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

JOURNAL OF COLLOID AND INTERFACE SCIENCE 179, 114–127 (1996)ARTICLE NO. 0193

Thermocapillary Bubble Migration at High Reynolds andMarangoni 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

Prandtl number (Eq. [3 ] ) . Since a characteristic bubbleThe thermocapillary migration of single bubbles at high Rey- velocity cannot be determined a priori, the velocity U0

nolds and Marangoni numbers has been investigated. Experiments derived from the tangential stress balance at the free sur-on the transient behavior of moving bubbles with Marangoni num- face is used for scaling the migration velocity in Eqs. [1]bers up to Mg Å 2500 were carried out in the Bremen drop tower.

and [2] :After a long heating period, a sufficiently linear temperature gradi-ent was established with paraffin liquids. For 4.74 s of stronglyreduced gravity the speed of the bubble migration was observed

Re Å U0rR

nwith U0 Å Z ÌsÌT ZrdT

dzr

R

h, [1]with a video camera. The temperature field was recorded by Pt100

temperature gauges and by differential interferometry. Numericalcalculations of the steady-state flow were performed and werefound to be in good correspondence with the experimental data. Mg Å U0rR

aÅ RerPr, [2]

The calculations for large Marangoni numbers show a slight butimportant correction of previous results. The specific advantagesof using a drop tower for such experiments result in multiple Pr Å n

a. [3]

experimental data on transient and quasisteady thermocapillarybubble migration, showing much better statistical reliability thanprevious microgravity experiments. q 1996 Academic Press, Inc.

Here, s denotes the surface tension, h the dynamic viscosity,Key Words: thermocapillary bubble dynamic; low gravity experi-n the kinematic viscosity, a the temperature diffusivity, Rments; high Reynolds and Marangoni numbers.the bubble radius, T the temperature, and z the coordinatein the direction of rising temperature.

Early theoretical investigations, in particular the classi-1. INTRODUCTION

cal work of Young et al. (4 ) , were concerned with thesteady-state motion under the assumption of negligible

Thermal gradients along the interface of fluid particlesconvective transport (Mg ! 1) and a low Reynolds num-

suspended in a continuous liquid phase lead to particle mi-ber Re ! 1. In other words, the paper of Young et al. (4 )

gration due to the temperature dependence of the interfacialrepresents the thermocapillary analogon to the classical

tension. This gravity-independent phenomenon is a mainStokes flow around a spherical drop (22, 23). Later, Subra-

cause of particle transport under conditions of reduced grav-manian (5) additionally took into account the convective

ity. It is also involved in the basic heat and mass transfertransport terms in the energy equation, such that the valid-

mechanisms of many technological processes (1) . The re-ity of the creeping flow solution was extended to high

cent literature documents a rising interest in the understand-Prandtl number liquids. The first numerical solution to the

ing and predictability of thermocapillary particle migration.general case of steady-state moving bubbles in liquids

Comprehensive surveys of this topic have been given bywith high Reynolds and Marangoni numbers was given

Subramanian (2) and by Wozniak et al. (3) .by Szymczyk (6) . His results present the dimensionless

The parameters characterizing thermocapillary bubblebubble velocity dependent on the Reynolds (Re õ 100)

migration, as revealed by the steady-state momentum andand the Prandtl numbers (Pr õ 100) . The maximum Mar-

energy equations, are the Reynolds and the Marangoniangoni number reached in these calculations was Mg Å

numbers (Eqs. [1] , [2 ] ) . Both are mutually coupled by the2000 for a liquid with Pr Å 100. Balasubramaniam andLavery (7) confirmed these calculations and extended the

1 To whom correspondence should be addressed. parameter range for Prandtl numbers Pr £ 1 up to the

1140021-9797/96 $18.00Copyright q 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

AID JCIS 4028 / 6g0d$$$$81 03-11-96 14:37:27 coidas AP: Colloid

115THERMOCAPILLARY BUBBLE MIGRATION

Reynolds numbers of Re Å 2000. Characteristic results arguments in favor of the drop tower: Investigations of highReynolds and Marangoni numbers require strong tempera-of these previous investigations and a comparison with

the present numerical calculations are given in Section 3. ture gradients, which lead to the fact that because of thetemperature dependence of the material properties, anThe first investigation of transient bubble motion valid in

the creeping flow limit was reported by Dill and Balasubra- asymptotic steady velocity never does exist. In that casethermocapillary bubble migration fundamentally is an un-maniam (8). They found that for Prandtl numbers Pr ú 1

the required time to reach the steady-state velocity is given steady event, but is able to develop a quasi-steady-state be-havior within the available time. From the scientific pointby the temperature diffusion timeof view the transient behavior is even more interesting thanthe steady state, but was not experimentally analyzed until

ta ÅR 2

a. [4] now. In this scientific situation the overwhelming advantage

of the drop tower becomes crucial in order to obtain a reliableand reproducible data base on transient bubble migration:Frequent availability of the experimental conditions leads toThe transient creeping flow of bubbles and drops was investi-

gated by Dill and Balasubramaniam (9) and independently a high rate of reproducibility and a reduction of the experi-mental scatter. This is particularly important compared toby Galindo et al. (10). Recently, Oliver and De Witt (11)

extended these predictions to the case of higher Marangoni experiments in space.Though the paper is mainly an experimental one, numeri-numbers (Mg à 200) by taking into account the convective

transport in the energy equation. The general case of the cal results for the steady-state migration velocity will bepresented, too. The reason is twofold:transient thermocapillary bubble motion at high Reynolds

and Marangoni numbers hitherto has not been solved theoret- —In any case the steady-state migration velocity servesically. as a reference value for the measured velocities. With respect

In contrast to the theoretical work documented in litera- to the recently predicted transient oscillations (Oliver andture, very few low-gravity experiments have been carried DeWitt (11)) , it will be particularly interesting if the mea-out up to date (2) . The most extensive data were presented sured values are always below the asymptotic steady velocityby Thompson and DeWitt (12), who used the drop tower or may even exceed it.at NASA Lewis Research Center with 5.2 s of free fall. —From the numerical point of view improved resultsThough they reported experimental conditions which in- compared to the currently available data are obtained andvolved strong convective effects, their results were in may be crucial for the theoretical analysis of the asymptoticagreement with theoretical data for negligible convection. A case of an infinite Marangoni number.discussion and a summary of this discrepancy as well as on The following comments are aimed at stressing explicitlyuncertainties caused by residual accelerations in the experi- some of the subsequent notations and to avoid possible con-ments were given by Subramanian (2). Further experiments fusion. The set of dimensionless parameters (Re, Mg) de-were carried out by Siekmann and Wozniak (13) during fined above is related to U0 and is meaningful for constantthe D1-Spacelab mission. They found the migration of five material parameters only. When the temperature dependencebubbles in silicon oil to be in good agreement with the of all material properties is taken into account, but U0 is stilltheoretical results of Szymczyk (6). The same results have the characteristic velocity, the notation (Re*, Mg*) willalso been documented by Nahle et al. (14). They reported be used. Furthermore, when the measured local migrationeight bubble runs, among which a stronger scattering of the velocity U rather than U0 is considered, the notation (Re*,results became obvious, showing the difficulties in per- Mg*) is introduced. Obviously, this last notation is the mostforming such experiments. Due to the low Reynolds numbers physical with respect to the real bubble migration. On thein these experiments (Re õ 0.4) , the results must be as- other hand, the other two are necessary and useful for thesigned to the creeping flow region. Reliable experimental comparison with theory and the comparison with literatureresults which make it possible to confirm the theory includ- data. Moreover, these different notations are necessary ining all inertia and convective effects still are missing. The order to present and emphasize the strong transient behaviorexperiments described in the following are intended to fill of bubble migration, which is an intrinsic feature of investi-this gap. gations at higher (Re, Mg) numbers. This is the main con-

It is important to emphasize the advantages and disadvan- cern of the present paper.tages of using a drop tower for this type of investigations.

2. EXPERIMENTSObviously, the short duration of the experiments, 4.7 s,2.1. Experimental Setup and Procedureseems to represent a disadvantage. The time to observe slow

migrations, as in the case of small Reynolds and Marangoni Though the low-gravity time in the drop tower is fairlyrestricted, there is ample time to prepare satisfactory temper-numbers, is clearly too short. However, there is a series of


116 TREUNER ET AL.

enabling control of the temperature field. The sensors werecalibrated with an accuracy of 0.1 K.

Bubble migration was recorded with a high-resolutionvideo camera (monochrome) and a Hi8 video recorder. Inorder to simplify evaluation, a real size grid was superim-posed on the observation by a beam-splitter. All the compo-nents were remotely controlled from the control room of thedrop tower by the timeline channels of the drop capsule.

After a heating period of approximately 2 h a sufficientlylinear temperature gradient was established in the test cell.The capsule’s drop, the bubble generation, and the retractionof the needles then were started by a timeline program. Sincethe time required for the bubble generation and the timescale of the transient bubble behavior depend on the bubblesize (Eq. [4]) , the bubbles were limited to diameters of 2mm. In this case (diameter, 2.0 mm), a total time of 0.5 swas needed for generation, so that 4.2 s was left to observethe migration.

Since the liquids wet the material of the injection needles(brass) , the liquid meniscus at the needle’s orifice was notstable during the heating period before the drop. The liquidscrept into the needles, thus causing a small liquid jet in themoment of bubble generation. Thus, in some of the firstexperiments, the migration after the bubble release wasslightly deviated in the horizontal direction (see Fig. 9) .The use of a shearing interferometer revealed this distur-bance, and the electronic controlling device, which drives theFIG. 1. Sketch of the test cell.syringes for injection, was then improved to permit manualcontrol of the liquid meniscus. In this way it was possibleto create purely vertical bubble motion in most of the experi-

ature conditions before the drop. A linear temperature field, mental runs.resulting in a stratified density field under normal gravity Three paraffin liquids, n-octane (C8H18) , n-decaneconditions, is a stable configuration and is restored within a (C10H22) , and n-tetradecane (C14H30) (Janssen Chimica) ,few seconds after a disturbance. As described in the follow- were used for the experiments. The injected gas was air anding, the easy access to the drop tower facility enables fast in some cases pure nitrogen to check if there were influencesresults and the possibility to improve and frequently repro- on the surface tension. Maximum temperature differencesduce experimental details. applied were in the range between 35 and 70 K, and the

Figure 1 shows the construction of the experimental appa- bubble diameters ranged between 0.5 and 2.0 mm.ratus. The optical cell with a square cross section was welded Two isothermal runs were carried out in order to checkfrom 8-mm-thick glass plates (Hellma GmbH, Mullheim/ if any residual accelerations were acting on the bubble. InBaden, Germany). The top is covered with a heating plate, these runs short bubble releases of about 0.2 s were observed,equipped with a filling and a degassing tube. In order to but the bubbles returned to their origin, forced by the dragallow evaporating bubbles to be released by the degassing of the retracting needles. The stationary buoyancy-inducedtube, the top plane is polished with slightly concave shape. migration velocity of a bubble with 1-mm diameter due toAt the bottom of the cell a cooling plate, powered by Peltier a residual acceleration of 1005 g0 is about 0.05 mm/s. Sinceelements, is installed with a set of two integrated injection the velocities observed in the experiments are on the orderneedles. After the drop of the capsule, two bubbles are gener- of magnitude of 10 mm/s, buoyancy forces are by far tooated by precision syringes and the needles are retracted by small to influence the thermocapillary migration.a small strike (2 mm) of the lifting magnets. The orifices ofthe needles have a diameter of 0.3 mm and enable the bub- 2.2. Evaluation of the Test Datables to be released with minimum turbulence. A set of tem-perature gauges (Pt100) is positioned within the liquid and The evaluation of the observed bubble migrations is ex-

plained in the example shown in Figs. 2 and 3. The measuredat the heating plate and the cooling plate (T1rrrT7), thus



to and after the drop of the capsule by the ring method(tensiometer K10T, Kruss GmbH). Minor differences in thesurface tension gradient D(Ìs /ÌT ) £ 0.01 mN/mK werefound in some cases after a retention time of 36 h in the testcell. Since the liquids used were charged 3 to 4 h before thedrop, the measured values for fresh fluids (Table 1) weretaken for the evaluation.

When the temperature dependence of all physical proper-ties is taken into account, the question of presentation of theresults arises. The problem is that the introduced velocityscale U0 and both parameters Re and Mg are no longerconstant, but dependent on temperature, and consequentlychange their values during bubble motion. For any compari-son with theoretical calculations or previous results from

FIG. 2. Temperature distribution in the cell. literature, the nondimensional presentation is crucial. Theoptimum approximation is to use the local values Re* andMg*. They are defined in such a way that the physical

temperature distribution in the cell is depicted in Fig. 2. properties are taken at the temperature corresponding to theSensor T1 is mounted in the heating plate, Sensors T2 to linear temperature profile (see Fig. 2) at the current bubbleT6 are located in the liquid, and T7 is mounted in the cooling position. The consequences of this presentation are shownplate. The temperatures measured in the liquid are overlain in Fig. 4, where in this dimensionless form the same bubbleby a linear regression, showing satisfactory initial conditions migration as in Fig. 3 is shown. Figures 3 and 4 describefor the experiment. In Fig. 3, the position of a single bubble

measured values of a single bubble, beginning with the as-is depicted versus the time after the bubble release. The

cent after the bubble release. During the total migrationbubble was generated at a height of 8 mm, was slightly

length of 40 mm, the effective Marangoni number increasesretracted by the needle’s strike, and then began its continuous

by about 60%, mainly because of the decrease in viscosityascent. In order to determine the bubble velocity, the mea-with increasing temperature and the corresponding increasesured position is overlaid by a regression of order 3. Theof U0 . The black circles are marking the time. Whereas invelocity then is determined by differentiation. It becomesthe dimensional presentation (Fig. 3) the velocity is linearlyobvious, in that case, that the velocity increases continu-increasing, the dimensionless velocity rises degressively dueously.to the increasing value of UYGB. Within 2 s, the PrandtlComparison of the experimental results with the theoreti-number changes from Pr Å 13.7 to Pr Å 10.8.cal data is advantageously done in a dimensionless represen-

Before presentation and discussion of more experimentaltation. The measured velocity is related to the velocity ofresults, the theoretical approach is described.Young et al. (4, Eq. [5]) , which has been determined analyt-

ically in the creeping flow limit

UYGB(Re ! 1) Å 0.5rU0 . [5]

As stated in the Introduction, the parameters governingthe thermocapillary bubble motion are the Reynolds (Eq.[1] ) and the Marangoni number (Eq. [2 ] ) . These dimen-sionless parameters include physical data which are tem-perature dependent, in particular, the dynamic viscosity.Especially for studying high Marangoni numbers, largetemperature differences are required. In these cases themoving bubble is exposed to a continuously rising temper-ature and accordingly to changing liquid properties. Therelevant physical properties of the liquids together withthe constitutive, temperature-dependent relations aregiven in Table 1. The corresponding coefficients for thedifferent liquids are taken from a database (15) . FIG. 3. Observed migration of an air bubble in n-decane; diameter

d Å 2 mm.The surface tension of the liquids was measured just prior


118 TREUNER ET AL.

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

Density r Å r0

rA1Skg

m3D with A Å S1 0 T

r2Dr

3

Viscosity h Å 10B 1 1003 (Pas) with B Å h0 / Sh1 1 1000T D

Thermal conductivity l Å K0 / (K1r1003)rT S WmrKD

Specific heat cp Å

Sc0 / c1r1003rT / c2r106

T 2/ c3r1006

rT 2 / c4r1009rT 3D 1 103

M S Jkg KD

Thermal diffusivity a Å l

rrcpSm2

s D Kinematic viscosity n Å h

r Sm2

s DSurface tension s Å s0rS1 0 T

TcritDs

1

1 1003 SNmD

Coefficient n-Octane n-Decane n-Tetradecane

r0 228.1 232.80 235.9r1 0.2548 0.2524 0.2582r2 568.8 618.50 692.4r3 0.2694 0.2857 0.2663h0 01.8818 01.937 02.0043h1 0.4738 0.5585 0.6897K0 0.216 0.206 0.196K1 00.296 00.25 00.199c0 06121.26 1867.3 509.02c1 42,465.05 016,520 0948.54c2 53.732 0 0c3 0104,961 56,760 2387.90c4 92,163.27 063,120 0s0 52.04 55.78 56.44s1 1.217 1.32 1.366M (g/mol) 114.23092 142.28468 198.3922Tcrit (K) 568.8 618.4 692.4

ÌsÌT (N/mK) 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 liquid interface. In order to calculate this steady velocity acode originally developed by A. Gelfgat from Technion

A full numerical simulation of the transient bubble motion Haifa (see, e.g., Gelfgat and Tanasawa (16)) was furtherincluding the temperature-dependent physical properties is developed and adapted to the general two-fluid problem ofnot yet available. Therefore, in order to improve the physical a migrating drop. Here only the case of a bubble will beunderstanding and to support the interpretation of experi- described; results for the drop will be published in a subse-mental results, two subproblems will be addressed. First, quent paper.numerical results for the steady asymptotic velocity are pre- The theoretical approach is based on the following as-sented. Second, some available results for the transient case sumptions:are summarized. —The flow is assumed to be incompressible and New-

tonian.3.1. Formulation of the Steady Problem

—Shape deformations are ignored; thus the bubble canbe treated as an object that remains spherical and hence theThe thermocapillary migration of a single gas bubble in

a liquid reaches an asymptotic, steady-state velocity if all normal stress balance at the bubble surface can be ignored.This presumes that the change of the interfacial tension Dsrelevant material properties are assumed to be temperature

independent except for the interfacial tension at the gas– along the surface is small compared to s itself.



The boundary conditions on the flow and temperaturefields are given as follows. Distant from the bubble, thevelocity and temperature are undisturbed. The boundary ofthe computational domain is taken as a circle with radiusR` . The following conditions are required:

n Å 0u`ez; T Å r cos(u) , for 0 £ u £ p

2

andÌT

ÌzÅ 1, for

p

2£ u £ p. [9]

The a priori boundary condition T Å r cos(u) all around thecomputational domain is replaced by the last condition forthe temperature derivative at the bubble rear. Such types of‘‘outflow’’ conditions are well known from wake simula-tions (17) and have proved useful for the present temperaturewake, too.

At the interface between bubble and external fluid, i.e.,at r Å 1, the following boundary conditions apply: no flowacross the interface, balance of the shear stress, and no heat

FIG. 4. Bubble migration in dimensionless presentation. flux across the surface. These conditions are expressed by

£r Å 0 [10]Under these assumptions the spherical bubble migratesparallel to the direction of the temperature gradient. Afteran initial nonsteady phase, the bubble reaches a constant sru Å

ÌÌr Snu

r D Å ÌT

Ìu[11]

migration velocity U` .It is convenient to use a reference frame traveling with

the bubble. The bubble center is taken as the origin of coordi- ÌT

ÌrÅ 0 [12]

nates, and the direction of the temperature gradient as thepolar axis. The flow, temperature, and pressure fields are

at r Å 1. The stress tensor component sru has been madepresupposed to be axisymmetric. Using spherical coordi-dimensionless in the same manner as the pressure. Due tonates (r , u, w) all quantities appear to be independent ofthe assumption of a spherical bubble a normal stress balancethe azimuthal angle w, and the problem can be treated asis not necessary.essentially two-dimensional.

Thus, the steady-state problem is settled by two indepen-The equations governing the steady case are the conserva-dent nondimensional parameters, namely, the Reynolds andtion of massthe Marangoni numbers of the matrix fluid. Because of Eq.[2] it is obvious that also the pair (Pr, Mg) can be chosendiv n Å 0, [6]as the determining set.

The bubble migration velocity U` in the steady state mustthe equations of motion (Navier–Stokes)be determined from the force balance. The net hydrodynamicforce F acting on the bubble volume is obtained by integrat-Re(nrÇ)n Å 0Çp / Ç2n , [7]ing the axial components of the normal and shear stressesof the external fluid over the bubble surface. This balanceand the energy transport equationcan be written as

Mg(u` / (nrÇ)T ) Å DT . [8]

F Å 2p *p

0Fnu(1 0 3 cos2u) 0 sin 2u

Ìnu

ÌuThe lengths are scaled in units of the bubble radius R ,velocities are sealed in units of U0 , and the pressure isscaled in units of U0h /R . The temperature T is scaled

0 sin2uÌnu

Ìr0 p cos u sin uG

rÅ1

du Å 0 [13]by (dT /dz )É`R and is considered relative to the currentposition of the bubble.


120 TREUNER ET AL.

was added to the problem. Convergence was supposed to bereached 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 1005 ,where f n is the value of the function f at the n th time stepof integration.

Since near the bubble surface large gradients are to beexpected, a fine grid near the surface and a coarse one faraway from the bubble are used. Nonregular meshes are em-ployed outside the bubble. Nodes in the r direction arestretched near the boundary of the bubble. Nodes are definedas half of the roots of a Chebyshev polynomial shifted tothe intervals [R , R`] .

Typical results for the stream function, vorticity, and tem-perature distribution around the steady migrating bubble areshown in Figs. 5 and 6. The anisotropy between the up- anddownstream directions is caused by convective transport. Itincreases with increasing Marangoni number.

The infinite domain outside the bubble is truncated in ther direction; i.e., the conditions at infinity are evaluated at afinite R` . In order to estimate how much the domain can betruncated without loss of physically relevant information,we performed some preliminary calculations for large Mar-angoni numbers, where thermal boundary layers are ex-pected, with different values of R` and examined the distur-bance of the external temperature gradient caused by theflow and heat transfer outside the bubble. The influence ofR` on the numerical result of the bubble migration velocityU` was studied. For example, in the case Pr Å 10 and Mg

FIG. 5. Temperature lines (top) and isolines of the perturbation fromuniform temperature gradient (bottom) for Pr Å 10 and different Marangoninumbers Mg.

and, finally, gives the condition to determine U` .

3.2. Numerical Method and Results for the Steady-StateCase

The problem formulated above was solved with the con-trol volume finite difference method using a predictor–cor-rector semi-implicit scheme for a straightforward time inte-gration. The time dependence is made artificially due tonumerical reasons only. For details about the numericalmethod see the paper of Gelfgat and Tanasawa (16). Conser-vative properties provided by the control volume method arevery important for correct calculation of the force balancein Eq. [13]. Straightforward integration in time was carriedout until convergence to a stationary solution was reached.To obtain the migration velocity u` in the same computa-tional process the formal equation

FIG. 6. Stream lines (top) and lines of equal vorticity (bottom) fordu

dt} ÉFÉ [14]

Pr Å 10 and different Marangoni numbers Mg.



in this paper do not decrease below the predicted asymptoticvalue given in Eq. [16]. This is particularly interesting sincethe boundary layer analysis of Crespo and Jimenez-Fernan-dez (18, 19) still contains a singularity at the rear stagnationpoint, and it seems to be an open question (as mentioned,e.g., by Subramanian (2)) whether the consideration of thissingularity will change the asymptotic value [16]. Here itcan be stated that the present numerical results seem to con-firm Eq. [16], whereas the numerical results available up tonow (6, 7) were in contradiction to the result [16].

3.3. Some Transient Results

Up to now the theoretical description covers the steadycase only. However, the experimental situation is stronglytransient due to the short migration time of 4.7 s and thefact that the bubbles start practically from rest. Of course,a numerical extension of the steady to the time-dependentsituation would be interesting, but is not available currently.In order to obtain some first transient results the creepingFIG. 7. Numerical results for the nondimensional migration velocity asflow limit (Re, Mg ! 1), which was also studied by Dill anda function of Pr and Mg.Balasubramaniam (8, 9) , was considered in (20). Typicalresults for the fluids used in the present experiments are

Å 1000, the bubble migration velocities for R` Å 5, 10, shown in Fig. 8 in physical units.20, and 30 are U` Å 0.2162, 0.2238, 0.2252, and 0.2262, It is interesting to mark the two different time scales, tn

respectively. In any case it is guaranteed that R` has no Å R 2 /n and ta Å R 2 /a , which define the transient behavior.influence on the results presented in the following. As a rough estimate it can be stated that after tn Å R 2 /n

The present calculations of the steady case were done the migration velocities reach about 70–80% of the finalprimarily to obtain some theoretical results for a precise value. After this quick velocity rise a rather long time Çta

experiment configuration. However, the question arises as is required for the final 20–30% of the velocity development.to what extent these calculations are identical to the known These aspects are discussed only qualitatively in the presentresults of Szymczyk (6) and Balasubramaniam and Lavery(7) . The present results for U` and the comparison to theliterature data are shown in Fig. 7.

At higher Marangoni numbers these curves show a distinctdifference from the literature data. The reason is obvious:The calculations of Balasubramaniam and Lavery (7) andSzymzcyk (6) were performed with a fixed value of R`

between 5 and 10, which becomes a too small computationaldomain if with increasing Marangoni number the tempera-ture trail reaches R` . The present results were always con-firmed by test calculations varying R` and the grid mesh.

This slight difference in the computational results be-comes very important if the asymptotic velocity for Mg r

` is considered. This asymptotic case was investigated byCrespo and Jimenez-Fernandez (18, 19) using a boundarylayer approach. They derived the asymptotic values

U`Å 0.445rUYGB for Mg r ` , Re ! 1 (Ref. (18)) [15]

FIG. 8. Unsteady thermocapillary bubble migration in the creeping flowU`Å 0.392rUYGB for Mg r ` , Re r ` (Ref. (19)) . [16]limit for the three liquids used. The bubble starts from rest and approachesits asymptotic velocity U` . R Å 1 mm, dT /dz Å 1 K/mm. The vertical

In contrast to the numerical results documented in the litera- lines mark the momentum time scales tn , respectively. The temperaturediffusion time scales ta are out of the figure (ta É 12.2 s in each case) .ture (6, 7) the results for the migration velocity U` presented


122 TREUNER ET AL.

context. They depend also on the various possible initial turbulence caused by the needles’ retraction, the two bubblesconditions, as discussed in (21). often started with different delay times. The resulting veloci-

Oliver and DeWitt (11) extended for the first time the ties, however, reached the same values in all the cases ob-transient description to higher values of Mg. They assumed served. The case depicted shows an additional small bubblethat Re ! 1 but allowed Marangoni numbers up to Mg Å beneath the regular one on the right side, which in the mo-200, and solved numerically. They found the interesting re- ment of the capsule’s drop was released unintentionally. Insult that oscillations in the transient velocity may occur for agreement with theory, when larger bubble radii rendera bubble starting from rest and approaching its final velocity higher velocities, the large bubble passes the small one veryU` . Lower Marangoni numbers (Mg õ 10) behave as an quickly.overdamped system, higher Marangoni numbers as an under- In all the experiments the bubble shape was controlled bydamped one with oscillations in velocity. However, it must visualization. It turns out that within an accuracy of at leastbe stressed that a direct application of these results to the 10% it was not possible to observe any deviation from thepresent measurements is not possible due to much higher spherical shape.values of Re and Mg in the experiments. Nevertheless, it The total numbers of bubble runs evaluated for the threegives some first hints for possible interpretation. liquids are presented in Figs. 10–12, respectively. The form

of presentation coincides with Fig. 4, which was described4. EXPERIMENTAL RESULTS above. In each run, time proceeds from left to right. Average

values for the Prandtl numbers Pr characterizing the differentliquids are given in the diagrams, where this average corre-Figure 9 shows a sequence of bubble positions observedsponds to the mean temperature in the cavity. In order toduring the experiments. At the upper right corner of each

picture, the time after the drop is marked. Due to the slight refer the measurements to a theoretical value, the steady-

FIG. 9. Example of observed bubble migration.



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.

AID JCIS 4028 / 6g0d$$4028 03-11-96 14:37:27 coidas AP: Colloid

124 TREUNER ET AL.

more complex behavior. The courses depicting the migrationmay show distinctive minima or maxima or may continuouslyincrease or decrease during the observation, even if they repre-sent the same parameter range. These effects are strongest forn-tetradecane and trigger the assumption that they increase withrising Prandtl number. Some curves of n-octane and n-tetrade-cane at lower Marangoni values even exceed the theoreticalsteady-state velocities.

Typical examples of the various behaviors observed are de-picted in Fig. 13 in dimensional presentation. Curves 13a and13b represent two bubble runs where the transient course differssignificantly instead of having nearly the same parameters. Case13c is the characteristic result of a bubble whose velocity in-creases continuously during the period of observation. After afast increase to a certain velocity, many bubbles even remainedto run with nearly constant speed, as in case 13d. Course 13e,finally, represents the case of n-tetradecane, for which the strong-est transient effects had been observed.

Since the three liquids used had different average Prandtlnumbers, the experimental data for each liquid on the aver-age reach different velocity levels, too. This result becomesmost distinctive in the presentation shown in Figs. 14–16.Here the Reynolds number Re * (defined with the local bub-ble velocity U) has been plotted versus the Marangoni num-ber Mg*. The actual Reynolds numbers of n-octane (Pr Å8.2) exceed the theoretical curve for Pr Å 10; the values of

FIG. 13. Observed migration behavior of several bubble runs.n-decane (Pr Å 12.1) stay below, as do the values of n-tetradecane (Pr Å 25.5) by a larger amount. Whereas the

state result is always given for the corresponding Prandtl Reynolds numbers of n-octane and n-decane increase duringnumber (solid line) . the observation time, the values for n-tetradecane appear to

The experimental results group into two different regions: decrease continuously, except in one case.Most of the runs show a continuous increase of the bubble The overall accuracy of the measurement depends mostmigration velocity in qualitatively good agreement with the tran- sensitively on the determination of the drop radius and thesient calculations of Fig. 8, but staying always below the calcu- surface tension gradient and is limited to an error of {5%.lated steady state by 5–20%. This group is observed mainly for Reconsideration of the applied evaluation method in lighthigh Marangoni numbers (Mg ú 1000). The other group, of the mentioned errors does not lead to a qualitative change

in the observed results.mainly at lower Marangoni numbers (100õMgõ 500), shows

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.



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 observed and may be explained by variations of the initialconditions caused by the generation process. While in mostof the runs depicted in Figs. 10 and 11 the bubbles start atThe complex behavior observed requires a comprehensivelow velocities, in a few cases they start with high valuesexplanation. In most of the bubble courses the maximumbecause they are released unintentionally at the moment ofvelocities reach average values of 75 to 95% of the steady-the capsule’s drop and accelerated by buoyancy. The strongstate velocities predicted by theory. This is true for smallvariation of the time-dependent behavior can be explainedbubbles reaching dimensionless times of 2.5ta during theneither by different initial conditions only nor by slight mea-observation time, as well as for bigger bubbles which dosuring errors. Bubble courses of Figs. 10–12, which start atnot exceed values of 0.3ta . Investigations on the transienthigh velocities, pass through a minimum, and then increasebehavior in the creeping flow limit published recently (10)in velocity again, must have different explanations. As men-yield about 90% of the terminal velocity even after 0.3ta .tioned above, new results of Oliver and De Witt (11) on theIn contrast to the theoretical description, where the terminaltransient behavior possibly provide an explanation. Whethervelocity is calculated for only one parameter set, the realtheir calculated oscillating transient behavior can explaintransient bubble migration is exposed to strongly varyingsome of the present measurements is currently not beingconditions. The comparison of the theoretical steady-stateinvestigated since theory and experiment cover different pa-velocities with the measurements thus reveals a principalrameter regions. Clarification on that point may be reacheddiscrepancy, in particular at high Marangoni numbers. Theonly by a numerical simulation which considers all transientmigrating bubbles try to accelerate to the local steady-stateand convective effects.velocities, which they never can reach. This qualitatively

An interesting comparison of the earlier experimental dataimplies that the measured velocities shall always be belowwith our numerical results is given in Fig. 17. The data docu-the theoretical values, even after several periods of ta .

Several strong difference in the transient behavior were mented by Thompson and De Witt (12) for three liquids (PrÅ

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


126 TREUNER ET AL.

FIG. 17. Experimental results documented in literature.

15, 125, and 137) have been given for the bubble’s Reynolds 6. CONCLUSION AND PROSPECTnumbers (defined with the measured velocity U) as a function

Results of drop tower experiments concerning thermocap-of the theoretical Reynolds number (defined with U0, in (12),illary bubble migration at high Reynolds and Marangonitermed Marangoni number). The experimental data for PrÅ 15numbers were presented. The observed complex transientappear to fit the theoretical steady-state curve for Pr Å 10,behavior was described and compared to steady-state calcu-whereas the values for Pr Å 125 and 137 clearly exceed thelations. Terminal velocities observed in the experiment seemtheoretical curve for Pr Å 100. Due to the documented experi-to confirm the theoretical values within a limit of 5 to 20%.mental conditions, the bubbles appeared to be exposed to residualThe discrepancies between theoretical steady-state consider-accelerations. If the measured velocities were influenced byations and real transient bubble migrations were discussedbuoyancy, the too high Reynolds numbers were evident. Furtherand lead to the necessity of fully transient models.data from Siekmann and Wozniak (13) are given for PrÅ 687

Numerical steady-state results presented here revealedand most of the data correlate very well with the theoreticalcorrections of previous results due to the use of a largervalues of Pr Å 1000.computational domain. The existence of an asymptotic ve-locity for infinite Marangoni numbers, as discussed in theliterature, seems to be confirmed by these calculations.

Investigations by means of shearing interferometry arecurrently carried out to reveal the temperature distributionaround the bubbles. Figure 18 shows the image of two bub-bles observed by horizontal beam separation. Differentialinterferometry visualizes deviations of constant refractiveindex gradients in the liquid which can be used for the evalu-ation of the temperature fields. The bubbles in Fig. 18 havediameters of 3 mm and move with a speed of approximately15 mm/s. Distinctively observable is a trial of bubbles show-ing similarity to the theoretically analyzed temperature fieldin Fig. 5. The investigations, together with quantitative eval-uations of the interferometer pictures, are continuing andwill be published in a forthcoming paper.

ACKNOWLEDGMENTSFIG. 18. Refractive index distribution of moving bubbles, observed by

a shearing interferometer with horizontal beam separation. Bubble size D This work was supported by the Deutsche Agentur fur Raumfahrtangele-genheiten (DARA) under Grants 50 QV 8717 and 50 WM 9213. TheÅ 3 mm.



authors appreciate very much the assistance of Dipl.-Ing. Peter Prengel, 10. Galindo, V., Gerbeth, G., Langbein, D., and Treuner, M., MicrogravitySci. Technol. 7(3) , 234 (1994).Dipl.-Ing. Dirk Lockowandt, cand.-Ing. Henrik Fricke, and cand. phys.

Fikret Sisman for their support on the experimental work. Special apprecia- 11. Oliver, D. L. R., and De Witt, K. J., J. Colloid Interface Sci. 164, 263(1994).tion is also expressed to Dr. Gelfgat of Technion Haifa (Israel) for providing

the basic numerical code. 12. Thompson, R. L., and De Witt, K. J., ‘‘Marangoni Bubble Motion inZero Gravity.’’ NASA Technical Memorandum TM 79250, 1979.

13. Siekmann, J., and Wozniak, G., Israel J. Tech. 23, 179 (1986).REFERENCES14. Nahle, R., Neuhaus, D., Siekmann, J., Wozniak, G., and Srulijes,

J., Z. Flugwiss. Weltraumforsch. 11, 211 (1987).1. ‘‘Proceedings of an RIT/ESA/SSC Workshop, Jarva Krog, Sweden15. TAPP, a Database of Thermochemical and Physical Properties, ES10–20 January 1984.’’ ESA SP-219.

Microware, 1992.2. Subramanian, R. S., in ‘‘Transport Processes in Drops, Bubbles and16. Gelfgat, A. Yu., and Tanasawa, I., Microgravity Sci. Technol. 8(1)Particles’’ (R. D. Chhabra and D. Dekee, Eds.) . Hemisphere, New

(1995).York, 1991.17. Fornberg, B., J. Fluid Mech. 98, 819 (1980).3. Wozniak, G., Siekmann, J., and Srulijes, J., Z. Flugwiss. Weltraum-18. Crespo, A., and Jimenez-Fernandez, J., in ‘‘IUTAM Symposium onforsch. 12, 137 (1988).

Microgravity Fluid Mechanics Bremen’’ (H. J. Rath, Ed.) , p. 405.4. Young, N. O., Goldstein, J. S., and Block, M. J., J. Fluid Mech. 6,Springer-Verlag, New York/Berlin, 1992.350 (1959).

19. Crespo, A., and Jimenez-Fernandez, J., ‘‘Thermocapillary Migration5. Subramanian, R. S., AIChE J. 27(4) , 646 (1981).of Bubbles: A Semi-Analytical Solution for Large Marangoni Num-6. Szymczyk, J. A., Ph.D. thesis, University of Essen, 1985.bers,’’ ESA SP-333, p. 193, 1992.7. Balasubramaniam, R., and Lavery, J. E., Numer. Heat Transfer A 16,

20. Galindo, V., and Gerbeth, G., Phys. Fluids A 5, 3290 (1993).175 (1989).21. Galindo, V., Gerbeth, G., Langbein, D., and Treuner, M., in ‘‘Proceed-8. Dill, L. H., and Balasubramaniam, R., Proc. AIP Conf. 197, 481

ings Drop Tower Days Bremen, July 1994,’’ p. 90.(1988).22. Hadamard, J., C. R. Acad. Sci., Paris, 1, 1735 (1911).9. Dill, L. H., and Balasubramaniam, R., Int. J. Heat Fluid Flow 13, 78

(1992). 23. Rybezynski, D., Bull. Acad. Sci., Cracovie, 1, 40 (1911).


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

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