Ind. Eng. Chem. Res. 1996,34,3411-3416
3411
Motion of Drops and Bubbles in Reduced Gravity R. Shankar Subramanian Department of Chemical Engineering, Box 5705, Clarkson University, Potsdam, New York 13699-5705
A brief summary is given of the research program of the author on the movement of drops and bubbles in a reduced gravity environment due to the action of capillary forces. 1. Introduction
It is a pleasure to congratulate Professors Bird, Stewart, and Lightfoot on the 35th anniversary of the publication of their classical textbook Transport Phenomena (1960). This book has served as an inspiration to me ever since 1969 when I first encountered it as a graduate student a t Clarkson taking the first transport phenomena course. I have continued to learn from it throughout my career and have used it on numerous occasions as a text. This paper is intended t o be a brief summary of the research program conducted by former and current students a t Clarkson on the motion of drops and bubbles under conditions when gravity is not a significant driving force. This is the situation existing aboard orbiting spacecraft as well as within space probes which are in near free fall under the action of the local gravitational field. Within such vehicles, to a first approximation all objects also are under the same free fall environment, and therefore appear weightless. In particular, the usual gravity-driven movement of particles, bubbles, or drops suspended in a fluid of different density would be negligible. Thus, one might expect to see such objects virtually stay in place unless one takes appropriate action. Fortunately, a temperature difference can be used to make bubbles or drops move. This motion is a direct consequence of the variation of interfacial tension with temperature which leads to a thermocapillary stress at the interface between the drop (“or bubble” will be omitted from hereon, but is implied) and the continuous phase. Because of this stress, the state of rest in the two fluids adjoining the interface is not tenable. The ensuing motion of the fluids usually leads to a propulsion of the drop, labeled “thermocapillary migration,” in the general direction of the warmer fluid. A variation of the composition of species on the interface also can lead to such capillary stresses and drop movement. However, a uniform temperature gradient is more straightforward to establish and maintain when compared t o a similar composition gradient. Therefore, virtually all existing experimental research on this subject has focused on thermocapillary migration. The literature on both experimental and theoretical research in this field up to approximately 1989 has been adequately reviewed by Wozniak et al. (1988) and Subramanian (1992). Given the purpose of this paper, I shall not present a review of either the earlier literature or that which has appeared since 1989. Bubbles and drops are similar fluid mechanically in that they share a mobile interface. The significant differences are simply those in the relevant ratios of physical properties of the dispersed phase to that of the continuous phase. For gas bubbles, one can usually consider the viscosity, density, and thermal conductivity t o be negligible when compared to similar properties in the surrounding liquid. Therefore, one needs to solve the governing transport equations only in one phase, 0888-5885/95/2634-3411$09.00/0
which simplifies the situation. Because of this, some of the early work on modeling was restricted to the case of gas bubbles. Besides plain scientific curiosity, why would one want to study the behavior of bubbles or drops in spacecraft? The movement of drops and bubbles is relevant to situations that are likely to arise in low gravity experiments. Liquid drops might be encountered during materials processing applications in reduced gravity such as the formation and solidification of alloys. Bubbles and drops will be involved in separation processes such as absorption or extraction that might be used for recycling purposes in long duration space voyages. During such excursions, one would have t o recycle oxygen as well as water. The exchange of matter between the drop and the surrounding fluid is facilitated by movement of the drop bringing it into contact with fresh fluid continuously. Therefore, if we understand how to make drops move, we can design suitable processes for such separations. Also, a dispersion of vapor bubbles might be expected in heat transfer fluids which undergo phase change. Gas bubbles are likely to be encountered in crystallization where dissolved gases are rejected at the interface and also in separation processes such as gas absorption. In most applications, a collection of drops or bubbles would be involved in which the individual members will influence the motion of each other, and also possibly coalesce leading to changes in size distributions over time. In addition to applications in space, one can envision situations on the Earth wherein this type of movement may be important. An example is when the drops are virtually of the same density as the suspending fluid. Also, when a bubble or drop is very small, of the order of 20-150 pm in radius, thermocapillary migration can be as important as gravity-induced motion. An example of this is an industrial process for forming certain alloys for automobile engine parts which uses vertical temperature gradients t o influence the dispersion of fine drops of one material in another (for details, see summary by Walter (1994)and article by Ratke (1994)). The objective of the research program at Clarkson has been to develop an understanding of the thermocapillary migration and interactions of drops and bubbles from first principles. Both theoretical predictions and experimental observations are necessary to achieve this objective. Predictions from models alone are inadequate because of the idealizations usually involved in formulating the models. Nevertheless, they can be a good guide in designing appropriate experiments and in interpreting the data. As will be noted later, the range of parameters that can be explored on the ground is limited, and one must go to flight experiments under reduced gravity conditions to study a larger range. Thermocapillary migration is unique and interesting because the driving force for motion, namely the temperature change over the interface, is influenced by the
0 1995 American Chemical Society
3412 Ind. Eng. Chem. Res., Vol. 34, No. 10,1995 motion itself. This leads to full coupling of the momentum and energy equations when convective energy transport is not negligible. Also, one finds that the disturbance flow created by a moving drop has interesting structural features that are not normally present in the corresponding problem wherein motion occurs due to a body force; furthermore, the decay rates of the disturbance flow with distance from the drop are different for the two cases. Finally, in thermocapillary migration, the drop is continually moving into fluid which is at a different temperature and therefore one must be concerned with the transient phenomena associated with this process and the variation of physical properties with temperature, especially when the motion is rapid. 2. Summary of Program
2.1. Experimental Work. When I first became interested in this topic in 1978, the literature was sparse both in theoretical predictions and in experimental observations. Nearly two decades prior to that time, Young et al. (1959) had published the results of pioneering studies demonstrating that air bubbles could be held motionless or even forced to "sink" in a liquid while in the Earth's gravitational field by the application of a judicious downward temperature gradient. They also had solved the relevant governing equations under conditions of negligible convective transport of momentum and energy t o predict how rapidly a bubble or drop would move when simultaneously subjected to the force of gravity and a vertical temperature gradient. Their principal result for the quasi-steady migration (vector) velocity (U)of a drop or bubble is reproduced below.
Here, g' is the density of the drop phase fluid, e the density of the continuous phase fluid, p the dynamic viscosity of the continuous phase fluid, g the (vector) acceleration due to gravity, R the drop radius, CJT the rate of change of interfacial tension with temperature, and VT, the (vector) temperature gradient imposed in the continuous phase fluid. The symbol a stands for the ratio of the viscosity of the drop phase to that of the continuous phase, and ,!I represents a similar ratio of thermal conductivities. For a gas bubble, the viscosity and thermal conductivity of the gas are both quite negligible compared t o the same properties of the surrounding liquid. Thus, one might set the property ratios, a and ,!I, to zero in the above result. Also, one might consider the density of the gas negligible compared to that of the continuous phase. Implicit in writing eq 1 is the assumption that the temperature gradient does not give rise to buoyant convection because such effects are not included in the prediction. This is the reason for the use of vertical temperature gradients in experimental work on thermocapillary migration. The experiments of Young et al. were focused on measuring the temperature gradients necessary to hold bubbles still. They did not report results on migration velocities. Therefore, R. Merritt, for his doctoral thesis (1988), built a test cell similar to that of Hardy (1979), who also had performed experiments analogous to those of Young et ul. while eliminating several sources of experimental error. Since the ratio of surface forces to
body forces varies inversely with the length scale of the drop, to observe the effects of thermocapillarity one must work with relatively small bubbles and drops, typically of radius ranging from 20 to 150 pm. This requires the use of magnification to obtain a sufficiently large image, and also the mastery of a technique for the introduction of a single bubble or drop a t a time. Video was the medium of choice for recording the images because of the modest needs it imposed on lighting. The images were initially recorded on reel-to-reel videotape, but gradually with the passage of time, the medium has evolved through 314 in. and 112 in. VHS videocassettes to the current standard of 112 in. S-VHSvideocassetttes. The video recorded during an experiment is later analyzed on a frame-by-frame basis to obtain data on bubble size and position as functions of time. Menitt performed experiments on air bubbles injected one a t a time into highly viscous silicone oils in which he had established suitable downward temperature gradients to force the bubbles t o move downward. He was careful not to exceed the critical Rayleigh number which would trigger intense buoyant convection in such an unfavorably stratified fluid. He confirmed from tracer studies performed independently that in fact the residual convection in the cell was negligible when compared to the bubble migration velocities. In Merritt's experiments, the Reynolds and PBclet numbers were both negligible. These quantities are defined in the usual way with the bubble radius as the length scale, a thermocapillary velocity scale defined below, and the appropriate physical properties of the continuous phase. Note that the definitions are good only when motion is predominantly driven by thermocapillarity. If a large gravitational contribution exists, one must change the velocity scale appropriately. The reference velocity used in thermocapillary motion, UO, can be obtained from scaling arguments as ug
=
l%d I V T 3 P
(2)
The Reynolds number Re and the PBclet number are defined below. When the above reference velocity for thermocapillary motion is used, the PBclet number is commonly identified as the Marangoni number, Mu.
RUO Re =V
M a =RVKO
(3) (4)
Here, v is the kinematic viscosity of the continuous phase and K is its thermal diffisivity. Menitt fitted his results to the prediction of Young et al. and obtained a gradient of surface tension with temperature that was consistent with the value reported by Hardy (1979) for a similar silicone oil. This confirmed the linear scaling of the thermocapillary velocity with the bubble radius. While Merritt did not directly verify the linear scaling with the applied temperature gradient evident from eq 1, this was accomplished by Barton (1990)in a later doctoral thesis on the motion of drops. Merritt went on to perform some experiments with bubbles moving toward the bottom wall of the test cell in a region where the motion was clearly affected by the wall. He observed behavior which runs counter to common intuition. Under certain conditions, a bubble
Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 3413 moved more rapidly when it was near the surface than a bubble of the same size would if located far away. The explanation is quite simple when one considers the motion of the bubble in this Stokes problem as a superposition of that due to thermocapillarity (downward) and that due to gravity (upward). The disturbance flow created by a bubble moving due to thermocapillarity is a potential dipole while the motion caused by a body force is a Stokeslet. The former decays as l P , where r is distance measured from the center of the bubble. The latter decays much more slowly, namely as llr. The disturbance flows are retarded by the surface leading t o more resistance being offered t o the motion of the bubble. Because of the different decay characteristics of the disturbance flows, the upward contribution to the velocity of the bubble (due to the body force) is reduced more substantially than the downward contribution (due to thermocapillarity), leading t o larger velocities downward than would be observed if the same bubble were located far away from the wall. Merritt found that when the body force contribution to the motion was properly subtracted, the results were consistent with predictions made by Meyyappan (1984)in his doctoral thesis dealing with theoretical problems involving interactions of a bubble with a boundary or with another bubble. The work initiated by R. Merritt has been continued by several graduate students over the past decade. Space does not permit a full accounting of all of their accomplishments. However, details may be found in their theses and their publications. Here, I shall only give a summary. When the group at Clarkson became interested in the thermocapillary migration of liquid drops, there was very little experimental work available on this subject. A NASA report by Lacy et al. (1982)detailed work performed by B. Facemire on the migration of drops rich in ethyl salicylate (ES) in diethylene glycol (DEG).The focus of these investigators was the study of miscibility gap systems in which a binary mixture is heated to a temperature above the critical solution point so that only a single liquid phase exists. However, upon cooling the mixture into the miscibility gap, one finds a suspension of fine drops rich in one component precipitating out of solution. The authors were studying such a system when they observed the ES-rich drops move upward in an upward temperature gradient even though ES is more dense than DEG. B. Facemire, who performed the migration studies, correctly inferred that this was a consequence of thermocapillarity. However, the measured velocities did not agree with the predictions of Young et al., probably due to the possibility of background convection in the system as well as interactions with other drops. The first student a t Clarkson t o study liquid drops was D. Fusco (19861,who observed drops of ethylene glycol as well as those of methanol moving in a silicone oil. Fusco found that the drops moved under isothermal conditions at velocities corresponding closer to the prediction of Stokes than those of Hadamard and Rybczyl’lski (see Happel and Brenner, 19651,which she took to be evidence of contamination of the interface by surface-active species. Surface activity displayed by trace contaminants is ubiquitous in any system involving two fluids in contact. It is commonly accepted that such surfactants are swept by motion of the fluid on the interface toward stagnation points and the resulting surfactant concentration gradient on the interface sets
up an interfacial tension gradient (Frumkin and Levich, 1947). This usually opposes the motion of the interface and therefore reduces its “mobility.” The situation is particularly serious when one considers capillaritydriven motion since the entire driving force resides on the interface and can be reduced drastically by such surfactant accumulation. The measurements made by Fusco in a temperature gradient were clouded by the existence of background free convection in the cell which was discovered only toward the end of the experimental runs. Therefore, she was unable to draw definitive conclusions regarding thermocapillary effects in the systems studied in her Master’s thesis. As a matter of interest, FUSCO’S work motivated another doctoral student, H. Kim, to study the effect of surfactants on thermocapillary migration when they are insoluble in the bulk. Kim, in his doctoral thesis (1988),provided the solution of a surfactant conservation equation on the surface of a drop in conjunction with the momentum and energy equations in both phases. His work proved useful to another graduate student, M. Nallani, who chose to interpret data she had obtained for her Master‘s thesis (1991)on methanol drops migrating in a silicone oil via a surfactant hypothesis. Nallani used Kim’s analysis of a drop possessing a stagnant cap of surfactants moving in a vertical temperature gradient to fit her data quite satisfactorily even though there was no direct evidence of such a cap from the experiments. In the mid 1980s,K. Barton began an experimental study of the thermocapillary motion of liquid drops as a doctoral student at Clarkson. After considering various systems, he selected the same pair that had been used a t NASA by Lacy et al. He made careful measurements of the motion of ethyl salicylate drops in an upward temperature gradient in diethylene glycol after triply distilling each fluid to purify it, and also conducted control experiments in which he added a known surface-active agent (Triton X-100) to both phases which arrested the thermocapillary motion. Barton’s results, reported in his thesis (1990)clearly demonstrated the correctness of the linear dependence of the thermocapillary contribution to the velocity on the applied temperature gradient and the drop radius predicted by Young et al. in eq 1. Later, Barton went on t o present clear evidence that when the thermocapillary effect opposes that of the body force, drops can move more rapidly near a surface than when far away; he developed an analysis which permitted him to predict the results observed in this situation. This was a more explicit demonstration than that provided earlier by Menitt because the air bubbles in Menitt’s experiments significantly changed size during a m,thereby making the interpretation of the data more difficult. More recently, H. Wei, in his doctoral thesis (1994), reported experimental results on interacting pairs of bubbles in a downward temperature gradient. He designed a system in which it was possible to record the motion as viewed from two orthogonal directions and reconstruct the three-dimensional trajectories followed by the bubbles. The most interesting observation made by Wei was that, when two bubbles move under the action of a downward temperature gradient, a large bubble moving downward can actually force a small bubble below it to move upward toward it. In the absence of the large bubble, the small bubble would move quite rapidly downward. This observation can be explained by considering the detailed structure of the flow field caused by the moving bubble. In his doctoral
3414 Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995
thesis, Wei also provided results from theoretical calculations for the problem of a chain of two or three bubbles moving in a temperature gradient under conditions of negligible Reynolds and PBclet numbers such that the line of centers is either parallel or perpendicular t o the applied temperature gradient. To perform these calculations, he used the method of boundary collocation modeled after Hassonjee et al. (1988). He found that when the bubbles were sufficiently far apart, the solution obtained by the method of reflections by Anderson (1985) was quite sufficient t o describe the influence of a given bubble on another. Wei went on to provide a painvise approximation for the interactions of three bubbles using Anderson's solution by reflections. Wei established that his observations on pairs of bubbles were supported by the predictions he made from theory. Experiments on the Earth are somewhat limited in scope due to interference from gravity. For instance, except when the densities are closely matched, there will be a substantial gravity-driven component of the motion. Since the motion influences the temperature difference over the interface which provides the driving force for thermocapillary movement, this confounds the interpretation of data except for situations of negligible Reynolds number wherein velocity fields can be superposed. Even when the densities are matched, it is possible to achieve this condition only a t one temperature. Therefore, some interference from gravity is always present. Furthermore, as one uses drops of larger sizes to obtain larger values of the Marangoni number, one must use larger test cells to minimize effects of interactions from the boundaries. This rapidly escalates free convection in the cell which commonly arises from horizontal temperature gradients caused by heat loss through the side walls. In view of the above restrictions, R. Balasubramaniam, C. E. Lacy, G. Wozniak, and I performed experiments on board the United States Space Shuttle Columbia in orbit during summer 1994 in an apparatus termed the Bubble, Drop, and Particle Unit (BDPU) which was controlled from the ground via telecommands. We were able to inject bubbles of air and drops of fluorinert as large as 15 mm in diameter into a silicone oil and observe and record their trajectories in temperature gradients ranging from 0.25 to 1.0 Wmm. Video records and a limited amount of cine film were obtained. The analysis of this data is under way. Perhaps the most interesting observation made was that a large drop trailing a relatively small drop may never reach it even though, when isolated, the larger drop will move more rapidly than the smaller one. 2.2. Theoretical Developments. Since 1978, graduate students at Clarkson also have made contributions in the theoretical arena. There were attempts made to extend the analysis of Young et aZ. to the case of snlall but nonnegligible Reynolds and Marangoni numbers via perturbation. I had examined this problem and pointed out in 1981 that the perturbation problems are singular in nature, very similar to those studied by Proudman and Pearson (1957) and Acrivos and Taylor (1962). N. Shankar studied two problems in his doctoral thesis (19841, one of which was the steady thermocapillary motion of an isolated bubble under conditions when the Reynolds number is negligible, but the Marangoni number takes on small t o moderate values. (The other problem studied by Shankar dealt with the motion of a gas bubble within a drop when the drop surface is subjected to a known, but arbitrary, axisymmetric
temperature distribution.) Highly viscous liquids with a large Prandtl number ( d ~ would ) yield conditions relevant to the model equations used by Shankar. He solved the energy equation along with the associated boundary conditions numerically by the method of finite differences while using an analytical solution of Stokes 'equations for the velocity field. His most interesting observation was that a separated reverse flow wake forms behind the moving bubble and that one can be found for any nonzero value of the Marangoni number, no matter how small. However, the strength of the flow in this wake was found to be significant only for moderate values of the Marangoni number. Later, D. Morton (1990), another doctoral student who experimentally studied the Stokes motion of compound drops (which are drops containing one or more droplets) and also constructed theoretical descriptions of such problems, found similar wakes behind compound drops undergoing thermocapillary migration. Barton also had discovered a similar physical feature when he solved the problem of the Stokes motion of a drop normal to a plane surface due to a temperature gradient. The fluid near the drop moved in the general direction of motion of the drop, but the far-field fluid was found to move in the opposite direction. Most recently, H. Wei demonstrated the appearance of such reverse flow regions in the case of two bubbles migrating in a temperature gradient under Stokes flow conditions. This feature is clearly universal in problems wherein a drop undergoes thermocapillary migration under conditions of negligible inertial effects. In fact, the results of Balasubramaniam and Lavery (1989), who carried out numerical calculations for the motion of an isolated bubble for a wide range of values of Reynolds and Marangoni numbers, indicate that similar behavior may be expected even when inertial effects are important. The explanation of this flow structure lies in the nature of the driving force for thermocapillary migration. The temperature field on the surface of a drop in an axisymmetric problem always can be expanded in a series of Legendre polynomials; in the nonaxisymmetric case, one must use associated Legendre functions but the basic explanation is similar. In the case of an isolated drop executing thermocapillary migration when both the Reynolds and the Marangoni numbers are negligible (the situation analyzed by Young et uZ.1, the surface temperature field is proportional to cos 8 where 8 is the polar angle measured from the forward stagnation streamline. This is, of course, the first Legendre polynomial, and the higher modes are simply not excited in this highly symmetric situation. The resulting thermocapillary stress, derived from the temperature gradient on the surface, drives a flow that is exactly the same as one from a potential dipole. This velocity field in the fluid decays as I/$, where r is distance measured from the center of the drop. When the symmetry of the above problem is disturbed in some way, the temperature field in general will consist of a sum of all Legendre polynomials, suitably weighted. The pure Legendre mode P,(cos 8 ) of the surface temperature field drives a flow in the fluid that is n-cellular, and which decays as l / P . Given this situation, it is clear that the flow arising from the surface temperature field that is proportional to the second Legendre polynomial will dominate the far-field structure, no matter how small its contribution. This flow is bicellular with fluid approaching the drop along the equator and being thrown out along the poles (the direction of the flow would be reversed were the sign of
Ind. Eng. Chem. Res., Vol. 34, No. 10, 1995 3416 this term reversed). This is the explanation of the ubiquitous separated reverse flow wakes predicted by the above students in their problems. In fact, as Wei points out in his thesis, it is possible to have a reverse flow region exist in front of a moving pair of bubbles, so that the region does not necessarily have to be a wake. Other fascinating flow structures have been described by Merritt in his thesis. These occur when gravitational and thermocapillary effects act simultaneously on a drop. For instance, when a bubble moves downward due to a temperature gradient opposing its buoyant rise, the far-field flow still is upward since it is dominated by the Stokeslet whose direction is upward. In the near field, the fluid is pushed ahead of the moving bubble in the downward direction and drawn in toward the bubble behind it. This leads to an opposition of flows along the stagnation streamlines. The flow region is divided into two by a spherical separatrix concentric with the bubble. When the bubble remains motionless, the separatrix collapses onto the surface of the bubble; it disappears when the bubble moves upward. This is the case when the flow is observed in a laboratory reference frame. When one travels with the moving bubble, one notes the exact opposite. In this situation, it is found that the flow is divided into two regions by a spherical separatrix when the bubble moves upward. When the thermocapillary effect dominates so as to force the bubble downward, the separatrix disappears. In the context of interactions among bubbles, M. Meyyappan studied pairs of bubbles in his doctoral thesis (1984), restricting his attention to Stokes flow and negligible values of the Marangoni number. He obtained results by specializing the general solutions in bispherical coordinates obtained by Jeffery (1912) and by Stimson and Jeffery (1926) for the Laplace and Stokes equations, respectively. Meyyappan made the remarkable observation that, when two bubbles are placed in a temperature gradient such that their line of centers is aligned with the gradient, the smaller of the two always will move more rapidly than when it is isolated, and the larger of the two always will move slightly less rapidly than when isolated. This leads to the logical conclusion that when two bubbles are precisely equal in size, each should move with the velocity it would have when isolated, a result confirmed numerically by Meyyappan. This result also holds when the bubbles are in any arbitrary orientation with respect to the temperature gradient vector. It is in contrast to the case of body force driven motion wherein one predicts in the equivalent situation that both bubbles would move more rapidly than when isolated. While Meyyappan made his observations from a numerical evaluation of the infinite series solutions, the solution obtained by Anderson (1985) by the method of reflections was found to display the same behavior to the order presented. More recently, Feullebois (1989) demonstrated analytically that the series solution indeed yields the results obtained by Meyyappan in the case of equal-sized bubbles, and Acrivos et al. (1990) have shown that this is a special case of a more general result that is valid for a collection of any number of equalsized bubbles. 2.3. Current Program. At present there is one doctoral student carrying on research in this field at Clarkson. X. Ma is studying the behavior of an isolated drop in a second liquid under conditions when the drop is suspended in place due t o the combined action of a
temperature gradient and gravity. He is canying out experiments on this subject and also developing theoretical predictions to compare with the observations. While some work has been done on similar systems in the past, the role of convective heat transport still remains to be explored. This is the objective of Ma’s research. In addition to the above doctoral student, Dr. Claud E. Lacy, a post-doctoral Research Associate, is conducting an experimental program on the influence of convective transport on the motion of drops and also interactions between pairs of drops. It is anticipated that the Marangoni number in these experiments might perhaps be as large as 50 while the Reynolds number will be negligible. In this situation, it is possible to explore in a limited fashion the role of convective transport of energy on thermocapillary migration here on the ground.
3. Concluding Remarks This has been a very short presentation of the work of several graduate students over the past 15 years on the subject of capillarity-driven motion of drops and bubbles. The problems chosen by these individuals demonstrate the importance of transport phenomena in the newly emerging field of microgravity science and applications. It is hoped that this will serve as an illustration of the foresight of Professors Bird, Stewart, and Lightfoot in introducing the subject of transport phenomena to a wide audience through the original publication of their book. The results of the research performed at Clarkson have been reported in several articles that have appeared in print over the years and any interested reader can contact me for representative reprints. Most of the articles arising from the work at Clarkson have been cited and discussed in my review (1992). Also, copies of the doctoral theses may be obtained directly from University Microfilms in Ann Arbor, MI, and those of the Master’s theses by writing t o me.
Acknowledgment The work described herein was supported by NASA’s Microgravity Sciences and Application Division through NASA Grants NAG3-1122 and NAG3-1470 from the Lewis Research Center to Clarkson University. I am most indebted to all of the graduate students whose work has been discussed above, as well as to Dr. Lacy, for helping me learn about physical features unique t o capillary migration problems. Also, I would like to express my gratitude to Dr. R. Balasubramaniam of NASA’s Lewis Research Center for numerous helpful discussions over the years.
Literature Cited Acrivos, A.; Taylor, T. D. Heat and Mass Transfer from Single Spheres in Stokes Flow. Phys. Fluids 1962,5,(4),387-394. Acrivos, A.; Jeffrey, D. J.; Saville, D. A. Thermocapillary and Electrophoretic Migration of Particles in Suspension. J.Fluid Mech. 1990,212,95-110. Anderson, J. L. Droplet Interactions in Thermocapillary Motion. Znt. J. Multiphase Flow 1985,11, (6), 813-824. Balasubramaniam, R.; Lavery, J. E. Numerical Simulation of Thermocapillary Bubble Migration Under Microgravity for Large Reynolds and Marangoni Numbers. Numer. Heat Transfer A 1989,16, 175-187.
3416 Ind. Eng. Chem. Res., Vol. 34,No. 10,1995 Barton, K. D. Thermocapillary Migration of Drops. Ph.D. Dissertation, Clarkson University, 1990. Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Tmnsprt Phenomena; Wiley: New York, 1960. Feuillebois, F. Thermocapillary Migration of Two Equal Bubbles Parallel to Their Line of Centers. J . Colloid Interface Sci. 1989, 131, (l),267-274. Frumkin, A. N.; Levich, V. G. Zh. Fiz. Khim. 1947, 21, 1183, as cited in Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1962. Fusco, D. L. Thermocapillary Droplet Migration. M.S. Thesis, Clarkson University, 1984. Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Prentice-Hall, Englewood Cliffs, NJ, 1965. Hardy, S. C. The Motion of Bubbles in a Vertical Temperature Gradient. J . Colloid Interface Sci. 1979, 69, (l), 157-162. Hassonjee, Q.;Ganatos, P.; Pfeffer, R. A Strong-Interaction Theory for the Motion of Arbitrary Three-Dimensional Clusters of Spherical Particles at Low Reynolds Number. J . Fluid Mech. 1988,197,l-37. Jeffery, G. B. On a Form of the Solution of Laplace’s Equation Suitable for Problems Relating to Two Spheres. Proc. R . SOC. 1912, A87, 109-120. Kim, H. S. Surfactant Effects on the Thermocapillary Migration of a Droplet. Ph.D. Dissertation, Clarkson University, 1988. Lacy, L. L.; Witherow, W. K.; Facemire, B. R.; Nishioka, G. M. Optical Studies of a Model Binary Miscibility Gap System. NASA TM-82494,1982. Merritt, R. M. Bubble Migration and Interactions in a Vertical Temperature Gradient. Ph.D. Dissertation, Clarkson University, 1988. Morton, D. S. The Motion of Compound Drops. Ph.D. Dissertation, Clarkson University, 1990. Meyyappan, M. Interaction Effects in Thermocapillary Bubble Migration. Ph.D. Dissertation, Clarkson University, 1984. Nallani, M. Thermocapillary Migration of Liquid Drops. M.S. Thesis, Clarkson University, 1991.
Proudman, I.; Pearson, J. R. A. Expansions at Small Reynolds Numbers for the Flow Past a Sphere and a Circular Cylinder. J . Fluid Mech. 1957,2, 237-262. Ratke, L. Bearings for Car Engines Made from Immiscible Alloys-a Spin off from Research in Space. Low G The INTOSPACE Journal for the Industrial Utilization of Microgravity 1994, 5, (2), 7-9. Shankar, N. Motion of Bubbles Due to Thermocapillary Effects. Ph.D. Dissertation, Clarkson University, 1984. Stimson, M.; Jeffery, G. B. The Motion of Two Spheres in a Viscous Fluid. Proc R . Soc. 1926,Alll, 110-116. Subramanian, R. S.Slow Migration of a Gas Bubble in a Thermal Gradient. AZChE J . 1981,27, (4),646-654. Subramanian, R. S. The Motion of Bubbles and Drops in Reduced Gravity. In Tmnsport Processes in Drops, Bubbles, and Particles; Chhabra, R., Dekee, D., Eds.; Hemisphere: New York, 1992;pp 1-42. Walter, H. U. Microgravity Research Spin off and Applications-Example: Marangoni Convection. Microgravity News from ESA 1994, 7, (l),2-4. Wei, H. Interactions of Bubbles in a Temperature Gradient. Ph.D. Dissertation, Clarkson University, 1994. Wozniak, G.; Siekmann, J.; Srulijes, J. Thermocapillary Bubble and Drop Dynamics Under Reduced Gravity-Survey and Prospects. 2. Flugwiss. Weltraumforsch. 1988, 12, 137-144. Young, N. 0.;Goldstein, J. S.; Block, M. J. The Motion of Bubbles in a Vertical Temperature Gradient. J . Fluid Mech. 1959, 6, 350-356. Received for review September 16, 1994 Accepted December 8,1994@
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* Abstract published in Advance ACS Abstracts, August 15, 1995.
