Thermocapillary Bubble Migration at High Reynolds and Marangoni Numbers under Low Gravity

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

179, 114–127 (1996)

0193

Thermocapillary Bubble Migration at High Reynolds and Marangoni Numbers under Low Gravity MATTHIAS TREUNER,* ,1 VLADIMIR GALINDO,† GUNTER GERBETH,† DIETER LANGBEIN,*

AND

HANS J. RATH *

*Z ARM, University of Bremen, D-28334 Bremen, Germany; and †Research Center Rossendorf, D-01314 Dresden, Germany Received May 26, 1995; accepted September 1, 1995

The thermocapillary migration of single bubbles at high Reynolds and Marangoni numbers has been investigated. Experiments on the transient behavior of moving bubbles with Marangoni numbers up to Mg Å 2500 were carried out in the Bremen drop tower. After a long heating period, a sufficiently linear temperature gradient was established with paraffin liquids. For 4.74 s of strongly reduced gravity the speed of the bubble migration was observed with a video camera. The temperature field was recorded by Pt100 temperature gauges and by differential interferometry. Numerical calculations of the steady-state flow were performed and were found to be in good correspondence with the experimental data. The calculations for large Marangoni numbers show a slight but important correction of previous results. The specific advantages of using a drop tower for such experiments result in multiple experimental data on transient and quasisteady thermocapillary bubble migration, showing much better statistical reliability than previous microgravity experiments. q 1996 Academic Press, Inc. Key Words: thermocapillary bubble dynamic; low gravity experiments; high Reynolds and Marangoni numbers.

1. INTRODUCTION

Thermal gradients along the interface of fluid particles suspended in a continuous liquid phase lead to particle migration due to the temperature dependence of the interfacial tension. This gravity-independent phenomenon is a main cause of particle transport under conditions of reduced gravity. It is also involved in the basic heat and mass transfer mechanisms of many technological processes (1). The recent literature documents a rising interest in the understanding and predictability of thermocapillary particle migration. Comprehensive surveys of this topic have been given by Subramanian (2) and by Wozniak et al. (3). The parameters characterizing thermocapillary bubble migration, as revealed by the steady-state momentum and energy equations, are the Reynolds and the Marangoni numbers ( Eqs. [1] , [ 2 ] ) . Both are mutually coupled by the 1

To whom correspondence should be addressed.

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Z Z

Ìs dT R r r , ÌT dz h

Re Å

U0rR n

Mg Å

U0rR Å RerPr, a

[2]

n . a

[3]

Pr Å

with U0 Å

[1]

Here, s denotes the surface tension, h the dynamic viscosity, n the kinematic viscosity, a the temperature diffusivity, R the bubble radius, T the temperature, and z the coordinate in the direction of rising temperature. Early theoretical investigations, in particular the classical work of Young et al. ( 4 ) , were concerned with the steady-state motion under the assumption of negligible convective transport ( Mg ! 1 ) and a low Reynolds number Re ! 1. In other words, the paper of Young et al. ( 4 ) represents the thermocapillary analogon to the classical Stokes flow around a spherical drop (22, 23). Later, Subramanian ( 5 ) additionally took into account the convective transport terms in the energy equation, such that the validity of the creeping flow solution was extended to high Prandtl number liquids. The first numerical solution to the general case of steady-state moving bubbles in liquids with high Reynolds and Marangoni numbers was given by Szymczyk ( 6 ) . His results present the dimensionless bubble velocity dependent on the Reynolds ( Re õ 100 ) and the Prandtl numbers ( Pr õ 100 ) . The maximum Marangoni number reached in these calculations was Mg Å 2000 for a liquid with Pr Å 100. Balasubramaniam and Lavery ( 7 ) confirmed these calculations and extended the parameter range for Prandtl numbers Pr £ 1 up to the

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0021-9797/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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Prandtl number ( Eq. [ 3 ] ) . Since a characteristic bubble velocity cannot be determined a priori, the velocity U0 derived from the tangential stress balance at the free surface is used for scaling the migration velocity in Eqs. [1] and [ 2 ] :

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Reynolds numbers of Re Å 2000. Characteristic results of these previous investigations and a comparison with the present numerical calculations are given in Section 3. The first investigation of transient bubble motion valid in the creeping flow limit was reported by Dill and Balasubramaniam (8). They found that for Prandtl numbers Pr ú 1 the required time to reach the steady-state velocity is given by the temperature diffusion time

ta Å

R2 . a

[4]

The transient creeping flow of bubbles and drops was investigated by Dill and Balasubramaniam (9) and independently by Galindo et al. (10). Recently, Oliver and De Witt (11) extended these predictions to the case of higher Marangoni numbers (Mg à 200) by taking into account the convective transport in the energy equation. The general case of the transient thermocapillary bubble motion at high Reynolds and Marangoni numbers hitherto has not been solved theoretically. In contrast to the theoretical work documented in literature, very few low-gravity experiments have been carried out up to date (2). The most extensive data were presented by Thompson and DeWitt (12), who used the drop tower at NASA Lewis Research Center with 5.2 s of free fall. Though they reported experimental conditions which involved strong convective effects, their results were in agreement with theoretical data for negligible convection. A discussion and a summary of this discrepancy as well as on uncertainties caused by residual accelerations in the experiments were given by Subramanian (2). Further experiments were carried out by Siekmann and Wozniak (13) during the D1-Spacelab mission. They found the migration of five bubbles in silicon oil to be in good agreement with the theoretical results of Szymczyk (6). The same results have also been documented by Na¨hle et al. (14). They reported eight bubble runs, among which a stronger scattering of the results became obvious, showing the difficulties in performing such experiments. Due to the low Reynolds numbers in these experiments (Re õ 0.4), the results must be assigned to the creeping flow region. Reliable experimental results which make it possible to confirm the theory including all inertia and convective effects still are missing. The experiments described in the following are intended to fill this gap. It is important to emphasize the advantages and disadvantages of using a drop tower for this type of investigations. Obviously, the short duration of the experiments, 4.7 s, seems to represent a disadvantage. The time to observe slow migrations, as in the case of small Reynolds and Marangoni numbers, is clearly too short. However, there is a series of

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arguments in favor of the drop tower: Investigations of high Reynolds and Marangoni numbers require strong temperature gradients, which lead to the fact that because of the temperature dependence of the material properties, an asymptotic steady velocity never does exist. In that case thermocapillary bubble migration fundamentally is an unsteady event, but is able to develop a quasi-steady-state behavior within the available time. From the scientific point of view the transient behavior is even more interesting than the steady state, but was not experimentally analyzed until now. In this scientific situation the overwhelming advantage of the drop tower becomes crucial in order to obtain a reliable and reproducible data base on transient bubble migration: Frequent availability of the experimental conditions leads to a high rate of reproducibility and a reduction of the experimental scatter. This is particularly important compared to experiments in space. Though the paper is mainly an experimental one, numerical results for the steady-state migration velocity will be presented, too. The reason is twofold: —In any case the steady-state migration velocity serves as a reference value for the measured velocities. With respect to the recently predicted transient oscillations (Oliver and DeWitt (11)), it will be particularly interesting if the measured values are always below the asymptotic steady velocity or may even exceed it. —From the numerical point of view improved results compared to the currently available data are obtained and may be crucial for the theoretical analysis of the asymptotic case of an infinite Marangoni number. The following comments are aimed at stressing explicitly some of the subsequent notations and to avoid possible confusion. The set of dimensionless parameters (Re, Mg) defined above is related to U0 and is meaningful for constant material parameters only. When the temperature dependence of all material properties is taken into account, but U0 is still the characteristic velocity, the notation (Re*, Mg*) will be used. Furthermore, when the measured local migration velocity U rather than U0 is considered, the notation (Re *, Mg * ) is introduced. Obviously, this last notation is the most physical with respect to the real bubble migration. On the other hand, the other two are necessary and useful for the comparison with theory and the comparison with literature data. Moreover, these different notations are necessary in order to present and emphasize the strong transient behavior of bubble migration, which is an intrinsic feature of investigations at higher (Re, Mg) numbers. This is the main concern of the present paper. 2. EXPERIMENTS

2.1. Experimental Setup and Procedure Though the low-gravity time in the drop tower is fairly restricted, there is ample time to prepare satisfactory temper-

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FIG. 1. Sketch of the test cell.

ature conditions before the drop. A linear temperature field, resulting in a stratified density field under normal gravity conditions, is a stable configuration and is restored within a few seconds after a disturbance. As described in the following, the easy access to the drop tower facility enables fast results and the possibility to improve and frequently reproduce experimental details. Figure 1 shows the construction of the experimental apparatus. The optical cell with a square cross section was welded from 8-mm-thick glass plates (Hellma GmbH, Mu¨llheim/ Baden, Germany). The top is covered with a heating plate, equipped with a filling and a degassing tube. In order to allow evaporating bubbles to be released by the degassing tube, the top plane is polished with slightly concave shape. At the bottom of the cell a cooling plate, powered by Peltier elements, is installed with a set of two integrated injection needles. After the drop of the capsule, two bubbles are generated by precision syringes and the needles are retracted by a small strike (2 mm) of the lifting magnets. The orifices of the needles have a diameter of 0.3 mm and enable the bubbles to be released with minimum turbulence. A set of temperature gauges (Pt100) is positioned within the liquid and at the heating plate and the cooling plate (T1rrrT7), thus

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enabling control of the temperature field. The sensors were calibrated with an accuracy of 0.1 K. Bubble migration was recorded with a high-resolution video camera (monochrome) and a Hi8 video recorder. In order to simplify evaluation, a real size grid was superimposed on the observation by a beam-splitter. All the components were remotely controlled from the control room of the drop tower by the timeline channels of the drop capsule. After a heating period of approximately 2 h a sufficiently linear temperature gradient was established in the test cell. The capsule’s drop, the bubble generation, and the retraction of the needles then were started by a timeline program. Since the time required for the bubble generation and the time scale of the transient bubble behavior depend on the bubble size (Eq. [4]), the bubbles were limited to diameters of 2 mm. In this case (diameter, 2.0 mm), a total time of 0.5 s was needed for generation, so that 4.2 s was left to observe the migration. Since the liquids wet the material of the injection needles (brass), the liquid meniscus at the needle’s orifice was not stable during the heating period before the drop. The liquids crept into the needles, thus causing a small liquid jet in the moment of bubble generation. Thus, in some of the first experiments, the migration after the bubble release was slightly deviated in the horizontal direction (see Fig. 9). The use of a shearing interferometer revealed this disturbance, and the electronic controlling device, which drives the syringes for injection, was then improved to permit manual control of the liquid meniscus. In this way it was possible to create purely vertical bubble motion in most of the experimental runs. Three paraffin liquids, n-octane (C8H18 ), n-decane (C10H22 ), and n-tetradecane (C14H30 ) (Janssen Chimica), were used for the experiments. The injected gas was air and in some cases pure nitrogen to check if there were influences on the surface tension. Maximum temperature differences applied were in the range between 35 and 70 K, and the bubble diameters ranged between 0.5 and 2.0 mm. Two isothermal runs were carried out in order to check if any residual accelerations were acting on the bubble. In these runs short bubble releases of about 0.2 s were observed, but the bubbles returned to their origin, forced by the drag of the retracting needles. The stationary buoyancy-induced migration velocity of a bubble with 1-mm diameter due to a residual acceleration of 10 05 g0 is about 0.05 mm/s. Since the velocities observed in the experiments are on the order of magnitude of 10 mm/s, buoyancy forces are by far too small to influence the thermocapillary migration. 2.2. Evaluation of the Test Data The evaluation of the observed bubble migrations is explained in the example shown in Figs. 2 and 3. The measured

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FIG. 2. Temperature distribution in the cell.

temperature distribution in the cell is depicted in Fig. 2. Sensor T1 is mounted in the heating plate, Sensors T2 to T6 are located in the liquid, and T7 is mounted in the cooling plate. The temperatures measured in the liquid are overlain by a linear regression, showing satisfactory initial conditions for the experiment. In Fig. 3, the position of a single bubble is depicted versus the time after the bubble release. The bubble was generated at a height of 8 mm, was slightly retracted by the needle’s strike, and then began its continuous ascent. In order to determine the bubble velocity, the measured position is overlaid by a regression of order 3. The velocity then is determined by differentiation. It becomes obvious, in that case, that the velocity increases continuously. Comparison of the experimental results with the theoretical data is advantageously done in a dimensionless representation. The measured velocity is related to the velocity of Young et al. (4, Eq. [5]), which has been determined analytically in the creeping flow limit UYGB (Re ! 1) Å 0.5r U0 .

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to and after the drop of the capsule by the ring method (tensiometer K10T, Kru¨ss GmbH). Minor differences in the surface tension gradient D( Ìs / ÌT ) £ 0.01 mN/mK were found in some cases after a retention time of 36 h in the test cell. Since the liquids used were charged 3 to 4 h before the drop, the measured values for fresh fluids (Table 1) were taken for the evaluation. When the temperature dependence of all physical properties is taken into account, the question of presentation of the results arises. The problem is that the introduced velocity scale U0 and both parameters Re and Mg are no longer constant, but dependent on temperature, and consequently change their values during bubble motion. For any comparison with theoretical calculations or previous results from literature, the nondimensional presentation is crucial. The optimum approximation is to use the local values Re* and Mg*. They are defined in such a way that the physical properties are taken at the temperature corresponding to the linear temperature profile (see Fig. 2) at the current bubble position. The consequences of this presentation are shown in Fig. 4, where in this dimensionless form the same bubble migration as in Fig. 3 is shown. Figures 3 and 4 describe measured values of a single bubble, beginning with the ascent after the bubble release. During the total migration length of 40 mm, the effective Marangoni number increases by about 60%, mainly because of the decrease in viscosity with increasing temperature and the corresponding increase of U0 . The black circles are marking the time. Whereas in the dimensional presentation (Fig. 3) the velocity is linearly increasing, the dimensionless velocity rises degressively due to the increasing value of UYGB . Within 2 s, the Prandtl number changes from Pr Å 13.7 to Pr Å 10.8. Before presentation and discussion of more experimental results, the theoretical approach is described.

[5]

As stated in the Introduction, the parameters governing the thermocapillary bubble motion are the Reynolds ( Eq. [1] ) and the Marangoni number ( Eq. [ 2 ] ) . These dimensionless parameters include physical data which are temperature dependent, in particular, the dynamic viscosity. Especially for studying high Marangoni numbers, large temperature differences are required. In these cases the moving bubble is exposed to a continuously rising temperature and accordingly to changing liquid properties. The relevant physical properties of the liquids together with the constitutive, temperature-dependent relations are given in Table 1. The corresponding coefficients for the different liquids are taken from a database ( 15 ) . The surface tension of the liquids was measured just prior

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FIG. 3. Observed migration of an air bubble in n-decane; diameter d Å 2 mm.

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TABLE 1 Physical Data of the Test Liquids (from the TAPP Data Base (14)) Density

rÅ

SD

S

r0 kg T with A Å 1 0 r1A m3 r2

D

r3

S

D

h1 1 1000 T W 03 Thermal conductivity l Å K0 / (K1r10 )rT mr K c2r106 03 c0 / c1r10 rT / / c3r1006rT 2 / c4r1009rT 3 1 103 T2 J Specific heat cp Å M kg K l m2 h m2 Thermal diffusivity a Å Kinematic viscosity n Å r s rrcp s T s1 N 03 Surface tension s Å s0r 1 0 1 10 Tcrit m

Viscosity

h Å 10B 1 1003 (Pas) with B Å h0 /

S D

S

SD S D

Coefficient

D

SD

SD

n-Octane

n-Decane

S D n-Tetradecane

r0 r1 r2 r3 h0 h1 K0 K1 c0 c1 c2 c3 c4 s0 s1 M (g/mol) Tcrit (K)

228.1 0.2548 568.8 0.2694 01.8818 0.4738 0.216 00.296 06121.26 42,465.05 53.732 0104,961 92,163.27 52.04 1.217 114.23092 568.8

232.80 0.2524 618.50 0.2857 01.937 0.5585 0.206 00.25 1867.3 016,520 0 56,760 063,120 55.78 1.32 142.28468 618.4

235.9 0.2582 692.4 0.2663 02.0043 0.6897 0.196 00.199 509.02 0948.54 0 2387.90 0 56.44 1.366 198.3922 692.4

Ìs (N/mK) ÌT

00.10 1 1003

00.10 1 1003

00.08 1 1003

Note. Values of (Ìs/ÌT) (N/mK) are our own measurements.

3. THEORETICAL ANALYSIS

A full numerical simulation of the transient bubble motion including the temperature-dependent physical properties is not yet available. Therefore, in order to improve the physical understanding and to support the interpretation of experimental results, two subproblems will be addressed. First, numerical results for the steady asymptotic velocity are presented. Second, some available results for the transient case are summarized. 3.1. Formulation of the Steady Problem The thermocapillary migration of a single gas bubble in a liquid reaches an asymptotic, steady-state velocity if all relevant material properties are assumed to be temperature independent except for the interfacial tension at the gas–

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liquid interface. In order to calculate this steady velocity a code originally developed by A. Gelfgat from Technion Haifa (see, e.g., Gelfgat and Tanasawa (16)) was further developed and adapted to the general two-fluid problem of a migrating drop. Here only the case of a bubble will be described; results for the drop will be published in a subsequent paper. The theoretical approach is based on the following assumptions: —The flow is assumed to be incompressible and Newtonian. —Shape deformations are ignored; thus the bubble can be treated as an object that remains spherical and hence the normal stress balance at the bubble surface can be ignored. This presumes that the change of the interfacial tension Ds along the surface is small compared to s itself.

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The boundary conditions on the flow and temperature fields are given as follows. Distant from the bubble, the velocity and temperature are undisturbed. The boundary of the computational domain is taken as a circle with radius R` . The following conditions are required: n Å 0u`ez;

T Å r cos( u ), for 0 £ u £ and

Under these assumptions the spherical bubble migrates parallel to the direction of the temperature gradient. After an initial nonsteady phase, the bubble reaches a constant migration velocity U` . It is convenient to use a reference frame traveling with the bubble. The bubble center is taken as the origin of coordinates, and the direction of the temperature gradient as the polar axis. The flow, temperature, and pressure fields are presupposed to be axisymmetric. Using spherical coordinates (r, u, w ) all quantities appear to be independent of the azimuthal angle w, and the problem can be treated as essentially two-dimensional. The equations governing the steady case are the conservation of mass [6]

the equations of motion (Navier–Stokes) Re( nrÇ) n Å 0Çp / Ç2n ,

[7]

and the energy transport equation Mg(u` / ( nrÇ)T ) Å DT.

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£r Å 0

sr u Å

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Ì Ìr

SD nu r

[10] Å

ÌT Ìu

[11]

ÌT Å0 Ìr

[12]

at r Å 1. The stress tensor component sr u has been made dimensionless in the same manner as the pressure. Due to the assumption of a spherical bubble a normal stress balance is not necessary. Thus, the steady-state problem is settled by two independent nondimensional parameters, namely, the Reynolds and the Marangoni numbers of the matrix fluid. Because of Eq. [2] it is obvious that also the pair (Pr, Mg) can be chosen as the determining set. The bubble migration velocity U` in the steady state must be determined from the force balance. The net hydrodynamic force F acting on the bubble volume is obtained by integrating the axial components of the normal and shear stresses of the external fluid over the bubble surface. This balance can be written as

[8]

The lengths are scaled in units of the bubble radius R , velocities are sealed in units of U0 , and the pressure is scaled in units of U0h / R . The temperature T is scaled by ( dT / dz )É` R and is considered relative to the current position of the bubble.

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ÌT p Å 1, for £ u £ p. Ìz 2

The a priori boundary condition T Å r cos( u ) all around the computational domain is replaced by the last condition for the temperature derivative at the bubble rear. Such types of ‘‘outflow’’ conditions are well known from wake simulations (17) and have proved useful for the present temperature wake, too. At the interface between bubble and external fluid, i.e., at r Å 1, the following boundary conditions apply: no flow across the interface, balance of the shear stress, and no heat flux across the surface. These conditions are expressed by

FIG. 4. Bubble migration in dimensionless presentation.

div n Å 0,

p 2

F Å 2p

*

p

0

F

nu(1 0 3 cos 2u ) 0 sin 2u

0 sin 2u

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du Å 0 rÅ1

[13]

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was added to the problem. Convergence was supposed to be reached when all the unknown functions—the fluid velocity, the pressure, the temperature, and the migration velocity— satisfy the stopping criterion \ f n /1 0 f n\ / \ f n\ õ 10 05 , where f n is the value of the function f at the nth time step of integration. Since near the bubble surface large gradients are to be expected, a fine grid near the surface and a coarse one far away from the bubble are used. Nonregular meshes are employed outside the bubble. Nodes in the r direction are stretched near the boundary of the bubble. Nodes are defined as half of the roots of a Chebyshev polynomial shifted to the intervals [R, R` ]. Typical results for the stream function, vorticity, and temperature distribution around the steady migrating bubble are shown in Figs. 5 and 6. The anisotropy between the up- and downstream directions is caused by convective transport. It increases with increasing Marangoni number. The infinite domain outside the bubble is truncated in the r direction; i.e., the conditions at infinity are evaluated at a finite R` . In order to estimate how much the domain can be truncated without loss of physically relevant information, we performed some preliminary calculations for large Marangoni numbers, where thermal boundary layers are expected, with different values of R` and examined the disturbance of the external temperature gradient caused by the flow and heat transfer outside the bubble. The influence of R` on the numerical result of the bubble migration velocity U` was studied. For example, in the case Pr Å 10 and Mg FIG. 5. Temperature lines (top) and isolines of the perturbation from uniform temperature gradient (bottom) for Pr Å 10 and different Marangoni numbers Mg.

and, finally, gives the condition to determine U` . 3.2. Numerical Method and Results for the Steady-State Case The problem formulated above was solved with the control volume finite difference method using a predictor–corrector semi-implicit scheme for a straightforward time integration. The time dependence is made artificially due to numerical reasons only. For details about the numerical method see the paper of Gelfgat and Tanasawa (16). Conservative properties provided by the control volume method are very important for correct calculation of the force balance in Eq. [13]. Straightforward integration in time was carried out until convergence to a stationary solution was reached. To obtain the migration velocity u` in the same computational process the formal equation du } É FÉ dt

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FIG. 6. Stream lines (top) and lines of equal vorticity (bottom) for Pr Å 10 and different Marangoni numbers Mg.

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121

in this paper do not decrease below the predicted asymptotic value given in Eq. [16]. This is particularly interesting since the boundary layer analysis of Crespo and Jimenez-Fernandez (18, 19) still contains a singularity at the rear stagnation point, and it seems to be an open question (as mentioned, e.g., by Subramanian (2)) whether the consideration of this singularity will change the asymptotic value [16]. Here it can be stated that the present numerical results seem to confirm Eq. [16], whereas the numerical results available up to now (6, 7) were in contradiction to the result [16]. 3.3. Some Transient Results

FIG. 7. Numerical results for the nondimensional migration velocity as a function of Pr and Mg.

Å 1000, the bubble migration velocities for R` Å 5, 10, 20, and 30 are U` Å 0.2162, 0.2238, 0.2252, and 0.2262, respectively. In any case it is guaranteed that R` has no influence on the results presented in the following. The present calculations of the steady case were done primarily to obtain some theoretical results for a precise experiment configuration. However, the question arises as to what extent these calculations are identical to the known results of Szymczyk (6) and Balasubramaniam and Lavery (7). The present results for U` and the comparison to the literature data are shown in Fig. 7. At higher Marangoni numbers these curves show a distinct difference from the literature data. The reason is obvious: The calculations of Balasubramaniam and Lavery (7) and Szymzcyk (6) were performed with a fixed value of R` between 5 and 10, which becomes a too small computational domain if with increasing Marangoni number the temperature trail reaches R` . The present results were always confirmed by test calculations varying R` and the grid mesh. This slight difference in the computational results becomes very important if the asymptotic velocity for Mg r ` is considered. This asymptotic case was investigated by Crespo and Jimenez-Fernandez (18, 19) using a boundary layer approach. They derived the asymptotic values

Up to now the theoretical description covers the steady case only. However, the experimental situation is strongly transient due to the short migration time of 4.7 s and the fact that the bubbles start practically from rest. Of course, a numerical extension of the steady to the time-dependent situation would be interesting, but is not available currently. In order to obtain some first transient results the creeping flow limit (Re, Mg ! 1), which was also studied by Dill and Balasubramaniam (8, 9), was considered in (20). Typical results for the fluids used in the present experiments are shown in Fig. 8 in physical units. It is interesting to mark the two different time scales, tn Å R 2 / n and ta Å R 2 /a, which define the transient behavior. As a rough estimate it can be stated that after tn Å R 2 / n the migration velocities reach about 70–80% of the final value. After this quick velocity rise a rather long time Çta is required for the final 20–30% of the velocity development. These aspects are discussed only qualitatively in the present

U` Å 0.445r UYGB for Mg r ` , Re ! 1 (Ref. (18)) [15] U` Å 0.392r UYGB for Mg r ` , Re r ` (Ref. (19)). [16] In contrast to the numerical results documented in the literature (6, 7) the results for the migration velocity U` presented

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FIG. 8. Unsteady thermocapillary bubble migration in the creeping flow limit for the three liquids used. The bubble starts from rest and approaches its asymptotic velocity U` . R Å 1 mm, dT/dz Å 1 K/mm. The vertical lines mark the momentum time scales tn , respectively. The temperature diffusion time scales ta are out of the figure ( ta É 12.2 s in each case).

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context. They depend also on the various possible initial conditions, as discussed in (21). Oliver and DeWitt (11) extended for the first time the transient description to higher values of Mg. They assumed that Re ! 1 but allowed Marangoni numbers up to Mg Å 200, and solved numerically. They found the interesting result that oscillations in the transient velocity may occur for a bubble starting from rest and approaching its final velocity U` . Lower Marangoni numbers (Mg õ 10) behave as an overdamped system, higher Marangoni numbers as an underdamped one with oscillations in velocity. However, it must be stressed that a direct application of these results to the present measurements is not possible due to much higher values of Re and Mg in the experiments. Nevertheless, it gives some first hints for possible interpretation. 4. EXPERIMENTAL RESULTS

Figure 9 shows a sequence of bubble positions observed during the experiments. At the upper right corner of each picture, the time after the drop is marked. Due to the slight

turbulence caused by the needles’ retraction, the two bubbles often started with different delay times. The resulting velocities, however, reached the same values in all the cases observed. The case depicted shows an additional small bubble beneath the regular one on the right side, which in the moment of the capsule’s drop was released unintentionally. In agreement with theory, when larger bubble radii render higher velocities, the large bubble passes the small one very quickly. In all the experiments the bubble shape was controlled by visualization. It turns out that within an accuracy of at least 10% it was not possible to observe any deviation from the spherical shape. The total numbers of bubble runs evaluated for the three liquids are presented in Figs. 10–12, respectively. The form of presentation coincides with Fig. 4, which was described above. In each run, time proceeds from left to right. Average values for the Prandtl numbers Pr characterizing the different liquids are given in the diagrams, where this average corresponds to the mean temperature in the cavity. In order to refer the measurements to a theoretical value, the steady-

FIG. 9. Example of observed bubble migration.

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FIG. 10. Dimensionless bubble velocity in octane vs Marangoni number. Mg* Å U0rR/a, — Steady-state theory.

FIG. 11. Dimensionless bubble velocity in decane vs Marangoni number. Mg* Å U0rR/a,— Steady-state theory.

FIG. 12. Dimensionless bubble velocity in tetradecane vs Marangoni number. Mg* Å U0rR/a,— Steady-state theory.

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FIG. 13. Observed migration behavior of several bubble runs.

state result is always given for the corresponding Prandtl number (solid line). The experimental results group into two different regions: Most of the runs show a continuous increase of the bubble migration velocity in qualitatively good agreement with the transient calculations of Fig. 8, but staying always below the calculated steady state by 5–20%. This group is observed mainly for high Marangoni numbers (Mg ú 1000). The other group, mainly at lower Marangoni numbers (100 õ Mg õ 500), shows

more complex behavior. The courses depicting the migration may show distinctive minima or maxima or may continuously increase or decrease during the observation, even if they represent the same parameter range. These effects are strongest for n-tetradecane and trigger the assumption that they increase with rising Prandtl number. Some curves of n-octane and n-tetradecane at lower Marangoni values even exceed the theoretical steady-state velocities. Typical examples of the various behaviors observed are depicted in Fig. 13 in dimensional presentation. Curves 13a and 13b represent two bubble runs where the transient course differs significantly instead of having nearly the same parameters. Case 13c is the characteristic result of a bubble whose velocity increases continuously during the period of observation. After a fast increase to a certain velocity, many bubbles even remained to run with nearly constant speed, as in case 13d. Course 13e, finally, represents the case of n-tetradecane, for which the strongest transient effects had been observed. Since the three liquids used had different average Prandtl numbers, the experimental data for each liquid on the average reach different velocity levels, too. This result becomes most distinctive in the presentation shown in Figs. 14–16. Here the Reynolds number Re * (defined with the local bubble velocity U) has been plotted versus the Marangoni number Mg*. The actual Reynolds numbers of n-octane (Pr Å 8.2) exceed the theoretical curve for Pr Å 10; the values of n-decane (Pr Å 12.1) stay below, as do the values of ntetradecane (Pr Å 25.5) by a larger amount. Whereas the Reynolds numbers of n-octane and n-decane increase during the observation time, the values for n-tetradecane appear to decrease continuously, except in one case. The overall accuracy of the measurement depends most sensitively on the determination of the drop radius and the surface tension gradient and is limited to an error of {5%. Reconsideration of the applied evaluation method in light of the mentioned errors does not lead to a qualitative change in the observed results.

FIG. 14. Reynolds versus Marangoni number in n-octane. Re * Å UrR/ n, Mg* Å U0rR/a. Steady-state theory: -r-, Pr Å 1; — , Pr Å 10; rrr, Pr Å 100.

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FIG. 15. Reynolds versus Marangoni number in n-decane. Steady-state theory: -r-, Pr Å 1; — , Pr Å 10; rrr, Pr Å 100.

5. DISCUSSION

The complex behavior observed requires a comprehensive explanation. In most of the bubble courses the maximum velocities reach average values of 75 to 95% of the steadystate velocities predicted by theory. This is true for small bubbles reaching dimensionless times of 2.5ta during the observation time, as well as for bigger bubbles which do not exceed values of 0.3ta . Investigations on the transient behavior in the creeping flow limit published recently (10) yield about 90% of the terminal velocity even after 0.3ta . In contrast to the theoretical description, where the terminal velocity is calculated for only one parameter set, the real transient bubble migration is exposed to strongly varying conditions. The comparison of the theoretical steady-state velocities with the measurements thus reveals a principal discrepancy, in particular at high Marangoni numbers. The migrating bubbles try to accelerate to the local steady-state velocities, which they never can reach. This qualitatively implies that the measured velocities shall always be below the theoretical values, even after several periods of ta . Several strong difference in the transient behavior were

observed and may be explained by variations of the initial conditions caused by the generation process. While in most of the runs depicted in Figs. 10 and 11 the bubbles start at low velocities, in a few cases they start with high values because they are released unintentionally at the moment of the capsule’s drop and accelerated by buoyancy. The strong variation of the time-dependent behavior can be explained neither by different initial conditions only nor by slight measuring errors. Bubble courses of Figs. 10–12, which start at high velocities, pass through a minimum, and then increase in velocity again, must have different explanations. As mentioned above, new results of Oliver and De Witt (11) on the transient behavior possibly provide an explanation. Whether their calculated oscillating transient behavior can explain some of the present measurements is currently not being investigated since theory and experiment cover different parameter regions. Clarification on that point may be reached only by a numerical simulation which considers all transient and convective effects. An interesting comparison of the earlier experimental data with our numerical results is given in Fig. 17. The data documented by Thompson and De Witt (12) for three liquids (Pr Å

FIG. 16. Reynolds versus Marangoni number in n-tetradecane. Steady-state theory: -r-, Pr Å 1; — , Pr Å 10; rrr, Pr Å 100.

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TREUNER ET AL.

FIG. 17. Experimental results documented in literature.

15, 125, and 137) have been given for the bubble’s Reynolds numbers (defined with the measured velocity U) as a function of the theoretical Reynolds number (defined with U0 , in (12), termed Marangoni number). The experimental data for Pr Å 15 appear to fit the theoretical steady-state curve for Pr Å 10, whereas the values for Pr Å 125 and 137 clearly exceed the theoretical curve for Pr Å 100. Due to the documented experimental conditions, the bubbles appeared to be exposed to residual accelerations. If the measured velocities were influenced by buoyancy, the too high Reynolds numbers were evident. Further data from Siekmann and Wozniak (13) are given for Pr Å 687 and most of the data correlate very well with the theoretical values of Pr Å 1000.

6. CONCLUSION AND PROSPECT

Results of drop tower experiments concerning thermocapillary bubble migration at high Reynolds and Marangoni numbers were presented. The observed complex transient behavior was described and compared to steady-state calculations. Terminal velocities observed in the experiment seem to confirm the theoretical values within a limit of 5 to 20%. The discrepancies between theoretical steady-state considerations and real transient bubble migrations were discussed and lead to the necessity of fully transient models. Numerical steady-state results presented here revealed corrections of previous results due to the use of a larger computational domain. The existence of an asymptotic velocity for infinite Marangoni numbers, as discussed in the literature, seems to be confirmed by these calculations. Investigations by means of shearing interferometry are currently carried out to reveal the temperature distribution around the bubbles. Figure 18 shows the image of two bubbles observed by horizontal beam separation. Differential interferometry visualizes deviations of constant refractive index gradients in the liquid which can be used for the evaluation of the temperature fields. The bubbles in Fig. 18 have diameters of 3 mm and move with a speed of approximately 15 mm/s. Distinctively observable is a trial of bubbles showing similarity to the theoretically analyzed temperature field in Fig. 5. The investigations, together with quantitative evaluations of the interferometer pictures, are continuing and will be published in a forthcoming paper. ACKNOWLEDGMENTS

FIG. 18. Refractive index distribution of moving bubbles, observed by a shearing interferometer with horizontal beam separation. Bubble size D Å 3 mm.

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This work was supported by the Deutsche Agentur fu¨r Raumfahrtangelegenheiten (DARA) under Grants 50 QV 8717 and 50 WM 9213. The

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THERMOCAPILLARY BUBBLE MIGRATION authors appreciate very much the assistance of Dipl.-Ing. Peter Prengel, Dipl.-Ing. Dirk Lockowandt, cand.-Ing. Henrik Fricke, and cand. phys. Fikret Sisman for their support on the experimental work. Special appreciation is also expressed to Dr. Gelfgat of Technion Haifa (Israel) for providing the basic numerical code.

REFERENCES 1. ‘‘Proceedings of an RIT/ESA/SSC Workshop, Ja¨rva Krog, Sweden 10–20 January 1984.’’ ESA SP-219. 2. Subramanian, R. S., in ‘‘Transport Processes in Drops, Bubbles and Particles’’ (R. D. Chhabra and D. Dekee, Eds.). Hemisphere, New York, 1991. 3. Wozniak, G., Siekmann, J., and Srulijes, J., Z. Flugwiss. Weltraumforsch. 12, 137 (1988). 4. Young, N. O., Goldstein, J. S., and Block, M. J., J. Fluid Mech. 6, 350 (1959). 5. Subramanian, R. S., AIChE J. 27(4), 646 (1981). 6. Szymczyk, J. A., Ph.D. thesis, University of Essen, 1985. 7. Balasubramaniam, R., and Lavery, J. E., Numer. Heat Transfer A 16, 175 (1989). 8. Dill, L. H., and Balasubramaniam, R., Proc. AIP Conf. 197, 481 (1988). 9. Dill, L. H., and Balasubramaniam, R., Int. J. Heat Fluid Flow 13, 78 (1992).

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10. Galindo, V., Gerbeth, G., Langbein, D., and Treuner, M., Microgravity Sci. Technol. 7(3), 234 (1994). 11. Oliver, D. L. R., and De Witt, K. J., J. Colloid Interface Sci. 164, 263 (1994). 12. Thompson, R. L., and De Witt, K. J., ‘‘Marangoni Bubble Motion in Zero Gravity.’’ NASA Technical Memorandum TM 79250, 1979. 13. Siekmann, J., and Wozniak, G., Israel J. Tech. 23, 179 (1986). 14. Na¨hle, R., Neuhaus, D., Siekmann, J., Wozniak, G., and Srulijes, J., Z. Flugwiss. Weltraumforsch. 11, 211 (1987). 15. TAPP, a Database of Thermochemical and Physical Properties, ES Microware, 1992. 16. Gelfgat, A. Yu., and Tanasawa, I., Microgravity Sci. Technol. 8(1) (1995). 17. Fornberg, B., J. Fluid Mech. 98, 819 (1980). 18. Crespo, A., and Jimenez-Fernandez, J., in ‘‘IUTAM Symposium on Microgravity Fluid Mechanics Bremen’’ (H. J. Rath, Ed.), p. 405. Springer-Verlag, New York/Berlin, 1992. 19. Crespo, A., and Jimenez-Fernandez, J., ‘‘Thermocapillary Migration of Bubbles: A Semi-Analytical Solution for Large Marangoni Numbers,’’ ESA SP-333, p. 193, 1992. 20. Galindo, V., and Gerbeth, G., Phys. Fluids A 5, 3290 (1993). 21. Galindo, V., Gerbeth, G., Langbein, D., and Treuner, M., in ‘‘Proceedings Drop Tower Days Bremen, July 1994,’’ p. 90. 22. Hadamard, J., C. R. Acad. Sci., Paris, 1, 1735 (1911). 23. Rybezynski, D., Bull. Acad. Sci., Cracovie, 1, 40 (1911).

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