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- Permanent Link:
- http://ufdc.ufl.edu/AA00022288/00001
## Material Information- Title:
- Dynamic interfacial properties and their influence on bubble motion
- Creator:
- Maruvada, Krishna S
- Publication Date:
- 1997
- Language:
- English
- Physical Description:
- xi, 82 leaves : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Bubbles ( jstor )
Film thickness ( jstor ) Fluids ( jstor ) Interfacial tension ( jstor ) Kinetics ( jstor ) Silicones ( jstor ) Surfactants ( jstor ) Thin films ( jstor ) Velocity ( jstor ) Viscosity ( jstor ) Bubbles ( lcsh ) Chemical Engineering thesis, Ph.D ( lcsh ) Dissertations, Academic -- Chemical Engineering -- UF ( lcsh ) Fluid dynamics ( lcsh ) Surface active agents ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph.D.)--University of Florida, 1997.
- Bibliography:
- Includes bibliographical references (leaves 80-81).
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Krishna S. Maruvada.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 028631706 ( ALEPH )
38855515 ( OCLC )
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DYNAMIC INTERFACIAL PROPERTIES AND THEIR INFLUENCE ON BUBBLE MOTION By KRISHNA S. MARUVADA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1997 ACKNOWLEDGMENTS During my stay at the University of Florida, I had the privilege to interact with many talented people who have influenced me in several ways. First among those is my research advisor Dr. Chang-Won Park whose enthusiasm for research and pursuit of excellence are admirable. Much of my learning is due to several insightful research discussions I had with him during my stay here. I wholeheartedly acknowledge the constant encouragement, guidance and support I have received from him during the course of this research work. I thank Dr. Ranga Narayanan, Dr. Lewis Johns, Dr. Dinesh Shah, Dr. Richard Dickinson and Dr. Wei Shyy for serving on my committee and providing helpful suggestions. I also thank all the members of our research group, Mr. Charlie Jacobson, Mr. Yongcheng Li, Ms. Keisha Wilson, Mr. John Bradshaw, for their cooperation and support. A task of this magnitude can not be completed without the support of my beloved family members; my parents, my wife, my brother, my sister and brother-in-law who have helped me in every way they can, to help convert my dream into a reality. With great happiness, I acknowledge their generous support and encouragement. Finally I would like to acknowledge financial support by the donors of the Petroleum Research fund and NASA, Microgravity Science and Applications Division. TABLE OF CONTENTS page ACKN OW LED GM EN TS .............................................................................................. ii LIST OF FIGU RES ....................................................................................................... v LIST OF SYM BOLS ........................................................................................................ vii ABSTRA CT ......................................................................................................................... x CHAPTERS 1 INTRODU CTION ......................................................................................................... 1 Background .................................................................................................................... 1 M otivation ...................................................................................................................... 4 Dissertation Outline ................................................................................................. 9 2 EX PERIM EN TS ...................................................................................................... 12 The Influence of Surfactants on Bubble M otion ................................................... 12 Experim ents ................................................................................................................. 18 Results and Discussion .......................................................................................... 23 Sum m ary and Conclusions ..................................................................................... 37 3 ESTIMATION OF BUBBLE VELOCITY ............................................................ 38 Introduction .................................................................................................................. 38 Theoretical Analysis .............................................................................................. 38 Com parison with experim ental results .................................................................... 46 Sum m ary ...................................................................................................................... 50 4 DISPLACEMENT OF A POWER-LAW FLUID BY INVISCID FLUID ............ 51 Background .................................................................................................................. 51 Basic Equations ...................................................................................................... 53 Scalings and Regions ............................................................................................... 54 Sm all param eter expansions ................................................................................... 60 Capillary Statics Region : ................................................................................. 60 Transition Region : ............................................................................................ 62 M atching Conditions ........................................................................................ 65 Stability Analysis ................................................................................................... 69 Sum m ary ...................................................................................................................... 76 5 CON CLU SION S ...................................................................................................... 77 REFEREN CES .................................................................................................................. 80 BIOGRA PH ICAL SKETCH ....................................................................................... 82 LIST OF FIGURES FIGURE PAGE 1.1 Schematic of a moving bubble in a Hele-Shaw Cell (a) View in the xy plane (b) Bubble in the xz plane ........................................................................................ 2 1.2 Various Bubble shapes observed by Kopf-Sill and Homsy (a) Near-Circle; (b) Flattened bubble; (c) Tanveer Bubble; (d) Ovoid; (e) Short-tail bubble; and (f) Long-tail bubble ................................................................................................. 5 1.3 Here U/V vs Ca (Data taken from experiment series 1,2, and 3 of Kopf-Sill and H om sy) ........................................................................................................................ 7 2.1 A water drop sitting on a Pyrex glass plate immersed in silicone oil .................. 22 2.2 Evolution of water drop shape with increasing capillary number. (a)Near circular drop; (b) flattened drop; (c) severely stretched drop; (d) ovoid; (e) transitional shape; and (f) short tail drop ........................................................ 24 2.3 U/V vs. Capillary number for air bubbles ............................................................ 26 2.4 U/V vs. Capillary number for small water drops ................................................. 31 2.5 U/V vs. Capillary number for large water drops ................................................ 33 2.6 U/V vs. Capillary number for Fluorocarbon oil drops ....................................... 35 3.1 Plan view and side view of an elliptic bubble and a control volume moving with the bubble ................................................................................................................... 42 3.2 Numerical value of the definite integral, Equation 3.11 for various values of the shape factor k ...................................................................................................... 45 3.3 Comparison of experimental results and the theoretical estimate for pressure driven flow (SDS concentration : LI 5% ; A 10%; x 20% of CMC) .................... 48 4.1 Schematic of an inviscid fluid displacing a power-law fluid in a Hele-Shaw cell. ..52 4.2 Schematic showing the scales and coordinate systems for different regions. (A: Transition region; B: Capillary-statics region) ............................................. 57 4.3 Comparison of interface profiles for Newtonian (n=l) and power-law (n = 0.5) flu ids ....................................................................................................................... 64 4.4 Plot of the Co-efficient Co as a function of the power-law index ...................... 67 4.5 Log (film thickness) vs Log Ca for a power-law fluid. (to = 100 cP; a = 30 dyns/cm; yo = I sec'; b = 0.06 cm) .................................. 68 4.6 Disturbance growth rate (8) as a function of wave number (k). (o= 100 cP, y. =1 s", n = 0.6, b = 0.06 cm, a = 30 dyns/cm, U= 0.05 cm/s) ....... 73 4.7 Wave number for the maximum growth rate as a function of the power-law index. (.o=100 cP, yo= 1 sec', U= 0.05 cm/s, a = 30 dyns/cm, b = 0.06 cm) .............. 74 LIST OF SYMBOLS a Dimension of the bubble in the flow directions A Planform area of the bubble b Dimension of the bubble transverse to the flow direction b Half-gap thickness as used in Chapter 4 c Constant that appears in Equation 3.5 CO, C1, C2 Constants that appear in Equation 4.27 Ca Capillary number Ca Ca based on the normal component of the bubble velocity D Domain encapsulating the bubble D Rate of strain tensor f(z) Equation of the interface in z = constant plane (in the plane of the cell) FD Drag force acting on the bubble h Gap thickness of the Hele-Shaw cell as used in Chapter 3. H Interface profile for the transition region in the scaled form h(x,z) Equation of the interface in y = constant plane (in the gap direction) I Constant as given in Equation 3.11 k Shape factor for the elliptic bubble k Exponent that appears in equations 4.9 4.14 K Consistency factor L Width of the Hele-Shaw cell 1 Location of origin in the transition region n Unit normal to the surface of the control volume n Power-law index n Unit normal to the interface p Depth averaged pressure filed p Pressure field in the power-law fluid q S S s t T til, t2 U U U* U, V, W V V Vavg X Greek Letters a (D 0 P U Scale factor for the elliptic cylindrical coordinate system Surface of the control volume = }, 2(Square of the shear rate) Shift factor Thickness of the wetting film Stress tensor Unit tangent vectors to the interface Velocity of the drop/bubble Velocity of the moving interface near the tip Unretarded bubble velocity in a buoyancy driven flow Components of velocities in the x,y,z directions Velocity field in the driving fluid away from the bubble (Chapter 3) Average velocity of the driving fluid (Chapter 3) Depth averaged velocity field in the driving fluid Scaled variable for the x-dimension Gauge function in the expansions Retardation factor as given by Equation 3.16 Constant that appears in Equation 4.24 Growth rate of the disturbance Small parameter (Ratio of natural length scales of Hele-Shaw cell) Velocity potential Shear viscosity of the power-law fluid Viscosity of the driving fluid Angle of inclination of Hele-Shaw cell Density of the driving fluid Equilibrium interfacial tension Stress tensor Complex potential Location of the edge of the control volume Directions in the elliptic cylindrical Coordinate system 0 Subscript x,y,z Superscript 0001etc Location of the bubble surface Stream function Partial differentiations with respect to the variables. "Over bar" indicates variable in the transition region Leading order, Order Ca etc.. Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DYNAMIC INTERFACIAL PROPERTIES AND THEIR INFLUENCE ON BUBBLE MOTION By Krishna S. Maruvada December 1997 Chairman: Dr. Chang-Won Park Major Department: Chemical Engineering The influence of surfactants on the motion of bubbles and drops in a Hele-Shaw cell is studied by carrying out experiments in a variety of fluid-fluid systems. Saffman- Taylor theory predicts the translational velocities and shapes of drops and bubbles, not influenced by surfactants. In order to differentiate the cases with and without surfactants, the motion of air bubbles and water drops driven by silicone oil was studied. A controlled amount of surfactant is added to the water drops in order to study its influence on the bubble motion. In the case of air bubbles in silicone oil, surfactants were likely to have negligible influence, although they may be present as contaminants. As expected, air bubbles were moving much faster and their shapes were also in accordance with the available theories for the surfactant free system. On the other hand, surfactant containing water drops moved with much smaller velocities and their shapes were also distorted. These observations are apparently consistent with those of Kopf-Sill and Homsy. When a surfactant containing drop is set in motion, the resulting spatial variation of surfactants at the interface induces Marangoni stresses along the interface, and consequently the drop surface may become rigid. If the surrounding fluid is the wetting phase, the net drag force experienced by the drop in the wetting film region is significantly high resulting in smaller drop velocities. Because the surface tension has a spatial variation, to keep the pressure inside the drop constant, the curvature adjusts itself resulting in a variety of bubble shapes. Assuming that the bubble surface is rigid due to surfactant influence, the translational velocity of an elliptic bubble is estimated. The result indicates that the bubble velocity can decrease by an order of magnitude compared to the prediction of Taylor and Saffman. The retarded bubble velocity is apparently in reasonable agreement with the experimental observations suggesting that the puzzling observations of Kopf-Sill and Homsy are likely to be due to surface active contaminants. Finally, two phase displacement of a shear thinning power-law fluid by an inviscid fluid is analyzed. The flow domain is divided into three different regions and a leading order solution to the flow field and the interface shape is obtained using the technique of matched asymptotic expansions. In the case of Newtonian fluids, capillary number uniquely determines the film thickness. On the other hand, in the case of shear thinning power-law fluids, the relationship is unique to a given fluid and flow conditions. Finally, the flow field away from the interface in the power-law fluid region is also solved assuming that the gradients of components of velocities is negligible in the directions normal to the gap direction. Unlike the Newtonian case, the depth-averaged velocity field is found be non-Darcian. A stability analysis is also carried out to determine the wave number with the maximum growth rate. The result indicates that the interface will be less sinuous for the power-law fluid than the Newtonian fluid. CHAPTER 1 INTRODUCTION 1.1 Background Hele-Shaw cell has been used as a model system to study various physical phenomena involving multiphase flows in porous media. Although the concept of Hele- Shaw cell is more than a century old, there has been a lot of interest in recent years in the motion of two phase flows in a Hele-Shaw geometry."5 The displacement of a more viscous liquid using a less viscous liquid and the related fingering phenomena have been extensively studied both experimentally and theoretically, using this model."5 When the fluid-fluid interface is a closed contour, the inner fluid is referred to as a drop or bubble depending upon whether the enclosed fluid is a liquid or gas. A typical Hele-Shaw cell consists of a thin gap between two flat plates. The gap thickness, which is of the order of a millimeter, is very small compared to the dimensions of the plates. The gap is filled with a liquid into which a secondary fluid is injected to form a drop or bubble and its motion can be studied. The injected drop or bubble is squeezed between the two plates and assumes a circular planform when it is stagnant because of the interfacial tension. If the surrounding fluid is the wetting phase, a thin film exists between the bubble surface and the solid boundary of the Hele-Shaw cell. The bubble can be set in motion either by applying a pressure gradient in a horizontal Hele- Shaw cell or by imposing a buoyancy force by tilting the cell. If the characteristic length \L~ $ t1qu~d s- __ _h ;gap Wetting film Stagnation Points (b) Figure 1.1 Schematic of a moving bubble in a Hele-Shaw cell : (a) View in the xy plane, (b) Bubble in the xz plane scale of the bubble in the transverse direction is larger than the gap thickness (a/h >>1, Figure 1.1), the dynamics of the bubble motion can be described by Hele-Shaw equations6 which are a two dimensional description of the gap-averaged velocity and pressure fields. This approximation is valid over the entire domain expect for a thin region (of the order of gap thickness) surrounding the rim of the bubble. The depth averaged equations resemble the Darcy approximation of the governing equations for the flow through porous media. Hence the Hele-Shaw cell can also be used to study flow through porous media. Taylor and Saffman' have solved the Hele-Shaw equations for the motion of bubbles in a thin gap with no side walls assuming negligible surface tension. Their solution predicts the bubble shape as a function of the area and bubble velocity. Furthermore they have concluded that for a freely suspended bubble, the bubble velocity (U) is always greater than the surrounding fluid average velocity (V) and the steady shapes are ellipses. Also if U < 2V these elliptic bubbles travel with their minor axis parallel to the flow direction and if U > 2V they travel with the major axis parallel to the flow direction. If U = 2V the bubble shapes are circles. Tanveer3 had incorporated the effect of finite surface tension to resolve the degeneracy of the Taylor-Saffman solution and obtained three solution branches. A stability analysis4 of these three solution branches indicate that only one of them (McLean-Saffinan branch) is stable. The inclusion of interfacial tension in the theoretical analysis does not alter Taylor-Saffman's conclusion about the relative velocities of the bubbles and the surrounding fluid. Bubble motion was experimentally studied be several researchers using a variety of fluid combinations.5-8 Maxworthy' had experimentally studied the motion of air bubbles in an oil filled Hele-Shaw cell under the influence of gravity. At smaller values of the inclination angles, stable circular bubbles or elliptic bubbles elongated in the flow direction are observed. However at higher angles of inclination, bubbles became unstable with severe shape distortions. When the bubble size relative to the transverse cell dimension (a/L) was larger than 0.1, the bubble velocity was slightly larger than the prediction of Taylor and Saffman.' With decreasing bubble size, however, the bubble velocity decreased to a value smaller than the predicted one. This discrepancy in the bubble velocity was apparently resolved by including the viscous dissipation around the bubble edge in calculating the bubble velocity. 1.2 Motivation In 1988, Kopf-Sill and Homsy6 conducted similar experiments to investigate the shapes and velocities of various size bubbles driven by a pressure gradient. They investigated the motion of air bubbles in a horizontal Hele-Shaw cell filled with a glycerin-water mixture. Unlike the observations of Maxworthy, a variety of unusual bubble shapes were observed as shown in Figure 1. 2. Besides the near-circular and elliptic bubbles, "Tanveer" bubbles, ovoids, long- and short-tail bubbles were observed. By tracing the locations of the leading and trailing edges of the bubbles for an extended period of time, they showed that these unusual shapes were not transient but steady. Furthermore, they found that at low capillary numbers the translational velocities of the bubbles (near-circle, flattened or elongated ellipses, Tanveer) were much smaller than (a) (b) (c) (d) (e) (f) Figure 1.2 Various Bubble shapes observed by Kopf-Sill and Homsy: (a) Near-Circle; (b) Flattened bubble; (c) Tanveer Bubble; (d) Ovoid; (e) Short-tail bubble; and (f) Long-tail bubble expected. Especially, the velocity of near-circular bubbles was about an order of magnitude smaller than the prediction of Taylor and Saffman. In Figure 1.3, the velocity data taken from Kopf-Sill and Homsy5 are given for three different bubble sizes. The capillary number in the x-axis is defined as p.V/a where t and V are the shear viscosity and the average velocity of the glycerin-water mixture, respectively, and ; is the surface tension. When Ca is small, interfacial tension is dominant resulting in a circular bubble shape. According to the theoretical prediction,' the velocity of a circular bubble relative to the average fluid velocity (i.e., UN) should be 2. Contrary to the prediction, the relative bubble velocity was observed to be only about 0.2. The very low bubble velocity at a small capillary number was common for all bubbles regardless of their size. As Ca was increased, the relative bubble velocity increased slightly until it jumped suddenly to a much higher value at a certain Ca. The sudden jump in velocity was apparently accompanied by a drastic change in bubble shape. While the similar trend was observed with all bubbles of different sizes, the sudden transition in the bubble shape and velocity occurred at a lower value of Ca for larger bubbles. They also reported a hysteresis in the transitional behavior. It should be noted that the Taylor-Saffman solution exists only when the relative bubble velocity U/V is greater than 1. Although the possibility of U/V < I cannot be ruled out for non-zero surface tension, Saffman and Tanveer'2 point out that such a low velocity of small bubbles cannot be explained by a simple inclusion of the surface tension A A 0 + Long tails 0 00 + Short tails o+ -A 0 Ovoids +... A4 0+ Z90 + lattened + + Near Circles 50 100 150 200 250 3 Ca x 104 A206cis 01.3cms +0.76crs Figure 1.3 Here U/V vs Ca (data taken from experiment series 1,2, and 3 of Kopf-Sill and Homsy) 2.50 2.00 1.50 U/V 1.00 0.50 0.00 effect and/or the wetting film effect. They suggested that the low bubble velocity might be due to a moving contact line dragging along the cell plates. Using an ad hoc boundary condition, they showed that a moving contact line could account for the very low bubble velocity. In Kopf-Sill and Homsy's experiment, however, the liquid phase wetted the solid surface and wetting films were always observed. Therefore, the presence of a moving contact line was unlikely. We may note that the surprising variety of the steady bubble shapes and the striking discrepancy in bubble velocities were observed only in a pressure driven flow. Unlike Kopf-Sill and Homsy, Maxworthy did not observe any unexpected shapes but only elliptic bubbles elongated in the flow direction. In his experiments, the bubble velocities were also in disagreement with the Taylor-Saffman theory. The disagreement, however, was not as great as in the Kopf-Sill-Homsy experiments and it could be explained by the inclusion of viscous dissipation at the bubble rim. In the work of Kopf- Sill and Homsy, the experiment of Eck and Siekmann7 is quoted in which the translational velocity of small bubbles were also much smaller than the predicted value. These observations apparently suggest that the pressure-driven and buoyancy-driven flows may be intrinsically different although Saffman and Taylor2 pointed out the equivalency between the two flows. The experiments of Kopf-Sill and Homsy involve several observations that are in need of explanation. These observations include (1) the velocity of small bubbles which appear to be an order of magnitude smaller than the theoretical prediction, (2) a surprising variety of steady bubble shapes, and (3) the sudden transition of bubble velocity and shape at a certain capillary number. Johann and Siekmann8 have measured the velocities of air bubbles in a water system containing steric acid. Their reported velocities also appear to be significantly retarded compared to the theoretical prediction. In all experimental studies described above, the Hele-Shaw cell was supposed to be clean and free from any contaminants. Many fluid flow systems, however, are susceptible to contamination and it seems that the perplexing observations of Kopf-Sill and Homsy may be explicable if we adopt the notion of the influence of surface-active contaminants on bubble motion. One of the objectives of this study is to provide a plausible explanation for the observations of Kopf-Sill and Homsy in the context of surfactant influence. 1.3 Dissertation Outline In order to investigate the influence of surfactants on the bubble motion, we conducted experiments using air bubbles, water drops and fluorocarbon oil drops in a Hele-Shaw cell filled with silicone oil. Chapter 2 presents our experimental findings and compares them with those of Kopf-Sill and Homsy. Air bubbles were chosen since the surface properties of an air-oil interface were less likely to be affected by surfactants which might be present as contaminants. It is well known that unlike the surface tension of water or the interfacial tension between water and a hydrocarbon, the surface tension of non-aqueous solutions is not affected significantly by surfactants.17 In case of water drops, a predetermined amount of surfactant (sodium dodecyl sulfate) was added so that the surfactant influence could be studied systematically. The motion of the water drops were found to be drastically retarded by the surfactant, whereas the air bubbles were moving much faster as we anticipated. When surface active substances are present in the system, they adsorb near the fluid-fluid interface. Under dynamic conditions, depending on the surfactant exchange kinetics, a spatial variation of surface concentration can arise resulting in Marangoni stresses.'8 As a consequence, the interface can behave as rigid surface.9'20 Hence in the thin film region, the drag increases significantly causing severe retardation of the bubble. Chapter 3 presents a theoretical calculation that predicts the extent of retardation and provides an analytic expression for the drop velocity at low surfactant concentrations. As the bulk surfactant concentration is increased, Marangoni effects are expected to be weaker and a complete remobilization is anticipated.2' Contrary to this expectation, the motion of a drop containing a high concentration of surfactant is also found to be retarded. Although it is not very clear as to what causes the retardation at high concentrations, it could possibly be due to the influence of interfacial viscosities. At high concentrations the interface gets completely saturated and the intrinsic viscosities of the surface can play a role in retarding the drop motion. An effort to validate this idea theoretically is only partially successful. Finally, Chapter 4 presents the analysis of the displacement flow of a shear thinning power-law fluid by an inviscid fluid in a Hele-Shaw geometry. This problem has practical applications in several methods of polymer processing such as injection molding using various gases. The technique of matched asymptotic expansions2 is used to determine the flow field and the shape of the interface. The flow domain is divided into three different regions and matching conditions between them provide the film thickness left behind by the advancing meniscus. Unlike the Newtonian case, the relationship between capillary number and film thickness is not unique and it varies from fluid to 11 fluid. For power-law fluids, the depth averaged flow field is non-Darcian. A linear stability analysis is also carried out to determine the dispersion relation for the shear thinning power-law fluid. Chapter 5 summarizes the findings of this work. CHAPTER 2 EXPERIMENTS In this chapter, the influence of surfactant on the bubble motion in a Hele-Shaw cell is described qualitatively, and the difference between pressure-driven and buoyancy- driven flows in the presence of surfactant is discussed. Subsequently, the results of the present experiments are presented to support the arguments described below. 2.1 The Influence of Surfactants on Bubble Motion As mentioned in Chapter 1, the bubble can be set in motion either by applying a pressure gradient in a horizontal Hele-Shaw cell or by imposing a buoyancy force by tilting the cell. When the bubble is driven by buoyancy through a stagnant fluid in a Hele- Shaw cell, the sweeping motion of the surrounding liquid in a bubble fixed coordinate system is always from the front to the trailing end of the bubble. If the bubble is driven by the surrounding liquid, on the other hand, two different flow situations may arise depending on the relative magnitude of the bubble velocity U to the average velocity of the surrounding fluid V. If U > V, the sweeping motion of the surrounding liquid is toward the back of the bubble as in the buoyancy driven flow. Whereas if U < V, the sweeping motion along the edge of the bubble is from the rear to the leading end of the bubble. In the absence of surface-active substances, the surface tension is constant and the two flow situations may be indifferent. They are, however, distinctly different if there exists a spatial variation of surface tension due to the presence of surfactant. When U > V, surfactants will be accumulated at the trailing end of the bubble thus lowering the surface tension at the back of the bubble. When U < V, the flow situation is more complicated. At the edge (or rim) of the bubble, the surrounding fluid is convected from the back to the front of the bubble. Thus, surfactants are convected toward the bubble front along the edge. In the thin film region between the wall and the bubble, on the other hand, the dominant convection is always toward the back of the bubble since the solid wall is moving backward relative to the bubble and the liquid film is very thin. Therefore, the surfactant distribution is complicated and may depend on the balance between the two competing effects. The studies of Park and Homsy9 and Burgess and Foster" indicate that stagnation points (or rings) are present around the bubble at the transition region between the thin film region and the cap region (i.e., points s in Figurel.1). These stagnation points are similar to those in the front and the back of the Brethertons bubble in a capillary tube." Although the location of the stagnation points may move toward the cap region when surfactants are present, the existence of the stagnation points (or rings) may result in three regions of surfactant accumulation; one near the front cap of the bubble and the other two in the thin film regions (top and bottom) near the back of the bubble. It should be noted that this complicated flow situation cannot occur in a buoyancy driven flow. Therefore, the pressure-driven and buoyancy-driven bubble motions can be different from one another if surfactants are present in the system. The most interesting bubble shapes that were observed by Kopf-Sill and Homsy may be the ovoids with the sharper end pointed in the flow direction and the bubbles with a long- or a short-tail. When the average liquid velocity V was low (i.e., at a low Ca where the surface tension is dominant), the bubbles were observed to be near circular. With increasing V, they became slightly deformed to take on the shape of an ovoid. As V was further increased, the bubble shape went through a sudden transition to the long-tail (or short-tail) shape. Interestingly, a hysteresis was observed in that the backward transition from long-tail to ovoid occurred at a lower value of V than the forward transition from ovoid to long-tail for the same bubble. One important aspect which should be noted is that the ovoids apparently moved with a velocity smaller than the average liquid velocity (i.e., U < V) whereas the velocities of the short- or long-tail bubbles were larger than V. That is, the sudden increase (or decrease in case of a reverse experiment) in the bubble velocity cuts across the UN=I line in Figure 1.3. Thus, the flow situations for the ovoids and the bubbles with a tail are different in that the direction of the sweeping motion is opposite. We may point out, however, that a group of four data points in Figure 1.3 (a=0.76 cm) does not follow this argument as their relative velocities were reported to be about 0.8 although the shapes were short-tail. It may be possible that this discrepancy is due to measurement error. When the surrounding liquid sweeps the bubble from back to front along the edge (i.e., when U < V), the local interfacial tension may be lower in the front region of the bubble due to the surfactant accumulation there. Since the pressure inside the bubble is constant, the radius of curvature at the bubble front should be smaller than that at the back to compensate for the lower interfacial tension. Therefore, assuming a constant curvature in the gap direction (i.e., the z-direction in Figure 1.1), an ovoid with the sharper end pointing in the flow direction is a plausible shape. The same argument may be applicable even in the absence of surfactants. In that case, however, the difference in curvature between the front and rear ends of the bubble will not be as large as the surfactant-influenced case. By applying the similar argument to the case of U > V, we may also speculate that a drastic curvature change in the rear part of the bubble (hence the long-tail or short-tail bubbles) is possible if surface-active substances are present. This qualitative explanation does not rule out the possibility that the long-tail bubbles can also exist in a buoyancy driven flow since U is always greater than V in that case. A probable reason why it was not observed in Maxworthy's experiment is that silicone oil was used instead of glycerine-water mixture in his experiments. As it was pointed out previously, the air-oil interface is less likely to be affected by surface-active contaminants. Under the assumption that surface-active contaminants were present, the overall behavior of a bubble observed by Kopf-Sill and Homsy may be explained as follows. When an air bubble is stagnant in a contaminated Hele-Shaw cell, its entire surface will be covered with surface-active contaminants. At a very low Ca, the low shear stress may induce only a small change in surfactant distribution. Thus the Marangoni effect may be global. Although the Marangoni stress may be small due to the small surface tension gradient, the shear stress is also small and the entire bubble surface may act as a rigid 24 surface. In the presence of the stagnation rings near the top and bottom wall (Figure 1.1), three separate regions of high surface concentration can coexist as pointed out previously. Along the edge of the bubble, surfactants may be swept toward the bubble front and consequently, the Marangoni effect may assist the bubble motion. In the thin film region at the top and bottom of the walls, on the other hand, surfactants are swept backward retarding the bubble motion. When Ca is small, the viscous stress along the bubble edge may not be as significant as that in the thin film region which is inversely proportional to the film thickness. Therefore the drag in the wetting film is dominant retarding the bubble motion significantly. Consequently, the relative bubble velocity (U/V) may remain much smaller than 1. Although the bubble motion can be severely retarded, the shape change may be insignificant due to the small spatial variation of the surface tension. Consequently, the bubble shape may be near-circular. As Ca is increased, the larger viscous stress along the bubble edge increases the surface tension gradient resulting in larger Marangoni stress. In the thin film region, on the other hand, the relative increase of the drag may not be as significant since the film thickness increases with the bubble velocity. Therefore, the relative bubble velocity may increase with increasing Ca although its rate of increase may not be very large. Furthermore, the surfactant concentration at the leading end of the bubble will also increase with Ca. Consequently, the surface tension will become lower at the leading end creating a larger curvature. Therefore, the ovoid with the sharper end pointing in the direction of the flow may be a favorable shape. With increasing Ca, the film thickness increases and the location of the stagnation rings may move toward the tip of the bubble edge. Eventually the two stagnation rings may collapse into one along the tip of the bubble edge." Then the three spearate regions of high surfactant concentration cannot coexist. Consequently, the large drag in the thin film region may decrease significantly inducing higher bubble velocity which may be greater than the average velocity of the surrouding fluid (i.e., UN > 1). Once the relative bubble velocity becomes greater than 1, all surfactants will be collected at the trailing end of the bubble lowering the surface tension there. Consequently, a larger curvature is required forming a high curvature tail. The size (or length) of the tail may vary depending upon the surfactant distribution at the rear end. In an extreme case where the bubble is large enough and the surfactant transport is limited by bulk diffusion, the leading end of the bubble may be virtually free from surfactant due to convection whereas the trailing end is full of surfactants.19'2'26 In such a case, a long cylindrical tail may be conceivable as a form of minimizing viscous dissipation while maintaining a large curvature for the pressure balance across the interface. The Marangoni effect is greater if the surface curvature is larger. Consequently, its strong influence may persist to a higher value of Ca if the bubble is smaller. Therefore, the critical value of Ca where the transition occurs may increase with decreasing bubble size. If the capillary number is decreased gradually from a large value to a small value, the reverse trend may be observed in which the bubble goes through a transition from a long- or short-tail bubble to an ovoid and eventually to a near-circle. The forward and backward transitions, however, are not necessarily reversible as the adsorption and desorption kinetics of surfactants may not be reversible. Thus, the hysteresis in the transitional behavior can occur if such a change is driven by the presence of surfactants. The argument presented above is rather simple and only qualitative. Nevertheless, it apparently accounts for most of the observations of Kopf-Sill and Homsy. In the following section, the results of the present experiments are given as a supporting evidence. 2.2 Experiments The Hele-Shaw cell consisted of two 1/2-in thick Pyrex glass plates separated by a rubber gasket of 0.9 mm or 1.8 mm in thickness. The effective cell dimension was 17.8 cm by 86.4 cm. A silicone oil with the measured viscosity and surface tension of 97 cp and 21 dyn/cm was used as the driving fluid. The cell had a deep groove channel at both ends so that a uniform distribution of the oil along the transverse direction could be ensured when it was injected. The cell also had an injection port at one end of the top plate so that an air bubble or a water drop could be introduced into the cell using a syringe without creating extra bubbles or drops. The planform diameter of the stagnant bubbles and drops was controlled to be at about 1.3 cm and 2.1 cm throughout the experimental study. These sizes were chosen to be about the same as those of Kopf-Sill and Homsy for direct comparison of the results. While the size variation among the small bubbles was small as they ranged between 1.3 cm and 1.4 cm, the size control was more difficult with larger bubbles and their size ranged between 2.2 cm to 2.5 cm. Water drops containing an organic surfactant at various levels of concentration were prepared by dissolving sodium dodecyl sulfate (SDS) in distilled water. Since the surfactant influence is more significant at low concentrations, the SDS concentration was set at 5%, 10%, 20%, and 100% of the critical micelle concentration (CMC), respectively. The CMC of SDS in water at 25'C is 8.2 mmol/I which is equivalent to 0.236% by weight. The interfacial tension measured at room temperature by the Wilhelmy plate method is listed in Table 2.1 at various SDS concentrations. As the present data for air- 21 water interface is found to be consistent with those reported in literature, the data for the silicone oil-water interface may also be reliable. It can be deduced from the data that the rate of change of interfacial tension is greater at a lower concentration indicating a stronger Marangoni effect at a lower concentration. Once an air bubble (or a water drop) of predetermined size was positioned at one end of the cell, the silicone oil was driven by a peristaltic pump at various flow rates to induce the bubble motion. The translational velocity and the shape of the bubble (or drop) were then recorded for each value of oil flow rate. For the prescribed dimensions, the values of both a/L and h/a were smaller than 0.14. Thus the influence of the side wall (i.e., the rubber gasket) may be small and the depth-averaged Hele-Shaw equations may be applicable to describe the bubble motion if the system is free from surfactants. In order to avoid an excessive pressure drop at high flow rates, most of the experiments were conducted with the thick gasket (1.8 mm) and only a part of the air bubble experiments used the thin gasket (0.9 mm) for comparison. The Bond number (Bo) which is defined as pgh2/C is a measure of the relative importance of gravity to surface tension, and a large Bo may indicate asymmetry of the bubble in the gap. In the experiments of Kopf-Sill and Homsy, Bo was as small as 0.1 whereas it was about 1.5 in Maxworthy's. In the present experiment, it was either 0.4 for the 0.9 mm thick gasket or 1.5 for the 1.8 mm thick gasket. The large Bo was caused not only by the thick gasket but also by the low interfacial tension. While the gravity effect might not be significant when Bo = 0.4, uncertainty existed when Bo was as large as 1.5. The experimental results for air bubbles, however, were indifferent in terms of both bubble velocity and shape to whether Bo was 0.4 or 1.5. Thus, the gravity effect might not have significant influence even when Bo = 1.5. In reporting the results in the following section, no distinction has been made between the data for Bo = 1.5 and 0.4 in order to avoid unnecessary complication. The size of air bubbles changed slightly during the experiment due to the pressure drop in the cell. Kopf-Sill and Homsy observed similar behavior and reported the volume change of air bubbles up to 20%. In the present experiment, not only was the gap thickness of the cell slightly larger but also the liquid viscosity was lower. Furthermore, the liquid flow rate was 0.75 cm3/sec when the capillary number was as high as 110. Consequently, the pressure drop was smaller and the change in the bubble volume was less than about 5%. Despite the change in the bubble volume, variation in the bubble velocity was not detectable. In case of water drops, neither the volume nor the velocity changed. The description in Section 2.1 regarding the surfactant influence on the bubble motion assumes the existence of a thin oil film between the wall and the moving bubble or drop. When oil is the wetting phase, the film thickness is uniform in the direction of bubble or drop motion (i.e., x direction in Figure 1.1) whereas it varies in the cross-stream direction (i.e., y direction in Figure 1.1). Therefore, it may be possible in principle to prove the existence of the thin film by an optical method which creates interference fringes. Our attempt to observe the interference fringes, however, was unsuccessful due to various difficulties involved with the large size of the cell. Thus we provide only a qualitative evidence to support the assumption. A photograph converted from a video image is shown in Figure 2.1 for a water drop of about 2 mm in diameter which is sitting on a Pyrex glass plate immersed in the silicone oil. In order to improve contrast, a blue food coloring was added to the water drop. Due to the difficulty in obtaining perfectly horizontal alignment for the contact angle measurement, the water drop was viewed with a small positive angle from the horizontal and a shadow of the drop was created in front by indirect back lighting. The lower part of the picture is the shadow of the drop and the bright line between the drop and its shadow represents the three-phase contact line. At the edge of the contact line, the apparent static contact angle is shown to be greater than 900 when measured from the water-drop side. This may suggest that silicone oil is the wetting phase under dynamic conditions. When a water drop is stationary in the Hele-Shaw cell at the beginning of each experiment, the water drop may have direct contact with the Pyrex glass wall creating a three-phase contact line. As we induced the drop motion by injecting the silicone oil into the cell, the trailing edge of the drop exhibited briefly a ragged contact line pinned on the wall. The contact line soon detached from the wall and the water drop took on a steady shape with a perfectly smooth edge around the entire drop (Figure 2.2). If a moving contact line existed, its motion might have been jerky at times unless the solid surface was perfectly homogeneous. In the present experiments, the drop motion was always smooth excluding the brief moment at the start of motion. In case of air bubbles, even the Figure 2.1 A water drop sitting on a Pyrex glass plate immersed in silicone oil. start-up motion was smooth without any evidence of contact line. These observations along with the observation on the apparent static contact angle seem to support the assumption regarding the existence of thin oil films. 2.3 Results and Discussion In Figure 2.2, shapes of a water drop containing surfactant are given sequentially with increasing capillary number. These pictures were converted from the video images of a water drop which was 2.5 cm in diameter and contained sodium dodecyl sulfate (SDS) at 20% of the critical micelle concentration (CMC). The direction of the flow is downward from top to bottom of the figure. In order to improve contrast a blue food coloring was added to the drop. It was confirmed by measurements that the food coloring did not affect the surface tension. All shapes were observed to be steady excluding (2.2e) which was a transitional shape from (2.2d) to (2.2f). Figure 2.2a is the image of a stationary drop at Ca = 0. Due to the typical distortion of a video image in the vertical direction, this circular drop appears to be slightly elliptic. As Ca was increased, the drop was elongated sideways (2.2b). Further increase in Ca resulted in further stretching of the drop in the transverse direction (2.2c). This shape is apparently asymmetric about the vertical axis (or the x-axis in Figure 1.1). Nevertheless, its dimensions remained unchanged and moved with a steady velocity. Hence a steady shape. It should be pointed out that the severely stretched drop shape (2.2c) occurred only when the drop was larger than about 2.1 cm in diameter and when it contained SDS at a concentration between 5% and 20% of CMC. It did not occur with the pure water drop nor with the one containing SDS at 100% CMC although their sizes were larger than (a) (b) (c) (d) (e) (f) Figure 2.2 Evolution of water drop shape with increasing capillary number. (a) Near circular drop; (b) flattened drop; (c) severely stretched drop; (d) ovoid; (e) transitional shape; and (f) short tail drop 2.1 centimeters. Furthermore, when Ca was increased further from the value for Figure 2.2c, the drop broke into two instead of showing a transition to an ovoid or short-tail. In order to obtain the steady shapes at a higher Ca (i.e., Figure 2.2d or 2.2f) with the large drops containing SDS at 5% to 20% CMC, the Ca had to be set to a large value from the beginning. Then the drop evolved directly from (2.2a) to (2.2d) or (2.2f) bypassing the severe sideways stretching (2.2c). In case of small drops (1.3 or 1.35 cm in diameter) or the large drops with 0% or 100% CMC, on the other hand, the severely stretched shape (2.2c) did not occur and the evolution of steady shapes was smooth from (2.2a) to (2.2f) with the gradual increase of Ca. It may be interesting to note that the drop shape (2.2c) has a negative curvature at the leading edge as in the case of "Tanveer" bubbles although the trailing edge also shows a negative curvature. The steady shape (2.2d) which occurred at a higher Ca than (2.2b) or (2.2c) is different from others as the curvature of its leading edge (i.e., the lower part of the drop in Figure 2.2d) is apparently larger than that of the trailing edge. We may call this shape "ovoid" with its sharper end pointing in the direction of flow which is equivalent to the sketch given in Figure 1.2d. The relative velocity (i.e., UN) of this drop was still smaller than 1. When Ca was further increased, the ovoid went through a rapid transition to a short-tail (2.2f) which moved faster than the average velocity of the surrounding fluid (i.e., UN > 1). This transition always occurred through a transitional shape given in Figure 2.2e. In Figure 2.3, 2.4, and 2.5, the bubble or drop velocity relative to the average velocity of the surrounding fluid is given as a function of capillary number Ca. Also 2.5 2.0 t A A A X A X O 1.5 A 0 UN/v 1.0 i mm m om i IN 0.5 .x 0.0 0 50 100 150 Cax 104 1 I.27 cms M 1.3 cms (Kopf-Sill) A2.54 cms x2.06 cms (Kopf-Sill) U/V vs Capillary number for air bubbles. Figure 2.3 specified in the figures are the steady shapes of the bubbles or drops classified according to Figure 2.2. The capillary number in these plots is defined as iV/a where p and V are the viscosity and the average velocity of the silicone oil respectively and a the equilibrium interracial tension. In case of air bubbles (Figure 2.3), a is 21 dyn/cm whereas a for the water drops (Figures 2.4 and 2.5) varies depending on the SDS concentration. The equilibrium values at each SDS concentrations are given in Table 2.1 and these respective values have been used in calculating the capillary number. The relative velocities of air bubbles described in Figure 2.3 are for two different sizes (a=l.27 cm and 2.54 cm). Overlaid in the figure are the data taken from Kopf-Sill and Homsy for similar size bubbles. In their experiments, both small and large bubbles experienced a sudden increase in the relative velocity accompanied by the shape change. The transition for the small bubble occurred when Ca was about 10-2 whereas it occurred at Ca z 5 X 104 for the large bubble. Such transitions, however, were not observed in the present experiment which covered the range of Ca between 2 X 10.3 and 1.1 X 102. The relative velocity was always larger than one for both bubbles, although the larger bubble was moving slightly faster than the smaller one. Furthermore, the bubble shapes were always near circular unlike the experiments of Kopf-Sill and Homsy. The lower limit of Ca (2 X 10") was set by the sensitivity of the pump whereas the upper limit (1.1 X 10"2) was determined by the maximum flow rate which was imposed by the pump capacity or the upper limit of pressure drop in the cell. The lower limit is well below the critical Ca of Kopf-Sill and Homsy at which the small bubble experienced the velocity/shape transition. The critical value for the large bubble (Ca -_ 5 X 10 4), on the other hand, was smaller than the lower limit. Thus, the present experiment could not have detected the transition if it had occurred at a very low Ca. It should be noted, however, that in the experiments of Kopf-Sill and Homsy, the velocity transition was always accompanied by the shape change and the bubbles with the relative velocity (U/V) greater than one had either a long- or a short-tail. In the present experiments, on the other hand, the bubble shapes were near-circular although their relative velocities (U/V) were greater than about 1.5. Thus, it is unlikely that these near-circular bubbles had experienced the sudden transition at a smaller Ca than the lower limit or they would experience it at a higher Ca than the upper limit. Considering the slight deviation from a circular shape and the viscous dissipation at the bubble edges which was pointed out by Maxworthy,5 the bubble velocities in the present experiments appear to be in reasonable agreement with the prediction of Taylor- Saffman. The distinct differences between the results of the present experiments and of Kopf-Sill and Homsy are apparently in accordance with the argument described in section 2.1 that they were probably caused by the influence of surface-active contaminants. In Figures 2.4 and 2.5, the results for water drops are given for two different sizes. In these experiments the 1.8 mm gasket was used to minimize the pressure in the cell at high flow rate, and the maximum attainable capillary number was determined by the capacity of the pump (Cole-Parmer, Masterflex Model 7520-25). Due to the variation of the equilibrium interfacial tension with the SDS concentration, the maximum capillary Table 2.1 Variation of the interfacial tension with sodium dodecyl sulphate concentration at 25C SDS concentration in water (% of CMC) air-w Interfacial Tension (dyn/cm) rater Silicone oil-water 72.8 58.7 53.8 41.7 33.0 38.8 34.3 32.1 25.9 13.7 number also varied accordingly for the given maximum flow rate. In case of the 100% CMC water drop, the equilibrium interfacial tension was as low as 13.7 dyn/cm whereas that of pure water and oil was 38.8 dyn/cm (Table 2.1). Consequently, the maximum capillary number for the former (3.7 X 10-2) was higher than that for the latter (1.3 X 10-2) by a factor of about 2.8. The relative velocities of the small water drops (a = 1.30 cm or 1.35 cm) are shown in Figure 2.4. The labels (a)-(f) indicate the drop shapes as indicated in Figure 2.2. Unlike the air bubbles, they were moving very slowly at low capillary numbers as in the observations of Kopf-Sill and Homsy. It is interesting to note that the water drops which were supposed to be pure were also moving very slowly at low Ca although they were expected to be as fast as the air bubbles in Figure 2.3. A plausible explanation for this behavior may be that the water drops still contained some surfactant despite the precaution we had taken to prevent contamination. When the SDS concentration was between 5% and 20% of CMC, the relative velocities were smaller than one even at the upper limit of Ca. Furthermore, the drop velocities were indistinguishable and the shapes were near-circular or elliptic (Figures 2.2a and 2.2b). Although it was expected that the relative velocities of these drops would increase with Ca and become eventually larger than one, the transition could not be observed as it might occur at a higher Ca than the attainable upper limit. In case of the "pure" drop or the drop with SDS at 100% CMC, on the other hand, the transition could be observed in that the drops took on the short-tail shape (Figure 2.2f) and moved with a relative velocity greater than one. Unlike the observations of Kopf-Sill 2.0 1.5 U/V1.0 0.5 0.0 .- (0 0 m (d) xx x [M ... * L (a) 200 300 400 Ca x 104 QO/ (13.5mm) n5% (135mm) A9.1*(13mm) x2-) (13mm) 00 /, (13.5m) Kopf-Sii (13 mm) U/V vs Capillary number for small water drops. Figure 2. 4 and Homsy, however, the change (or the transition) was gradual rather than abrupt. Although it is not clear what caused this difference in the transitional behavior, it may be partly due to the differences between the two experiments. In the current experiment of water drop and silicone oil, the surfactant was present inside the water drop whereas in case of Kopf-Sill and Homsy's experiment of air bubble and aqueous solution, the surfactants (or surface-active contaminants) might be present in the aqueous solution outside the bubble. Therefore, the mechanism for surfactant transport to the bubble or drop surface could be different. Furthermore, considering the differences in the fluid properties, quantitative match between the two experiments may not be expected. These differences could be also the reason why long-tail drops were not observed in the present study. The large water drops showed the similar trend as the small ones in that the relative velocities were smaller than one at low Ca (Figure 2.5). Unlike the case of small drops, however, the transitional behavior is evident for all water drops in Figure 2.5. This observation is consistent with the trend in Figure 1.3 in that the critical Ca for the velocity/shape transition decreased with increasing bubble size. As we mentioned in section 2.1, the surfactant influence (or the Marangoni effect) should be greater if the surface curvature is larger. Consequently, the strong influence of surfactant may persist to a higher value of Ca if the drop is smaller. Apparently, the critical Ca for the large drops was smaller than the upper limit of Ca of the present experiments whereas that for the small ones was larger than the upper limit. The variation in the drop shape was also 2.0 13 0 0 1.5 XX .1u X X &X U/N 1.0 ... ... .. X d DXA... (b)) 0.5 _6() c )(a) 0 .0 . 0 100 200 300 400 Ca x 104 0/.cmc (25mm) g 5% c (24mm) & 91% cc (21.5mm)X 20% cmc (23.5mm)' 0 100/ cmc (24m) Ann (20.6mm) U/V vs Capillary number for large water drops. Figure 2. 5 consistent in that it was near-circular, elliptic or ovoid when U/V < 1, whereas all drops became short-tailed when UV > 1. Figure 2.5 also indicates that the critical Ca for the velocity and shape transition increases with increasing SDS concentration with the exception of the drop at 100% CMC. Although only a quantitative analysis will provide proper answer to such a trend, a qualitative explanation may be given as follows. For most organic surfactants the adsorption kinetics is fast. Therefore, if the surfactant concentration is low, its transport may be limited by bulk diffusion. At a reasonably high Ca, the surfactant distribution may be localized due to convection and consequently, the Marangoni effect is also restricted to the local region. As long as the surface concentration gradient is smaller than the maximum allowable value before the monolayer collapse for the given surfactant, the area covered with the surfactant may grow with increasing bulk concentration. Consequently, the Marangoni effect may also increase with the bulk concentration. If the bulk concentration is higher than a certain limit, however, surfactant transport may not be limited by bulk diffusion any more and the surface concentration gradient will start to decrease. Thus the surfactant influence will also diminish with further increase in bulk concentration. It seems that the critical bulk concentration for the present experiment at which the surfactant influence starts to decrease may be between 20% and 100% of CMC. In an extreme case, if the bulk concentration is very high, the surface may be remobilized due to the lack of surface concentration gradient. We have also conducted an experiment with a water drop which contained SDS at 10 times the CMC to investigate whether the remobilization would 2.0 A A A 1.6 3A 0 0 1.2 o 0 U/V 0.8 0.4 0.0 0 200 400 600 800 Ca x 104 06ram 09mm A 13m Figure 2.6 U/V vs Capillary number for Fluorocarbon oil drops. occur at that concentration. The result, however, almost coincided with that for 100% CMC drop in Figure 2.5 in that the relative drop velocity was still very low at small Ca. Finally, an experiment is conducted using an oil-oil system. Although, surface active substances are present as contaminants in this system, they are not expected to adsorb at the interface hence their influence on the drop motion will be insignificant. Fluorocarbon oil is chosen because it is immiscible with silicone oil. The interfacial tension was measured by Wilhelmy plate method and is found to be close to 5 dyns/cm. Pressure gradient is imposed to study the motion of a Fluorocarbon oil drop placed in Hele-Shaw cell filled with silicone oil. Figure 2.6 presents the normalised velocity of fluorocarbon oil drop as a function of capillary number. A nominal value of 5 dyns/cm is used in calculating the capillary number. These results indicate that the normalised velocity is always greater than one and the drop velocity is close to the predictions of Taylor- Saffman theory. Unlike the water drops, the shape of the fluorocarbon oil drop is always found to be circular. These results reinforce the idea that the retardation experienced by drops or bubbles in an aqueous environment is due to the influence of surfactants. 2.4 Summary and Conclusions Under the assumption that surface-active contaminants were the primary reason for the interesting behaviors of the bubbles in a Hele-Shaw cell observed by Kopf-Sill and Homsy, similar experiments were conducted using the fluid combinations of air-silicone oil and water-silicone oil. These systems were chosen to delineate the surfactant influence systematically. In case of air bubbles in silicone oil for which the surfactant influence was likely to be insignificant, the relative velocities were observed to be as high as the Taylor-Saffman prediction. Furthermore, the unusual bubble shapes such as long- or short-tail bubbles were not observed. In case of water drops which contained predetermined amount of an organic surfactant (sodium dodecyl sulfate), on the other hand, very low translational velocities as well as the unusual shapes of Kopf-Sill and Homsy were observed. Although it is impossible to confirm the presence of surfactants in the past experiments, the present results are consistent with those of Kopf-sill and Homsy and it appears that most of the observations by them may be due to the influence of surface- active contaminants. CHAPTER 3 ESTIMATION OF BUBBLE VELOCITY 3.1 Introduction This chapter presents a theoretical calculation for the translational velocity of a bubble which is retarded by the surfactant influence. In order to simplify the analysis, the bubble shape is assumed a priori to be elliptic. It is also assumed that the surrounding fluid wets the solid surface thereby forming a thin film between the bubble and the plates of the Hele-Shaw cell. For a relatively small bubble (i.e., b/L << 1) yet large enough to neglect the edge effect (h/b << 1), the entire bubble surface tends to be rigid due to the surface active substance. Consequently, a large drag arises in the thin film region thus retarding the bubble motion significantly. The calculated velocity of the bubble is apparently in reasonable agreement with our own experimental observations and also with those of Kopf-Sill and Homsy.6 In addition, the present calculation provides an explanation for the evolution of the bubble shape with increasing capillary number. 3.2 Theoretical Analysis A bubble or drop in a Hele-Shaw cell assumes a circular plan form when it is stagnant because of the interfacial tension. Once the bubble is set in motion, its shape deviates from the circular plan form and takes on a steady shape depending on the flow conditions (Figure 1.1). In the absence of surface tension effect, the analysis by Taylor and Saffman' predicts an elliptic shape if the bubble size is much smaller than the width of the cell (i.e., b/L << 1). If surface active substances are present, the bubbles are not necessarily elliptical and may take on various interesting shapes depending on the flow conditions6. Nevertheless, we assume an elliptic plan form since the shape distortion is small at low capillary numbers. While the bubble shape should be determined as a part of solution in a rigorous analysis, the present analysis is an approximation in which the bubble shape is assumed a priori. Unlike a full three-dimensional analysis that may be possible only numerically, the present analysis provides an analytic description for the translational velocity of the bubble. As indicated in Figure 1.1, the two principal axes of the elliptic bubble are assumed to be aligned with the flow and the transverse directions. The surrounding fluid wets the solid surface thus forming a thin liquid film between the plates and the bubble. In the presence of surface active substances, the Marangoni effect resulting from the surface tension gradient may complicate the flow field near the bubble. When the bubble is small, however, it may be simplified since the entire bubble surface may become rigid9-2". Here we consider the case of a small bubble in which the entire bubble surface is assumed to be rigid. The flow field slightly away from the bubble is known to be parabolic in the xz-plane (i.e., across the gap). Thus, the velocity field can be given as1 v 4z ) 3.1 v l5avg 1- h 2 Here vavg is the gap (or depth)-averaged velocity field that satisfies the following Hele- Shaw equations: 40 V"Vavg = 0 3.2 Vp = v 3.3 Here p is the gap-averaged pressure field, t is the fluid viscosity and h is the gap of the Hele-Shaw cell. It is apparent from Eqns. (3.2) and (3.3), that the average flow field in the xy-plane can be described as a potential flow. In an elliptic cylindrical coordinate system depicted in Figure 3.1, the complex potential Q for the flow past an elliptic cylinder in a bubble fixed frame of reference is given by6 Cl=ID+i = -b(V-(k+l)Vk2 ,7-1 k+I 2 [k+ k- ]. where y = + i i1 and k the shape factor defined as k = a/b. V and U are the average velocity of the surrounding fluid and the bubble, respectively. From the gradient of the velocity potential cD or the stream function ', the velocity components in and ti directions are given as 2q Lk+1lj- q Vg=_C(VU)k+lF e' e- l UC 3.6 I1 2q Lk+l+ k-1 sn -U sirq Here c = a2- b2j and q is the scale factor for the elliptic cylindrical coordinate system defined as q2 = c2 (sinh2 +sin2Tl). As it was pointed out previously, Eqns (3.5) and (3.6) are valid only in the region which is O(h) distance away from the bubble rim. In the thin film region between the plates and the bubble, the flow field is represented by a Couette flow if the bubble surface is rigid. In this case the thickness of the wetting film can be given as10,20,23,27 = h pun_ 3 t = 1.337 (2Ca,,)3 where Ca,, 3.7 2 " Here Can is the capillary number based on the component of the bubble velocity normal to the bubble surface in the xy-plane, and a the equilibrium interfacial tension. In the absence of surfactant, the bubble surface is stress-free and the film thickness can be expressed by the same equation but with the constant 2 in the parenthesis replaced by 1. Equations (3.1), (3.5), (3.6) and the Couette flow along with the film thickness described by Equation (3.7) represent the three-dimensional velocity field around the translating bubble excluding the small region of O(h) in the immediate vicinity of the bubble rim. Using this velocity field, an analytic description of the translational velocity of the elliptic bubble can be obtained from the x-directional momentum balance on a control volume surrounding the bubble. An integral form for the x-directional momentum balance for a domain D which encapsulates the bubble and moves with the same velocity U as the bubble may be written as (Figure 3.1) d fpvdV = -f {p[n. (v U)]v, + (np n. c)dS 3.8 dtD S This integral is zero at steady state and the surface integral can be evaluated using the V V U STop SBubble film region I x Sedge 2a S Bottom Plan view and side view of an elliptic bubble and a control volume moving with the bubble Figure 3.1 prescribed velocity field over the four separate surfaces as shown in Figure 3.1; top, bottom, edge of the control volume D, and the bubble surface. On the bubble surface, v = U and the integral over the bubble surface represents the drag force (FD) acting on the bubble: f (np- n. -)xds = -FD 3.9 Sbbbk On the top and bottom surfaces of D, the first term of the integrand in Equation (3.8) is zero, since (v U) is perpendicular to n. The second term is evaluated separately for the film region directly above (or below) the bubble and the region which is O(h) distance away from the bubble rim to give (np- n "t)xdS 12 c2 {(e2 0" V 3.10 SUt + S h 4 4 _(e-2'_ e-2 #V _U)e2o+ U] 161pU T(Al 2.122hCa (Al As noted in Figure 3.1, 4= and = represent the edge (or rim) of the control volume and the bubble, respectively. The last term in Equation (3.10), which is the contribution from the Couette flow in the thin film region, is zero if the bubble surface is clean and stress-free. Here A is the plan form area of the bubble (i.e., A = cab) and Ca the capillary number based on the bubble velocity U. I is a constant given as a definite integral which accounts for the film thickness variation in the transverse direction (i.e., y-direction): 1_r ,+ (k2 -1)y2]-[1 3.11 Once k(= a/b) is specified, I can be determined numerically and the results are given in Figure 3.2 for various values of k. When the bubble is circular (i.e., k = 1), I = 0.91. Finally at the edge of the control volume, the only non-zero contribution to the surface integral is due to the pressure which is given as Jnp) ds = l2 2 (v-U)+ Ve2_ -e2 U+ (VU)k+1 3.12 h 4 1k-I k - From Equation (3.9), (3.10) and (3.12), the final form of the x-directional momentum balance for a large control volume (i.e., -+oo) is obtained as h 4 k16Uk+ Ib 3k"1 2.122hCaI For a freely suspended bubble with zero inclination angle (i.e., 0 = 0 in Fig. 1), FD= 0 and Equation (3.13) reduces to 2 U =(k +I)V Ca3 3.14 Ca 3 + 0.2kI When the bubble is driven by gravity only, V = 0 and FD = -Apg(7rabh)sin0, since the drag force is balanced by the buoyancy force. Thus, 2 U=kU 2Ca where U* Aph gsin0 3.15 CaI + 0.2kI 1.5 1.3 0.9 0.7 0.5 0.0 1.5 2.0 Shape factor (k) Figure 3.2 Numerical value of the definite integral, Equation 3.11 for various values of the shape factor k. When the bubble is driven by both pressure gradient and buoyancy, Equation (3.14) and (3.15) are combined to give a general expression for an elliptic bubble as 2 U=8 [(k + 1)V + kU*] where = Ca3 3.16 Cai + 0.2kI As Figure 3.2 indicates, I is 0(1) unless k is extremely large. Thus 0 is O(Ca3). This order of magnitude decrease in the bubble velocity is due to the large drag in the thin film region where the bubble surface is rigid. Since this drag is proportional to the film thickness, the bubble velocity is also proportional to Cda. In the absence of Marangoni effect, on the other hand, the bubble surface in the thin film region is stress free. In this case, the expression for the bubble velocity is equivalent to setting 03 = 1 in Equation (3.16) since the last term in Equation (3.13) is absent. Thus the result of Taylor and Saffman is recovered in which U = 2 V for a circular bubble, U > 2 V and U < 2 V for an elliptical bubble elongated in the flow direction (i.e., k > 1) and in the transverse direction (k < 1), respectively. 3.3 Comparison with Experimental Results Retarded motion of aqueous drops containing sodium dodecyl sulfate (SDS) in an oil- filled Hele-Shaw cell was reported in the previous chapter for a pressure driven flow. In Figure 3.3 the present predictions for ellipses with various aspect ratios are compared with experimental results for a drop which is 13 mm in diameter (b/L = 0.04). The drop contained SDS at 5, 10 and 20% of the critical micelle concentration. At these concentrations of SDS, the data sets are indistinguishable from one another and show reasonable agreement with current predictions for ellipses that are elongated in the transverse direction (i.e., k < 1). While the drops in the experiment were not exactly elliptical (Figure 2.2), they were nearly elliptical and elongated in the transverse direction. The present analysis assumes the bubble shape a priori and consequently, the evolution of bubble shape with increasing capillary number cannot be predicted. However, an interesting observation may be made in Figure 3.3 When the drop was stagnant at Ca = 0, its shape was circular (i.e., k = 1.0). As it was pushed by the surrounding fluid, the k value decreased initially from 1.0 prior to exhibiting an increasing trend with increasing Ca. The initial decrease in the k value occurred at a very low Ca and in some cases, the drops simply elongated in the transverse direction without moving until the average velocity (V) of the surrounding fluid reached a certain value. Unfortunately, not all k values are available for the experimental data in Figure 3.3, and those that are available have been specified in the figure. Although a quantitative agreement is not expected between the observed and calculated k values, it is interesting to note that they are not very far apart. It should be pointed out that the retardation factor 3 in Equation (3.13) is independent of bubble size suggesting that the same level of retardation may occur regardless of the bubble size as long as their shapes are identical. One of the simplifying assumptions in the present study, however, is that the entire bubble surface is rigid. While this assumption is plausible for small bubbles, it may fail for large bubbles since a stress-free mobile surface region may emerge in the front part of the thin film region". U/V 150 0.2 0.3 0.5 1 .0 200 Ca x I 04 Comparison of experimental results and the theoretical estimate for pressure driven flow (SDS concentration: 5% ; A 10%; x 20% ofcMc) Saffman-Taylor Theory k=1.0 k=-0.4 A a ,A I X x .... K ...k-- .3 Figure 3.3 Consequently, big bubbles may move with a higher velocity than the present estimate. Furthermore, as pointed out in the previous chapter, bubbles which are large enough to have a mobile surface region for a given surfactant concentration, may experience a transition at a certain value of Ca in that the bubble velocity becomes greater than the average velocity of the surrounding fluid (i.e., U > P). Such a transition, then, will make it possible for the bubble to take on an unusual shape such as long- or short-tails as observed previously.6 This argument, of course, is only qualitative and remains to be proven rigorously. Experiments with air bubbles in an oil-filled Hele-Shaw cell have shown that the bubbles move with a velocity comparable to the predictions of Taylor and Saffman. In addition, they are always elongated in the flow direction unlike the bubbles influenced by surfactants. This difference in the steady shapes between clean bubbles and surfactant- laden bubbles may also be explained by the current study at least qualitatively. When two identical elliptic bubbles move with the same velocity but with different orientation (i.e., one moving with its longer axis in the flow direction and the other with its shorter axis in the flow direction), our calculation indicates that the prolate bubble experiences smaller drag force (or smaller energy dissipation) than the oblate one in the absence of surfactant effect. In case of surfactant-laden bubbles, on the other hand, the prolate bubble results in a larger energy dissipation than the oblate one due to the larger drag in the thin film region. Thus, the elongation in the flow direction (i.e., prolate bubble) may be a preferable shape for clean bubbles whereas the oblate shape is preferable for surfactant-laden ones. 3.4 Summary A theoretical calculation has been presented to estimate the translational velocity of bubbles or drops in a Hele-Shaw cell under the influence of surface active substances. Assuming that the solid wall is wet by the surrounding fluid and that the bubble shape is elliptical, an analytic expression has been derived for the bubble velocity. The calculated velocity is apparently in reasonable agreement with our own experimental findings and also with those of Kopf-Sill and Homsy. This result along with our experimental findings suggest that most of the observations by Kopf-Sill and Homsy are probably due to the influence of surface active substance which may be present in the system as contaminants. CHAPTER 4 DISPLACEMENT OF A POWER-LAW FLUID BY INVISCID FLUID 4.1 Background Two phase displacement flow is a problem of industrial importance in areas such as enhanced oil recovery, separations etc., and has been studied by various researchers in the past4. When the displacing fluid is of low viscosity, then instabilities can result causing viscous fingering. Park and Homsy have theoretically analyzed the two-phase displacement of Newtonian fluids in a Hele-Shaw geometry using the technique of matched asymptotic expansions.'5 Analytic expressions for both film thickness and pressure drop and their dependence on capillary number were derived. When the displaced fluid is a shear thinning fluid, these results can be very different. In this chapter, the displacement of a power-law fluid by an inviscid fluid (i.e. air/gas) in a Hele-Shaw cell is analyzed using the same technique of matched asymptotic expansions. The present analysis is aimed at deriving the film thickness and understanding the stability issue involved in this process. This problem has practical applications in the general area of polymer processing and specifically in applications such as gas assisted injection molding. Figure 4. 1 is a schematic of an inviscid fluid displacing a power-law fluid in a Hele-Shaw geometry. It is assumed that the displaced fluid wets the wall and there will be a thin film left behind after the displacement. The displacement is slow enough to ignore Re ion x = f(z) Region II Region III x Fluid 2 (air/gas) ) 2b Fluid 1 (Power-law fluid) z Figure 4. 1 Schematic of an inviscid fluid displacing a power-law fluid in a Hele-Shaw cell the effects of inertia and the gap thickness is small enough to ignore the effects of gravity. Under these assumptions, the interface remains symmetrical about the center plane. The tip of the interface is assumed to be moving with a constant velocity U and the analysis is carried out in a frame of reference moving with the tip velocity. 4.2 Basic Equations The governing equations relevant to the problem are as given below. VOv=0 4.1 Vp = T](S)V2v + 2D e Vil 4.2 Where D = (Vv+VvT) 4.3a S= 2(D :D) 4.3b n-I ri=KS 2 4.3c K is called the consistency factor and n is called the power law index. Boundary conditions are u=-U, v,w=0 at y=b 4.4a nev=0 tTn=0 at y =h(x,z) 4.4b t2oTon=0 noTn=c(VOn Here rl is the viscosity of the power-law fluid which depends on the shear rate and (u,v,w) are the components of velocities in the (x,y,z) directions respectively. n, t,, and t2 are the unit normal and unit tangential vectors to the interface, cr is the interfacial tension and T is the stress tensor. The origin for the reference frame is placed at the tip which is moving with a velocity U. b and h(xz) are the half gap thickness and the location of the interface respectively. f(z) is the projection of the tip of the interface onto the xz plane. Since the interface is symmetrical about the center plane, only the bottom half (i.e. y = -h(x, z))is considered for the analysis. The following two dimensionless parameters appear during the scaling of the above problem. b K___b_-_ 6 = and Ca = L a c is a parameter which represents the ratio of the two characteristic length scales of Hele- Shaw geometry and capillary number, Ca, represents the ratio of viscous to interfacial tension forces. A complete solution to this problem is very complicated to obtain, and only an asymptotic solution using the above mentioned two small parameters is attempted in the following sections. 4.3 Scalings and Regions As shown in Figure 4. 1, the fluid domain is divided into three different regions; Constant film thickness region (Region I), front meniscus region (Region II), and power- law fluid region (Region III). The flow profile in Region III can be obtained by straight- forward integration of the governing equations and is given in the next section. In the constant film region the flow field resembles plug flow in the moving reference frame. The flow profile in Region II can be understood only after a rigorous analysis as described below. The following scalings are used to non-dimensionalize the equations. (u,v,w)-(U,U,cU) (x,y,z) (b,b,L ) -a ) p t-e S (Ub)e Using the above scalings, the Equations 4.1- 4.4 are written as given below. n-I px = Ca S (Ux3 + Uyy U, +V y +9 2Wz =0 n-3 f 2uxSx + (Uy +* V)Sy +6 2 (U" + WX)Sz I py : + Vyy +6 2Vz ) + n-3 S 2 2+ + p =Ca S2(wx + Wy +62 n Sn {2 3-(1 + W ,)Sx + 2} Whe2 v + wy)SY + 2wzS (U w 2) + (uy + Vx)' + +w ) 62(U. ) Where S = 2(u2 + c v +Wx 6(Z+W at y = -1, at y = -h(x,z), I +(hkv -hv.) 4.6a 4.6b 4.6c u=-l, v=w=O uh. + v + ewh = 0 4.8a 4.8b n-1 (h + 62h { h (U + h.u) 4.8c + V)+g-:( v: +w)U =0 n-1 -h,(l++2h)-2h(l+h2)+2e2hh 2CaS 2-A Ap = +- (1+h + 2h2)i (1+h+e2h ) 4.8d where A=[-h(u + V 2 h v(v + W)_ 2hx h(u. + wx) vy h ehw,] The subscripts x,y,z in the above equation represent partial differentiation of the respective variables. To solve the above set of equations, we can begin by setting both , and Ca equal to zero. The resulting equations indicate that pressure is hydrostatic and is independent of (x,y,z). The shape of the interface is a portion of a circle which can be obtained by solving the normal stress condition. Hence this sub-region (Outer region of Region II) may be referred to as capillary statics region where pressure and interfacial tension are both important. In order to match the solution smoothly to the constant film region (Region I), there should be a second sub-region (Inner region of Region II) where the interface profile is smoothly matched to a constant value. This may be called as transition region. In the transition region both pressure and viscous effects are important and the variables need to be rescaled accordingly. Under the lubrication approximation, the scalings relevant for this region can be derived by balancing the pressure and viscous terms in the momentum equation and by balancing the pressure and interfacial tension forces in the normal stress condition. These scalings are different from the ones used for ly III III Figure 4. 2 Schematic showing the scales and coordinate systems for different regions. (A: Transition region; B: Capillary-statics region) the Newtonian case and they depend upon the power-law index and capillary number as mentioned below. _)(v wheek u, v, w ,w where k = - =W),Cak 2n+1 Ix z=x+1 + y Cak Ca2k ) -h ; p = Ca4kS ca2k P P; S=aS The variables in the transition region are denoted by an overbar. As indicated in Figure 4.2, the origin for the transition region is x = -1 and y = -1. The unknown I will be determined later using the matching conditions. Using the newly scaled variable, the governing equations valid for the transition region are given below. uX + Vy + C2Cak w = 0 [ ca2ku ( - -n- nI-n-3 p-= S 2 +U- + \+E2Ca4k u 2 4.9 4.10a _ Y=n-I py; = Ca2k-S 2 n-I cakp =S 2 n-3 + nICa22 2 3 Ca2k (Cak uz + wx )SX +1 n 2 S2~ a -+- z y)- +22Ca4k wZSz 4.10b 4.1 Oc Where S = at y = O, at y -z), u=-1, v=w=0 uh v + 2Cak whz = 0 '(akhzuy- hxw)+ 2Ca'*kh-x hz-(s2Ca +Ca2k(Cak -- + -)(i 2Ca2kjKJ + ca3k (h _x =z 0 Ca 2k (h2+,2Ca2kh2J hz(Ca3k vz 2 Ca Ca - 2 - hx(u + Ca 2k VX)c 2Ca k h (Ca 3k VZ +w) 4.12c + 2Ca4k-hJ + e2Ca2kh(1 + Ca2kh) 2e2Ca4kh-hzh (1 + Ca2kh2 + 62Ca4k-h2)2 n-I S 2 + Ca2kV ) + 2Cak (Ca3k + Wy c 2Ca 3kh;-(Ca k + )-v; _ Ca2k-h2- _4 Ca5k-i2w-J x z n-I S2 4.11 4.12a 4.12b n-i S 2 +2 Ca2k Ap= 2Ca-2kn+k+l I + Ca2k-h2 + ,2Ca4k-h 2 .1 2d 4.4 Small Parameter Expansions To obtain the solution, all the variables (u, w, p, h etc..) are expanded as given below. p=(p00 + CaapOl+ Ca2aP02 +...)+82(p1O+Caapll+ Ca2apI2 +...) +. 4.13 The choice of gauge function, a is obtained by comparing all the exponents that appear in the governing equations and boundary conditions and choosing the least common multiple of them so that matching is possible at every order. For example, the choice of a few gauge functions are as given below. a ..2nk ifn= 1; etc. 4.14 nk if n = I ;I;-etc. Since the gauge function depends on the power-law index, only the leading order solution is attempted in the following sub-sections. Once the choice of gauge function is known, higher order corrections can be obtained following the procedure of Park and Homsy. Consequently, in the present analysis the dynamic pressure jump across the interface can not be determined since it appears at a higher order. 4.4.1 Capillary Statics Region : Substituting 4.13 into 4.5-4.8, the leading order governing equations and boundary conditions for the capillary statics region are as given below. T 00 =0 uO + =y 4.13 00 00 004 PX PY PZ 4.14 Boundary conditions, u00 = -1, v oo = 0 at y = -1 4.15a at y = -h(x,z) uh + v = 0 4.15b ApOO -h 4.15c (1+ hxo2 )2 [(h uyoo +oo + 2h~hu + h u.o2 (u? + w ) + (h2v hv)] 0 4.15d h UO+ vOO) + 2h n ( V, h,(uoo V0] 4.15e As evident from the above equations, the viscous force term in the momentum equations is negligible and the pressure is independent of (x, y, z). The shape of the interface can be obtained by integrating the normal stress balance and is given as below. =oo 00 [1 {Apoo(x -f(z))+ 1}12]2 4.16 The two boundary conditions that were used in the integration and which apply at all orders are hx -oo as x f(z) 4.17 h = 0 at x = f(z) The solution 4.16 indicates that the tip of the interface is a portion of a circle modulated in the z-direction by f(z). This solution contains one unknown Apo, which will be determined using the matching conditions. Equation 4.16 is not uniformly valid in the entire region and it needs to matched with the constant film region (Region I). 4.4.2 Transition Region Variables in Equations 4.9-4.12 are expanded using 4.13 and the leading order terms are gathered to obtain the governing equations for the transition region. -00 -00 4.18 ux +Vy =0 0(-oo,, 4.19a 0 Y 4.19b pyOO = 0 -00 4.19c -0000 0=w + (n-1) -- -wy Uy --00 -00 00 4.20 at y=O u =-1, v =w 0 at y h oo(x ) -ooo =0O 4.21a aty= hx, z) u hX -00_0 = 4.1 -00 -00 42l Ap = hoo 4.21 b 00 u7 =0 4.21c By solving these equations and matching with the constant film thickness region (Region I) the velocity profile can be determined. Once the velocity field is known, then normal stress along with kinematic condition results in an evolution equation for the shape of the interface as given below. n -oo = 2n +I n0oo -t xxX nh i 4.22 When n is set equal to one, the above equation reduces to the Newtonian case (Park and -00 Homsy 1984). t represents the leading order approximation for the constant film thickness which will be determined later using the matching conditions. For numerical integration, the above equation is transformed into a canonical form by using the following transformations. -00 H00 = 4.23 -00 t x+s n+2 X where n+2 4.24 (t 00) 3 s is an arbitrary shift factor which is determined by higher order matching conditions. With these transformations, the evolution equation is written as follows. H00 =(2n+l (H00 1) 4.25 The condition for matching with region I now becomes H -- 1 as X -- -oo 4.26 Equation 4.25 can be numerically integrated, using small slope and small curvature as initial conditions. A 4thorder Runge-Kutta routine is used to do the numerical integration. The shape of the interface depends on the power-law index and it is plotted in Figure 4.3 for the case where n = 0.5 along with the profile for a Newtonian case (i.e. n = 1.0). Since as X -+ +o the interface profile H also goes to infinity, it is possible to approximate the profile using a quadratic form as given below. H0(x,z) = 1COX2 + Cj(z) X + C2(z) 4.27 2 Equation 4.27 can be written in terms of the original scalings as given below. 5000 4000 N Power-law n=0.5 3000 nOO 2000 nl 1000 0 0 20 40 60 80 x Figure 4. 3 Comparison of interface profiles for Newtonian (n=l) and power-law (n=0.5) fluids. 2t(z-C 002,6-1 0 2,8-1 -, +1 4.28 The unknown t will be determined by the following matching conditions. 4.4.3 Matching Conditions The solution in the transition region has to be matched with the solution in the capillary statics region using the following equation. Jim { Ca2 z =im h(x,z) 4.29 This limits are to be interpreted in terms of the matching principle by Van Dyke (1964). By expanding h(x,z) about x = -1 using Taylor series expansion, rewriting the expansion in inner variables and comparing it term by term with the left-hand side, matching conditions for each order can be determined as given below. Higher order matching conditions can be derived once the choice of ac is known. OOCao) hoo(_o,Z) = 1 4.30 0cak) hxO(-10,z)=0 4.31 OCa2k h. 000 3 h0(-_0' Z) = _C0(j) 4.32 Physically these conditions mean that the outer static solution meets the wall with zero slope and the curvature matches with that of transition region solution. Solving the above equations, the following unknowns in the solution can be determined. A00 = I 4.33a 0 = 1-(z) 4.33b -00 3 t = (Co)2,,+1 4.33c The above solution determines the location of the origin and also the film thickness to a leading order. Since the film thickness is stretched by Ca2k, the dimensionless film thickness is given by 3[U~bn n 4.34 too= (Co)21 K -bI 2n+1 Figure 4.4 displays the dependence of the constant Co on the power-law index n. As n -> 1 the constant C. approaches 1.337 which is the value for the Newtonian case. In Equation (4.34), if K is replaced by go /7 o where go is apparent viscosity and 70 is a reference shear rate, then the film thickness is given as follows. 32(1-n) 2n 4.35 to = (C0) 2nLl Ca,2n+' where Ca' atg Figure 4.5 is a plot of Equation (4.35) which is a straight line on a log-log scale. All the variables influence the intercept while only the power law-index influences the slope of the straight line plot. In the Newtonian case, all the fluids have a master curve (i.e. the plot for n = 1) whereas each power-law fluid has a separate curve uniquely defined for the given operating conditions. 2.5 0 2.0 C0 o. '01 1.5 0 1.337 for Newtonian Fluid 0 1.0 0 0.2 0.4 0.6 0.8 Power Law Index (n) Figure 4. 4 Plot of the Coefficient Co as a function of the power-law index 0~ 7 -6 -5 -4 -3 -2 -1 *A A X X X x X A A x x n = 1.0 n = 0.7 A n = 0.5 X n = 0.3 Log Ca Figure 4. 5 Log (film thickness) vs Log Ca for a power-law fluid. ( Io = 100 cP; a = 30 dyns/cm; y, = 1 sec'; b = 0.06 cm) 4.5 Stability Analysis In order to carry out the stability analysis, it is important to derive the flow field in the power-law fluid region. Although far away from the interface, the flow is only along the x-direction, near the interface a two dimensional flow field is considered. The solution to this region is fairly straight forward. It is assumed that the gradient of components of velocities in xz direction are negligible and only the gradients in the gap direction (i.e. y direction) are important. The simplified governing equations and boundary conditions are given below. Px = K S2 where S= Uy 2y y 4.36a n-1 p = K S-2-Wy y 4.36b py = 0 4.36c at y=b u=w=0 4.37a at y=O Uy,Wy = 0 4.37b The above set of equations can be solved to obtain the velocity field as given below. n 21-- (n n+l + U. =, n I + P 2 2 y n b n4 .3 8 a n [ 2 n [PX2]K r ~ .8 W n Pz 1-] n n-bn Wn +IPx l 2, b 4.38b The components of velocity are averaged in the gap direction (i.e. y direction) and the resulting velocity profile is written in the vector form as given below. -v nb (b P11-I v 2n+ 1K) p 4.40 When n = 1, the above equation reduces to Darcy's law. However, in the case of power- law fluids, it may be noted that the average velocity field is non-Darcian. The interface is assumed to be flat across the gap and moving with a constant velocity U. Then, the average velocity in the power-law fluid, ug, can be related to the velocity of the interface, U, as given below. The average velocity in the lateral direction is zero. ~ where b film thickness 4.41 b The location of the interface is denoted by x = rio which is a function of time and without loss of generality the pressure inside the inviscid fluid may be set equal to zero. Now, equation (4.41) is substituted in equation (4.40) and the steady state solution to the depth averaged pressure field can be obtained as given by the following equation. p S = _((2n + 1)_ U K( 1. ) nb ) b -)o 4.42 Equations (4.41) and (4.42) constitute the base state solution or the steady state solution for the stability analysis. The stability of this solution is analyzed following the standard procedure for the linear stability analysis. The steady state velocity and pressure field is perturbed by introducing disturbances in all the physical variables as given below. u = U + U' w= w' p=pS +p, x = 776 + i'(z, t) 4.43 Here the superscript prime indicates that it is a perturbation from the steady state solution. By substituting in equation (4.40), the relation between the perturbed velocity and pressure can be written as follows. I = bU[(2n+l)TU]-P'x 4.44a w' bU [(2n+l)TU -nP' 4.44b K [ nb 4.4 According to the linear stability theory, the perturbation in the velocity field can be written as follows. u'= y(x) exp[ikz+ 8t] 4.45 Here k is the wave number and 8 is the growth rate of the disturbance. Using equation (4.45) and the continuty equation (i.e. u + w' = 0), an expression for w' can be obtained. By substituting u', w' in equation (4.44) and eliminating pressure the following ordinary differential equation is obtained. y" nk2Y =0 4.46 Far away from the interface, into the power-law fluid region, the disturbance dies out. Hence the above differential equations can be solved assuming that as x -+ 0, T + 0. The resulting solution is as given below. y(x)= a exp(-nk2 x) 4.47 Now, the components of the perturbed velocity, u', w', are determined to a constant. By substituting them in equation (4.44) and integrating, the pressure field in the power-law fluid region can be determined. As indicated in equation (4.43), the instantaneous location of the interface is given by x, which can be differentiated to give instantaneous velocity in the x-direction as mentioned below. (x r76)t = 17t = U'x= o +rf 4.48 Equation (4.48) can be integrated with respect to time to obtain an expression for the perturbed location of the interface (i.e. rq'). The pressure jump across the interface can be written as Pi P' = r zz 4 The constant ;'/4 appearing in equation (4.48) was first derived by Park and Homsy (1984) as an 0(82) correction that accounts for the curvature change in the z-direction. Substituting the deviation variables in the above boundary condition, the dispersion relation for the growth rate of the disturbance can be written as follows. 15 kI_ nb b____k 4.50 T=n (2n+1) U 4K .50 When n is set equal to one, the dispersion relation for the Newtonian case is recovered (Saffman & Taylor 1958). Figure 4.6 is a graph of 8 as a function of the wave number. If 8 is positive then the disturbance grows at an exponential rate and the interface is unstable to small disturbances. If 6 is negative, then the disturbance is damped at an exponential rate and the motion is stable to small disturbances. As is evident from the graph, there is an optimum value for the wave number for which the growth rate is maximum and beyond which value, the growth rate monotonically decreases. So 5 = 0 0.040 0.035 0.030 0.025 Decreasing n 0.020 0.015 SO~lO /./ \\ kcut-.9 o0.010 0.005 k 0.000 -0.005 -0.010 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Wave number k Figure 4.6 Disturbance growth rate (8) as a function of wave number (k). (It = 100 cP, y'=1 s-', n = 0.6, b = 0.06 cm c = 30 dyns/cm, U = 0.05 cm/s) 0.8 0.7 0.6 0.5 0.4 0.0 0.2 0.4 0.6 0.8 Power-law index (n) Figure 4.7 Wave number for the maximum growth rate as a function of the power-law index. (0.=100 cP, y-= 1 sec1, U = 0.05 cm/s, a = 30 dyns/cm, b = 0.06 cm) represents the cut-off wave number whose expression is as given below. kc 4K J(2n+)t 451 ut- nb Hence the wave numbers greater than the cut-off wave numbers die out while the smaller ones grow at different growth rates. The wave number with the maximum growth rate can be obtained by differentiating 8 with respect to k and setting it equal to zero and it is related to the cut-off wave number as given below. k. ktff 4.52 In equation (4.51), the consistency index K can be replaced by apparent viscosity (g.) and a reference shear rate(y) as K = ,Uo/yn-I, then equation (4.52) can be plotted for various power-law fluids as given in Figure 4.7. While making this comparison, it should be kept in mind that the reference shear rate should be high enough so that power-law model is applicable to all the fluids being compared. Because, one of the limitations of power-law model is that it can not describe the viscosity behavior at low shear rates. The result in Figure 4.7 indicates that, the wave number for maximum growth rate (kn,) decreases as the power-law index reduces. In other words, the wave length for the maximum growth (km) is greater for power-law fluids. To experimentally observe this wave length, the width of the Hele-Shaw cell (L) should be larger than Xm. Hence, for fluids with similar viscosity at a constant reference shear rate, it may be concluded that the interface will be less sinuous for a power-law fluid than for a Newtonian fluid. 4.6 Summary The displacement of a power-law fluid by an inviscid fluid is analyzed and a leading order solution is obtained using the technique of matched asymptotic expansions. For slow displacement, the shape of the interface in the capillary statics region is a portion of a circle which is smoothly matched to a parabolic profile in the transition region. Under the lubrication approximation, the shape of the interface in the transition region is obtained by numerically integrating the evolution equation. Matching conditions with the capillary statics region determine the film thickness of Region I. The flow field in Region III is determined and found to be non-Darcian. Assuming that the interface is flat, the steady state solution for the velocity and pressure field is determined. Stability analysis for this steady state solution is carried out to determine the wave number with the maximum growth rate. The result indicates that the interface will be less sinuous for power-law fluids than Newtonian fluids. CHAPTER 5 CONCLUSIONS The experimental findings of Kopf-Sill and Homsy on the motion of air bubbles in a Hele-Shaw cell filled with Glycerin-water mixture have raised the following issues. " The velocities of small air bubbles were found to be an order of magnitude smaller than the theoretical predictions of Saffman and Taylor. " A variety of unusual steady bubble shapes and a sudden transition of bubble velocity and shape at a critical capillary number were observed. The present study explains the above observations in the context of surface active organic contamination. To study the influence of surfactants, experiments were conducted both using air bubbles and water drops containing surfactant in an oil-filled Hele-Shaw cell. While the motion of air bubbles is in agreement with the predictions of Saffman-Taylor theory, the velocity of water drops is an order of magnitude smaller than the predicted value. Unlike the air bubbles, the water drops exhibited a variety of shape transitions similar to the ones observed by Kopf-Sill and Homsy. In the presence of surfactants, the bubble motion can induce a surface concentration variation resulting in Marangoni stresses along the drop interface. These stresses can cause the drop surface to be rigid. In the presence of a wetting film, the dissipation in the thin film region can be significantly high and consequently the velocity of the water drops is decreased by an order of magnitude. The observed shapes can also be explained using the same argument. The motion of the bubble creates a spatial variation of surface tension. In order to keep the pressure constant, the curvature normal to the gap direction adjusts itself resulting in a variety of shapes. Theoretical modeling was done for an elliptic bubble assuming that the entire surface is rigid. The theoretical predictions agree with the experimentally observed velocities. This calculation also suggests that a sideways elongated shape is the preferred one in the presence of surfactants. In Chapter 4, the displacement of a shear thinning power-law fluid by an inviscid fluid is analyzed. The flow domain is divided into three different regions and a leading order solution is obtained using the technique of matched asymptotic expansions. Since the displacement is very slow, in the capillary statics region the shape of the interface is nearly circular. Under the lubrication approximation, the interface shape in the transition region is obtained by numerically integrating an evolution equation. Matching condition with the capillary statics region provides an expression for the film thickness in the constant film thickness region. In the case of Newtonian fluids, irrespective of individual physical properties, capillary number uniquely determines the film thickness. On the other hand, in the case of shear thinning power-law fluids, the relationship is unique to a given fluid and flow conditions. Just as in the Newtonian case, as the Capillary number is increased, the film thickness increases linearly on logarithmic scale. However, at a given capillary number, fluids with a lower power-law index have a smaller film thickness. The flow field away from the interface in the power-law fluid region is also solved assuming that the gradients of components of velocities are negligible in the directions normal to the gap direction. Unlike the Newtonian case, the depth averaged velocity field is found 79 be non-Darcian. A stability analysis is carried out for a flat interface and a dispersion relation is derived for the growth rate as a function of wave number. The result indicates that the wave number for the maximum growth rate decreases as the power-law index decreases. In other words, the dominant wave length increases as the fluid becomes more and more non-Newtonian. Hence, the interface will be less sinuous for the power-law fluid REFERENCES 1 G. I. Taylor and P. G. Saffman, "A note on the motion of bubbles in a Hele-Shaw cell and porous medium," Q.J.Mech.Appl.Math. 12, 265 (1959) 2 P.G. Saffman, and G.I.Taylor, "The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid," Proc. R. Soc. London Ser. A 245, 312 (1958). 3 S. Tanveer, "The effect of surface tension on the shape of a Hele-Shaw cell bubble," Phys.Fluids 29, 3537 (1986) 4 S. Tanveer, "New Solutions for steady bubbles in a Hele-Shaw cell," Phys.Fluids 30, 651 (1987) 5 T. Maxworthy, "Bubble formation, motion, and interaction in a Hele-Shaw cell," J.Fluid.Mech. 173, 95 (1986) 6 A. R. Kopf-Sill and G. M. Homsy, "Bubble motion in a Hele-Shaw cell," Phys.Fluids 31, 18 (1988) 7 W. Eck and J. Siekmann, "On bubble motion in a Hele-Shaw cell, a possibility to study two-phase flows under reduced gravity," Ing. Arch 47, 153 (1978) 8 W. Johann and J. Siekmann, "Migration of a bubble with adsorbed film in a Hele- Shaw cell," Acta Astronautica 5, 687 (1978) 9 C.-W. Park and G. M. Homsy, "Two-phase displacement in Hele-Shaw cells: Theory," J.Fluid.Mech. 139, 291 (1984) 10 D. A. Reinelt, "Interface conditions for two-phase displacement in Hele-Shaw cells," J.FluidMech. 183, 219 (1987) 11 S. Tanveer, and P.G. Saffman, "Stability of Bubbles in a Hele-Shaw Cell," Phys. Fluids 30, 2624 (1987). 12 P.G. Saffman, and S.Tanveer, "Prediction of bubble velocity in a Hele-Shaw cell: Thin film and contact angle effects," Phys. Fluids A 1, 219 (1989). 13 D. Burgess, and M.R.Foster, "Analysis of the boundary conditions for a Hele- Shaw bubble," Phys. Fluids A, 2, 1105 (1990). 14 G.M.Homsy, "Viscous Fingering in porous media," Ann.Rev.Fluid Mech.,19:271- 311 (1987) 15 C.W.Park, and G.M.Homsy, "The instability of long fingers in Hele-Shaw Flows," Phys. Fluids 28(6), 1583-1585(1985) 16 H.Lamb, Hydrodynamics, 6th ed. (Cambridge University Press, Cambridge, 1932), pp.581-584 17 S.Ross, and I.D.Morrison, Colloidal Systems and Interfaces, (John Wiley & Sons, New York, 1988). 18 M.J.Rosen, Surfactants and Interfacial Phenomena, 2nd ed., (John Wiley & Sons, New York, 1989). 19 R. E. Davis and A. Acrivos, "The influence of surfactants on the creeping motion of bubbles," Chem. Eng. Sci. 21, 681 (1966) 20 C.-W. Park, "Influence of soluble surfactants on the motion of a finite bubble in a capillary tube," Phys.Fluids A 4, 2335 (1992) 21 K.J.Stebe, S.Y.Lin, and C.Maldarelli, "Remobilizing surfactant retarded fluid particle interface. I. Stress-free conditions at the interfaces of micellar solutions of surfactants with fast sorption kinetics," Phys. Fluids A 3, 3 (1991). 22 M.Van Dyke, Perturbation Methods in Fluid Mechanics, (Academic Press, New York, 1964) 23 F. P. Bretherton, "The motion of long bubbles in tubes" J.FluidMech. 10, 166 (1961) 24 B.Levich, Physicochemical Hydrodynamics, (Prentice Hall Englewood Cliffs, NJ, 1962). 25 G.I.Taylor, "Deposition of viscous fluid on the wall of a tube," J. Fluid Mech. 10, 161 (1961). 26 S.S.Sadhal, and R.E.Johnson, "Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film-exact solution," J. Fluid Mech. 126, 237 (1983). 27 J. Ratulowski and H.-C. Chang, "Marangoni effects of trace impurities on the motion of long gas bubbles in capillaries," J. Fluid. Mech. 210, 303 (1990) BIOGRAPHICAL SKETCH Krishna Maruvada was born in Andhra Pradesh, India. He got his bachelors degree in chemical engineering from Birla Institute of Technology & Science (BITS), Pilani, India, in 1990. Subsequently he worked for 2 years for a chemical company called Sriram Fibers, at New Delhi as a projects engineer. Later he joined BITS, Pilani and served as a lecturer for one year. In 1993, he joined the graduate program at the University of Florida, Gainesville, and started working toward his Ph.D. degree. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Chang-*j Park, Chairman Associate'lrofessor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Dinesh 0. Shah Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ranga Narayanan Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Lewis Johns 4 Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Wei Shyy Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Richard B. Dickinson Assistant Professor of Chemical Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1997 Winfred M. Phillips Dean, College of Engineering Karen A. Holbrook Dean, Graduate School |

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MARUVADA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1997 ACKNOWLEDGMENTS During my stay at the University of Florida, I had the privilege to interact with many talented people who have influenced me in several ways. First among those is my research advisor Dr. Chang-Won Park whose enthusiasm for research and pursuit of excellence are admirable. Much of my learning is due to several insightful research discussions I had with him during my stay here. I wholeheartedly acknowledge the constant encouragement, guidance and support I have received from him during the course of this research work. I thank Dr. Ranga Narayanan, Dr. Lewis Johns, Dr. Dinesh Shah, Dr. Richard Dickinson and Dr. Wei Shyy for serving on my committee and providing helpful suggestions. I also thank all the members of our research group, Mr. Charlie Jacobson, Mr. Yongcheng Li, Ms. Keisha Wilson, Mr. John Bradshaw, for their cooperation and support. A task of this magnitude can not be completed without the support of my beloved family members; my parents, my wife, my brother, my sister and brother-in-law who have helped me in every way they can, to help convert my dream into a reality. With great happiness, I acknowledge their generous support and encouragement. Finally I would like to acknowledge financial support by the donors of the Petroleum Research fund and NASA , Microgravity Science and Applications Division. li TABLE OF CONTENTS page ACKNOWLEDGMENTS ii LIST OF FIGURES v LIST OF SYMBOLS vii ABSTRACT x CHAPTERS 1 INTRODUCTION 1 Background 1 Motivation 4 Dissertation Outline 9 2 EXPERIMENTS 12 The Influence of Surfactants on Bubble Motion 12 Experiments 18 Results and Discussion 23 Summary and Conclusions 37 3 ESTIMATION OF BUBBLE VELOCITY 38 Introduction 38 Theoretical Analysis 38 Comparison with experimental results 46 Summary 50 4 DISPLACEMENT OF A POWER-LAW FLUID BY INVISCID FLUID 51 Background 51 Basic Equations 53 Scalings and Regions 54 Small parameter expansions 60 Capillary Statics Region : 60 in Transition Region : 62 Matching Conditions 65 Stability Analysis 69 Summary 76 5 CONCLUSIONS 77 REFERENCES 80 BIOGRAPHICAL SKETCH 82 IV FIGURE LIST OF FIGURES PAGE 1.1 Schematic of a moving bubble in a Hele-Shaw Cell (a) View in the xy plane (b) Bubble in the xz plane 2 1.2 Various Bubble shapes observed by Kopf-Sill and Homsy (a) Near-Circle; (b) Flattened bubble; (c) Tanveer Bubble; (d) Ovoid; (e) Short-tail bubble; and (f) Long-tail bubble 5 1.3 Here U/V vs Ca (Data taken from experiment series 1,2, and 3 of Kopf-Sill and Homsy) 7 2.1 A water drop sitting on a Pyrex glass plate immersed in silicone oil 22 2.2 Evolution of water drop shape with increasing capillary number. (a)Near circular drop; (b) flattened drop; (c) severely stretched drop; (d) ovoid; (e) transitional shape; and (f) short tail drop 24 2.3 U/V vs. Capillary number for air bubbles 26 2.4 U/V vs. Capillary number for small water drops 31 2.5 U/V vs. Capillary number for large water drops 33 2.6 U/V vs. Capillary number for Fluorocarbon oil drops 35 3.1 Plan view and side view of an elliptic bubble and a control volume moving with the bubble 42 3.2 Numerical value of the definite integral, Equation 3.11 for various values of the shape factor k 45 3.3 Comparison of experimental results and the theoretical estimate for pressure driven flow (SDS concentration : â–¡ 5% ; A 10%; x 20% of CMC) 48 4.1 Schematic of an inviscid fluid displacing a power-law fluid in a Hele-Shaw cell. ..52 4.2 Schematic showing the scales and coordinate systems for different regions. (A: Transition region; B: Capillary-statics region) 57 4.3 Comparison of interface profiles for Newtonian (Â«=1) and power-law (n = 0.5) fluids 64 4.4 Plot of the Co-efficient C0 as a function of the power-law index 67 4.5 Log (film thickness) vs Log Ca for a power-law fluid. (p0 = 100 cP; ct = 30 dyns/cm; y0 = 1 sec'1; b = 0.06 cm) 68 4.6 Disturbance growth rate (6) as a function of wave number (k). (p0 = 100 cP, y0 =1 s'1, n - 0.6, b = 0.06 cm , ct = 30 dyns/cm, U= 0.05 cm/s) 73 4.7 Wave number for the maximum growth rate as a function of the power-law index. (po=100 cP, y0= 1 sec'1, U = 0.05 cm/s, a = 30 dyns/cm, b = 0.06 cm) 74 vi LIST OF SYMBOLS a A b b c QÂ» Câ€ž C2 Ca Caâ€ž D D m FD h H h(x,z) I k k K L / n n n P P Dimension of the bubble in the flow directions Planform area of the bubble Dimension of the bubble transverse to the flow direction Half-gap thickness as used in Chapter 4 Constant that appears in Equation 3.5 Constants that appear in Equation 4.27 Capillary number Ca based on the normal component of the bubble velocity Domain encapsulating the bubble Rate of strain tensor Equation of the interface in z = constant plane (in the plane of the cell) Drag force acting on the bubble Gap thickness of the Hele-Shaw cell as used in Chapter 3. Interface profile for the transition region in the scaled form Equation of the interface in y = constant plane (in the gap direction) Constant as given in Equation 3.11 Shape factor for the elliptic bubble Exponent that appears in equations 4.9 - 4.14 Consistency factor Width of the Hele-Shaw cell Location of origin in the transition region Unit normal to the surface of the control volume Power-law index Unit normal to the interface Depth averaged pressure filed Pressure field in the power-law fluid VÃœ q s s s t T *1Â» *2 u u u* u,v,w V V ^avg X Greek Letters a P P 5 8 O q p e p CT X Q r Scale factor for the elliptic cylindrical coordinate system Surface of the control volume = y (Square of the shear rate) Shift factor Thickness of the wetting film Stress tensor Unit tangent vectors to the interface Velocity of the drop/bubble Velocity of the moving interface near the tip Unretarded bubble velocity in a buoyancy driven flow Components of velocities in the x,y,z directions Velocity field in the driving fluid away from the bubble (Chapter 3) Average velocity of the driving fluid (Chapter 3) Depth averaged velocity field in the driving fluid Scaled variable for the x-dimension Gauge function in the expansions Retardation factor as given by Equation 3.16 Constant that appears in Equation 4.24 Growth rate of the disturbance Small parameter (Ratio of natural length scales of Hele-Shaw cell) Velocity potential Shear viscosity of the power-law fluid Viscosity of the driving fluid Angle of inclination of Hele-Shaw cell Density of the driving fluid Equilibrium interfacial tension Stress tensor Complex potential Location of the edge of the control volume Directions in the elliptic cylindrical Coordinate system viii Â¥ Subscript x,y,z Superscript Â°Â°,01,etc... Location of the bubble surface Stream function Partial differentiations with respect to the variables. â€œOver barâ€ indicates variable in the transition region Leading order, Order Caâ€œ etc.. IX Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DYNAMIC INTERFACIAL PROPERTIES AND THEIR INFLUENCE ON BUBBLE MOTION By Krishna S. Maruvada December 1997 Chairman: Dr. Chang-Won Park Major Department: Chemical Engineering The influence of surfactants on the motion of bubbles and drops in a Hele-Shaw cell is studied by carrying out experiments in a variety of fluid-fluid systems. Saffman- Taylor theory predicts the translational velocities and shapes of drops and bubbles, not influenced by surfactants. In order to differentiate the cases with and without surfactants, the motion of air bubbles and water drops driven by silicone oil was studied. A controlled amount of surfactant is added to the water drops in order to study its influence on the bubble motion. In the case of air bubbles in silicone oil, surfactants were likely to have negligible influence, although they may be present as contaminants. As expected, air bubbles were moving much faster and their shapes were also in accordance with the available theories for the surfactant free system. On the other hand, surfactant containing water drops moved with much smaller velocities and their shapes were also distorted. These observations are apparently consistent with those of Kopf-Sill and Homsy. When a surfactant containing drop is set in motion, the resulting spatial variation of surfactants at x the interface induces Marangoni stresses along the interface, and consequently the drop surface may become rigid. If the surrounding fluid is the wetting phase, the net drag force experienced by the drop in the wetting film region is significantly high resulting in smaller drop velocities. Because the surface tension has a spatial variation, to keep the pressure inside the drop constant, the curvature adjusts itself resulting in a variety of bubble shapes. Assuming that the bubble surface is rigid due to surfactant influence, the translational velocity of an elliptic bubble is estimated. The result indicates that the bubble velocity can decrease by an order of magnitude compared to the prediction of Taylor and Saffman. The retarded bubble velocity is apparently in reasonable agreement with the experimental observations suggesting that the puzzling observations of Kopf-Sill and Homsy are likely to be due to surface active contaminants. Finally, two phase displacement of a shear thinning power-law fluid by an inviscid fluid is analyzed. The flow domain is divided into three different regions and a leading order solution to the flow field and the interface shape is obtained using the technique of matched asymptotic expansions. In the case of Newtonian fluids, capillary number uniquely determines the film thickness. On the other hand, in the case of shear thinning power-law fluids, the relationship is unique to a given fluid and flow conditions. Finally, the flow field away from the interface in the power-law fluid region is also solved assuming that the gradients of components of velocities is negligible in the directions normal to the gap direction. Unlike the Newtonian case, the depth-averaged velocity field is found be non-Darcian. A stability analysis is also carried out to determine the wave number with the maximum growth rate. The result indicates that the interface will be less sinuous for the power-law fluid than the Newtonian fluid. xi CHAPTER 1 INTRODUCTION 1.1 Background Hele-Shaw cell has been used as a model system to study various physical phenomena involving multiphase flows in porous media. Although the concept of Hele- Shaw cell is more than a century old, there has been a lot of interest in recent years in the motion of two phase flows in a Hele-Shaw geometry.1'15 The displacement of a more viscous liquid using a less viscous liquid and the related fingering phenomena have been extensively studied both experimentally and theoretically, using this model.2,14'15 When the fluid-fluid interface is a closed contour, the inner fluid is referred to as a drop or bubble depending upon whether the enclosed fluid is a liquid or gas. A typical Hele-Shaw cell consists of a thin gap between two flat plates. The gap thickness, which is of the order of a millimeter, is very small compared to the dimensions of the plates. The gap is filled with a liquid into which a secondary fluid is injected to form a drop or bubble and its motion can be studied. The injected drop or bubble is squeezed between the two plates and assumes a circular planform when it is stagnant because of the interfacial tension. If the surrounding fluid is the wetting phase, a thin film exists between the bubble surface and the solid boundary of the Hele-Shaw cell. The bubble can be set in motion either by applying a pressure gradient in a horizontal Hele- Shaw cell or by imposing a buoyancy force by tilting the cell. If the characteristic length 1 gap (a> ViW , m the xy 3 scale of the bubble in the transverse direction is larger than the gap thickness (a/h Â»1, Figure 1.1), the dynamics of the bubble motion can be described by Hele-Shaw equations'6 which are a two dimensional description of the gap-averaged velocity and pressure fields. This approximation is valid over the entire domain expect for a thin region (of the order of gap thickness) surrounding the rim of the bubble. The depth averaged equations resemble the Darcy approximation of the governing equations for the flow through porous media. Hence the Hele-Shaw cell can also be used to study flow through porous media. Taylor and Saffman' have solved the Hele-Shaw equations for the motion of bubbles in a thin gap with no side walls assuming negligible surface tension. Their solution predicts the bubble shape as a function of the area and bubble velocity. Furthermore they have concluded that for a freely suspended bubble, the bubble velocity (U) is always greater than the surrounding fluid average velocity (V) and the steady shapes are ellipses. Also if U < 2V these elliptic bubbles travel with their minor axis parallel to the flow direction and if U > 2V they travel with the major axis parallel to the flow direction. If U = 2V the bubble shapes are circles. Tanveer5 had incorporated the effect of finite surface tension to resolve the degeneracy of the Taylor-Saffman solution and obtained three solution branches. A stability analysis4 of these three solution branches indicate that only one of them (McLean-Saffman branch) is stable. The inclusion of interfacial tension in the theoretical analysis does not alter Taylor-Saffman's conclusion about the relative velocities of the bubbles and the surrounding fluid. 4 Bubble motion was experimentally studied be several researchers using a variety of fluid combinations.5'8 Maxworthy5 had experimentally studied the motion of air bubbles in an oil filled Hele-Shaw cell under the influence of gravity. At smaller values of the inclination angles, stable circular bubbles or elliptic bubbles elongated in the flow direction are observed. However at higher angles of inclination, bubbles became unstable with severe shape distortions. When the bubble size relative to the transverse cell dimension (a/L) was larger than 0.1, the bubble velocity was slightly larger than the prediction of Taylor and Saffman.1 With decreasing bubble size, however, the bubble velocity decreased to a value smaller than the predicted one. This discrepancy in the bubble velocity was apparently resolved by including the viscous dissipation around the bubble edge in calculating the bubble velocity. 1.2 Motivation In 1988, Kopf-Sill and Homsy6 conducted similar experiments to investigate the shapes and velocities of various size bubbles driven by a pressure gradient. They investigated the motion of air bubbles in a horizontal Hele-Shaw cell filled with a glycerin-water mixture. Unlike the observations of Maxworthy, a variety of unusual bubble shapes were observed as shown in Figure 1. 2. Besides the near-circular and elliptic bubbles, "Tanveer" bubbles, ovoids, long- and short-tail bubbles were observed. By tracing the locations of the leading and trailing edges of the bubbles for an extended period of time, they showed that these unusual shapes were not transient but steady. Furthermore, they found that at low capillary numbers the translational velocities of the bubbles (near-circle, flattened or elongated ellipses, Tanveer) were much smaller than (a) (c) (b) (d) (e) \ \ I (f) Figure 1.2 Various Bubble shapes observed by Kopf-Sill and Homsy: (a) Near-Circle; (b) Flattened bubble; (c) Tanveer Bubble; (d) Ovoid; (e) Short-tail bubble; and (f) Long-tail bubble 6 expected. Especially, the velocity of near-circular bubbles was about an order of magnitude smaller than the prediction of Taylor and Saffman. In Figure 1.3, the velocity data taken from Kopf-Sill and Homsy5 are given for three different bubble sizes. The capillary number in the x-axis is defined as pV/a where p and V are the shear viscosity and the average velocity of the glycerin-water mixture, respectively, and a is the surface tension. When Ca is small, interfacial tension is dominant resulting in a circular bubble shape. According to the theoretical prediction,' the velocity of a circular bubble relative to the average fluid velocity (i.e., U/V) should be 2. Contrary to the prediction, the relative bubble velocity was observed to be only about 0.2. The very low bubble velocity at a small capillary number was common for all bubbles regardless of their size. As Ca was increased, the relative bubble velocity increased slightly until it jumped suddenly to a much higher value at a certain Ca. The sudden jump in velocity was apparently accompanied by a drastic change in bubble shape. While the similar trend was observed with all bubbles of different sizes, the sudden transition in the bubble shape and velocity occurred at a lower value of Ca for larger bubbles. They also reported a hysteresis in the transitional behavior. It should be noted that the Taylor-Saffman solution exists only when the relative bubble velocity U/V is greater than 1. Although the possibility of U/V < 1 cannot be ruled out for non-zero surface tension, Saffman and Tanveer'2 point out that such a low velocity of small bubbles cannot be explained by a simple inclusion of the surface tension 7 CaxlO4 ^2.06cms Ql.3cms ^.0.76 cms Figure 1.3 Here U/V vs Ca (data taken from experiment series 1,2, and 3 of Kopf-Sill and Homsy) 8 effect and/or the wetting film effect. They suggested that the low bubble velocity might be due to a moving contact line dragging along the cell plates. Using an ad hoc boundary condition, they showed that a moving contact line could account for the very low bubble velocity. In Kopf-Sill and Homsy's experiment, however, the liquid phase wetted the solid surface and wetting films were always observed. Therefore, the presence of a moving contact line was unlikely. We may note that the surprising variety of the steady bubble shapes and the striking discrepancy in bubble velocities were observed only in a pressure driven flow. Unlike Kopf-Sill and Homsy, Maxworthy did not observe any unexpected shapes but only elliptic bubbles elongated in the flow direction. In his experiments, the bubble velocities were also in disagreement with the Taylor-Saffman theory. The disagreement, however, was not as great as in the Kopf-Sill-Homsy experiments and it could be explained by the inclusion of viscous dissipation at the bubble rim. In the work of Kopf- Sill and Homsy, the experiment of Eck and Siekmann7 is quoted in which the translational velocity of small bubbles were also much smaller than the predicted value. These observations apparently suggest that the pressure-driven and buoyancy-driven flows may be intrinsically different although Saffman and Taylof pointed out the equivalency between the two flows. The experiments of Kopf-Sill and Homsy involve several observations that are in need of explanation. These observations include (1) the velocity of small bubbles which appear to be an order of magnitude smaller than the theoretical prediction, (2) a surprising variety of steady bubble shapes, and (3) the sudden transition of bubble velocity and 9 shape at a certain capillary number. Johann and Siekmann8 have measured the velocities of air bubbles in a water system containing steric acid. Their reported velocities also appear to be significantly retarded compared to the theoretical prediction. In all experimental studies described above, the Hele-Shaw cell was supposed to be clean and free from any contaminants. Many fluid flow systems, however, are susceptible to contamination and it seems that the perplexing observations of Kopf-Sill and Homsy may be explicable if we adopt the notion of the influence of surface-active contaminants on bubble motion. One of the objectives of this study is to provide a plausible explanation for the observations of Kopf-Sill and Homsy in the context of surfactant influence. 1.3 Dissertation Outline In order to investigate the influence of surfactants on the bubble motion, we conducted experiments using air bubbles, water drops and fluorocarbon oil drops in a Hele-Shaw cell filled with silicone oil. Chapter 2 presents our experimental findings and compares them with those of Kopf-Sill and Homsy. Air bubbles were chosen since the surface properties of an air-oil interface were less likely to be affected by surfactants which might be present as contaminants. It is well known that unlike the surface tension of water or the interfacial tension between water and a hydrocarbon, the surface tension of non-aqueous solutions is not affected significantly by surfactants.17 In case of water drops, a predetermined amount of surfactant (sodium dodecyl sulfate) was added so that the surfactant influence could be studied systematically. The motion of the water drops were found to be drastically retarded by the surfactant, whereas the air bubbles were moving much faster as we anticipated. 10 When surface active substances are present in the system, they adsorb near the fluid-fluid interface. Under dynamic conditions, depending on the surfactant exchange kinetics, a spatial variation of surface concentration can arise resulting in Marangoni stresses.18 As a consequence, the interface can behave as rigid surface.19 20 Hence in the thin film region, the drag increases significantly causing severe retardation of the bubble. Chapter 3 presents a theoretical calculation that predicts the extent of retardation and provides an analytic expression for the drop velocity at low surfactant concentrations. As the bulk surfactant concentration is increased, Marangoni effects are expected to be weaker and a complete remobilization is anticipated.21 Contrary to this expectation, the motion of a drop containing a high concentration of surfactant is also found to be retarded. Although it is not very clear as to what causes the retardation at high concentrations, it could possibly be due to the influence of interfacial viscosities. At high concentrations the interface gets completely saturated and the intrinsic viscosities of the surface can play a role in retarding the drop motion. An effort to validate this idea theoretically is only partially successful. Finally, Chapter 4 presents the analysis of the displacement flow of a shear thinning power-law fluid by an inviscid fluid in a Hele-Shaw geometry. This problem has practical applications in several methods of polymer processing such as injection molding using various gases. The technique of matched asymptotic expansions22 is used to determine the flow field and the shape of the interface. The flow domain is divided into three different regions and matching conditions between them provide the film thickness left behind by the advancing meniscus. Unlike the Newtonian case, the relationship between capillary number and film thickness is not unique and it varies from fluid to 11 fluid. For power-law fluids, the depth averaged flow field is non-Darcian. A linear stability analysis is also carried out to determine the dispersion relation for the shear thinning power-law fluid. Chapter 5 summarizes the findings of this work. CHAPTER 2 EXPERIMENTS In this chapter, the influence of surfactant on the bubble motion in a Hele-Shaw cell is described qualitatively, and the difference between pressure-driven and buoyancy- driven flows in the presence of surfactant is discussed. Subsequently, the results of the present experiments are presented to support the arguments described below. 2.1 The Influence of Surfactants on Bubble Motion As mentioned in Chapter 1, the bubble can be set in motion either by applying a pressure gradient in a horizontal Hele-Shaw cell or by imposing a buoyancy force by tilting the cell. When the bubble is driven by buoyancy through a stagnant fluid in a Hele- Shaw cell, the sweeping motion of the surrounding liquid in a bubble fixed coordinate system is always from the front to the trailing end of the bubble. If the bubble is driven by the surrounding liquid, on the other hand, two different flow situations may arise depending on the relative magnitude of the bubble velocity U to the average velocity of the surrounding fluid V. If U > V, the sweeping motion of the surrounding liquid is toward the back of the bubble as in the buoyancy driven flow. Whereas if U < V, the sweeping motion along the edge of the bubble is from the rear to the leading end of the bubble. In the absence of surface-active substances, the surface tension is constant and the two flow situations may be indifferent. They are, however, distinctly different if there exists a spatial variation of surface tension due to the presence of surfactant. 12 13 When U > V, surfactants will be accumulated at the trailing end of the bubble thus lowering the surface tension at the back of the bubble. When U < V, the flow situation is more complicated. At the edge (or rim) of the bubble, the surrounding fluid is convected from the back to the front of the bubble. Thus, surfactants are convected toward the bubble front along the edge. In the thin film region between the wall and the bubble, on the other hand, the dominant convection is always toward the back of the bubble since the solid wall is moving backward relative to the bubble and the liquid film is very thin. Therefore, the surfactant distribution is complicated and may depend on the balance 9 between the two competing effects. The studies of Park and Homsy and Burgess and Foster' indicate that stagnation points (or rings) are present around the bubble at the transition region between the thin film region and the cap region (i.e., points s in Figure 1.1). These stagnation points are similar to those in the front and the back of the 23 Brethertons bubble in a capillary tube. Although the location of the stagnation points may move toward the cap region when surfactants are present,20 the existence of the stagnation points (or rings) may result in three regions of surfactant accumulation; one near the front cap of the bubble and the other two in the thin film regions (top and bottom) near the back of the bubble. It should be noted that this complicated flow situation cannot occur in a buoyancy driven flow. Therefore, the pressure-driven and buoyancy-driven bubble motions can be different from one another if surfactants are present in the system. The most interesting bubble shapes that were observed by Kopf-Sill and Homsy may be the ovoids with the sharper end pointed in the flow direction and the bubbles with a 14 long- or a short-tail. When the average liquid velocity V was low (i.e., at a low Ca where the surface tension is dominant), the bubbles were observed to be near circular. With increasing V, they became slightly deformed to take on the shape of an ovoid. As V was further increased, the bubble shape went through a sudden transition to the long-tail (or short-tail) shape. Interestingly, a hysteresis was observed in that the backward transition from long-tail to ovoid occurred at a lower value of V than the forward transition from ovoid to long-tail for the same bubble. One important aspect which should be noted is that the ovoids apparently moved with a velocity smaller than the average liquid velocity (i.e., U < V) whereas the velocities of the short- or long-tail bubbles were larger than V. That is, the sudden increase (or decrease in case of a reverse experiment) in the bubble velocity cuts across the U/V=l line in Figure 1.3. Thus, the flow situations for the ovoids and the bubbles with a tail are different in that the direction of the sweeping motion is opposite. We may point out, however, that a group of four data points in Figure 1.3 (a=0.76 cm) does not follow this argument as their relative velocities were reported to be about 0.8 although the shapes were short-tail. It may be possible that this discrepancy is due to measurement error. When the surrounding liquid sweeps the bubble from back to front along the edge (i.e., when U < V), the local interfacial tension may be lower in the front region of the bubble due to the surfactant accumulation there. Since the pressure inside the bubble is constant, the radius of curvature at the bubble front should be smaller than that at the back to compensate for the lower interfacial tension. Therefore, assuming a constant curvature in the gap direction (i.e., the z-direction in Figure 1.1), an ovoid with the 15 sharper end pointing in the flow direction is a plausible shape. The same argument may be applicable even in the absence of surfactants. In that case, however, the difference in curvature between the front and rear ends of the bubble will not be as large as the surfactant-influenced case. By applying the similar argument to the case of U > V, we may also speculate that a drastic curvature change in the rear part of the bubble (hence the long-tail or short-tail bubbles) is possible if surface-active substances are present. This qualitative explanation does not rule out the possibility that the long-tail bubbles can also exist in a buoyancy driven flow since U is always greater than V in that case. A probable reason why it was not observed in Maxworthy's experiment is that silicone oil was used instead of glycerine-water mixture in his experiments. As it was pointed out previously, the air-oil interface is less likely to be affected by surface-active contaminants. Under the assumption that surface-active contaminants were present, the overall behavior of a bubble observed by Kopf-Sill and Homsy may be explained as follows. When an air bubble is stagnant in a contaminated Hele-Shaw cell, its entire surface will be covered with surface-active contaminants. At a very low Ca, the low shear stress may induce only a small change in surfactant distribution. Thus the Marangoni effect may be global. Although the Marangoni stress may be small due to the small surface tension gradient, the shear stress is also small and the entire bubble surface may act as a rigid 24 surface. In the presence of the stagnation rings near the top and bottom wall (Figure 1.1), three separate regions of high surface concentration can coexist as pointed out previously. Along the edge of the bubble, surfactants may be swept toward the bubble 16 front and consequently, the Marangoni effect may assist the bubble motion. In the thin film region at the top and bottom of the walls, on the other hand, surfactants are swept backward retarding the bubble motion. When Ca is small, the viscous stress along the bubble edge may not be as significant as that in the thin film region which is inversely proportional to the film thickness. Therefore the drag in the wetting film is dominant retarding the bubble motion significantly. Consequently, the relative bubble velocity (U/V) may remain much smaller than 1. Although the bubble motion can be severely retarded, the shape change may be insignificant due to the small spatial variation of the surface tension. Consequently, the bubble shape may be near-circular. As Ca is increased, the larger viscous stress along the bubble edge increases the surface tension gradient resulting in larger Marangoni stress. In the thin film region, on the other hand, the relative increase of the drag may not be as significant since the film thickness increases with the bubble velocity. Therefore, the relative bubble velocity may increase with increasing Ca although its rate of increase may not be very large. Furthermore, the surfactant concentration at the leading end of the bubble will also increase with Ca. Consequently, the surface tension will become lower at the leading end creating a larger curvature. Therefore, the ovoid with the sharper end pointing in the direction of the flow may be a favorable shape. With increasing Ca, the film thickness increases and the location of the stagnation rings may move toward the tip of the bubble edge. Eventually the two stagnation rings may collapse into one along the tip of the bubble edge.'5 Then the three spearate regions of high surfactant concentration cannot coexist. Consequently, the large drag in the thin 17 film region may decrease significantly inducing higher bubble velocity which may be greater than the average velocity of the surrouding fluid (i.e., U/V > 1). Once the relative bubble velocity becomes greater than 1, all surfactants will be collected at the trailing end of the bubble lowering the surface tension there. Consequently, a larger curvature is required forming a high curvature tail. The size (or length) of the tail may vary depending upon the surfactant distribution at the rear end. In an extreme case where the bubble is large enough and the surfactant transport is limited by bulk diffusion, the leading end of the bubble may be virtually free from surfactant due to convection whereas the trailing end is full of surfactants. 19'2Â°â€™â€˜'6 In such a case, a long cylindrical tail may be conceivable as a form of minimizing viscous dissipation while maintaining a large curvature for the pressure balance across the interface. The Marangoni effect is greater if the surface curvature is larger. Consequently, its strong influence may persist to a higher value of Ca if the bubble is smaller. Therefore, the critical value of Ca where the transition occurs may increase with decreasing bubble size. If the capillary number is decreased gradually from a large value to a small value, the reverse trend may be observed in which the bubble goes through a transition from a long- or short-tail bubble to an ovoid and eventually to a near-circle. The forward and backward transitions, however, are not necessarily reversible as the adsorption and desorption kinetics of surfactants may not be reversible. Thus, the hysteresis in the transitional behavior can occur if such a change is driven by the presence of surfactants. The argument presented above is rather simple and only qualitative. Nevertheless, it apparently accounts for most of the observations of Kopf-Sill and Homsy. In the 18 following section, the results of the present experiments are given as a supporting evidence. 2.2 Experiments The Hele-Shaw cell consisted of two 1/2-in thick Pyrex glass plates separated by a rubber gasket of 0.9 mm or 1.8 mm in thickness. The effective cell dimension was 17.8 cm by 86.4 cm. A silicone oil with the measured viscosity and surface tension of 97 cp and 21 dyn/cm was used as the driving fluid. The cell had a deep groove channel at both ends so that a uniform distribution of the oil along the transverse direction could be ensured when it was injected. The cell also had an injection port at one end of the top plate so that an air bubble or a water drop could be introduced into the cell using a syringe without creating extra bubbles or drops. The planform diameter of the stagnant bubbles and drops was controlled to be at about 1.3 cm and 2.1 cm throughout the experimental study. These sizes were chosen to be about the same as those of Kopf-Sill and Homsy for direct comparison of the results. While the size variation among the small bubbles was small as they ranged between 1.3 cm and 1.4 cm, the size control was more difficult with larger bubbles and their size ranged between 2.2 cm to 2.5 cm. Water drops containing an organic surfactant at various levels of concentration were prepared by dissolving sodium dodecyl sulfate (SDS) in distilled water. Since the surfactant influence is more significant at low concentrations, the SDS concentration was set at 5%, 10%, 20%, and 100% of the critical micelle concentration (CMC), respectively. The CMC of SDS in water at 25Â°C is 8.2 mmol/I which is equivalent to 0.236% by weight. The interfacial tension measured at room temperature by the Wilhelmy plate 19 method is listed in Table 2.1 at various SDS concentrations. As the present data for air- water interface is found to be consistent with those reported in literature,26 the data for the silicone oil-water interface may also be reliable. It can be deduced from the data that the rate of change of interfacial tension is greater at a lower concentration indicating a stronger Marangoni effect at a lower concentration. Once an air bubble (or a water drop) of predetermined size was positioned at one end of the cell, the silicone oil was driven by a peristaltic pump at various flow rates to induce the bubble motion. The translational velocity and the shape of the bubble (or drop) were then recorded for each value of oil flow rate. For the prescribed dimensions, the values of both a/L and h/a were smaller than 0.14. Thus the influence of the side wall (i.e., the rubber gasket) may be small and the depth-averaged Hele-Shaw equations may be applicable to describe the bubble motion if the system is free from surfactants. In order to avoid an excessive pressure drop at high flow rates, most of the experiments were conducted with the thick gasket (1.8 mm) and only a part of the air bubble experiments used the thin gasket (0.9 mm) for comparison. The Bond number (Bo) which is defined as pgh^/a is a measure of the relative importance of gravity to surface tension, and a large Bo may indicate asymmetry of the bubble in the gap. In the experiments of Kopf-Sill and Homsy, Bo was as small as 0.1 whereas it was about 1.5 in Maxworthy's. In the present experiment, it was either 0.4 for the 0.9 mm thick gasket or 1.5 for the 1.8 mm thick gasket. The large Bo was caused not only by the thick gasket but also by the low interfacial tension. While the gravity effect 20 might not be significant when Bo = 0.4, uncertainty existed when Bo was as large as 1.5. The experimental results for air bubbles, however, were indifferent in terms of both bubble velocity and shape to whether Bo was 0.4 or 1.5. Thus, the gravity effect might not have significant influence even when Bo = 1.5. In reporting the results in the following section, no distinction has been made between the data for Bo = 1.5 and 0.4 in order to avoid unnecessary complication. The size of air bubbles changed slightly during the experiment due to the pressure drop in the cell. Kopf-Sill and Homsy observed similar behavior and reported the volume change of air bubbles up to 20%. In the present experiment, not only was the gap thickness of the cell slightly larger but also the liquid viscosity was lower. Furthermore, the liquid flow rate was 0.75 cm^/sec when the capillary number was as high as 110. Consequently, the pressure drop was smaller and the change in the bubble volume was less than about 5%. Despite the change in the bubble volume, variation in the bubble velocity was not detectable. In case of water drops, neither the volume nor the velocity changed. The description in Section 2.1 regarding the surfactant influence on the bubble motion assumes the existence of a thin oil film between the wall and the moving bubble or drop. When oil is the wetting phase, the film thickness is uniform in the direction of bubble or drop motion (i.e., x direction in Figure 1.1) whereas it varies in the cross-stream direction (i.e., y direction in Figure 1.1). Therefore, it may be possible in principle to prove the existence of the thin film by an optical method which creates interference fringes. Our attempt to observe the interference fringes, however, was unsuccessful due 21 to various difficulties involved with the large size of the cell. Thus we provide only a qualitative evidence to support the assumption. A photograph converted from a video image is shown in Figure 2.1 for a water drop of about 2 mm in diameter which is sitting on a Pyrex glass plate immersed in the silicone oil. In order to improve contrast, a blue food coloring was added to the water drop. Due to the difficulty in obtaining perfectly horizontal alignment for the contact angle measurement, the water drop was viewed with a small positive angle from the horizontal and a shadow of the drop was created in front by indirect back lighting. The lower part of the picture is the shadow of the drop and the bright line between the drop and its shadow represents the three-phase contact line. At the edge of the contact line, the apparent static contact angle is shown to be greater than 90Â° when measured from the water-drop side. This may suggest that silicone oil is the wetting phase under dynamic conditions. When a water drop is stationary in the Hele-Shaw cell at the beginning of each experiment, the water drop may have direct contact with the Pyrex glass wall creating a three-phase contact line. As we induced the drop motion by injecting the silicone oil into the cell, the trailing edge of the drop exhibited briefly a ragged contact line pinned on the wall. The contact line soon detached from the wall and the water drop took on a steady shape with a perfectly smooth edge around the entire drop (Figure 2.2). If a moving contact line existed, its motion might have been jerky at times unless the solid surface was perfectly homogeneous. In the present experiments, the drop motion was always smooth excluding the brief moment at the start of motion. In case of air bubbles, even the 22 Figure 2.1 A water drop sitting on a Pyrex glass plate immersed in silicone oil. 23 start-up motion was smooth without any evidence of contact line. These observations along with the observation on the apparent static contact angle seem to support the assumption regarding the existence of thin oil films. 2.3 Results and Discussion In Figure 2.2, shapes of a water drop containing surfactant are given sequentially with increasing capillary number. These pictures were converted from the video images of a water drop which was 2.5 cm in diameter and contained sodium dodecyl sulfate (SDS) at 20% of the critical micelle concentration (CMC). The direction of the flow is downward from top to bottom of the figure. In order to improve contrast a blue food coloring was added to the drop. It was confirmed by measurements that the food coloring did not affect the surface tension. All shapes were observed to be steady excluding (2.2e) which was a transitional shape from (2.2d) to (2.2f). Figure 2.2a is the image of a stationary drop at Ca = 0. Due to the typical distortion of a video image in the vertical direction, this circular drop appears to be slightly elliptic. As Ca was increased, the drop was elongated sideways (2.2b). Further increase in Ca resulted in further stretching of the drop in the transverse direction (2.2c). This shape is apparently asymmetric about the vertical axis (or the x-axis in Figure 1.1). Nevertheless, its dimensions remained unchanged and moved with a steady velocity. Hence a steady shape. It should be pointed out that the severely stretched drop shape (2.2c) occurred only when the drop was larger than about 2.1 cm in diameter and when it contained SDS at a concentration between 5% and 20% of CMC. It did not occur with the pure water drop nor with the one containing SDS at 100% CMC although their sizes were larger than 24 (c) (d) (e) (f) Figure 2.2 Evolution of water drop shape with increasing capillary number, (a) Near circular drop; (b) flattened drop; (c) severely stretched drop; (d) ovoid; (e) transitional shape; and (f) short tail drop 25 2.1 centimeters. Furthermore, when Ca was increased further from the value for Figure 2.2c, the drop broke into two instead of showing a transition to an ovoid or short-tail. In order to obtain the steady shapes at a higher Ca (i.e., Figure 2.2d or 2.2f) with the large drops containing SDS at 5% to 20% CMC, the Ca had to be set to a large value from the beginning. Then the drop evolved directly from (2.2a) to (2.2d) or (2.2f) bypassing the severe sideways stretching (2.2c). In case of small drops (1.3 or 1.35 cm in diameter) or the large drops with 0% or 100% CMC, on the other hand, the severely stretched shape (2.2c) did not occur and the evolution of steady shapes was smooth from (2.2a) to (2.2f) with the gradual increase of Ca. It may be interesting to note that the drop shape (2.2c) has a negative curvature at the leading edge as in the case of â€œTanveerâ€ bubbles although the trailing edge also shows a negative curvature. The steady shape (2.2d) which occurred at a higher Ca than (2.2b) or (2.2c) is different from others as the curvature of its leading edge (i.e., the lower part of the drop in Figure 2.2d) is apparently larger than that of the trailing edge. We may call this shape "ovoid" with its sharper end pointing in the direction of flow which is equivalent to the sketch given in Figure 1.2d. The relative velocity (i.e., U/V) of this drop was still smaller than 1. When Ca was further increased, the ovoid went through a rapid transition to a short-tail (2.2f) which moved faster than the average velocity of the surrounding fluid (i.e., U/V > 1). This transition always occurred through a transitional shape given in Figure 2.2e. In Figure 2.3, 2.4, and 2.5, the bubble or drop velocity relative to the average velocity of the surrounding fluid is given as a function of capillary number Ca. Also 26 Cax 104 01-27 cms â– 1.3 cms (Kopf-Sill) Â£2.54 cms x2.06 cms (Kopf-Sill) Figure 2.3 U/V vs Capillary number for air bubbles. 27 specified in the figures are the steady shapes of the bubbles or drops classified according to Figure 2.2. The capillary number in these plots is defined as pV/c where p and V are the viscosity and the average velocity of the silicone oil respectively and ct the equilibrium interfacial tension. In case of air bubbles (Figure 2.3), ct is 21 dyn/cm whereas ct for the water drops (Figures 2.4 and 2.5) varies depending on the SDS concentration. The equilibrium values at each SDS concentrations are given in Table 2.1 and these respective values have been used in calculating the capillary number. The relative velocities of air bubbles described in Figure 2.3 are for two different sizes (a=1.27 cm and 2.54 cm). Overlaid in the figure are the data taken from Kopf-Sill and Homsy for similar size bubbles. In their experiments, both small and large bubbles experienced a sudden increase in the relative velocity accompanied by the shape change. The transition for the small bubble occurred when Ca was about 102 whereas it occurred -4 at Ca Â« 5 X 10 for the large bubble. Such transitions, however, were not observed in the . -3 -2 present experiment which covered the range of Ca between 2X10 and 1.1 X 10 . The relative velocity was always larger than one for both bubbles, although the larger bubble was moving slightly faster than the smaller one. Furthermore, the bubble shapes were always near circular unlike the experiments of Kopf-Sill and Homsy. The lower limit of Ca (2 X 10 ) was set by the sensitivity of the pump whereas the upper limit (1.1 X 10 ) was determined by the maximum flow rate which was imposed by the pump capacity or the upper limit of pressure drop in the cell. The lower limit is well below the critical Ca of Kopf-Sill and Homsy at which the small bubble experienced 28 -4 the velocity/shape transition. The critical value for the large bubble (Ca Â« 5 X 10 ), on the other hand, was smaller than the lower limit. Thus, the present experiment could not have detected the transition if it had occurred at a very low Ca. It should be noted, however, that in the experiments of Kopf-Sill and Homsy, the velocity transition was always accompanied by the shape change and the bubbles with the relative velocity (U/V) greater than one had either a long- or a short-tail. In the present experiments, on the other hand, the bubble shapes were near-circular although their relative velocities (U/V) were greater than about 1.5. Thus, it is unlikely that these near-circular bubbles had experienced the sudden transition at a smaller Ca than the lower limit or they would experience it at a higher Ca than the upper limit. Considering the slight deviation from a circular shape and the viscous dissipation at the bubble edges which was pointed out by Maxworthy,5 the bubble velocities in the present experiments appear to be in reasonable agreement with the prediction of Taylor- Saffman. The distinct differences between the results of the present experiments and of Kopf-Sill and Homsy are apparently in accordance with the argument described in section 2.1 that they were probably caused by the influence of surface-active contaminants. In Figures 2.4 and 2.5, the results for water drops are given for two different sizes. In these experiments the 1.8 mm gasket was used to minimize the pressure in the cell at high flow rate, and the maximum attainable capillary number was determined by the capacity of the pump (Cole-Parmer, Masterflex Model 7520-25). Due to the variation of the equilibrium interfacial tension with the SDS concentration, the maximum capillary 29 Table 2.1 Variation of the interfacial tension with sodium dodecyl sulphate concentration at 25Â°C SDS concentration in water Interfacial Tension (dyn/cm) (% of CMC) air-water Silicone oil-water 0 72.8 38.8 5 58.7 34.3 10 53.8 32.1 20 41.7 25.9 100 33.0 13.7 30 number also varied accordingly for the given maximum flow rate. In case of the 100% CMC water drop, the equilibrium interfacial tension was as low as 13.7 dyn/cm whereas that of pure water and oil was 38.8 dyn/cm (Table 2.1). Consequently, the maximum capillary number for the former (3.7 X 102) was higher than that for the latter (1.3 X 10 ) by a factor of about 2.8. The relative velocities of the small water drops (a = 1.30 cm or 1.35 cm) are shown in Figure 2.4. The labels (a)-(f) indicate the drop shapes as indicated in Figure 2.2. Unlike the air bubbles, they were moving very slowly at low capillary numbers as in the observations of Kopf-Sill and Homsy. It is interesting to note that the water drops which were supposed to be pure were also moving very slowly at low Ca although they were expected to be as fast as the air bubbles in Figure 2.3. A plausible explanation for this behavior may be that the water drops still contained some surfactant despite the precaution we had taken to prevent contamination. When the SDS concentration was between 5% and 20% of CMC, the relative velocities were smaller than one even at the upper limit of Ca. Furthermore, the drop velocities were indistinguishable and the shapes were near-circular or elliptic (Figures 2.2a and 2.2b). Although it was expected that the relative velocities of these drops would increase with Ca and become eventually larger than one, the transition could not be observed as it might occur at a higher Ca than the attainable upper limit. In case of the "pure" drop or the drop with SDS at 100% CMC, on the other hand, the transition could be observed in that the drops took on the short-tail shape (Figure 2.2f) and moved with a relative velocity greater than one. Unlike the observations of Kopf-Sill 31 Cax 104 *0% (13.5mm) a 5% (13 5mm) A 9 1% (13mm) x 20% (13mm) D 100% (13.5mm) , Kopf-Sill (13 mm) Figure 2. 4 U/V vs Capillary number for small water drops. 32 and Homsy, however, the change (or the transition) was gradual rather than abrupt. Although it is not clear what caused this difference in the transitional behavior, it may be partly due to the differences between the two experiments. In the current experiment of water drop and silicone oil, the surfactant was present inside the water drop whereas in case of Kopf-Sill and Homsy's experiment of air bubble and aqueous solution, the surfactants (or surface-active contaminants) might be present in the aqueous solution outside the bubble. Therefore, the mechanism for surfactant transport to the bubble or drop surface could be different. Furthermore, considering the differences in the fluid properties, quantitative match between the two experiments may not be expected. These differences could be also the reason why long-tail drops were not observed in the present study. The large water drops showed the similar trend as the small ones in that the relative velocities were smaller than one at low Ca (Figure 2.5). Unlike the case of small drops, however, the transitional behavior is evident for all water drops in Figure 2.5. This observation is consistent with the trend in Figure 1.3 in that the critical Ca for the velocity/shape transition decreased with increasing bubble size. As we mentioned in section 2.1, the surfactant influence (or the Marangoni effect) should be greater if the surface curvature is larger. Consequently, the strong influence of surfactant may persist to a higher value of Ca if the drop is smaller. Apparently, the critical Ca for the large drops was smaller than the upper limit of Ca of the present experiments whereas that for the small ones was larger than the upper limit. The variation in the drop shape was also 33 Cax 104 â™¦ 0% cmc (25mm) a 5% cmc (24mm) ^9.1% cmc (21.5mm) ^20% cmc (23.5mm) q 100% cmc (24mm) a Ann (20.6mm) Figure 2. 5 U/V vs Capillary number for large water drops. 34 consistent in that it was near-circular, elliptic or ovoid when U/V < 1, whereas all drops became short-tailed when U/V > 1. Figure 2.5 also indicates that the critical Ca for the velocity and shape transition increases with increasing SDS concentration with the exception of the drop at 100% CMC. Although only a quantitative analysis will provide proper answer to such a trend, a qualitative explanation may be given as follows. For most organic surfactants the adsorption kinetics is fast. Therefore, if the surfactant concentration is low, its transport may be limited by bulk diffusion. At a reasonably high Ca, the surfactant distribution may be localized due to convection and consequently, the Marangoni effect is also restricted to the local region. As long as the surface concentration gradient is smaller than the maximum allowable value before the monolayer collapse for the given surfactant, the area covered with the surfactant may grow with increasing bulk concentration. Consequently, the Marangoni effect may also increase with the bulk concentration. If the bulk concentration is higher than a certain limit, however, surfactant transport may not be limited by bulk diffusion any more and the surface concentration gradient will start to decrease. Thus the surfactant influence will also diminish with further increase in bulk concentration. It seems that the critical bulk concentration for the present experiment at which the surfactant influence starts to decrease may be between 20% and 100% of CMC. In an extreme case, if the bulk concentration is very high, the surface may be remobilized due to the lack of surface concentration gradient. We have also conducted an experiment with a water drop which contained SDS at 10 times the CMC to investigate whether the remobilization would 35 Figure 2.6 U/V vs Capillary number for Fluorocarbon oil drops. 36 occur at that concentration. The result, however, almost coincided with that for 100% CMC drop in Figure 2.5 in that the relative drop velocity was still very low at small Ca. Finally, an experiment is conducted using an oil-oil system. Although, surface active substances are present as contaminants in this system, they are not expected to adsorb at the interface hence their influence on the drop motion will be insignificant. Fluorocarbon oil is chosen because it is immiscible with silicone oil. The interfacial tension was measured by Wilhelmy plate method and is found to be close to 5 dyns/cm. Pressure gradient is imposed to study the motion of a Fluorocarbon oil drop placed in Hele-Shaw cell filled with silicone oil. Figure 2.6 presents the normalised velocity of fluorocarbon oil drop as a function of capillary number. A nominal value of 5 dyns/cm is used in calculating the capillary number. These results indicate that the normalised velocity is always greater than one and the drop velocity is close to the predictions of Taylor- Saffman theory. Unlike the water drops, the shape of the fluorocarbon oil drop is always found to be circular. These results reinforce the idea that the retardation experienced by drops or bubbles in an aqueous environment is due to the influence of surfactants. 2.4 Summary and Conclusions Under the assumption that surface-active contaminants were the primary reason for the interesting behaviors of the bubbles in a Hele-Shaw cell observed by Kopf-Sill and Homsy, similar experiments were conducted using the fluid combinations of air-silicone oil and water-silicone oil. These systems were chosen to delineate the surfactant influence systematically. In case of air bubbles in silicone oil for which the surfactant influence was likely to be insignificant, the relative velocities were observed to be as high 37 as the Taylor-Saffman prediction. Furthermore, the unusual bubble shapes such as long- or short-tail bubbles were not observed. In case of water drops which contained predetermined amount of an organic surfactant (sodium dodecyl sulfate), on the other hand, very low translational velocities as well as the unusual shapes of Kopf-Sill and Homsy were observed. Although it is impossible to confirm the presence of surfactants in the past experiments, the present results are consistent with those of Kopf-sill and Homsy and it appears that most of the observations by them may be due to the influence of surface- active contaminants. CHAPTER 3 ESTIMATION OF BUBBLE VELOCITY 3.1 Introduction This chapter presents a theoretical calculation for the translational velocity of a bubble which is retarded by the surfactant influence. In order to simplify the analysis, the bubble shape is assumed a priori to be elliptic. It is also assumed that the surrounding fluid wets the solid surface thereby forming a thin film between the bubble and the plates of the Hele-Shaw cell. For a relatively small bubble (i.e., b/L Â« 1) yet large enough to neglect the edge effect (h/b Â« 1), the entire bubble surface tends to be rigid due to the surface active substance. Consequently, a large drag arises in the thin film region thus retarding the bubble motion significantly. The calculated velocity of the bubble is apparently in reasonable agreement with our own experimental observations and also with those of Kopf-Sill and Homsy.6 In addition, the present calculation provides an explanation for the evolution of the bubble shape with increasing capillary number. 3.2 Theoretical Analysis A bubble or drop in a Hele-Shaw cell assumes a circular plan form when it is stagnant because of the interfacial tension. Once the bubble is set in motion, its shape deviates from the circular plan form and takes on a steady shape depending on the flow conditions (Figure 1.1). In the absence of surface tension effect, the analysis by Taylor and Saffman1 predicts an elliptic shape if the bubble size is much smaller than the width of 38 39 the cell (i.e., b/L Â« 1). If surface active substances are present, the bubbles are not necessarily elliptical and may take on various interesting shapes depending on the flow conditions6. Nevertheless, we assume an elliptic plan form since the shape distortion is small at low capillary numbers. While the bubble shape should be determined as a part of solution in a rigorous analysis, the present analysis is an approximation in which the bubble shape is assumed a priori. Unlike a full three-dimensional analysis that may be possible only numerically, the present analysis provides an analytic description for the translational velocity of the bubble. As indicated in Figure 1.1, the two principal axes of the elliptic bubble are assumed to be aligned with the flow and the transverse directions. The surrounding fluid wets the solid surface thus forming a thin liquid film between the plates and the bubble. In the presence of surface active substances, the Marangoni effect resulting from the surface tension gradient may complicate the flow field near the bubble. When the bubble is small, however, it may be simplified since the entire bubble surface may become rigid19 20. Here we consider the case of a small bubble in which the entire bubble surface is assumed to be rigid. The flow field slightly away from the bubble is known to be parabolic in the xz-plane (i.e., across the gap). Thus, the velocity field can be given as1,7 3.1 Here vavg is the gap (or depth)-averaged velocity field that satisfies the following Hele- Shaw equations: 40 V-Vavg = 0 vP = 12p 2 avg 3.2 3.3 Here p is the gap-averaged pressure field, p is the fluid viscosity and h is the gap of the Hele-Shaw cell. It is apparent from Eqns. (3.2) and (3.3), that the average flow field in the xy-plane can be described as a potential flow. In an elliptic cylindrical coordinate system depicted in Figure 3.1, the complex potential Q for the flow past an elliptic cylinder in a bubble fixed frame of reference is given by16 n = Â® + /Â¥ = -b(V-U)(k + l)jk2 -1 !~-gL..Â£-L~! 3.4 where y = Â£, + i r\ and k the shape factor defined as k- a/b. V and U are the average velocity of the surrounding fluid and the bubble, respectively. From the gradient of the velocity potential Â® or the stream function T, the velocity components in Â£, and r\ directions are given as c(V-U) k+\\ _e^_ 2 q [& + 1 JLÂ±1 k-1 Uc cos p + â€” sinh ^ cosp q 3.5 Uc yt +1T eâ€™ Ã©~^ 1 wl. vavg = -c(V- Ãœ) 1 1- i sin rj â€” â€” cosh % sin r| â€™i V â€™ 2q |_Jfc + l k-lj q S 3.6 Here c = ^|a2 - b2 and q is the scale factor for the elliptic cylindrical coordinate system defined as q2 = c2 (sinh2^+sin2r|). As it was pointed out previously, Eqns (3.5) and (3.6) are valid only in the region which is 0(h) distance away from the bubble rim. 41 In the thin film region between the plates and the bubble, the flow field is represented by a Couette flow if the bubble surface is rigid. In this case the thickness of the wetting film can be given as10-20-23,27 / = 1.337-(2Coâ€ž)f where Ca=^ 3.7 2V â€™ a Here Can is the capillary number based on the component of the bubble velocity normal to the bubble surface in the jcy-plane, and a the equilibrium interfacial tension. In the absence of surfactant, the bubble surface is stress-free and the film thickness can be expressed by the same equation but with the constant 2 in the parenthesis replaced by 1. Equations (3.1), (3.5), (3.6) and the Couette flow along with the film thickness described by Equation (3.7) represent the three-dimensional velocity field around the translating bubble excluding the small region of 0(h) in the immediate vicinity of the bubble rim. Using this velocity field, an analytic description of the translational velocity of the elliptic bubble can be obtained from the x-directional momentum balance on a control volume surrounding the bubble. An integral form for the x-directional momentum balance for a domain D which encapsulates the bubble and moves with the same velocity U as the bubble may be written as (Figure 3.1) ^ IpvxdV = -J jp[n â€¢ (v - U)]v, + (np - n â€¢ t)x}fZS 3.8 This integral is zero at steady state and the surface integral can be evaluated using the 42 Figure 3.1 Plan view and side view of an elliptic bubble and a control volume moving with the bubble 43 prescribed velocity field over the four separate surfaces as shown in Figure 3.1; top, bottom, edge of the control volume D, and the bubble surface. On the bubble surface, v = U and the integral over the bubble surface represents the drag force (FD) acting on the bubble: Ã (np-n-t)xdS = -Fd 3.9 $hlihhle On the top and bottom surfaces of D, the first term of the integrand in Equation (3.8) is zero, since (v - U) is perpendicular to n. The second term is evaluated separately for the film region directly above (or below) the bubble and the region which is 0(h) distance away from the bubble rim to give J (np- Slop + S bottom n-x) dS = - >x 12p nc2 ~h 4~ fe-2Z.â€˜ _e-25.) (V-U)e2^ +U \6\iU ( AI\ V / 2A22hCaJ \ n ) 3.10 As noted in Figure 3.1, and represent the edge (or rim) of the control volume and the bubble, respectively. The last term in Equation (3.10), which is the contribution from the Couette flow in the thin film region, is zero if the bubble surface is clean and stress-free. Here A is the plan form area of the bubble (i.e., A = 7tab) and Ca the capillary number based on the bubble velocity U. I is a constant given as a definite integral which accounts for the film thickness variation in the transverse direction (i.e., y-direction): 44 /=] [i+(*2-iy ]*[i-/]> 3.11 Once *(= a/b) is specified, I can be determined numerically and the results are given in Figure 3.2 for various values of k. When the bubble is circular (i.e., k= 1), /= 0.91. Finally at the edge of the control volume, the only non-zero contribution to the surface integral is due to the pressure which is given as \(*P),dS ~U)+VeÂ«-e u u+(v-u) k + \ k-1 3.12 From Equation (3.9), (3.10) and (3.12), the final form of the x-directional momentum balance for a large control volume (i.e., -*oo) is obtained as F \2\i ttc fc + 1 D h 4 \ k-\ (V-U) + U k -1 k +1 + (V-U) 2 I 16pf/(ab/) 3.13 k~X\ 2A22hcJ For a freely suspended bubble with zero inclination angle (i.e., 0 = 0 in Fig. 1), FD= 0 and Equation (3.13) reduces to U = (k + \)V- 2 Ca1 3.14 Ca3 +0.2*7 When the bubble is driven by gravity only, V = 0 and Fj) = -Apg(7rab/i)sin0, since the drag force is balanced by the buoyancy force. Thus, U = kU 2 Ca1 Ca3 +0.2*7 , TI* Ap* gsin0 where U = 12p 3.15 45 Figure 3.2 Numerical value of the definite integral, Equation 3.11 for various values of the shape factor k. 46 When the bubble is driven by both pressure gradient and buoyancy, Equation (3.14) and (3.15) are combined to give a general expression for an elliptic bubble as t/ = /?[(* + l)F+ *{/*] where /? = 2 Ca1 3.16 Ca3 + 0.2 Â¿7 As Figure 3.2 indicates, / is 0(1) unless k is extremely large. Thus P is 0(Ccr/3). This order of magnitude decrease in the bubble velocity is due to the large drag in the thin film region where the bubble surface is rigid. Since this drag is proportional to the film thickness, the bubble velocity is also proportional to Co2'3. In the absence of Marangoni effect, on the other hand, the bubble surface in the thin film region is stress free. In this case, the expression for the bubble velocity is equivalent to setting P = 1 in Equation (3.16) since the last term in Equation (3.13) is absent. Thus the result of Taylor and Saffman is recovered in which U = 2 E for a circular bubble, U > 2 V and U < 2 E for an elliptical bubble elongated in the flow direction (i.e., k > 1) and in the transverse direction {k < 1), respectively. 3.3 Comparison with Experimental Results Retarded motion of aqueous drops containing sodium dodecyl sulfate (SDS) in an oil- filled Hele-Shaw cell was reported in the previous chapter for a pressure driven flow. In Figure 3.3 the present predictions for ellipses with various aspect ratios are compared with experimental results for a drop which is 13 mm in diameter (b/L = 0.04). The drop contained SDS at 5, 10 and 20% of the critical micelle concentration. At these concentrations of SDS, the data sets are indistinguishable from one another and show 47 reasonable agreement with current predictions for ellipses that are elongated in the transverse direction (i.e., k < 1). While the drops in the experiment were not exactly elliptical (Figure 2.2), they were nearly elliptical and elongated in the transverse direction. The present analysis assumes the bubble shape a priori and consequently, the evolution of bubble shape with increasing capillary number cannot be predicted. However, an interesting observation may be made in Figure 3.3 When the drop was stagnant at Ca = 0, its shape was circular (i.e., k = 1.0). As it was pushed by the surrounding fluid, the k value decreased initially from 1.0 prior to exhibiting an increasing trend with increasing Ca. The initial decrease in the k value occurred at a very low Ca and in some cases, the drops simply elongated in the transverse direction without moving until the average velocity (V) of the surrounding fluid reached a certain value. Unfortunately, not all k values are available for the experimental data in Figure 3.3, and those that are available have been specified in the figure. Although a quantitative agreement is not expected between the observed and calculated k values, it is interesting to note that they are not very far apart. It should be pointed out that the retardation factor P in Equation (3.13) is independent of bubble size suggesting that the same level of retardation may occur regardless of the bubble size as long as their shapes are identical. One of the simplifying assumptions in the present study, however, is that the entire bubble surface is rigid. While this assumption is plausible for small bubbles, it may fail for large bubbles since a stress-free mobile surface region may emerge in the front part of the thin film region20. 48 2.0 - Saffman-Taylor Theory k= 1.0 1.6 - 1 2 U/V ~ k=0.4 k 0.8 - â€” â€”* * ~ ~~ ZZ*â€” â€”*-* A m Â¥l| â– â– X X 0.4 k=0.3 0.0 El1 1 i.l i ; O 50 100 150 200 Cax 10M Figure 3.3 Comparison of experimental results and the theoretical estimate for pressure driven flow (SDS concentration: 5% ; A 10%; x 20% of CMC) 49 Consequently, big bubbles may move with a higher velocity than the present estimate. Furthermore, as pointed out in the previous chapter, bubbles which are large enough to have a mobile surface region for a given surfactant concentration, may experience a transition at a certain value of Ca in that the bubble velocity becomes greater than the average velocity of the surrounding fluid (i.e., U > V). Such a transition, then, will make it possible for the bubble to take on an unusual shape such as long- or short-tails as observed previously.6 This argument, of course, is only qualitative and remains to be proven rigorously. Experiments with air bubbles in an oil-filled Hele-Shaw cell have shown that the bubbles move with a velocity comparable to the predictions of Taylor and Saffman. In addition, they are always elongated in the flow direction unlike the bubbles influenced by surfactants. This difference in the steady shapes between clean bubbles and surfactantÂ¬ laden bubbles may also be explained by the current study at least qualitatively. When two identical elliptic bubbles move with the same velocity but with different orientation (i.e., one moving with its longer axis in the flow direction and the other with its shorter axis in the flow direction), our calculation indicates that the prolate bubble experiences smaller drag force (or smaller energy dissipation) than the oblate one in the absence of surfactant effect. In case of surfactant-laden bubbles, on the other hand, the prolate bubble results in a larger energy dissipation than the oblate one due to the larger drag in the thin film region. Thus, the elongation in the flow direction (i.e., prolate bubble) may be a preferable shape for clean bubbles whereas the oblate shape is preferable for surfactant-laden ones. 50 3.4 Summary A theoretical calculation has been presented to estimate the translational velocity of bubbles or drops in a Hele-Shaw cell under the influence of surface active substances. Assuming that the solid wall is wet by the surrounding fluid and that the bubble shape is elliptical, an analytic expression has been derived for the bubble velocity. The calculated velocity is apparently in reasonable agreement with our own experimental findings and also with those of Kopf-Sill and Homsy. This result along with our experimental findings suggest that most of the observations by Kopf-Sill and Homsy are probably due to the influence of surface active substance which may be present in the system as contaminants. CHAPTER 4 DISPLACEMENT OF A POWER-LAW FLUID BY INVISCID FLUID 4.1 Background Two phase displacement flow is a problem of industrial importance in areas such as enhanced oil recovery, separations etc., and has been studied by various researchers in the past14. When the displacing fluid is of low viscosity, then instabilities can result causing viscous fingering. Park and Homsy have theoretically analyzed the two-phase displacement of Newtonian fluids in a Hele-Shaw geometry using the technique of matched asymptotic expansions.15 Analytic expressions for both film thickness and pressure drop and their dependence on capillary number were derived. When the displaced fluid is a shear thinning fluid, these results can be very different. In this chapter, the displacement of a power-law fluid by an inviscid fluid (i.e. air/gas) in a Hele-Shaw cell is analyzed using the same technique of matched asymptotic expansions. The present analysis is aimed at deriving the film thickness and understanding the stability issue involved in this process. This problem has practical applications in the general area of polymer processing and specifically in applications such as gas assisted injection molding. Figure 4. 1 is a schematic of an inviscid fluid displacing a power-law fluid in a Hele-Shaw geometry. It is assumed that the displaced fluid wets the wall and there will be a thin film left behind after the displacement. The displacement is slow enough to ignore 51 52 Figure 4. 1 Schematic of an inviscid fluid displacing a power-law fluid in a Hele-Shaw cell 53 the effects of inertia and the gap thickness is small enough to ignore the effects of gravity. Under these assumptions, the interface remains symmetrical about the center plane. The tip of the interface is assumed to be moving with a constant velocity U and the analysis is carried out in a frame of reference moving with the tip velocity. 4.2 Basic Equations The governing equations relevant to the problem are as given below. VÂ» v = 0 4.1 V/? = ri(5')V2v + 2D â€¢ Vr| 4.2 Where D = -(Vv + Vvr) 4.3a 2 v > S = 2(D :D) 4.3b n-1 ri = K S 2 4.3c K is called the consistency factor and n is called the power law index. Boundary conditions are u = -U, v, w = 0 at y = Â±b 4.4a n â€¢ v = 0 t| *TÂ« n = 0 t2 *T* n = 0 n â€¢ T â€¢ n = a(V â€¢ n) Here r\ is the viscosity of the power-law fluid which depends on the shear rate and (u,v,w) are the components of velocities in the (x,y,z) directions respectively, n, tâ€ž and t2 â€¢ at y = Â±h(x,z) 4 4b 54 are the unit normal and unit tangential vectors to the interface, a is the interfacial tension and T is the stress tensor. The origin for the reference frame is placed at the tip which is moving with a velocity U. b and h(x,z) are the half gap thickness and the location of the interface respectively. f(z) is the projection of the tip of the interface onto the xz plane. Since the interface is symmetrical about the center plane, only the bottom half (i.e. y = -h(x,z))is considered for the analysis. The following two dimensionless parameters appear during the scaling of the above problem. 8 and Ca = KUnb n 11-Â« s is a parameter which represents the ratio of the two characteristic length scales of Hele- Shaw geometry and capillary number, Ca, represents the ratio of viscous to interfacial tension forces. A complete solution to this problem is very complicated to obtain, and only an asymptotic solution using the above mentioned two small parameters is attempted in the following sections. 4.3 Scalings and Regions As shown in Figure 4. 1, the fluid domain is divided into three different regions; Constant film thickness region (Region I), front meniscus region (Region II), and power- law fluid region (Region III). The flow profile in Region III can be obtained by straightÂ¬ forward integration of the governing equations and is given in the next section. In the constant film region the flow field resembles plug flow in the moving reference frame. The flow profile in Region II can be understood only after a rigorous analysis as described below. The following scalings are used to non-dimensionalize the equations. 55 (u,v,w)~(U,U,eU) (x,y,z)~(b, b,L ) S~(U/bf Using the above scalings, the Equations 4.1- 4.4 are written as given below. Ux + vy+ S wz = 0 4.5 px=Ca n-1 S 2 (uâ€ž+Uyy+B2Ua) 1 n~3 + ^ÃTj 2Â«a+(^+vxK + e2(uz+wx)Sz 4.6a Pv=Ca eV/ 2 \ Â«-1 [(â€œy + + ^vyS> S 2 (vâ€ž + v.â€ž+e vâ€žJ-t S 2 xi ^ zz +E l(vz + wy)Sz 4.6b pz=Ca w-l 5 2 (> xx rr>y 22 v n-3 (m2 +wj Sx + 1] 2 2 1V*+HA +2e2M'i5jJ 4.6c Where 5 = 2[u2x + v2 + e4w2) + + s2(w2 + wx)2 + s2(v2 + wyj at y = -1, m = -1, v = w = 0 at y = -/2(x, z), uhx + v + t\vhz = 0 4.7 4.8a [(A", â€œ h*Wy)~ 2hA(^W: - Â«,) - (w. + - i2A){ _ |+(/i.vx -^v.) = 0 4.8b 56 /I-1 (*,!+A2) -hx(u2 + vx) + 2- h](ux -vy) + e2h][e2w1 - v,) 1 5 2 -e h,\v. + wyj + e2hxh:(u: + wx) J + hx(uy + vx) + Â£2h!(v1 + wy] = - K(\ + g2^2) - + /?') + 2Â£2KKK , 2 CaS 2 A (\ + h2x+e2h2) â– + (\ + h2x+e2h2) 4.8d where - + v,) - f2/ir(vJ + vv>( j â€” e2hxh.(u, + w,)- v, - /i2ut - Â£-4/?_;V The subscripts x,y,z in the above equation represent partial differentiation of the respective variables. To solve the above set of equations, we can begin by setting both s and Ca equal to zero. The resulting equations indicate that pressure is hydrostatic and is independent of (x,y,z). The shape of the interface is a portion of a circle which can be obtained by solving the normal stress condition. Hence this sub-region (Outer region of Region II) may be referred to as capillary statics region where pressure and interfacial tension are both important. In order to match the solution smoothly to the constant film region (Region I), there should be a second sub-region (Inner region of Region II) where the interface profile is smoothly matched to a constant value. This may be called as transition region. In the transition region both pressure and viscous effects are important and the variables need to be rescaled accordingly. Under the lubrication approximation, the scalings relevant for this region can be derived by balancing the pressure and viscous terms in the momentum equation and by balancing the pressure and interfacial tension forces in the normal stress condition. These scalings are different from the ones used for 57 Figure 4. 2 Schematic showing the scales and coordinate systems for different regions. (A: Transition region; B: Capillary-statics region) 58 the Newtonian case and they depend upon the power-law index and capillary number as mentioned below. w,â€”w where k = Cak ) 2n +1 x + l y +1 ^ Cak â€™ Ca2k â€™ Z > * =Â¿4; ?-/>; 5=caÂ«s The variables in the transition region are denoted by an overbar. As indicated in Figure 4.2, the origin for the transition region is x = -l and y = -1. The unknown / will be determined later using the matching conditions. Using the newly scaled variable, the governing equations valid for the transition region are given below. ux +vy + e2Cak wz = 0 4.9 _n-1 S 2 'Ca2kuÃ± i n-3 + "-'s- 2 ca2kuxSx + Uyy +[uy + Cci2k Vx v+8 Cci Uzz J Z +z2Ca2k(CakÃ¼z + Wx)Szy 4.10a _ _n-1 p- = Ca2kS 2 (Ca2kvÃ± ) '(Uy+Ca2kv-x)s~x i n-3 l n ~ * Ca2k +v^ + 2 V yS y 2 K+e2Ca4k +e2 Cak Sz{Ca3k vj + 4.10b \Ca2kwÃ± ] Ca2k[Cak uz + w'xjs,x + n-\ + "-â€˜sT 2 Cakp-=S 2 |Caikv~z + [+e2Ca4*>Vzz J +2 e2Ca4kw-2Sz 4.10c 59 at y at y n- s~: Ap = Where S = 2Ca2k(^ux +v2y + e4Ca2k wf] + {uy +Ca2kvx j +e2Ca2k(Cak Uz + wx j + z2[Caikv~z + = 0, = h(x,z}, u = -1, v = w = 0 uhx - v + e2Cak wh~z = 0 -i â€¢ [CakhzUy - hxwy^ + 2Caik hxh:{z2Ca2k wz - w*) +Ca2k(CakÃ¼z+Wx){hÃ-t2Ca2khTj + Ca3k (h-2vx-hxVz) = 0 4.11 4.12a 4.12b n-1 5 2 (h2+Â£2Ca2kh2) hx[uy +Ca2kvx)j V x z) + Â£2Cak hzÃCa2k vz +Wy\ + 2 Ca 2* + f2Ca2* h-7[e2Cak - Vy) + s Ca hxhz^Ca Wz + Wx^ -hx[uy +Ca2k vxj- Â£2Cak hz[ca3k vz + = 0 4.12c h^c(l + Â£2CaAk h fj + s2Ca2k h7z[l + Ca2k h2-j - 2e2Ca4k hxhz h7z 3 ,2Ai;2 ^ ^2^4* 2 \ + CaAKh-+Â£ACa^h < X Z <,12d _n-\ 2Ca~2kn+k+X S~Y hx(uy + Ca v* j + Â£2Cakhz[Ca2 vz + Wyj [\ + Ca2kh^+Â£2Ca4kh ix\â€œy â€” Â£ Ca hxhz ^Ca uz + wx j â€” vy -Ca2kh2-ux-Â£*Ca5kh-Wz X z 60 4.4 Small Parameter Expansions To obtain the solution, all the variables (u, w, p, h etc..) are expanded as given below. -( .00 a 01 p +Ca p + Ca2ap02 + â€¢ M .10 a 1 p + Ca p + Ca2ap12 4.13 The choice of gauge function, a , is obtained by comparing all the exponents that appear in the governing equations and boundary conditions and choosing the least common multiple of them so that matching is possible at every order. For example, the choice of a few gauge functions are as given below. a = Ink ifn = â€” etc. 4.14 2 4 6 = nk if Â« = etc. 3 5 7 Since the gauge function depends on the power-law index, only the leading order solution is attempted in the following sub-sections. Once the choice of gauge function is known, higher order corrections can be obtained following the procedure of Park and Homsy. Consequently, in the present analysis the dynamic pressure jump across the interface can not be determined since it appears at a higher order. 4.4.1 Capillary Statics Region : Substituting 4.13 into 4.5-4.8, the leading order governing equations and boundary conditions for the capillary statics region are as given below. M,oo+v;Â°=o 4.13 â€ž00 _ 00 â€ž00 Px ~ Py = Pz = 0 4.14 61 Boundary conditions, w00 = -1, v00 = w00 = 0 at y = -1 4.15a . 7 00/ \ 00 7 00 , 00 n at y = -h [x,z) u hx +v =0 4.15b V- 4.15c (l + hff [(/*â€œÂ«â€œ - /i>;Â°)+2 hÂ°otÃxÂ°+hf +wÂ°xÂ°)+(/iÂ°ovÂ°Â° - /iÂ°yÂ°)]=o 4.15d [- + Â»r)+ 2hf (â€žÂ» - vâ€œ)+ + vf)] = 0 4.15e As evident from the above equations, the viscous force term in the momentum equations is negligible and the pressure is independent of (x, y, z). The shape of the interface can be obtained by integrating the normal stress balance and is given as below. h00 = 1 Ap00 L l-{VÂ°(*-/(z)) + l}2J 4.16 The two boundary conditions that were used in the integration and which apply at all orders are K ->-Â°o as x->/(z) 417 h = 0 at x = /(z) The solution 4.16 indicates that the tip of the interface is a portion of a circle modulated in the z-direction by f(z). This solution contains one unknown Ap00, which will be determined using the matching conditions. Equation 4.16 is not uniformly valid in the entire region and it needs to matched with the constant film region (Region I). 62 4.4.2 Transition Region Variables in Equations 4.9-4.12 are expanded using 4.13 and the leading order terms are gathered to obtain the governing equations for the transition region. -oo Ux -oo + Vy = 0 4.18 00 Px (-00"'] ={u> J 4.19a 00 Py v y y = 0 4.19b -oo a Ã 1 \Uyy 0 = Wyy + (w-l)=55-W, 4.19c Uy at y = 0 -00 , -oo â€”oo u =-l, v = w =0 4.20 - yOOx v at y = h (x, z) â€”ooâ€”oo -oo u hx - v =0 4.2la â€”00 -00 A p = hxx 4.21 b _00 Uy =0 4.21c By solving these equations and matching with the constant film thickness region (Region I) the velocity profile can be determined. Once the velocity field is known, then normal stress along with kinematic condition results in an evolution equation for the shape of the interface as given below. Jr-- rlXXX â€” 2n +1 V n -oo -oo h -t h 2/1+1 00 " 4.22 When n is set equal to one, the above equation reduces to the Newtonian case (Park and Homsy 1984). t represents the leading order approximation for the constant film thickness which will be determined later using the matching conditions. For numerical 63 integration, the above equation is transformed into a canonical form by using the following transformations. Hoo -00 / 4.23 X = x + s where [3 = n + 2 4.24 s is an arbitrary shift factor which is determined by higher order matching conditions. With these transformations, the evolution equation is written as follows. H oo XXX (2n + \\ â€( S3 O o ->)' \ n j o o S: 2n+l 4.25 The condition for matching with region I now becomes //00-> 1 as X â€”Â»-oo 4.26 Equation 4.25 can be numerically integrated, using small slope and small curvature as initial conditions. A 4th-order Runge-Kutta routine is used to do the numerical integration. The shape of the interface depends on the power-law index and it is plotted in Figure 4.3 for the case where n = 0.5 along with the profile for a Newtonian case (i.e. n = 1.0). Since as X â€”> +oo the interface profile H00 also goes to infinity, it is possible to approximate the profile using a quadratic form as given below. Â«"M = \coX2 + C,(z) X + C2(z) 4.27 Equation 4.27 can be written in terms of the original scalings as given below. 64 Figure 4.3 Comparison of interface profiles for Newtonian (Â«=1) and power-law (n=0.5) fluids. 65 -oo h (x,z) = Câ€ž -2 X + -oo2/*-' It C.S -00 2/*~' -OO*'"1 V2/ t x + cy c.s _ -oo + â€”\rr + C,t -002/*-' ' -00^â€œ' ' ~2 V21 t ^ 4.28 -oo . The unknown t will be determined by the following matching conditions. 4.4.3 Matching Conditions The solution in the transition region has to be matched with the solution in the capillary statics region using the following equation. Jim jl - Ca^k h(x,z^> = Um h(x,z) 4-^9 xâ€”>00 ' ' xâ€”>-l This limits are to be interpreted in terms of the matching principle by Van Dyke (1964). By expanding h(x,z) about x = -/ using Taylor series expansion, rewriting the expansion in inner variables and comparing it term by term with the left-hand side, matching conditions for each order can be determined as given below. Higher order matching conditions can be derived once the choice of a is known. hm(-lÂ°,z) = 1 4.30 sÂ°Caâ€˜) Af(-/\z) = 0 4.31 eÂ°&2â€˜) Aâ€œ(-/V) = -Câ€ž(iâ€œ)' 3 4.32 Physically these conditions mean that the outer static solution meets the wall with zero slope and the curvature matches with that of transition region solution. Solving the above equations, the following unknowns in the solution can be determined. 66 A/0 = i 4.33a /Â° = 1 - f(z) 4.33b -00 , s 3 1 = (Co)2â€+l 4.33c The above solution determines the location of the origin and also the film thickness to a leading order. Since the film thickness is stretched by Ca2k, the dimensionless film thickness is given by t oo KUnb'-n (7 2 2w+l 4.34 Figure 4.4 displays the dependence of the constant Co on the power-law index n. As n-> 1 the constant C0 approaches 1.337 which is the value for the Newtonian case. In Equation (4.34), if K is replaced by p0/y0 where p0 is apparent viscosity and y0 is a reference shear rate, then the film thickness is given as follows. ,oo = (co) 3 2n+l Â¿>YoFo 2Â«+l 2n Ca'2n+l where Ca' = IÂ¿oU 4.35 Figure 4.5 is a plot of Equation (4.35) which is a straight line on a log-log scale. All the variables influence the intercept while only the power law-index influences the slope of the straight line plot. In the Newtonian case, all the fluids have a master curve (i.e. the plot for n = 1) whereas each power-law fluid has a separate curve uniquely defined for the given operating conditions. 67 2.5 2.0 C0 1.5 1.0 o < 1.337 for Newtonian Fluid 0.2 0.4 0.6 0.8 Power Law Index (n) Figure 4. 4 Plot of the Coefficient C0 as a function of the power-law index 68 Figure 4. 5 Log (film thickness) vs Log Ca for a power-law fluid. (p0 = 100 cP; a = 30 dyns/cm; y0 = 1 sec'1; b = 0.06 cm) 69 4.5 Stability Analysis In order to carry out the stability analysis, it is important to derive the flow field in the power-law fluid region. Although far away from the interface, the flow is only along the x-direction, near the interface a two dimensional flow field is considered. The solution to this region is fairly straight forward. It is assumed that the gradient of components of velocities in x,z direction are negligible and only the gradients in the gap direction (i.e. y direction) are important. The simplified governing equations and boundary conditions are given below. 71-1 PX = K S 2 Uy 9 9 where S = uy + wy y 4.36a 71-1 pz=K S 2 wy y 4.36b 4.36c at y = Â±b at y = 0 u= w= 0 4.37a 4.37b The above set of equations can be solved to obtain the velocity field as given below. u = y " -b " 71+A 4.38a 4.38b 70 The components of velocity are averaged in the gap direction (i.e. y direction) and the resulting velocity profile is written in the vector form as given below. = - nb 2/1 + 1 |V/?| n S/p 4.40 When n = 1, the above equation reduces to Darcyâ€™s law. However, in the case of power- law fluids, it may be noted that the average velocity field is non-Darcian. The interface is assumed to be flat across the gap and moving with a constant velocity U. Then, the average velocity in the power-law fluid, uavz, can be related to the velocity of the interface, U, as given below. The average velocity in the lateral direction is zero. uavg = U 7 where 7 = b - film thickness 4.41 The location of the interface is denoted by x = r|0 which is a function of time and without loss of generality the pressure inside the inviscid fluid may be set equal to zero. Now, equation (4.41) is substituted in equation (4.40) and the steady state solution to the depth averaged pressure field can be obtained as given by the following equation. ps=- (2n + l)?t/ nb K( \ 4.42 Equations (4.41) and (4.42) constitute the base state solution or the steady state solution for the stability analysis. The stability of this solution is analyzed following the standard procedure for the linear stability analysis. The steady state velocity and pressure field is perturbed by introducing disturbances in all the physical variables as given below. 71 u = U + u' w=w' p = ps+p' x = 7]0 + rj'(z,t) 4.43 Here the superscript prime indicates that it is a perturbation from the steady state solution. By substituting in equation (4.40), the relation between the perturbed velocity and pressure can be written as follows. u = -- bU nK (2n + \)7U nb w = -- bU_ K (2n + l)7U nb 4.44a 4.44b According to the linear stability theory, the perturbation in the velocity field can be written as follows. u' = exp [ikz + 8 /] 4.45 Here k is the wave number and 8 is the growth rate of the disturbance. Using equation (4.45) and the continuty equation (i.e. u'x+w'z= 0), an expression for w' can be obtained. By substituting w' in equation (4.44) and eliminating pressure the following ordinary differential equation is obtained. vj/" - nk2\\i = 0 4.46 Far away from the interface, into the power-law fluid region, the disturbance dies out. Hence the above differential equations can be solved assuming that as x -Â» oo, T â€”> 0. The resulting solution is as given below. i|/(x) = a exp^-ylnk2 x 4.47 Now, the components of the perturbed velocity, u',w', are determined to a constant. By substituting them in equation (4.44) and integrating, the pressure field in the power-law 72 fluid region can be determined. As indicated in equation (4.43), the instantaneous location of the interface is given by x, which can be differentiated to give instantaneous velocity in the jc-direction as mentioned below. (x - r/0). = n't =u'\ , 4.48 V 'Oft 11 I X = T]g + T] Equation (4.48) can be integrated with respect to time to obtain an expression for the perturbed location of the interface (i.e. rj'). The pressure jump across the interface can be written as f f f P\- P2=â€”Â°' Hzz 4.49 The constant k/4 appearing in equation (4.48) was first derived by Park and Homsy (1984) as an 0(e2) correction that accounts for the curvature change in the z-direction. Substituting the deviation variables in the above boundary condition, the dispersion relation for the growth rate of the disturbance can be written as follows. 5 nb box# (2n + \)7U\ 4 K 4.50 When n is set equal to one, the dispersion relation for the Newtonian case is recovered (Saffman & Taylor 1958). Figure 4.6 is a graph of 5 as a function of the wave number. If 8 is positive then the disturbance grows at an exponential rate and the interface is unstable to small disturbances. If 8 is negative, then the disturbance is damped at an exponential rate and the motion is stable to small disturbances. As is evident from the graph, there is an optimum value for the wave number for which the growth rate is maximum and beyond which value, the growth rate monotonically decreases. So 8 = 0 73 Wave number k (n=0.3) q (n=06) _^_(n=I O) Figure 4.6 Disturbance growth rate (5) as a function of wave number (k) . (|i0 = 100 cP, y0=l s'1, n = 0.6, b = 0.06 cm , a = 30 dyns/cm, U = 0.05 cm/s) 74 Figure 4.7 Wave number for the maximum growth rate as a function of the power-law index. (po=100 cP, y0= 1 sec'1, U = 0.05 cm/s, a = 30 dyns/cm, b = 0.06 cm) 75 represents the cut-off wave number whose expression is as given below. kcut-off 4 K (2Â« + l)7t/| y nab nh j 4.51 Hence the wave numbers greater than the cut-off wave numbers die out while the smaller ones grow at different growth rates. The wave number with the maximum growth rate can be obtained by differentiating 8 with respect to k and setting it equal to zero and it is related to the cut-off wave number as given below. K = 'â€™cut-off ~7T 4.52 In equation (4.51), the consistency index K can be replaced by apparent viscosity (p0) and a reference shear rate(y0) as K = ju0/yÂ£~l, then equation (4.52) can be plotted for various power-law fluids as given in Figure 4.7. While making this comparison, it should be kept in mind that the reference shear rate should be high enough so that power-law model is applicable to all the fluids being compared. Because, one of the limitations of power-law model is that it can not describe the viscosity behavior at low shear rates. The result in Figure 4.7 indicates that, the wave number for maximum growth rate ik-max) decreases as the power-law index reduces. In other words, the wave length for the maximum growth (Xmax) is greater for power-law fluids. To experimentally observe this wave length, the width of the Hele-Shaw cell (L) should be larger than X,max. Hence, for fluids with similar viscosity at a constant reference shear rate, it may be concluded that the interface will be less sinuous for a power-law fluid than for a Newtonian fluid. 76 4.6 Summary The displacement of a power-law fluid by an inviscid fluid is analyzed and a leading order solution is obtained using the technique of matched asymptotic expansions. For slow displacement, the shape of the interface in the capillary statics region is a portion of a circle which is smoothly matched to a parabolic profile in the transition region. Under the lubrication approximation, the shape of the interface in the transition region is obtained by numerically integrating the evolution equation. Matching conditions with the capillary statics region determine the film thickness of Region I. The flow field in Region III is determined and found to be non-Darcian. Assuming that the interface is flat, the steady state solution for the velocity and pressure field is determined. Stability analysis for this steady state solution is carried out to determine the wave number with the maximum growth rate. The result indicates that the interface will be less sinuous for power-law fluids than Newtonian fluids. CHAPTER 5 CONCLUSIONS The experimental findings of Kopf-Sill and Homsy on the motion of air bubbles in a Hele-Shaw cell filled with Glycerin-water mixture have raised the following issues. â€¢ The velocities of small air bubbles were found to be an order of magnitude smaller than the theoretical predictions of Saffman and Taylor. â€¢ A variety of unusual steady bubble shapes and a sudden transition of bubble velocity and shape at a critical capillary number were observed. The present study explains the above observations in the context of surface active organic contamination. To study the influence of surfactants, experiments were conducted both using air bubbles and water drops containing surfactant in an oil-filled Hele-Shaw cell. While the motion of air bubbles is in agreement with the predictions of Saffman-Taylor theory, the velocity of water drops is an order of magnitude smaller than the predicted value. Unlike the air bubbles, the water drops exhibited a variety of shape transitions similar to the ones observed by Kopf-Sill and Homsy. In the presence of surfactants, the bubble motion can induce a surface concentration variation resulting in Marangoni stresses along the drop interface. These stresses can cause the drop surface to be rigid. In the presence of a wetting film, the dissipation in the thin film region can be significantly high and consequently the velocity of the water drops is decreased by an order of magnitude. The 77 78 observed shapes can also be explained using the same argument. The motion of the bubble creates a spatial variation of surface tension. In order to keep the pressure constant, the curvature normal to the gap direction adjusts itself resulting in a variety of shapes. Theoretical modeling was done for an elliptic bubble assuming that the entire surface is rigid. The theoretical predictions agree with the experimentally observed velocities. This calculation also suggests that a sideways elongated shape is the preferred one in the presence of surfactants. In Chapter 4, the displacement of a shear thinning power-law fluid by an inviscid fluid is analyzed. The flow domain is divided into three different regions and a leading order solution is obtained using the technique of matched asymptotic expansions. Since the displacement is very slow, in the capillary statics region the shape of the interface is nearly circular. Under the lubrication approximation, the interface shape in the transition region is obtained by numerically integrating an evolution equation. Matching condition with the capillary statics region provides an expression for the film thickness in the constant film thickness region. In the case of Newtonian fluids, irrespective of individual physical properties, capillary number uniquely determines the film thickness. On the other hand, in the case of shear thinning power-law fluids, the relationship is unique to a given fluid and flow conditions. Just as in the Newtonian case, as the Capillary number is increased, the film thickness increases linearly on logarithmic scale. However, at a given capillary number, fluids with a lower power-law index have a smaller film thickness. The flow field away from the interface in the power-law fluid region is also solved assuming that the gradients of components of velocities are negligible in the directions normal to the gap direction. Unlike the Newtonian case, the depth averaged velocity field is found 79 be non-Darcian. A stability analysis is carried out for a flat interface and a dispersion relation is derived for the growth rate as a function of wave number. The result indicates that the wave number for the maximum growth rate decreases as the power-law index decreases. In other words, the dominant wave length increases as the fluid becomes more and more non-Newtonian. Hence, the interface will be less sinuous for the power-law fluid 1 2 3 4 5 6 7 8 9 10 11 12 13 REFERENCES G. I. Taylor and P. G. Saffman, â€œA note on the motion of bubbles in a Hele-Shaw cell and porous medium,â€ Q.J.Mech.Appl.Math. 12, 265 (1959) P.G. Saffman, and G.I.Taylor, â€œThe penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid,â€ Proc. R. Soc. London Ser. A 245,312 (1958). S. Tanveer, â€œThe effect of surface tension on the shape of a Hele-Shaw cell bubble,â€ Phys.Fluids 29, 3537 (1986) S. Tanveer, â€œNew Solutions for steady bubbles in a Hele-Shaw cell,â€ Phys.Fluids 30, 651 (1987) T. Maxworthy, â€œBubble formation, motion, and interaction in a Hele-Shaw cell,â€ J.Fluid.Mech. 173, 95 (1986) A. R. Kopf-Sill and G. M. Homsy, â€œBubble motion in a Hele-Shaw cell,â€ Phys. Fluids 31,18 (1988) W. Eck and J. Siekmann, â€œOn bubble motion in a Hele-Shaw cell, a possibility to study two-phase flows under reduced gravity,â€ Ing. Arch 47, 153 (1978) W. Johann and J. Siekmann, â€œMigration of a bubble with adsorbed film in a Hele- Shaw cell,â€ Acta AstronÃ¡utica 5,687(1978) C.-W. Park and G. M. Homsy, â€œTwo-phase displacement in Hele-Shaw cells: Theory,â€ J.Fluid.Mech. 139, 291 (1984) D. A. Reinelt, â€œInterface conditions for two-phase displacement in Hele-Shaw cells,â€ J.Fluid.Mech. 183, 219 (1987) S. Tanveer, and P.G. Saffman, â€œStability of Bubbles in a Hele-Shaw Cell,â€ Phys. Fluids 30, 2624(1987). P.G. Saffman, and S.Tanveer, â€œPrediction of bubble velocity in a Hele-Shaw cell: Thin film and contact angle effects,â€ Phys. Fluids A 1, 219 (1989). D. Burgess, and M.R.Foster, â€œAnalysis of the boundary conditions for a Hele- Shaw bubble,â€ Phys. Fluids A, 2, 1105 (1990). 80 14 15 16 17 18 19 20 21 22 23 24 25 26 27 81 G.M.Homsy, â€œViscous Fingering in porous media,â€ Ann.Rev.Fluid Mech.,\9:21\- 311(1987) C.W.Park, and G.M.Homsy, â€œThe instability of long fingers in Hele-Shaw Flows,â€ Phys.Fluids 28(6), 1583-1585(1985) H.Lamb, Hydrodynamics, 6th ed. (Cambridge University Press, Cambridge, 1932), pp.581-584 S.Ross, and I.D.Morrison, Colloidal Systems and Interfaces, (John Wiley & Sons, New York, 1988). M.J.Rosen, Surfactants and Interfacial Phenomena, 2nd ed., (John Wiley & Sons, New York, 1989). R. E. Davis and A. Acrivos, â€œThe influence of surfactants on the creeping motion of bubbles,â€ Chem. Eng. Sci. 21, 681 (1966) C.-W. Park, â€œInfluence of soluble surfactants on the motion of a finite bubble in a capillary tube,â€ Phys. Fluids A 4, 2335 (1992) K.J.Stebe, S.Y.Lin, and C.Maldarelli, "Remobilizing surfactant retarded fluid particle interface. I. Stress-free conditions at the interfaces of micellar solutions of surfactants with fast sorption kinetics," Phys. Fluids A 3, 3 (1991). M.Van Dyke, Perturbation Methods in Fluid Mechanics, (Academic Press, New York, 1964) F. P. Bretherton, â€œThe motion of long bubbles in tubesâ€ J.Fluid.Mech. 10, 166 (1961) B.Levich, Physicochemical Hydrodynamics, (Prentice Hall Englewood Cliffs, NJ, 1962). G.I.Taylor, â€œDeposition of viscous fluid on the wall of a tube,â€ J. Fluid Mech. 10, 161 (1961). S.S.Sadhal, and R.E.Johnson, "Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film-exact solution," J. Fluid Mech. 126,237(1983). J. Ratulowski and H.-C. Chang, â€œMarangoni effects of trace impurities on the motion of long gas bubbles in capillaries,â€ J.Fluid.Mech. 210, 303 (1990) BIOGRAPHICAL SKETCH Krishna Maruvada was born in Andhra Pradesh, India. He got his bachelors degree in chemical engineering from Birla Institute of Technology & Science (BITS), Pilani, India, in 1990. Subsequently he worked for 2 years for a chemical company called Sriram Fibers, at New Delhi as a projects engineer. Later he joined BITS, Pilani and served as a lecturer for one year. In 1993, he joined the graduate program at the University of Florida, Gainesville, and started working toward his Ph.D. degree. 82 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ã¡?^L-7-7-1 Chang-X^aM Park, Chairman Associate 'Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Dinesh O. Shah Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ranga Narayanan Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. /V. - * Lewis Johns Professor of Chemical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Wei Shyy Professor of Aerospace Engineering, Mechanics and Engineering Science I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Richard B. Dickinson Assistant Professor of Chemical Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. December, 1997 4 0 V Winfred M. Phillips Dean, College of Engineering Karen A. Holbrook Dean, Graduate School xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E7NAWSAV0_ZVFO80 INGEST_TIME 2014-06-23T21:46:17Z PACKAGE AA00022288_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES |