Bond ('Ebtvos), Weber and particle Reynolds numbers tested were 0.3 - 300 , l - 250, and 600 - 13,000, respectively. 127 128 5.2 Background For a solid sphere moving slowly through a quiescent fluid the magnitude of the terminal velocity, luTI, of the sphere is well approximated by Stokes’ law (see p. 57, Bird, et al. 1961): |_____Ps “prlgzg Iu r|=-’——— , 5.1 18”! ( ) where p, and pf are the densities of the solid sphere and fluid phase, respectively. I is the diameter of the solid sphere; u, is the viscosity of the fluid; and, g is the acceleration due to gravity (g=980 cm/sz). Eq.(S. 1) is valid for the creeping flowing assumption, which for this system occurs when the particle Reynolds number, (I uT p,)/ [1,, is less than about 0.1. In studying the terminal velocity of immiscible bubbles and drops through quiescent fluids, Mersmann (1983) expressed Stokes’ law using the following nondirnensional form: . 2 .. (dv) (5.2) where the dimensionless terminal velocity, u; , is defined as pz ”3 u' E u —-’——— , (5.3) T {Ala-Mg] 129 and the dimensionless diameter of the solid sphere, d; , is defined as; l/3 d; E ([loflf’s—"zpflfij . (5,4) M In an effort to describe non Stokesian behavior, Mersmann rewrote Eq.(5.2) as . f . 2 11,13 =1_2.3_I%.22.(c1v 1“,) , (5.5) and attempted to find dimensionless functions fI and f2, relating to a class of geometries, such that the experimental data for bubbles in liquids, droplets in gases, and droplets in liquids all fall on a single curve when plotted as u; fl versus d; f2 regardless of the particle Reynolds number. Mersmann found that the empirical relationships describing the terminal velocity were 1.43 u; = 0.15(d:,) , circulating spheres (5.6) and . . 1/2 . uT = 0.714(dv) . spherical caps. (5.7) Equations Eq.(5.1) - (5.7) may be modified to include the effects of a constant acceleration by replacing g with g y , where ‘y is defined by (see also Eq.(3. 19)) 130 (5.8) (5.9) where the dimensionless terminal velocity and dimensionless particle diameter for the accelerating frameare p2 l/3 U' E u ' , (5.10) T {Ala-mgr] and I — ' ”3 D; E [(0. Amp. gr) . (5.11) The relationships found by Mersmann for spherical droplets with internal circulation and for spherical caps with 7:1 can be expressed as 143 u; = 01503;) ' , circulating spheres (5.12) and 131 s11”" ”Q (I) 9% / {Gr 39 V1 rnbkmi (ghuaflflnng on ' "3: :3 O- p» ‘y‘ 10 s?“”*““ C395 mks .13) | 00 1000 l 1 1 mm, 1983 Cknufifin°“Cfi‘Agni 1 690M 132 1/2 U; = O.714(D:,) , spherical caps. (5.13) Figure 5.1 is a plot of Eqs. (5.9), (5.12) and (5.13) for Stokes’ law, completely circulating spheres, and spherical caps. These curves should serve as bounds for experimentally observed flows of fluid particles in accelerating frames. Mendelson (1967) postulated an analogy between the propagation velocity of progressive waves at the interface between two immiscible fluids and the terminal velocity of dispersed fluid particles. It was shown in Chapter 3, that the progressive wave velocity at the interface between two immiscible inviscid fluids could be described in a dimensionless form as (see Eq.(3.129)) We = Bo +1 where the Weber number, We, is defined as 2 . . we 5 (pH + pl. )1“ [z] mertralforce kzo and, where the Bond number is defined by BO E- (pH - pL )g‘}I [___] buoyancyforce kzo interfacial tension ’ interfacial tension ' (5.14) (5.15) (5.16) In Eq.(5.15), c represents the wave speed for a stable, inviscid interface (see Eq.(3.122)). Mendelson found that Eq.(5.14) gave a good estimate of the terminal velocity of fluid particles when the wavelength, A ( =21t/k), was replaced with the perimeter of the 133 maximum cross section of an equivalent sphere of equal volume and c was replaced by uT. Thus, if rp denotes the radius of a spherical drop or bubble, then the wavenumber k in Eqs. (5.15) and (5.16) is replaced by 1/rp: 122-kg —) 27rr. (5.17) The substitution of Eq.(5.17) into the Eqs.(5.14) and (5.15) yields the following modified definitions of the particle Weber number and the particle Bond number: . (pa+pr)r..U% We 5 (5.18) O' and _ 2 30' 500" ””537 . (5.19) a With this ad hoc assumption, Eq.(5.14) suggests that We' = 80' +1 . (5.20) Figure 5.2 is a plot of Eq.(5.20) which illustrates the functionality of the particle Weber number with the particle Bond number as proposed by Mendelson and employed hereinafter for accelerating environments. 134 awe—a2 233 - 595.2 deem .m> eon—Eaz .5903 N6 835 com 82 8. 2 _ 3 85 - dd .qq-qq - q «.141. q q 11. «la-«ii .—.° _ mc— {03 Tenure? H m 8— m 82 135 5.3 Apparatus An accelerometer was built to observe the growth rates of disturbances of a flat interface between two immiscible fluids. The apparatus is identical to that used in Chapter 4 with the addition of a sample introduction tube. This tube allowed the lighter phase to be introduced into the accelerometer prior to acceleration. A schematic of the elevator shaft and modified elevator is shown in Figure 5.3. The elevator used in these experiments was identical to that described in Chapter 4 with the exception of a 1/2 inch threaded hole drilled through the top and bottom of the elevator. Three feet of flexible tubing was attached to the bottom hole to allow introduction of the second fluid phase. After the elevator was filled, the top hole was plugged with a rubber stopper intentionally leaving a small air bubble. This air bubble was compressed during the introduction of the second phase at the beginning of the experiment. The position of the drops and bubbles were determined by noting the top, bottom, right and left position of the edges. An equivalent radius was determined as the average of the differences of the values. The x-y position of the particle was determined as the half distance between the values. The velocity of the particle was taken as the change in position relative to a fixed position within the moving accelerometer, typically taken as the bottom edge. The terminal velocity of the particle was estimated as the average of the resulting velocities. 5 .4 Experimental Procedure The fluids tested were kerosene and air dispersed in water. The fluid pairs tested were those used in Chapter 4 with the exception of the water/Freon pair. The physical 136 Plexiglass Guide Chamba' Filled Sample Container To Pressure Reservoir Sample%nug:duction Deoeleration (butaina Figure 5 .3 Experimental Apparatus 137 properties (i.e., viscosity, interfacial tension, and density) of the fluids tested were measured and are summarized in Table 4.1. The experimental protocol was identical to that used in Chapter 4 with the following exceptions: 1) The second phase was introduced into the container by pressurizing the filled sample introduction tube (see Figure 5.3) until a bubble or drop formed and began to rise within the stationary accelerometer. 2) When the drop or bubble neared the center of the container, the video camera was turned on and the pressure was released from the pressure reservoir to the elevator shaft by manually opening the valve. 5.5 Results and Discussion The video frames for experiment 0814-01 (see Table 5.1) are shown in Figure 5 .4. The experiment consisted of videographing a single drop of kerosene rising in a stationary elevator filled with water. The video was recorded at 125 frames per second. The frames 310, 330, 350, and 370 are displayed with a time difference of 160 ms between 20 frames. The kerosene was injected into the water from below as outlined in the experimental procedure of this chapter. The sequence of four frames show that the kerosene dr0p remained approximately spherical throughout the experiment. This was typical of the inertial frame experiments. The video frames for experiment 0815-05 (see Table 5.1) are shown in Figure 5.5 . The experiment was performed with the elevator being accelerated at 12.6 g. The video was recorded at 2000 frames per second. The images shown were enhanced using Adobe Photoshop® prior to printing. The frames 235, 245, 255 and 265 are displayed with a time 138 d u an .553 E n05 5:883— :73de EoEtunxm he moEEm 82> 1m ouswfi own can emu on we GEE. own 0mm dmm an e ofiflm 139 w QS u as .553 5 £89 2585M 8-28 eoaeaxm .8 seem 82> on use". an mm on 2 m8 .oEfi. mom mmN mg 28 oEEm 140 difference of 5 ms between 10 frames. The kerosene drops were injected into the water from below and allowed to rise prior to the acceleration. The frames show the changing shape throughout the experiment. No stable shape for these drops was observed. This was typical of the moderately accelerated kerosene-water frame experiments. The video frames for the air-water experiment 0815-01(see Table 5.1) are displayed in Figure 5.6. The experiment was performed with the elevator being accelerated at 46.3 g. The image was recorded at 2000 frames per second. The bubbles were injected and allowed to rise prior to the acceleration. The images shown were enhanced using Adobe Photoshop‘1° prior to printing. The frames 210, 230, 240 and 250 are displayed with a time difference of 10 ms between each frame. The images show that the bubbles have a shape resembling spherical caps. This was typical of the air-water experiments. Figure 5.7 illustrates the three types of fluid particle shapes observed. The spherical shape was observed for kerosene-water drops in the inertial frame experiments (i.e., aF=0). The spherical cap shape was observed for all the air-water bubbles. The accelerated kerosene-water drops had unstable deformed shapes. The frame acceleration, aF, was determined from the video images as outlined in the experimental section of Chapter 4. Figure 5.8 illustrates this procedure for experiment 0815-01 and is typical of the experimental data analyzed. Plotted in Figure 5.8 on the x- axis is the time in milliseconds and on the y axis the square root of the elevator displacement Also shown is a least square fit of the data. The frame acceleration was determined from the slope of the least squares curve fit. The error associated with the acceleration was estimated as twice the standard deviation of the data relative to the least square approximation. The fluid particle velocity was determined by the slope of the least squares fit of the particle displacement with time. Figure 5.9 is a plot of the relative position of the air bubble for experiment 0815-01, which was typical of the experiments analyzed. The error 141 m «.9. u as can? 5 235 a... 8-28 .5525 a. seam 82> on 25E dm on 2 o 2. .oEE. dun dmm emu EN 5&8.“— 142 G 95 555:9.” , ’ seam 225m 22m 8:030 E 83E 3 53525 An 585:9. 143 mm :83»: 8-28 asaeaxm 58:. m> eoEmom .8555 we 80% Bang wd charm m8 .58: ON 3 2 m o E H w as. u s o Au - Haven Gnu I. ..I e BEE/“53f 144 . $298 $28 aoaeaxm - 56E. .m> “5832.55 o_ou.5m 395 ad BewE m8 6:5 E 2 E Q 2 w c v N O -.q-14q.~_-qd-+uquaqua-dqqd-uu..u-uW-quuquq 86 a ”52.5 3 us; LUIIIIAIIIIILIIIIllllllllllll a a 2 turn ‘ruauraoeldsrq anqng In N O M 145 reported is two times the standard deviation of the sample data relative to the regression curve. The result of all of the experiments are listed in Table 5.1. The width and height of the particle was monitored on a frame by frame basis and converted to length by the conversion factor determined for each experiment as outlined in Chapter 4. The diameter of the fluid particle was determined to be the average of these values. The results with error approximations are summarized in Table 5.1. The results of the video experiments are also listed in Table 5.1. The parameters D: and U; are also listed in Table 5.1 and are shown plotted in Figure 5.10. The errors reported for the Bond and Weber numbers were calculated using the procedure recommended by Holman and Gajda (1989) as outlined in Appendix D. Each axis is logarithmic with a range of 0.1 to 1000. The actual calculated value is represented by the graphical symbol with its explanation detailed in the legend with error bars. Since the error for the inertial kerosene/water experiments are small, the error bars for these experiments have been removed for clarity. Also plotted are Eqs.(5.12) and (5.13) representing the limits proposed by Mersmann (1983) for completely circulating spheres and spherical caps, respectively. The observed shapes for each experimental group are also displayed in Figure 5.10. The results for the spherical cap data are slightly outside of the limits proposed by Mersmann. Figure 5.11 is a plot of the results of this work and the results reported by Peebles and Garber (1953) for comparison. No approximations of error are shown in the figure since none were reported by the authors. Note that although most of the results for Peebles and Garber reside within the limits of the spherical cap and circulating spheres, some stray outside of the spherical cap limit, similar to that of the air/water experiments. Table 5.2 is a tabulation of the calculated values for the parameters associated with the wave analogy. These results are plotted in Figure 5.12. The figure shows that the experiments were conducted for a range of particle Bond numbers of almost five orders of magnitude resulting in particle Weber numbers varying over three orders of magnitude. 146 Table 5.1 Experimental Results Experiment aF', g 2 x rp, cm uT, cm/s D; U: 815-01“) 46.3 :1: 2.7 0.7 :1: 0.05 160 :1: 8 534 d: 45 21 :1: 8 815-03“) 16 21:1.3 1.1 :1: 0.05 100:1:6 581 i21 19:1:5 815-02-1“) 25 :1: 1.5 0.8 :1: 0.05 100 :1: 6 494 :1: 22 16 :1: 7 815-02-2“) 25:1:1.5 l.2:1:0.05 110 :1: 6 741 :1: 45 18 :t 7 815-054(2) 12.6 :1: 0.9 1.3 :1: 0.05 20 :1: 1.5 335 :1: 32 8 :1: 1 815-05-2‘2’ 12.6 :1: 0.9 1.1 i 0.05 20 :1: 1.5 382 :1: 26 8 :1: 1 816-014(2) 17.4 i 1.2 1.0 :1: 0.05 25 :1: 1.5 289 :1: 27 9 :1: l 816-01-2‘2’ 17.4 :t 1.2 0.9 :1: 0.05 25 :t 1.5 260 i 6 9 11 0814—Ola) O 0.58 :1: 0.05 11.15 i: 0.5 66 i 7 10 :1: 1 0814-020) 0 0.71 i 0.05 11.51 :1: 0.5 81 :1: 7 10 :1: 1 0814-03-1‘2) 0 0.84 :1: 0.05 11.52 :1: 0.5 96 :1: 7 10 :1: l 0814-03-2‘2) 0 0.68 :1: 0.05 12.80 i 0.5 77 :1: 7 11 i l 0814—03-3‘2) 0 0.60 :1: 0.05 11.82 :1: 0.5 68 :1: 7 10 :1: l (”Air-Water (”Kerosene-Water 147 100 - ' Spherical Cap Completely / 1 Circulating f 1 Spheres Eq.(5.12) L [rT 10 _ 1 1 L r Spherical Caps Eq.(5.13) l L. A 4 A . r... 4. u l 10 100 1000 D‘v Figure 5.10 Generalized Correlation of Mersmann,]983 (A- Air/Water; O -KersosenelWater; O- KersosenelWater, 7:1) 100 I 148 Completely Circulating Spheres A 1 11111 10 100 1000 D'v Figure 5.11 Generalized Correlation of Mersmann, 1983 (A -Air/Water;O-Kersosene/Water;O-KerosenelWater, Fl; 0 -Peebles and Garber, 1953) 149 Table 5.2 Experimental Results - Wave Analogy Exp. No.t an, g 2 x r,, cm 11,, cm/s 80* We“ l 815-01") 46.3:2.7 0.7:0.05 160:8 293:42 242:26 11 ,2:(:1:970| 815-03") 16:1.3 1. 1:0.05 100:6 60:11 74:10 1 1,000:830| 815-02-1‘" 25:1.5 0.8:0.05 100:6 51:13 54:7 8,000:700 I 815-02-2‘” 25:1.5 1.2:0.05 110:6 114:19 98:10 13,200:900 815-05-1‘" 12.6:0.9 1.3:0.05 20:1.5 21:8 7:1 2,600:220 815-05-2‘2’ 12.6:0.9 l.1:0.05 20:1.5 15:6 6:1 2,200:200 816-01-1‘2’ 17.4:1 .2 1.0:0.05 25:1.5 17:7 9:1 2,500:180 816-01-2‘2’ 17.4:1.2 0.9:0.05 25:1.5 14:6 8:1 2,250:60 0814-01‘” 0 0.58:0.05 11. 15:0.5 0.35:0.17 1.03:0.13 647:60 0814-02‘2’ O 0.71:0.05 11.51:0.5 0.53:0.24 1.35:0.15 817:70 0814034” 0 0.84:0.05 l 1.52:0.5 0.74:0.31 1.59:0.17 968:70 0814-03-2‘2’ 0 0.68:0.05 12.80:0.5 0.48:0.22 1.58:0.18 870:70 0814-03-3‘2’ 0 0.60:0.05 l 1.82:0.5 0.38:0.18 1.20:0.15 709:70 "’Air—Water “Kerosene-Water 150 CL. .buagcofiov—6u8ugcome3n $53382- «V .3552 28m n> 83:5: 803 a; 293m Om 5.9 _ d .dldd‘ q IG‘GIIJ I G ICII'Id I J Ilflddd I 1 I did. ...... tum... 05825 .. . . . O ens—m .. 000 ... Euro—Em ..1/ v.30 282% c— 03 m8. 151 In Figure 5.13, Eq.(5.18) is superimposed upon the results. The air/water experiments and the kerosene/water experiments show good agreement with the predictions based on the analogy. The kerosene/water inertial experiments are just outside the region predicted by the analogy. To compare the results with that of other researchers, Figure 5 .14 shows data reported by Peebles and Garber (1953) and Habner (1953) for air bubbles rising various stagnant liquids. Those experiments were conducted in inertial frames, (i.e., aF=0). No approximations of error are plotted since none were reported. In Figure 5.10 all the experimental results fall within the range proposed by Mersmann except those of the unstable configuration of the accelerated kerosene/water experiments. This indicates that the experimental terminal velocity is lower than that predicted by the Mersmann correlation for the given dimensionless drop diameter. This may be attributed to energy losses in the constant deformation of the drop and thereby lowering the translational velocity of the drop. The wave analogy illustrated in Figure 5.14 does show good agreement with inertial kerosene/water experiments and the accelerated air/water experiments but predicts a lower Weber number for the accelerated kerosene/water experiments. It has been proposed by Marrucci et al. (1970) and Meneri (1995) in their application of the wave analogy, that the density of the dispersed phase (bubbles or drops) should be ignored when calculating the particle Weber number. They reasoned that the dispersed phase does not contribute to the inertial forces, in contrast with the original classical wave problem developed in Chapter 3 (see Eq.(3.130). Using this logic, the particle Weber number described by Eq.(5.16) is modified such that A_nacaaaeasfiuéguises—-383}? 3 8:52 28 .2, 8:52 .303 2 .m 2:5 ...Om 2: 2 _ ..o 8... 152 JI-Idl d I I .ddd‘u q 1 G CI‘GI‘ d d d dd. 111114 4 1 7.81.»? . m 9: . Anna—.8980 Ea guinea—-0 u _n%.§a3\o=383—-0 “buagxocooeov—‘cofizuz- $ .352 25m .9 .352 .303 z .n 2:5 ...om 82 R: 2 _ 3 8.: 1...... . 1.1.... r ...... . . ....... . . 14.4.. —.o o M I 3 U O— ...o? ~+iomfl103 U m8— U :8— 154 We 5 , (5.19) where pd is the dispersed phase density. Eq.(5.18) becomes We" = Bo. +1 . (5.20) Figure 5.15 is a plot of the experimental results along with the data of Peebles and Garber with the modified Weber number. Although this ad hoc assumption shows better agreement with that of the inertial frame kerosene/water experiments, significant differences are noted for the accelerated kerosene/water experiment Note that there was essentially no change in the air/water experiment. This is because for the air/water system, pL << pH and therefore We" 5 We'. The accelerated kerosene/water results show lower than predicted values of We". This is consistent with that of the Mersmann prediction. The implication is that the unstable shape of the accelerating kerosene/water experiments may contribute to the poor prediction of the wave analogy. The wave analogy is only an ad hoc empirical means to correlate experimental data. What is implied from the development of the linearized theory is that the inertial, viscous, interfacial, and acceleration forces are all in quasi-equilibrium at the interface. These forces are also in equilibrium for the case of the fluid particle moving through the continuous fluid phase. For a marginally stable system the interface oscillates in a direction parallel to the direction of the acceleration. The propagation wave velocity may be viewed as the velocity of a disturbance, of a particular wave length, along the interface of two immiscible fluids. Recall that this velocity does not include any nonlinear effects and may be viewed as the 155 limiting velocity for which a disturbance may propagate along the interface. If the bubble or drop is viewed as a disturbance, its terminal velocity may be only limited by the wave analogy. 5.6 Conclusions The terminal velocity of fluid particles in an accelerating fiame was investigated. Experiments were conducted on systems of water and kerosene dispersed in a continuous phase of water. The experimental apparatus was built to videograph the fluid particles as they moved through the continuous phase. The fluid particles were observed to have two stable shapes, spherical and bubble cap, and a continuously oscillating non-stable shape. The spherical shape was observed for the kerosene/water system in an inertial frame. The bubble cap shape was observed for the accelerated air/water system. The unstable shape was observed for the accelerated kerosene/water system. The experimental results for the stable shapes fit between the limits of the correlation of Mersmann(l983) modified to account for the accelerated frame. The experimental results for the unstable kerosene/water drops fell outside of the limits, having terminal velocities lower than predicted by the correlation. This was attributed to the constant changing of shape of the drop thereby lowering its transitional energy. The wave analogy, modified to include the acceleration effects, also showed good agreement with the experimental results for the accelerated fiame experiments but predicted lower terminal velocities for the inertial frame kerosene/water experiments. The wave analogy was modified by neglecting the density of the dispersed phase. As was suggested by Marrucci et al.(l970) and Meneri(l995), this approach was found to agree well with 156 stable fluid particle experiments. The modified wave analogy predicted higher terminal velocities for the unstable kerosene/water drops similar to the Mersmann correlation. Although the author knows of no physical justification for the wave analogy, the modified version, which has been extended in this work to include the effect of acceleration, was shown to provide a good correlation of the terminal velocity for stable fluid/fluid particles. For unstable fluid particle shapes, however, both the terminal velocity predictions of Mersmann and the modified wave analogy yield terminal velocity predictions that were too high. CHAPTER 6: CONCLUSIONS The terminal velocity of fluid particles in accelerating fields was investigated in this work. The investigation was divided into three major areas: a linear stability analysis of a flat interface between two immiscible, viscous fluids; an experimental investigation of the growth rate of disturbances on a flat interface between two immiscible fluids in a constant accelerating environment; and, an exploratory experimental investigation of the terminal velocity of dispersed fluid particles in a constant accelerating field. The results obtained confirmed some previous findings obtained by other researchers and expanded the experimental/theoretical knowledge base related to Rayleigh-Taylor instability. The interface between two immiscible inviscid fluids was shown to be unstable to arbitrary infinitesimal disturbances if (p2 -— pl)(a,. — g) < 0, where p2 is the density of the fluid layer situated above another fluid of density p I. A stable interface is obtained for (p2 — p,)(aF — g) > O. Physically, a light fluid over a heavy fluid is unstable if the acceleration of the interface satisfies the inequality aF> g and stable if aF< g. A heavy fluid over a light fluid is unstable for aFg. These results have been understood for well over a century and represent the most elemental predictions of the general class of problems know as Rayleigh-Taylor instability. 157 158 This work, which complements the earlier results of Reid(l96l),investigated the stability of commonly encountered fluids for which vH at VL. The earlier theoretical work of Reid (1961) and of Chandrasekhar (1961) was restricted to vH = vL. Apparently more recent generalizations of the classical Rayleigh-Taylor problem have continued with this earlier simplification (see, esp., Kull, 1991). The influence of vH / vL $1 on the dynamic behavior of stable and unstable interfaces was quantified by calculating the eigenvalues associated with the temporal behavior of the interface. The property ratio vH / vL has a significant quantitative and qualitative effect on the wave numbers associated with the rotational component of the eigenfunctions normal to the fluid/fluid interface. This discovery associated with the classical Rayleigh-Taylor problem is new with this research. The mathematical formulation of the problem as a superposition of rotational and irrotational disturbances (see Eq.(3.20)) allowed the nonlinear dispersion equation to be rewritten as a multiplicative product between a complex viscous modulus and an inviscid spectrum (see Eq.(3.110)). This technique permitted a convenient evaluation of the viscous influence on the eigenvalues for stable and unstable interfaces. The complex growth rates were examined relative to the viscous scaling, X = s/(v,k2), and relative to an inviscid scaling, 4), E s/sc with sc defined by Eq.(3.100). The dimensionless growth rate, either X or (1),, depends on five dimensionless groups: a, the stability group; or", the fractional density ratio of the heavy phase; B", the functional viscosity ratio of the heavy phase; M0, the Morton number; and, the dimensionless wave number (1),. The dimensionless rotational complex wave numbers Yl (2 ml / k) and Y2 (5 m2 / k) also depend on these five groups. The dimensionless growth rates of disturbances entering the interface of two accelerating immiscible fluids was experimentally investigated. Experiments were conducted on three fluid-fluid systems: air-water, kerosene-water and Freon-water. The 159 growth rates were analyzed by decomposing the disturbance into its harmonic components by taking the Fourier transform and then monitoring the growth rate of individual modal components. This allowed multiple wave number disturbances to be analyzed during each experiment. The experimental apparatus was built in order to videograph the interface between two immiscible fluids while being accelerated. The dimensions of the container were chosen to agree with experiments conducted by Lewis(1950) and Emmons et al. (1962). The combination of the physical dimensions of the container and discrete Fourier transform operation, however, limited the investigation of the wavenumbers above the critical wave number to the accelerating kerosene/water experiment set. The dimensionless growth rates were found to be consistently lower than that predicted by linear stability theory. The observed experimental results followed the trend anticipated by linear stability theory inasmuch as a maximum dimensionless growth rate occurs near ¢2=0.6 and approaches zero for ¢2=0 and ¢2=l. The terminal velocity of fluid particles in an accelerating frame was investigated. Experiments were conducted on systems of air and kerosene dispersed in a continuous phase of water. The experimental apparatus was built to videograph the fluid particles as each moved through the continuous phase. The fluid particles were observed to have two stable shapes: spherical and bubble cap. Continuously oscillating non-stable shapes were also observed. The spherical shape was obtained for the kerosene/water system in the inertial fiame experiments. The bubble cap shape was observed for the accelerated air/water system. The unstable shape was observed for the accelerated kerosene/water system. The experimental results for the stable shapes fit between the limits of the correlation of Mersmann(1983) modified to account for the accelerated frame, while the experimental results for the unstable kerosene/water drops fell outside of the limits, having terminal velocities lower than predicted by the correlation. 160 A modified analogy to progressive waves supported by flat interfaces was used to correlate the experimental results for the accelerated frame experiments. This approach, however, predicted lower terminal velocities for the inertial frame kerosene/water experiments. The wave analogy was modified by neglecting the density of the dispersed phase as recommended by Marrucci et al.(1970) and by Meneri(l995). This modification yielded terminal velocity predictions that agreed well with stable fluid particle experiments. The modified wave analogy also predicted higher terminal velocities for the unstable kerosene/water drops similar to that of the Mersmann correlation. The modified version of the wave analogy has been extended in this work to include the effect of acceleration and was shown to predict the terminal velocity for stable fluid/fluid particles. For the unstable fluid particle shapes, however, the terminal velocity predictions of the Mersmann correlation and the modified wave analogy yielded temrinal velocity predictions that where too high. Summary Observations 1. Inviscid Theory: Stable Interfaces. The condition (p2 - p,)(aF - g) > O is sufficient for the stability of the interface between two fluids. For (p2 - p,)(aF - g) < 0, there exists a marginal wavenumber l:M (Eq.(3.73)) for which interfacial tension stabilizes the interface for disturbances with k Z k”. . This marginal wavenumber is independent of viscosity and is a direct result of the increasing normal force due to interfacial tension with increasing curvature of the interface. For stable inviscid interfaces, the eigenvalues are pure imaginary numbers. The stable disturbance at an inviscid interface is therefore represented by a standing wave. In the absence of viscosity, the velocity field is irrotational. 161 2. Inviscid Theory: Unstable Interfaces. The condition (oz-p,)(aF—g)<0 is necessary for instability. For an inviscid interface satisfying the above condition, disturbances with wavenumber < kM are unstable yielding eigenvalues that are real numbers and represent an exponentially growing disturbance with no oscillatory behavior. Moreover, unlike when interfacial tension is absent, there exists a mode of maximum instability for which the amplitude of the disturbance grows fastest. 3. Viscous Theory: Marginal Stability. The marginal state for viscous fluids represents a stationary state, i.e. the real part of the growth rate equals zero. The marginal wave number that determines this state is identical to that for the inviscid result. This shows that viscosity plays no role in determining the marginal wavenumber. 4. Viscous Theory: Unstable Interfaces. Viscosity retards the growth rate of an otherwise unstable interface with its effectiveness being greatest in the region of maximum growth rate. The effect is small for dimensionless wave numbers where the curvature is low (i.e. near ¢2=0) and the velocity gradients are small. The viscosity is also unimportant where interfacial tension dominates (i.e., ¢2=1). This effect is also a function of the acceleration parameter 7, increasing significantly as 7 increases several orders of magnitude. Although the impact of viscosity is small for the three systems investigated it is anticipated to be much larger for systems in which the two phase viscosity length scale (see Eq.(3.96)) is much larger. 162 The spatial decay of the velocity disturbance normal to the interface is strongly influenced by viscosity. Its magnitude, in general, depends on the side of the interface for fluids with different viscosities. For systems in which the two fluids had different kinematic viscosities, the depth of the vorticity sublayer was greatest in those fluids with the higher kinematic viscosity. This spatial nonuniformity of vorticity near the interface may provide an initial state which triggers nonlinear phenomena associated with the entrainment of one fluid by another. 5. Viscous Theory: Stable Interfaces. Unlike the case with the unstable interface, the dimensionless spatial decay of the vorticity is nearly equal on either side of the interface. For systems of fluids with different kinematic viscosities, it was observed that the viscous sublayer was smaller for the fluid with the higher kinematic viscosity. This observation was just the opposite of that described above for the unstable case. For the three systems studied the viscous effects are only apparent for very small wave lengths, and the progressive wave velocity is well approximated by the inviscid theory. Apparently the cause for this is the relatively low value of M0 for each of the systems studied. It is anticipated that for fluids with higher values of Mo, viscosity can, in principle, completely suppress the progressive wave velocity for small wavelength disturbances by completely dampening the oscillatory behavior of the interface. 6. Experimental Growth Rates. In general the dimensionless growth rates of the experimental fluid/fluid pairs were lower than those predicted by the linear stability theory. However, the dimensionless growth rates followed the same trend as suggested by the theory: rising initially and, after going through a maximum for «p, = 0.6 decreasing to zero as d), —9 1. For the kerosene-water system, complete stability was observed for 4), =1, a phenomena validated by this research for the first time. 163 7. Terminal Velocity of Fluid Particles in Accelerating Frames. The empirical correlations developed by Mersmann(l983) when modified to include the effect of an accelerating frame predicted the limits of the terminal velocity for stable drops and for bubbles. The unstable drops fell outside the bounds of the Mersmann correlations. The wave analogy expanded to include the effects of accelerating frames predicts the terminal velocity of fluid particles when modified to neglect the dispersed fluid density. The analogy predicts a terminal velocity too high for unstable shapes. CHAPTER 7: RECOMMENDATIONS FOR FURTHER RESEARCH Theoretical and experimental findings of this research have raised many questions regarding the stability of accelerating interfaces and the terminal velocity of fluid particles in accelerating frames. Additional experiments outlined below are proposed to answer some of the question raised by this research. New experiments are also suggested which may provide additional understanding of the relationship between progressive waves supported by flat interfaces and the terminal velocity of drops and bubbles. Experiments to test stability for wave lengths greater than kc The low resolution of the digital video equipment and the poor lighting techniques employed in this research limited the smallest wave number increment of the projected interface to approximately 1.35 cm". As a result, this restricted access to the critical wave number for a given experiment. Digital equipment with high resolution will allow a finer division of wavenumber space and hence allow the investigation of growth rates associated 164 165 with higher wave numbers. Special attention should be given to the lighting of the apparatus. During the experimental analysis of this work it became clear that the lighting arrangements could have been improved. Additional care should be taken to eliminate glare on the video images, which lowers the number of fiames available for analysis. The use of a polarized lens should also be considered. Viscous eflects near the interface The linearized theory suggests that the spatial dampening of the rotational component of the velocity to be fundamentally different for the stable and unstable cases. This should be investigated experimentally. This may prove to be very difficult to do since it will require velocity measurements to be measure over very small distances while the interfaces are being accelerated. Experiments to test the limitations of the wave equations Inspection of the terminal velocity prediction using the inviscid wave analogy equation implies that very high terminal velocities occur for very small fluid particles. This is obviously not true for small particles at low Reynolds number, i.e., the Stokes’ law regime. The limitation of the inviscid wave analogy should be investigated by measuring the terminal velocity of smaller radius drops. These results can then be compared to that predicted by Stokes’ Law. The experiments should identify the Reynolds number lower limit bounds of the utility of the wave analogy. APPENDIX A: INTERFACIAL FORCE BALANCE APPENDIX A: INTERFACIAL FORCE BALANCE A.1 Background In order to formally express the force at the interface, it is necessary to consider the nature of the forces which act at the interface. According to a simple interface description which involves only interfacial tension, these consist of two kinds: First, there are the bulk pressure and stresses which act on the faces of the interfacial element and produce a net effect that is proportional to the surface area; and second there is a force due to the interfacial tension which acts in the plane of the interface at the edges of the surface element and is specified via the magnitude of the interfacial tension as a force per unit length. If it is assumed that these are the only forces acting in the interface, a pointwise form for the force equilibrium condition can be derived. Prior to doing this it is useful to recall the physical origin of interfacial tension. From a thermodynamic point of view, the interfacial tension is introduced as a measure of the free energy per unit surface area. Thus, an increase in the area of the interface requires 166 167 an increase in the fiee energy of the system. The necessity of doing work to create new interfacial area is a consequence of the fact that the molecules in the immediate vicinity of the interface experience a net force which tends to pull them back into the bulk liquid. In a macroscopic or continuum theory, a way to include the required rate of work for changes in the interfacial area is to assume the existence of a force per unit length which acts on the edges of an interfacial element. The magnitude of this force is o, the interfacial tension. A.2 Mathematical Formulation The force due to interfacial tension, which acts in the plane of the interface at the edges of the surface element, is specified via the magnitude of the interfacial tension as a force per unit length. To express the force equilibrium condition in a mathematical form, a force balance is considered on an arbitrary surface element of a fluid interface, which is denoted as A. A sketch of this surface element is shown in Figure A.1 The unit normal to the interface at any point in A is denoted as n, such that n is positive when pointing outward into fluid 2. Let t be a unit vector that is tangent to the boundary curve C at each point. With this convention, the interfacial tension force at the point can be described as 012. The force equilibrium condition applied to surface element A requires that: H(l“’—l‘2’)-ndA+Iobdl=0 (A.1) C A Although this expression is a satisfactory statement of the force balance at an interface, it is not particularly useful in this form because it is an overall balance on a macroscopic element of the interface. In order to be useful with the 168 Fluid 2 Interfacial Surface Element A Figure A.l Arbituary Surface Element A 169 differential Navier-Stokes equation which applies in the two contiguous fluid phases, a point wise condition on the surface forces is needed. For this purpose, it is necessary to convert the line integral to a surface integral on A. It can be seen fiorn Figure A.1 that b may be represented as the vector product between t and 11. Therefore japd1=jagxgdr (A.2) C C or, equivalently, jagdr=-jr-[ag-g]dr . (A.3) C C Eq.(A.3) expresses the vector product in terms of the permutation triadic g . A generalization of Stokes’ theorem for dyadics (see page 66, Morse and Feshbauch, 1953) gives the following result It-[on-g]dlsHg-[VA(0g-g)]dA . (A.4) C A Equating Eq.(A.3) to Eq.(A.4) and expanding the vector operation on the right-hand side of Eq.(A.4) yields the following representation for the line integral 10§d1=1 [V(O'n) - _q - 9V - (09)]dA (A.5) 170 The right-hand side of Eq.(A.5) can be further simplified using the identities (V9) - n s Q and n-gsl,thus: 12 V0 , (A.6) and II E O'QV - g , (A.7) inasmuch as V0 - I! a 0. Substituting Eq.(A.6) and (A.7) into (A5) and in turn back into Eq.(A.l), it follows that: 11111“) 'l(l)1‘fl+V°-Gn(V-a)]dA= Q - (A.8) Because A is an arbitrary surface element in the interface, the integrand must be zero everywhere on the surface, i.e., (3" — mg + (Vo) - an(V ~11) = _Q . (A9) at each point on the interface. Eq.(A.9) is the differential stress balance reported by Lea] (1992). APPENDIX B: ROTATIONAL WAVENUMBERS APPENDIX B: ROTATIONAL WAVENUMBERS The rotational wavenumber is defined by Eq.(3.25), m2 = k2 + (B-l) 9— . v In general, m and s are complex numbers. Therefore, with m=mR+im, , (B-2) and s=sR +isl , (8.3) It follows from Eq.(B.1) that 171 172 m; -m,2 =1.2 +S—R (13.4) and sr 2m,mR = 17 . (B5) The above two equations can be solved for Y a mR /k and YI a rnI I k with the result that 1+x : 1+x 2+x2 Y:=( R) \/(2 R) 1 (B6) and 1 X R where XR a 5R /(vk2) and XI a s1 /(Vk2) . The physical condition on the disturbance for |z| —-> oo requires YR > O for the light and heavy phases. Because \[(1+x,,)’ +xf 2(1+x,,) , (8.8) it follows from (3.6) that for any combination of (XR, X!) the value of YR is given by 173 1 x 1 x 2 x2 Y =+\/(+ ”HR; R) + ' . (13.9) R Thus, YR > 0 always exists, the implication by Chandrasekhar to the contrary notwithstanding. APPENDIX C: INTERFACIAL BOUNDARY CONDITION APPENDIX C: INTERFACIAL BOUNDARY CONDITION The normal component of the interfacial force balance (see Eq.(3.l)) reduces to the result expressed by Eq.(3.55): an (a -g)z -p +2 ) (pl F I d M 31 Fluidl@z, (C.1) all z 'Ovrzrzr '1' (p2 (aF '— g)zr _ Pd '1' 2’12 I) - Fluidzaz. The z—component of the velocity within each fluid is given by Eq.(3 .42) as [afk’ exp(+kz) + a?) exp(+mzz)]exp(st) Fk (x, y), z < 21 u"" = (C2) 2 [agk’ exp(-kz) + a?) exp(-m,z)]exp(st) Fk(x,y), z > 21 . 174 175 Differentiating Eq.(C.2) with respect to z yeilds [kafk’ exp(+kz) + mza‘z") eXp(+mzz)]exp(st) Fk(x, y), z < zl . (C3) [—kagk’ exp(—kz) + —m,a‘4'° exp(-—m,z)]exp(st) Fk(x, y), z > 21 a1“) az _ The kth mode of the interfacial position zI can be related to kth mode of the velocity uz by rearranging Eq.(3.56): (k) (k) ll 11 22k) = [_) = {—J . (C.4) S Fluidl® 2. S Fluid2®zl Substituting Eq.(C.2) into (C.4) yeilds '(agk) +3310) ' -———exp(st) Fk (x, y), Flurd 2 2"") =< (C5) (.9) ma“) . ———exp(st)Fk(x, y), Flurdl . s L Eq.(C.5) together with the disturbance pressure given by Eq.(3.44), Eq.(C.3) for the velocity gradient given by Eq.(C.3), and the Laplacian of the planform function, Fk(x,y), can be used to rewite Eq.(C.1). It follows that 176 )(agk’ + am) pr(ar= ‘ g s m m _E_ls agk)+ZS/J1(- ka3 -1ma4) k s (1:) +3“) +a(k) +a(k) = +0181 2 ‘ ) (C6) 5 (al 0:) + a9”) + P2 (8; - g)——- kaflr) +m atzk) +-- p__s_2 a“) +2srrz( 2 ) . k s as well as a§"’ +890: a“) +a"" +a("’ +3210, (C7) 2 and also ago + 2190:2100 + a?) :agk) + a?) (C8) A similar rearrangement can also be made on Eq.(3 .49) yielding kaflt) + “12390— _ 1‘31” + m2a(2k)— kagk) _ 11123:“ (cg) 2 177 and likewise (k) (k) (k) (k) (k, m _ kal + mza2 - ka3 - mza Substituting the relationships expressed by Eqs.(C.7) - (C.lO) into Eq.(C.6), gives Eq.(3.57) in Chapter 3. rem-"’7‘ APPENDIX D: COMPUTER PROGRAM AND RESULTS FOR THE EIGENVALUES OF RAYLEIGH-TAYLOR VISCOUS INTERFACES APPENDIX D: COMPUTER PROGRAM AND RESULTS FOR THE EIGENVALUES OF RAYLEIGH-TAYLOR VISCOUS INTERFACES Figure 3.5 illustrates the solution methodology applied to Eq.(3.90), the nonlinear dispersion equation for the eigenvalues associated with the temporal behavior of the interface. Mathematica® was employed as a means to find the complex conjugate roots to Eq.(3.90) for a specified set of physical properties which determine the five independent groups: a, on", BH, M0, and $2- Table D.1 compares the program notation with the nomenclature used in Chapter 3. Table D2 gives the listing of the Mathematica" algorithm, which was implemented on an Apple Macintosh" IIci. The physical property data were read into the program using the module function. The routine FindRoot is performed to find the complex conjugate roots of Eq.(3.90): 178 179 F(X)=O . (D.1) The default initial guess is used for all calculations, i.e., Xo=1+i . (13.2) For a given physical example, the program finds XR and XI for a range of wave numbers. Figure D.3 is an example output from the program. The result file was converted into a text file using Microsoft Word”. This allowed all of the headers to be removed and the real and imaginary parts of the eigenvalue to be separated. The tabulated results are copied and pasted into a spreadsheet for postprocessing. Tables D.4 - D.9 give the results of the calculations for air/water, kerosene/water, and water/freon interfaces. 180 Table D.1 Program Notation Program Chapter 3 Description a a see Eq.(3.87) af a, Frame acceleration, cm/s2 alpha] (1, Density ratio of Fluid 1 alpha2 0‘2 Density ratio of Fluid 2 b 1) see Eq.(3.88) betal B. Viscosity ratio ofFluidl beta2 B. Viscosity ratio ofFluid2 g g gravitational acceleration, cmls2 i - Iteration counter k k wavenumber, cm" km k“ Marginal wavenumber, cm" mul 11: Viscosity of Fluid 1, poise mu2 liq Viscosity of Fluid 2, poise phi2 92 Dimensionless wavenumber rhol Pi Density of Fluid 1, grrrlcm3 rh02 Pz Density of Fluid 2, gin/cm3 s 8 Growth rate, 5" sigma 0 Interfacial tension, dynes/cm yl Y, see Eq.(3.81) y2 Y2 see Eq.(3.82) 1 8 1 Table D2 Program Listing: Eigenvalues for Rayleigh-Taylor Interfaces oqnstph12_J x_]:- Modulel {mul-0.01,mu2-0.00018,rh01-1.,rhoZ-0.0013,81gna-74,af= 1078,9-980., km,k,alphal,alpha2,bata1,bata2,a,b,y1,y2,part1,part2, part3,£), Ian I (-(rh02-rhol) (af-g) / sigma “0.5; k I phiz km; alphal . rhol / (rhol+rh02); alpha2 a 1-a1pha1; batal - mul / (mul-tuna); botaz - 1-bata1; a I (-(rh02-rh01) (at-g) - sigma k‘2)/(9 (rho1+rh02)); b I (rhol+rh02) / (nu1+nu2) (alk‘3)“0.5; y1 = (1 + alphallbctal x)‘0.5; y2 :- (1 + alpha2/bata2 x)*0.5; part-.1 - (betaz -bata1) (y1-1) + (alphal x) ; part-.2 a ( bataz -bata1) (yz - 1) - (alpha2 x) ; part3 - (y2 -1) alphal + (y1- 1) alpha2 ; fa-abcz + :02 - 4 (part1 part2 I part3) I solutionSat[waveN_, iteration_]:- Module [ (t, xx) , Do I s :- rindRoot [aqua [wavaNlfi [i] ] , xx] --0, (xx, 1}] ; ta OpanAppandmtz-ingaoinl"growth" , ToString [rate] ] ] ; WriteString[t, 'xt" , ToString [i] , "J a "] ; Writelt, s] ; Closelt], {1, iteration} I] 182 Table D2 (Cont) wach-{0.0001,0.001, 0.01,0.02.0.03,0.04,0.05.0.06,0.07,0.08,0.09,0.1, 0.11,0.12,0.13.0.14,0.15,0.16,0.17,0.18,0.19,0.2, 0.21.0.22,0.23.0.24,0.25,0.26,0.27,0.28,0.29,0.3, 0.31,0.32,0.33,0.34,0.35,0.36,0.37,0.38,0.39,0.4, 0.41,0.42,0.43,0.44,0.45,0.46,0.47,0.48,0.49,0.5, 0.51,0.52.0.53.0.54,0.55,0.56,0.57,0.58,0.59.0.6, 0.61,0.62,0.63,0.64,0.65,0.66,0.67,0.68,0.69,0.7, O.71,0.72,0.73,0.74,0.75,0.76,0.77,0.78,0.79,0.8, 0.81,0.82,0.83,0.84,0.85,0.86,0.87,0.88,0.89,0.9, 0.91.0.92,0.93,0.94,0.95,0.96,0.97,0.98,0.99, 0.999,0.9999} solution80t[wavan,103] l 83 Table D.3 Example Output of F indRoot Algorithm 8(1) I {xxSZ -> -0.00349463538318433827 + 0.7037743880855087366*I} 3(2] I {xx$2 -> -0.006061524503227770982 + 0.701495897083748201*I} 8(3] I (“$2 -> -0.008430966182289182773 + 0.699490521005080408*I} 8(4] I (”$2 -> -0.01070249118884186656 «r- 0.6976401324202754035*I} 8(5] I {16:32 -> -0.01291601318086407014 + 0.6958938467350702269*I} 8(6] I (“$2 -> -0.01509207246054889939 + 0.6942235983042916933*I} 8(7) I {xxSZ -> -0.0172427073584238629 «r- 0.6926117477645162835*I} 8(8] I (“$2 -> -0.01937556483173214622 «r- 0.6910462576001811051*I} 8(9] I {xx$2 -> -0.02149577341450720379 + 0.6895184316657899311*I) 8(10] I {xx$2 -> -0.02360690732111857786 +- 0.6880217193188314462*I} 8(11] I {xxsz -> -0.04455591653497065918 + 0.6739987319866751589*I} 8(12] I {amSZ -> -0.06550482929708137603 + 0.6605916747682567071*I} x(13] I (”$2 -> -0.08645974640443807393 + 0.6471523066536550167*I} x(14] I {xxSZ -> -O.1073238779702685318 0.6334354567713358236*I} 8(15] I (“$2 -> -0.1280067629547486844 + 0.6193350213383856172*I} 8(16] I {xx$2 -> -O.1484425835055486155 0.6048064761417052929*I} 8(17] I (”$2 -> -0.1685931460317796635 0.5898338132484939606*I} x(18] I {xxSZ -> -O.1884471901807635692 + 0.5744113587496411013*I} 3(19] I {xx$2 -> -0.2080184267853502935 + 0.5585315499285860586*I} 3(20] I {xxsz -> -O.227342792895286087 «r- 0.5421754011582650367*I) 8(21] I (“$2 -> -0.2464750092690362077 + 0.5253043211485464525*I} 8(22] I {xx$2 -> -0.2654845200058967314 + 0.5078526532508033054*I} x(23] I {xxSZ -> -0.2844509968490219433 0.4897204860012332124‘11 3(24] I (”$2 -> -0.3034596959014123036 0.4707661341685978844*I} 8(25] I {xxSZ -> -0.3225970103471980295 + 0.4507972658293782861*I} + + + + q. 184 Table D.3 (Cont) 8(26] I {xx$2 -> -O.341946548427666663 «r- 0.4295588776059582545*I} 8(27] I {xxSZ -> -0.3615859870534184658 0.4067149344838144538*I} '1- 8(28] I (”$2 -> -0.3815848333450570127 + O.3818177765272355611*I} 8(29] I {xxSZ -> -0.4020031020455771201 0.3542534263454238917*I} x[30] I {xxSZ -> -0.422890814304114074 + 0.3231360718803673384*I} x(31] I {xxSZ -> -0.4442881584485344014 0.2870821006774124252*I} 3(32] I {xxSZ -> -O.4662261281186867886 0.243641857151881704*I} 8(33] I (10:52 -> -0.4887274598784531492 0.187412581306428348*I} 8(34] I (”$2 -> -0.5118077193441183116 0.09786210220816936146*I} 8(35] I {xxSZ -> -O.4035694004145670024 6.520632775712645017*10*-20*I} ... ... ... ... 185 Table D4 Eigenvalues for Unstable Air/Water Interfaces y = 0.1 11, ab’ x Y1 Y2 M o, 9°1 0.1 6.15E+08 2.48E+04 158.809 42.690 -r.7015+05 0.315 0.315 0.2 7.46E+07 8.63E+03 93.700 25.199 -4.86E+04 0.438 0.438 0.3 2.101-:+07 4.57E+03 68.208 18.356 -2.351-:+04 0.522 0.522 0.4 8.16E+06 2.85E+03 53.877 14.511 -1.38E+04 0.579 0.580 0.5 3.73E+06 1.93E+03 44.297 11.943 -8.96E+03 0.612 0.612 0.6 1.84E+06 1.35E+03 37.129 10.025 -6.09E+O3 0.619 0.620 0.7 9.24E+05 9.59E+02 31.245 8.452 -4.l9E+03 0.596 0.597 0.8 4.37E+05 6.59E+02 25.904 7.028 -2.80E+O3 0.535 0.537 0.9 1.6ZE+05 4.00E+02 20.203 5.514 -l.64E+03 0.411 0.414 7:] .0 o, ab’ x Y1 Y2 M 6, 11". 0.1 1.95E+08 1.39E+04 119.087 32.018 -853F.+04 0.315 0.315 0.2 2.36E+07 4.85E+03 70.261 18.907 2.511304 0.438 0.438 0.3 6.63E+06 2.57E+03 51.143 13.778 -1.2315+04 0.522 0.522 0.4 2.58E+06 1.60E+03 40.396 10.899 -7.321~:+03 0.579 0.580 0.5 1.18E+06 1.08E+03 33.211 8.977 -4.781~:+03 0.611 0.612 0.6 5.83E+05 7.61E+02 27.835 7.542 -3.26E+03 0.618 0.620 0.7 2.92£+05 5.39E+02 23.421 6.368 -2.25E+03 0.595 0.597 0.8 1.38E+05 3.70E+02 19.415 5.306 -1.51E+03 0.534 0.537 0.9 5.121~:+04 2.24E+02 15.137 4.181 -8.85E+02 0.410 0.414 186 Table D.4 (Cont) F10 62 ab’ x Y1 Y2 M 4;. 9°. 0.1 6.15E+07 7 .84E+03 89.299 24.018 -4.34E+04 0.315 0.315 0.2 7.46E+06 2.73E+03 52.683 14.191 -1.31E+04 0.438 0.438 0.3 2.10E+06 1.45E+03 38.346 10.350 -6.53E+03 0.522 0.522 0.4 8.16E+05 9.01E+02 30.285 8.196 -3 .91E+03 0.578 0.580 0.5 3.73E+05 6.09E+02 24.896 6.760 -2.57E+03 0.610 0.612 0.6 1.84E+05 4.27E+02 20.863 5.689 -1.76E+03 0.617 0.620 0.7 9.24E+04 3.02E+02 17.552 4.814 -1.21E+03 0.594 0.597 0.8 4.37E+04 2.07E+02 14.546 4.026 -8.13E+02 0.532 0.537 0.9 1.62E+04 1.25E+02 1 1.336 3.195 -4.77E+02 0.407 0.414 3:100 42 ab2 X Y1 Y2 M 91 9°: 0.1 1.9sra+07 4.4rr~:+03 66.961 18.021 -2.2sra+04 0.314 0.315 0.2 2.36B+06 1.53E+03 39.500 10.659 -6.97E+03 0.438 0.438 0.3 6.63E+05 8.12E+02 28.747 7.785 -3.50E+03 0.521 0.522 0.4 2.58E+05 5.06E+02 22.701 6.176 -2.11E+03 0.577 0.580 0.5 l.lSE+05 3.4lE+02 18.658 5.106 -l.3SE+03 0.609 0.612 0.6 5.83E+04 2.39E+02 15.632 4.310 -9.4SE+02 0.615 0.620 0.7 2.92E+04 1.69E+02 13.148 3.662 -6.55E+02 0.591 0.597 0.8 1.38E+04 1.1615402 10.892 3.081 -4.37E+02 0.528 0.537 0.9 5.12E+03 6.98E+01 8.482 2.475 -2.55E+02 0.403 0.414 y=1000 0, ab2 X Y1 Y2 M 0, 0°, 0.1 6.15E+06 2.48E+03 50.208 13.527 -l.18E+04 0.314 0.315 0.2 7.46E+05 8.62E+02 29.613 8.016 -3.73E+03 0.437 0.438 0.3 2.10E+05 4.56E+02 21.547 5.870 -l.88E+03 0.520 0.522 0.4 8.16E+04 2.84E+02 17.01 1 4.672 -1. 14E+03 0.576 0.580 0.5 3.73E+04 1.91E+02 13.978 3.878 -7.47E+02 0.606 0.612 0.6 1.84E+04 1.34E+02 1 1.707 3.290 -5.1 1E+02 0.61 1 0.620 0.7 9.24E+03 9.4313+01 9.842 2.815 -3.52£+02 0.586 0.597 0.8 4.37E+03 6.43E+01 8.148 2.392 -2.34E+02 0.522 0.537 0.9 1.62E+03 3.85E+01 6.339 1.957 -1.35E+02 0.396 0.414 187 Table D5 Eigenvalues for Unstable Kerosene/Water Interfaces y: 0.1 ¢, ab’ x Y1 Y2 M 6, 4°. 0.1 1.72908 1.30904 130.105 101.379 -1.49B+06 0.313 0.315 0.2 2.08907 4.53903 76.655 59.732 -3.06F.+05 0.435 0.438 0.3 5.85906 2.39903 55.730 43.429 -1.18E+05 0.517 0.522 0.4 2.28906 1.491903 43.967 34.265 -5.85E+04 0.572 0.580 0.5 1.04E+06 1.00903 36.105 28.140 325904 0.603 0.612 0.6 5.14905 7.03902 30.224 23.559 -1.92904 0.608 0.620 0.7 2.58905 4.96902 25.397 19.799 115904 0.584 0.597 0.8 1.22905 3.40902 21.017 16.388 -6.56E+03 0.522 0.537 0.9 4.52904 2.05902 16.344 12.751 3.131903 0.399 0.414 y=l.0 o, ab: x I Y1 Y2 M 6. 9°. 0.1 5.28908 7.22904 | 306.127 238.533 193907 0.989 0.315 0.2 6.40907 2.50904 | 180.260 140.459 -3.95E+06 1.372 0.438 0.3 1.80907 1.32904 ' 130.981 102.061 -l.52E+06 1.630 0.522 0.4 7.00906 8.22903 103.278 80.476 4.47905 1.801 0.580 0.5 3.20906 5.54903 84.759 66.046 4.14905 1.896 0.612 0.6 1.58906 3.87903 70.903 55.250 2.43905 1.910 0.620 0.7 7.93905 2.73903 59.528 46.388 4.441905 1.832 0.597 0.8 3.75905 1.87903 49.203 38.344 -8.17E+04 1.635 0.537 0.9 1.39905 1.12903 38.182 29.757 -3.84E+04 1.246 0.414 188 Table D5 (Cont) y=10 :12 ab’ x Y1 Y2 M 4;, 6°, 0.1 5.28908 2.28905 543.842 423.759 -l.08E+08 3.122 0.315 0.2 6.40907 7.89904 320.009 249.350 2.20907 4.324 0.438 0.3 1.80907 4.16904 232.369 181.061 -8.45E+06 5.130 0.522 0.4 7.00906 2.58904 183.098 142.671 «4.149% 5.662 0.580 0.5 3.20906 1.74904 150.159 117.004 2.29906 5.950 0.612 0.6 1.58906 1.21904 125.51 1 97.799 -1.34E+06 5.986 0.620 0.7 7.93905 8.54903 105.274 82.031 2.91905 5.732 0.597 0.8 3.75905 5.82903 86.901 67.716 -4.46E+05 5.101 0.537 0.9 1.39905 3.49903 67.282 52.429 2.08905 3.870 0.414 7:100 11, ab’ x Y1 Y2 M 8, 9". 0.1 5.28908 7.19905 965.828 752.567 -6.05E+08 9.847 0.315 0.2 6.40907 2.48905 567.768 442.401 - 1 .23908 13.61 1 0.438 0.3 1.80907 1.31905 41 1.899 320.950 470907 16.1 18 0.522 0.4 7.00906 8.10904 324.263 252.665 2.29907 17.758 0.580 0.5 3.20906 5.44904 265.667 207.007 -1.26E+07 18.625 0.612 0.6 1.58906 3.79904 221.815 172.838 -7.35E+06 18.697 0.620 0.7 7.93905 2.66904 185.803 144.778 -4.33E+06 17.856 0.597 0.8 3.75905 1.81904 153.098 1 19.295 2.42906 15.834 0.537 0.9 1.39905 1.08904 1 18.155 92.067 - r . 12906 1 1.936 0.414 y=1000 62 ab’ x Y1 Y2 M 6, 9°. 0.1 5.28908 2.27906 1714.463 1335.899 -3.3SE+09 31.028 0.315 0.2 6.40907 7.81905 1006.535 784.286 -6.84E+08 42.777 0.438 0.3 1.80907 4.10905 729.298 568.265 2.60908 50.529 0.522 0.4 7.00906 2.53905 573.397 446.788 -1.27E+08 55.529 0.580 0.5 3.20906 1.70905 469.136 , 365.548 -6.94E+07 58.080 0.612 0.6 1.58906 1.18905 391.086 304.732 4.02907 58.121 0.620 0.7 7.93905 8.24904 326.970 254.774 2.35907 55.297 0.597 0.8 3.75905 5.57904 268.709 209.377 4.31907 48.779 0.537 0.9 1.39905 3.28904 206.402 160.828 -5.93E+06 36.424 0.414 189 Table D6 Eigenvalues for Unstable Water/Freon Interfaces y = 0.1 02 ab2 X Y1 Y2 M 0, 0°, 0.1 ' 1.27E+08 1.12E+04 105.619 106.623 -1.14E+06 0.313 0.315 0.2 1.54E+07 3.90E+03 62.226 62.818 -2.36E+05 0.435 0.438 0.3 4.34E+06 2.06E+03 45.239 45.669 -9.1 1E+04 0.517 0.522 0.4 1.69E+06 1.28E+03 35.690 36.030 —4.51E+04 0.572 0.580 0.5 7.72905 8.64902 29.308 29.587 2.51904 0.602 0.612 0.6 3.81905 6.05902 24.534 24.767 -1.48E+04 0.608 0.620 0.7 1.91E+05 4.27E+02 20.617 20.812 -8.88E+03 0.583 0.597 0.8 9.05E+04 2.92E+02 17.062 17.223 -5.08E+03 0.521 0.537 0.9 3.35E+04 1.76E+02 L 13.270 13.395 -2.43E+03 0.398 0.414 7:10 92 ab’ X Y1 Y2 M 4): ¢°r 0.1 4.03E+07 6.31E+03 79.145 79.898 4.83E+05 0.313 0.315 0.2 4.88E+06 2. 19E+03 46.607 47.050 -9.96E+04 0.434 0.438 0.3 1.37E+06 1.15903 33.869 34.191 -3.86E+04 0.515 0.522 0.4 5.34905 7.18902 26.710 26.964 -1.91904 0.569 0.580 0.5 2.44905 4.83902 21.925 22.133 -l.06E+04 0.599 0.612 0.6 1.21905 3.38902 18.346 18.520 -6.29E+03 0.603 0.620 0.7 6.0SE+04 2.38E+02 15.410 15.556 -3.77E+03 0.579 0.597 0.8 2.86E+04 1.63902 12.746 12.866 -2.16E+03 0.516 0.537 0.9 1.06E+04 9.78E+01 9.906 9.999 -l.03E+03 0.393 0.414 190 Table D6 (Cont) y=10 11:, ab’ x Y1 Y2 M 9. 9°. 0.1 1.27907 3.54903 59.294 59.857 2.04905 0.312 0.315 0.2 1.54906 1.23903 34.895 35.226 6.211904 0.432 0.438 0.3 4.34905 6.46902 25.344 25.585 2.63904 0.513 0.522 0.4 1.69905 4.01902 19.977 20.167 2.09903 0.566 0.580 0.5 7.72904 2.70902 16.391 16.546 -4.52E+03 0.594 0.612 0.6 3.81904 1.88902 13.709 13.838 2.67903 0.598 0.620 0.7 1.91904 1.32902 11.508 11.617 2.60903 0.572 0.597 0.8 9.05903 9.02901 9.513 9.603 2.19902 0.509 0.537 0.9 3.35903 5.40901 7.388 7.457 6.41902 0.385 0.414 y=100 62 ab’ X Y1 Y2 M 6. 9°. 0.1 4.03906 1.99903 44.408 44.830 2.63904 0.311 0.315 0.2 4.88905 6.86902 26.113 26.361 2.78904 0.430 0.438 0.3 1.37905 3.61902 18.954 19.133 6.93903 0.509 0.522 0.4 5.34904 2.24902 14.931 15.072 2.43903 0.561 0.580 0.5 2.44904 1.50902 12.243 12.359 2.92903 0.588 0.612 0.6 1.21904 1.04902 10.234 10.331 2.14903 0.590 0.620 0.7 6.05903 7.33901 8.587 8.667 6.82902 0.563 0.597 0.8 2.86903 4.97901 7.094 7.161 2.92902 0.499 0.537 0.9 1.06903 2.95901 5.507 5.557 2.88902 0.375 0.414 F1000 92 ab2 X Y1 Y2 M 91 9°. 0.1 1.27906 1.11903 33.246 33.562 2.65904 0.310 0.315 0.2 1.54905 3.83902 19.530 19.715 2.57903 0.427 0.438 0.3 4.34904 2.01902 14.164 14.298 2.94903 0.504 0.522 0.4 1.69904 1.24902 11.149 11.255 2.46903 0.554 0.580 0.5 7.72903 8.31901 9.137 9.223 2.17902 0.579 0.612 0.6 3.81903 5.77901 7.633 7.705 4.85902 0.579 0.620 0.7 1.91903 4.03901 6.402 6.461 2.91902 0.550 0.597 0.8 9.05902 2.72901 5.288 5.336 2.67902 0.485 0.537 0.9 3.35902 1.60901 4.106 4.142 -8.02901 0.361 0.414 191 Table D7 Eigenvalues for Stable Air/Water Interfaces y=0.1 k,cm" Re(X) I Im(X) Im(s),s'I c,crer c°,cmls 0.1 2.12900] 3.09904 3.14900 31.381 31.382 0.5 2.29900] 3.00903 7.62900 15.244 15.246 1 2.15900] 1.29903 1.31901 13.099 13.102 5 2.01900] 3.88902 9.86901 19.715 19.725 10 2.98900] 2.69902 2.73902 27.345 27.364 50 2.90900] 1.19902 3.04903 60.705 60.804 100 2.86900] 8.44901 8.58903 85.763 85.973 500 2.7790(fl 3.76901 9.55904 190.921 192.229 1000 2.72900] 2.64901 2.69905 268.896 271.853 y=1.0 k,cm" Re(X) Im(X) Im(s),s" c,cmls c°,cmls 0.1 4.04900 9.73904 9.89900 98.902 98.904 0.5 2.56900 8.78903 2.23901 44.627 44.630 1 2.30900 3.19903 3.24901 32.421 32.425 5 2.03900 4.67902 1.19902 23.759 23.770 10 2.98900 2.84902 2.89902 28.908 28.927 50 2.90900 1.20902 3.04903 60.849 60.949 100 2.86900 8.44E+01 8.58E+03 85.814 86.024 500 2.77900 3.76901 9.55904 190.926 192.234 1000 2.72900 2.64901 2.69905 268.898 271.855 192 Table D7 (Cont) 7:10 k, cm" Re(X) Im(X) Im(s),s" c,cm/s c°,cmls 0.1 -5.68E+00 3.08E+05 3.13E+01 312.651 312.655 0.5 -3.06E+00 2.75E+04 7.00E+01 139.944 139.950 1 -2.60E+00 9.76E+03 9.92E+01 99.232 99.239 5 -2.11E*00 9.48E+02 2.41E+02 48.200 48.212 10 -2.02E+00 4.07E+02 4.14E+02 41.410 41.431 50 -1.90E+00 1.23E+02 3.11E+03 62.276 62.375 100 -l.86E+00 8.49E+01 8.63E+03 86.324 86.534 500 -1.77E+00 3.76E+01 9.55E+04 190.972 192.280 IMO -l.72E+00 2.65E+01 2.69E+05 268.914 271.871 7:100 cm’ Re(X) Im(X) Im(s),s" c,c s c°,cmls 0.1 -8.83E—05 9.99E+00 9.89E+01 988.660 988.667 0.5 -4.51E-04 9.99E+00 221902 442.175 442.185 1 -1.01E-03 9.99E+00 3.13E+02 312.749 312.761 5 -8.26E—03 1.01E+01 7.06E+02 141.114 141.133 10 -2.18E-02 1.04E+01 1.03E+03 102.510 102.536 50 -2.20E-01 1.70E+01 3.75E+03 75.070 75.167 100 —6.07E-01 2.92E+01 9.13E+03 91.268 91.476 500 -6.42E+00 1.37E+02 957904 191.430 192.737 1000 -1.76E+01 2.72E+02 2.69E+05 269.077 272.032 193 Table D8 Eigenvalues for Stable Kerosene/Water Interfaces y = 0.1 k, cm" Re(X) 1m(x) Im(8),s" c, chE c°, cm/s 0.1 -2.95E+01 6.92E+03 8.98E-01 8.982 9.019 0.5 -1.00E+01 7.66E+02 2.49E+00 4.972 5.035 1 -7.26E+00 3.93E+02 5.09E+00 5.092 5.183 5 -4.57E+00 1.47E+02 4.76E+01 9.528 9.807 10 -3.87E+00 1.03E+02 1.33E+02 13.315 13.783 50 -2.67E+00 4.50E+01 l .46E+03 29. 191 30.759 100 -2 .29E+00 3. 15E+01 4.09E+03 40.857 43.497 500 -1.61E+00 1.36E+01 4.42E+04 88.385 97.260 1000 ~1.40E+00 9.45E+00 123905 122.546 137.546 y=1.0 k, cm‘ Re(X) Im(X) Im(s) ,s" c, crn/s c°, cm/s 0.1 -5.19E+01 2.17E+04 2.82E+00 28.155 28.222 0.5 -1.59E+01 1.98E+03 6.44E+00 12.874 12.976 1 -9.94E+00 7.55E+02 9.79E+00 9.793 9.919 5 -4.72E+00 1.58E+02 5.11E+01 10.222 10.511 10 -3.90E+00 1.05E+02 1.36E+02 13.567 14.040 50 -2.67E+00 4.50E+01 1.46E+03 29.214 30.782 100 -2.29E*00 3.15E+01 4.09E+03 40.865 43.505 500 -1.61E+00 1.36E+01 4.42E+04 88.385 97.261 1000 -l.40E+00 9.45E+00 1239-05 122.546 137.546 194 Table D8 (Cont.) y=10 k, cm' Re(X) Im(X) m(s) ,s‘ c,cm/s c°,cmls 0.1 -9.21E+01 6.86E+04 8.90E+00 89.032 89.151 0.5 -2.78E+01 6.14E+03 1.99E+01 39.805 39.983 1 -l.67E+01 2.18E+03 2.83E+01 28.309 28.522 5 -5.75E+00 2.40E+02 7.78E+01 15.566 15.922 10 ~4.20E+00 1.22E+02 1.59E+02 15.879 16.391 50 -2.68E+00 4.54E+01 1.47E+03 29.439 31.014 100 -2.29E+00 3.16E+01 4.09E+03 40.945 43.587 500 -1.61E+00 1.36E+01 4.42E+04 88.392 97.268 1000 -l.4OE+00 9.45E+00 1.23E+05 122.549 137.549 7:100 k, cm Re(X) Im(X) m(s) ,s' c,cm/s c°,cmls 0.1 -1.63E+02 2.17E+05 2.82E+01 281.677 281.889 0.5 -4.91E+01 1.94E+04 6.29E+01 125.784 126.101 1 -2.93E+01 6.85E+03 8.8990] 88.869 89.246 5 -9.07E+00 6.24E+02 2.02E+02 40.463 41.034 10 -5.71E+00 2.36E+02 3.07E+02 30.659 31.365 50 -2.77E+00 4.87E+01 1.58E+03 31.610 33.240 100 -2.31E+00 3.22E+01 4.17E+03 41.733 44.400 500 -l.6lE+00 1.36E+01 4.42E+04 88.463 97.341 1000 -1.40E+00 9.45E+00 1.23E+05 122.573 137.575 195 Table D9 Eigenvalues for Stable Water/Freon Interfaces y=0.1 cm“ Re(X) Im(X) Im(s),s‘I c, cm/s c°,cmls 0.1 -4.09E+01 1.46E+04 1.48E+00 14.755 14.796 0.5 -l.30E+Ol 1.40E+03 3.55E+00 7.095 7.159 1 -8.59E+00 5 .95E+02 6.02E+00 6.024 6.107 5 -4.83E+00 1.75E+02 4.42E+01 8.837 9.065 10 -4.08E+00 1.20E+02 1.22E+02 12.183 12.562 50 -2.83E+00 5.27E+01 1.33E+03 26.635 27.904 100 -2.43E+00 3.69E+01 3.73E+03 37.314 39.454 500 -1.72E+00 1.60E+01 4.05E+04 80.977 88.216 1000 -1.50E+00 1.1 1E+01 1.12E+05 112.475 124.757 y=l.0 k, cm'I Re(X) Im(X) Im(s),s'l c,cm/s c°,cmls 0.1 -7.24E+01 4.60E+04 4.66E+00 46.565 46.638 0.5 -2.20E+01 4.14E+03 1.05E+01 20.926 21.035 1 - l .34E+01 1.50E+03 1.51E+01 15.130 15.262 5 -5.29E+00 2.13E+02 5.38E+01 10.762 11.013 10 -4.19E+00 1.28E+02 1.29E+02 12.928 13.318 50 -2.83E+00 5.28E+01 1.34E+03 26.704 27.974 100 -2.43E+00 3.69E+01 3.73E+03 37.339 39.479 500 -1.72E+00 1.60E+01 4.05E+04 80.980 88.219 1000 -1.50E+00 1.11E+01 1.12E+05 112.476 124.757 196 Table D9 (Con’t) y=10 k,cm" Re(X) Im(X) Im(s),s'I c,cm/s c°,cmE 0.1 -1.28E+02 1.46E+05 1.47E+01 147.305 147.435 0.5 -3.87E+01 1.30E+04 3.29E+01 65.798 65.991 1 -2.32E+01 4.60E+03 4.66E+01 46.557 46.788 5 -7.44E+00 4.40E+02 1.11E+02 22.280 22.639 10 -4.98E+00 1.86E+02 1.88E+02 18.843 19.313 50 -2.86E+00 5.41E+01 1.37E+03 27.379 28.665 100 -2.43E+00 3.71E+01 3.76E+03 37.579 39.726 500 -1.72E+00 1.60E+01 4.05E+04 81.001 88.241 1000 -1.50E+00 1.11E+01 1.12E+05 112.483 124.765 7:100 k, cm7 Re(X) Im(X) Im(s),s7 c,cm/s c°,c s 0.1 -2.28E+02 4.61E+05 4.66E+01 465.984 466.214 0.5 -6.85E+01 4.12E+04 1.04E+02 208.171 208.515 1 -4.09E+01 1.45E+04 1.47E+02 147.072 147.482 5 -l.25E+01 1.30E+03 3.30E+02 65.904 66.520 10 -7.67E+00 4.70E+02 4.75E+02 47.520 48.262 50 -3.12E+00 6.61E+01 1.67E+03 33.411 34.827 100 -2.50E+00 3.94E+01 3.99E+03 39.907 42.116 500 —1.73E+00 1.61E+01 4.06E+04 81.214 88.462 1000 -1.50E+00 1.11E+01 1.13E+05 112.558 124.844 APPENDIX E: DATA ERROR ANALYSIS APPENDIX E: DATA ERROR ANALYSIS E.l Error Analysis Procedure The analysis performed on the experimental data to determine the errors associated with it was done using the recommendations outlined by Holman and Gajda (1989). The experimental results fall into three specific categories: 1) results from data measured directly; 2) results from a collection of data; and, 3) results from calculations involving other experimental data. The procedures in which the estimated error is calculated for each category is outlined below. Measured Data Density, interfacial tension, and drop radius were measured directly. The value reported is that of the average of the measured values. The estimated error is twice the calculated sample standard deviation. This represents a 95 % confidence level if the distribution of error is assumed to be normal. 197 198 Regression Analysis Results Drop terminal velocity and frame acceleration were calculated from regression analysis. The result reported is derived from the slope of the least squares curve fit. The estimated error of this result is twice the standard error of the calculated SIOpe parameter. This also represents a confidence level of 95%. Calculated Results The Weber number and Bond number were calculated using the uncertainty analysis suggested by Holman and Gajda (1989, see pg. 42). This procedure is best described by the following example. Suppose R is a function of the independent variables a,,a2,a3, ..... an. Thus R=R(a,,a2,a3...,an) . (E.l) Let dR be the reported error in the result and d,,d2,...d,, be the reported error in the independent variables. The error associated with the derived result is: 1 an 2 an 2 an 2 5 dR —[[§a—l d,) +1—872— d2] +...+[aan (1,] ] (E.2) 199 No error has been associated with results derived from mathematical operations such as Fourier transforms. APPENDIX F: RAW ACCELERATION DATA - FLAT INTERFACE APPENDIX F: RAW ACCELERATION DATA - FLAT INTERFACE Raw experimental data used to calculate the acceleration of the elevator for the flat interface experiments is presented in Tables F.l - F.7. Each table lists the pressure used to accelerate the elevator, the video recording rate, and the length scale. The time listed in each table was calculated based on the frame rate. The column label y pix refers to the pixel location of the top of the elevator at the specified time. The column label zF is the displacement of the elevator relative to time equal to zero. 200 Table F .1 Raw Acceleration Data - Experiment 0213-03 (Air/Water) Pressure = 30 psig 201 Recording rate = 2000 fps Scale = 0.59 mm/pixel Frame No. Time, ms ,Jrix 21,, mm 1390 0 63 0 1395 2.5 63 0 1400 5.0 64 0.59 1405 7.5 68 2.95 1410 10.0 75 7.08 1415 12.5 84 12.39 1420 15.0 96 19.47 1425 17.5 110 27.73 1430 20.0 123 35.40 1435 22.5 142 46.61 1440 25.0 160 57.23 1445 27.5 179 68.44 1450 30.0 199 80.24 1455 32.5 219 92.04 202 Table F.2 Raw Acceleration Data - Experiment 0213-07 (Air/Water) Pressure = 15 psig Recording rate = 2000 fps Scale = 0.59 mm/pixel Frame No. Time, ms y, Jix 2,, mm 950 0 97 0 960 5 97 0 970 10 97 0 980 15 99 1.33 990 20 105 4.67 1000 25 120 13.33 1010 30 142 26.67 1020 35 171 43.34 1030 40 197 58.67 1040 45 228 77.34 1050 50 275 104.67 1060 55 313 127.34 1070 60 358 154.01 1080 65 395 175.81 203 Table F.3 Raw Acceleration Data - Experiment 0215-03 (Air/Water) Pressure = 20 psig Recording rate = 2000 fps Scale = 0.58 mm/pixel Frame No. Time, ms y, pix 2,, mm 2290 0 56 0.00 2300 5 56 0.00 2310 10 60 2.31 2320 15 67 6.35 2330 20 82 15.01 2340 25 105 28.29 2350 30 137 46.76 2360 35 181 72.16 2370 40 237 104.49 2380 45 268 122.38 204 Table F.4 Raw Acceleration Data - Experiment 0412-01 (Kerosene/Water) Pressure = 85 psig Recording rate = 2000 fps Scale = 0.58 mm/pixel Frame No. Time, ms y, pix 2,, mm 1730 0 9 0 1740 5 13 2.36 1750 10 28 11.21 1760 15 66 33.63 1770 20 123 67.26 1780 25 191 107.38 205 Table F.5 Raw Acceleration Data - Experiment 0412-04 (Kerosene/Water) Pressure = 75 psig Recording rate = 2000 fps Scale = 0.59 mm/pixel Frame No. Time, ms y, pix 2,, mm 980 0 5 0 990 5 7 1.18 1000 10 14 5.61 1010 15 33 16.82 1020 20 61 33.30 1030 25 105 59.00 206 Table E6 Raw Acceleration Data - Experiment 0412—05 (Kerosene/Water) Pressure = 45 psig Recording rate = 2000 fps Scale = 0.59 mm/pixel Frame No. Time, ms y, pix 2,, mm 660 0 3 0.00 670 5 4 0.79 680 10 9 3.74 690 15 22 11.21 700 20 42 22.83 710 25 74 41.69 207 Table F.7 Raw Acceleration Data - Experiment 0512-03 (W ater/Freon) Pressure = 90 psig Recording rate = 2000 fps Scale = 0.59 mm/pixel Frame No. Time, ms y, pix 2,, mm 150 0 9 0 160 5 13 2.36 170 10 33 14.16 180 15 66 33.63 190 20 134 73.75 200 25 227 128.62 APPENDIX G: PROJECTED SURFACE DATA APPENDIX G: PROJECTED SURFACE DATA The experimental data of the projected interface for the experiments is presented in Tables 6.1 - 67 Each table lists the relative vertical value of the interface(mrn) at the listed horizontal value(mm) for the given time listed(ms). Tables 6.8 - G.14 lists the resulting Fast Fourier Transform (FFI‘) calculated amplitudes for the listed wave numbers and times. 208 209 ”Na n.5— ad: _.mm_ ad: edem— NS: wdfi QR: 9m: «.2: QNN— finn— ¢._m_ NAN. 569 n.6— m.m© Qmw fin» ed” Qua ad» can Eva ads 9.2. «.2. v.5 *8 Q? 5.3 as” :6— fioe 0N” v.3 :3 5.2. a. E 565 You has. «:2. v.3. Wm» fin» 2.» EN» flaw ed— 03 3.5 fins was méh 02. N65 12. ”.3 ads. ado 0.: can 3k. 005 9.3. Wm— ». _c .60 _.$ 0.2. ads N. K mac 1% five ado ode N65 N. m. was m. :. WE. m."— adm 08 QNo «.3 OS 3% 93 RS aim ado v.8 five ode NS ..ne ode ...: mém _dm 9% <3 o.$ 5.8 can fiwm 5mm NSm QR mdc ado 5N0 v.3 ham e...— wdv w.~m odn mém mom m.mm 06m 9mm vdm c. _ n v. _ m mém _Km v.3 mém Yen m.» 9:. _dm 2% v. _ m o. _ n _. _ m mdm 03.. 0.2.. N? 5.9. _._m fimm 9mm a. _ m 5cm ms adv v.3. ”.3. 99. 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The column label y pix refers to the pixel location of the top of the elevator at the specified time. The column label zF is the displacement of the elevator relative to time equal to zero. 223 224 Table 11.1 Raw Acceleration Data - Experiment 815-01 (Air/Water) Pressure = 90 psig Recording rate = 2000 fps Scale = 0.59 mrn/pixel Frame No. Time, ms y, JDIX 2,, mm 1830 0 42 0 1825 2.5 45 1.77 1820 5 47 2.95 1815 7.5 50 0 1810 10 54 2.36 1805 12.5 62 7.08 1800 15 73 13.57 1795 17.5 91 24.19 1790 20 113 37.17 1785 22.5 141 53.69 1780 25 172 71.98 1775 27.5 210 94.4 225 Table H.2 Raw Acceleration Data - Experiment 815-02 (Air/Water) Pressure = 50 psig Recording rate = 2000 fps Scale = 1.14 mm/pixel Frame No. Time, ms y, pix 2,, mm 660 0 39 0 670 5 39 0 680 10 41 2.28 690 15 44 5.7 700 20 51 13.68 710 25 64 28.5 720 30 83 50.16 730 35 108 78.66 740 40 140 115.14 750 45 179 159.6 760 50 223 209.76 226 Table H3 Raw Acceleration Data - Experiment 815-03 (Air/Water) Pressure = 60 psig Recording rate = 2000 fps Scale = 0.50 mm/pixel Frame No. Time, ms y, pix 2,, mm 580 0 37 0 585 2.5 47 5 590 5 59 11 595 7.5 72 17.5 600 10 87 25 605 12.5 104 33.5 610 15 122 42.5 615 17.5 142 52.5 620 20 163 63 625 22.5 186 74.5 630 25 209 86 635 27.5 235 99 645 32.5 294 128.5 650 35 323 143 655 37.5 357 160 660 40 394 178.5 665 42.5 431 197 670 45 473 218 227 Table H.4 Raw Acceleration Data - Experiment 815-05 (Kerosene/Water) Pressure = 50 psig Recording rate = 2000 fps Scale = 0.70 mmlpixel Frame No. Time, ms y, pix 21,, mm 630 0 9 0 640 5 13 2.78 650 10 24 10.44 660 15 37 19.50 670 20 54 31.33 680 25 76 46.66 690 30 102 64.76 700 35 132 85.66 710 40 168 110.73 720 45 207 137.89 228 Table H.5 Raw Acceleration Data - Experiment 816-01 (Kerosene/Water) Pressure = 65 psig Recording rate = 2000 fps Scale = 0.70 mm/pixel Frame No. Time, ms y, pix an m 500 0 26 0.00 510 5 28 1.39 520 10 28 1.39 530 15 34 5.57 540 20 46 13.93 550 25 65 27.16 560 30 91 45.27 570 35 124 68.25 580 40 162 94.71 590 45 » 209 127.45 APPENDIX I: RAW POSITION DATA - FLUID PARTICLES APPENDIX 1: RAW POSITION DATA - FLUID PARTICLES Raw experimental data used to calculate the acceleration of the elevator for the fluid particle experiments is presented in Tables 1.1 - 1.5. Each table lists the pressure used to accelerate the elevator, the video recording rate, and the length scale. The time listed in each table was calculated based on the frame rate. In Tables 1.1-1.5 the column labels top, bottom, left, and right refer to the pixel position of the fluid particle. The column label “Disp” is the displacement of the fluid particle relative to time equal to zero. 229 230 Table 1.1 Raw Particle Position Data - Experiment 815-01 (Air/Water) Pressure = 90 psig Recording rate = 2000 fps Scale = 0.59 mm/pixel Frame Top,pix Bot,pix Left,pix Right, pix Ref, pix Disp, mm 1825 186 198 89 101 200 7.8 1820 170 180 88 100 184 9.4 1815 154 164 89 101 172 13.3 1810 138 148 90 102 161 18.0 1805 122 132 91 103 150 22.8 1800 106 114 90 102 135 25.1 1795 88 98 90 102 124 31.4 231 Table 1.2 Raw Particle Position Data - Experiment 815-02 (Air/Water) Pressure = 50 psig Recording rate = 2000 fps Scale = 1.14 mm/pixel a) Upper Bubble Frame Top Bot Left Right Ref, pix Disp, mm 170 112 122 66 80 188 5.2 180 98 108 70 84 173 6.3 190 88 98 78 90 160 8.9 200 72 84 86 100 147 12.0 210 55 65 88 102 132 15.2 220 38 47 88 103 120 16.7 230 22 32 88 104 114 20.9 b) Lower Bubble Frame Top Bot Left Right Ref, pix Disp, mm 170 146 156 66 80 188 4.9 180 130 140 65 78 173 5.9 190 112 122 66 80 160 8.3 200 98 108 70 84 147 l 1.3 210 88 98 78 90 132 14.2 220 72 84 86 100 120 15 .7 230 55 65 88 102 114 19.6 232 Table 1.3 Raw Particle Position Data - Experiment 815-03 (Air/Water) Pressure = 60 psig Recording rate = 2000 fps Scale = 0.50 mm/pixel Frame Top,pix Bot,pix Left,pix Right, pix Ref, pix Disp, mm 680 178 188 75 90 188 4.9 690 162 172 71 85 173 5.9 700 146 156 66 80 159 8.3 710 130 140 65 78 146 11.3 720 112 122 66 80 131 14.2 730 98 108 70 84 119 15.7 740 88 98 78 90 113 19.6 233 Table 1.4 Raw Particle Position Data - Experiment 815-05 (Kerosene/Water) Pressure = 50 psig Recording rate = 2000 fps Scale = 0.70 mm/pixel a) Upper Drop Frame Top Bot Left Right Ref, pix Disp, mm 660 47 60 90 106 54 1.0 670 32 42 90 104 38 1.2 675 22 34 88 102 30 1.7 680 12 28 88 104 22 2.3 685 6 22 88 103 17 2.8 b) Lower Drop Frame Top Bot Left Right Ref, pix Disp, mm 660 77 90 85 101 54 0.9 670 62 72 85 99 38 1.0 675 52 64 83 97 29 1.5 680 42 58 83 99 22 2.0 685 36 52 83 98 17 2.5 234 Table 1.5 Raw Particle Position Data - Experiment 816-01 (Kerosene/Water) Pressure = 65 psig Recording rate = 2000 fps Scale = 0.70 mm/pixel a) Upper Drop Frame Top Bot Left Right Ref, pix Disp, mm 660 108 122 82 96 55 1.2 670 94 108 86 100 38 1.4 675 86 98 87 100 30 2.1 680 78 88 88 100 23 2.8 685 69 78 90 102 18 3.5 b) Lower Drop Frame Top Bot Left Right Ref, pix Disp, mm 660 138 152 77 91 55 1.1 670 124 138 81 95 38 1.3 675 116 128 82 95 30 1.9 680 108 1 18 83 95 23 2.6 685 99 108 85 97 17 3.3 LIST OF REFERENCES LIST OF REFERENCES Adamson, A. 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