The motion of droplets in a vertical temperature gradientMichael Hähnel, Volkmar Delitzsch, and Helmut Eckelmann Citation: Physics of Fluids A: Fluid Dynamics (1989-1993) 1, 1460 (1989); doi: 10.1063/1.857323 View online: http://dx.doi.org/10.1063/1.857323 View Table of Contents: http://scitation.aip.org/content/aip/journal/pofa/1/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Directional motion of evaporating droplets on gradient surfaces Appl. Phys. Lett. 101, 064101 (2012); 10.1063/1.4742860 Motion of droplets along thin fibers with temperature gradient J. Appl. Phys. 91, 4751 (2002); 10.1063/1.1459099 Collective effects of temperature gradients and gravity on droplet coalescence Phys. Fluids A 5, 1602 (1993); 10.1063/1.858837 Motion of aerosol particles in a temperature gradient Phys. Fluids 18, 144 (1975); 10.1063/1.861119 Vertical Temperature Gradients in a Liquid Helium I Bath Rev. Sci. Instrum. 41, 348 (1970); 10.1063/1.1684514

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The motion of droplets in a vertical temperature gradient Michael Hiihnel,a) Volkmar Delitzsch, and Helmut Eckelmann I~~ti:utfur Angewandte Mechanik and Stromungsphysik der Georg-August-Universitat Gottingen, D-3400 Gottmgen, Federal Republic a/Germany

(Received 31 October 1988; accepted 23 May 1989)

T~e thermocapiUary migration of single water droplets in butyl benzoate was investigated usmg the Plateau configuration. The droplets were injected below the level of equal density in t?e stab.ly stratified butyl benzoate. Because of buoyancy and Marangoni force, the droplets nse until these forces balance each other. For seven different temperature gradients the final position of four to eight single droplets was determined. The reciprocal diameter was found to be a linear function of the deflection from the height of equal density. A systematic deviation from the theory of Young, Goldstein, and Block [J. Fluid Mech. 6, 350 (1959)] was observed. This deviation was attributed to the convective heat transport inside and outside the droplet not covered by the theory. The experimental data could be fitted by introducing an effective temperature gradient that decreases with the square of the applied gradient.

I. INTRODUCTION

The investigation of the migration of droplets in liquids is very important for basic research as well as for material sciences and chemical engineering. Especially, ever since mi­crogravity experiments on the production of highly pure ma­terials began, transport phenomena independent of gravity have been of great interest. Such a phenomenon has already been described by Marangoni. 1-3 The so-called Marangoni convection arises in the interface of two liquids from the local variation of the interfacial tension. The flow in the in­terface is then transferred to the adjacent liquid layers on both sides by viscosity. In the case of a droplet embedded in a liquid, the droplet experiences a force, and, in the absence of other forces (e.g., buoyancy), migrates in the direction of decreasing interfacial tension. The variation of the interfa­cial tension can be caused by concentration andlor by tem­perature gradients and is in the latter case also caned ther­mocapiHarity.

Young, Goldstein, and Block4 were the first to investi­gate the thermocapillary migration of bubbles in liquids. They observed bubble migration in a vertical liquid column heated from below. To avoid free convective flow, the Ray­leigh number was kept small by the choice of appropriate experimental parameters. They succeeded in holding the bubbles stationary or driving them downward against buoy­ancy. A linear relation between the bubble diameter and the temperature gradient at which the bubbles were held station­ary was found. Furthermore, they presented a theory that, although neglecting the convective terms, could explain their experimental results. The theory of Young, Goldstein, and Block was extended by Subramanian,5,6 who admitted an additional convective heat transport but neglected gravi­ty. Besides an additional correction term, his results agree with the theory of Young, Goldstein, and Block for zero gravity. In a further publication, Subramanian 7 calculated the force acting on a fluid droplet in an unbounded fluid

., Present address: Institut fUr Meereskunde der Universitat Hamburg, D-2000 Hamburg 54, Federal Republic of Germany.

medium and found this force to be proportional to the inte­gral of the interfacial tension gradient over the droplet sur­face. After the pioneering work of Young, Goldstein, and Block, the Marangoni convection of bubbles in liquids was subject to many experimental investigations. 8-! i The investi­gation of the Marangoni convection of droplets in liquids is, however, more difficult and thus performed less often. Chifu et al. 12 observed the flow on the surface of a drop (freely suspended in a liquid) by injecting a small quantity of a dyed surfactant into the interface. The velocity of the spreading surfactant front was measured and compared with the values resulting from a theoretical model also presented by the au­thors. In a rocket space experiment, Langbein and Heide13

investigated the migration of droplets due to temperature and concentration gradients. Using a two-component sys­tem with miscibility gap (methanol!cyclohexane) and cool­ing it down below the temperature where cyclohexane sepa­rates, they found a migration of the separated droplets in the direction of the warmer region. The influence of thermoca­pillarity on the migration of water droplets in butyl benzoate and fluorobenzene was investigated by Delitzsch, Eckel­mann, and Wuest. 14

-16 The two components of both liquid

systems are immiscible and have the same density at 32°C and 50 °C, respectively. Water droplets injected into either the stably stratified butyl benzoate or fluorobenzene are sorted out dependent on their diameters in a manner that smaller droplets migrate to higher temperatures than do larger ones. The final droplet position is determined by the balance of the Marangoni (proportional to the surface) and gravity force (proportional to the volume). Under similar experimental conditions, Wozniak!7 observed the motion of si.ngle paraffin oil droplets in a solution of ethanol and water and confirmed the findings of Delitzsch, Eckelmann, and Wuest.

The goal of the present investigation is to provide more quantitative data about the thermocapillary migration of single water droplets in butyl benzoate. It can be understood as a continuation of the work of Delitzsch, Eckelmann, and Wuest; details which cannot be presented here can be found in Delitzsch and Eckelmannl8 and Hiihnel. I9

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II. THEORETICAL. CONSIDERATION

To our knowledge, the first theoretical treatment of thermocapillary migration of bubbles in a liquid was pub­lished by Young, Goldstein, and Block.4 They calculated the velocity of bubble migration in a temperature field with a homogeneous vertical temperature gradient. In principle, their considerations are valid for the motion of droplets as well, and consequently their results can be applied to our problem. The calculations of Young, Goldstein, and Block are based on continuity, Navier-Stokes, and energy equa­tions. Assuming a steady, creeping flow the partial time de­rivatives as well as the convection terms can be neglected. By using the solutions of the axisymmetric problem given by Hadamard20 and Rybczinsky2! and under consideration of the boundary conditions of the flow field, Young, Goldstein, and Block obtained the migration velocity w of the droplets as W = D2g(p - pi) e# + ,u' )

6p, (2# + 3p,')

x DA.,

_ dy dT dT dz

(2j.t + 3p')(2A., + A') (1)

Here D is the droplet diameter, g is the gravitational aced· eration, dy/dTis the temperature coefficient of the interfa­cial tension, dT / dz is the temperature gradient at infinity, p, and pi are the dynamic viscosities, and A and A ' are the ther­mal conductivities ofthe continuous phase and of the droplet phase (primed), respectively. The migration velocity is the sum oftwo terms of which the first represents the veiocity of a droplet without thermocapillary effects. The second term describes the velocity caused by Marangoni forces only. Equation (1) is considerably simplified when Marangoni and gravity forces balance each other. Then the migration velocity vanishes and for the reciprocal droplet diameter we obtain

(2)

Provided that the temperature range is small, the reciprocal diameter can be assumed to vary linearly with temperature,

D-1=(dY dT) i(aT+b) . dT dz '

(3)

where a and b are material constants. Assuming a linear temperature field and setting

T= dT (; _!l.-, (4) dz a

the reciprocal diameter remains only a function of the dis­tance (; between the droplet and the level of equal density,

D-! = (:~r-la(;. (5)

Since Eq. (5) is independent of the temperature gradient, an droplets of equal diameter should experience the same de­flection from the level of equal density. Thus this equation can be used for an easy verification of the calculations of Young. Goldstein, and Block applied to a liquid two-compo-

1461 Phys. Fluids A, Vol. 1, No.9, September 1989

nent system. In addition, the temperature coefficient of the interfacial tension of two liquids of approximately equal den­sity can also be determined. by this formula. This is of great importance as it is very difficult to measure this quantity directly.

III. EXPERIMENTAL SETUP

Since thermocapillary forces are in general negligible with respect to buoyancy forces, the only successful possibil­ity to study thermocapillary migration of droplets under gravity exists in the application of the Plateau configuration, explained by means of Fig. 1. Two immiscible liquids, both of them possessing the same density at an experimentally realizable temperature (e.g., at room temperature) are used. The density p of the continuous phase has to decrease strong· er with temperature than the density pi of the droplet liquid. By heating the continuous phase from above and cooling it from below a stably stratified temperature field can be real­ized. A droplet inserted into the continuous phase of density p will then be driven by buoyancy forces, Fb , in the direction of the point of equal density (p = pi, T= To). Consequent­ly, in equilibrium all droplets independent of their diameter should be localized at the temperature level of equal density. As a result of Marangoni convection, however, an additional interfacial force acts on the droplets. The final position of a droplet is thus determined by the balance of the Marangoni and gravity forces, both of which depend on the droplet radi­us but in a different manner. Since the interfacial force is proportional to the droplet surface area (0:: D 2) and the gravitational force to the droplet volume ( IX D 3), smaller droplets will be displaced farther from the level of equal den­sity than larger ones. The direction of the displacement is determined by the sign of the temperature coefficient of the interfacial tension; the system considered here possesses a negative temperature coefficient and thus the equilibrium level of Marangoni and gravity forces will be above the level of equal density.

A. Test cell

The phenomenon to be investigated here is based on the temperature dependence of the interfacial tension which is sensitive to contaminations. Therefore it is very important to work with the greatest possible cleanness while handling test substances and facility. This has been taken into account

conlin.uous phase T + AT z,T /

/ jroplet - -::::. - - - - - - - - - - - - - - - --.()

I Fb< oT I : -------------0------

I I T~ I: __ F..:': D F b = 0 I

T

FIG. I. Principle of the Plateau configuration. Buoyancy force Fb acting on a droplet due to density difference (p - p') at various heights z in the test cell subject to a vertical temperature gradient.

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Pel element

Cu cylinder

PI 100

droplet generator

FIG. 2. Vertical cut through the test cell with the droplet generator on the floor window (schematically). Dimensions are given in millimeters.

while preparing and performing the experiments. Figure 2 shows a vertical cut through the test cell that is basically the same as used by Delitzsch and Ecke!mann. 18 The main ele~ ment of the cell is a vertically arranged copper cylinder con~ taining the continuous phase liquid. To achieve a well-de­fined temperature gradient in the continuous phase, a high heat flux through the cylinder is employed, The cylinder is heated from above and cooled from below by two copper blocks, which themselves are kept at constant temperature by means of Peltier elements. The parts of the cylinder con~ taining the two opposite windows are thickened to anow a flush mounting of the plane window panes in order to mini­mize the perturbation of the temperature field. An addi­tional window on the fioor of the test cell serves for observing a possible horizontal droplet migration. All windows are double glazed to obtain a better thermal insulation. The cyl­inder is insulated by a Teflon block, which, in addition, is also used to seal the test cell and to support the temperature sensors. For temperature measurement and control of the continuous phase, nine vertically arranged Pt-lOO resistors

40

~-:-l [mm] >< 0 ,.

30 /r ~~( / )Of .

20

~.~J J A

() o x

28 30 32 34 36 38 T rOc]

FIG, 3. Vertical temperature distributions in the test cell for the first set of measurements. The lines were obtained by linear regression. Here. X: 0.83 'Clem; 0: 1.36 'C/cm; 0: 2.07 ·C/cm.

1462 Phys. Fluids A, Vol. 1, No.9, September 1989

are used which extend into the copper cylinder through a slit. Before installing the sensors, they were calibrated so that a mutual accuracy of ± 0,05 ·C is achieved. The tem­perature distribution was monitored by a computer. As an example, Fig. 3 shows three of the seven different vertical temperature distributions at which measurements were car~ ried out. The maximum deviation from a linear temperature profile in the area of observation (10-30 mm) is below 0.1 ·C.

BoTestsubstances The test cell was filled with butyl benzoate, which was

chosen as the continuous phase liquid. Using water as drop­let phase, a well-suited two-component liquid system with a strongly temperature~dependent interfacial tension is ob­tained. Since interfacial tensions are extremely sensitive to contaminations, bidistilled water was used. It was fined di­rectly from the distillery into a 15 yr old glass bottle that served as a supply container for the experiments.

Butyl benzoate is an organic ester that has the same density as water at 32 ·C. The purity, miscibility, and chemi­cal reactivity of both the pure and the mutually saturated test liquids were investigated by gas chromatographic analy­sis. It was found that the purity of the butyl benzoate de­pends strongly on the supplier. For the final experiments the purest available butyl benzoate was used. The unsaturated phase showed a purity of99.9%; in the saturated phase 0.6% water was found. No contamination could be detected in the corresponding water samples. Furthermore, no reaction products were found in both saturated phases so that a chemical reactivity can be excluded. Thus for the experi­ments, solubility effects (e.g., Marangoni forces due to con­centration gradients) do not play any role.

Co Droplet generation

A special arrangement was developed to enable a distur­bance-free insertion of single droplets into the continuous phase liquid. The main element, the droplet generator shown in Fig. 4, is positioned on the bottom of the test cell. It con­sists of a little Tefton block with three channels of 0.2 mm diameter. By means of a conical expansion of the vertical channel droplets of various size can be generated, The two horizontal channels lead into the vertical one at different heights and are connected with gastight microliter syringes. The syringes are mounted on micrometer slide devices al­lowing a liquid supply in steps of 0.01 pL.

PTFE hoses

Teflon ---._---

continuous phase

rt>2

FIG. 4. Vertical cut through the droplet generator (schematically). Dimen­sions are given in millimeters.

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A droplet is generated by first inserting a determined amount of droplet liquid through the upper channel into the conical part of the vertical channel. In a second step contin­uous phase liquid is pushed through the lower channel to shear off the droplet and to carry it into the outer fluid. Since the generator is made of Teflon, droplets easily detach and are then freely suspended over the bottom in the center ofthe test cel1. With the dimensions given in Fig. 4, single droplets in the range of 0.3-3 mm diam can be produced.

D. Experimental course

At the beginning, the test cell, the droplet generator, and the liquid supply have to be cleaned, assembled, and fined very carefully. Then the temperature field can be built up until the desired temperature is obtained and the measuring range is inside the field of view. When the temperature field is steady, the measurement can be started by inserting an amount of droplet liquid according to the desired diameter. To minimize the temperature difference between droplet and continuous phase, a certain time depending on the diam­eter (several minutes) has to elapse before detaching the droplet. Detachment and rising of the droplet can be ob­served through the vent on top or through the window in the bottom ofthe test cell, After the droplet has entered the field of view, it is projected on a screen where its diameter and its trajectories can be determined. After the droplet has reached its final position it is sucked off cautiously with a syringe. A new droplet can be inserted 15 to 20 min later when the perturbation has decayed. This waiting time is much longer when a new temperature gradient has to be adjusted.

As a demonstration of the Marangoni forces, Fig. 5 shows how the spatial arrangement oftwo droplets of ditfer-

FIG. S. Time sequence of the spatial arrangement of two water droplets ( 1.7 mm and 0.6 mm diameter) in butyl benzoate. Here dT Idz = 2.07 'Clem.

1463 Phys. Fluids A, Vol. 1, No.9, September 1989

ent diameters progresses with time. The smaller droplet had to be inserted first in order for both droplets to appear at the same time in the field of view. At the beginning of the se­quence, the large droplet rises faster and finally overtakes the smaller one. The larger one, however, quickly reaches its final position and remains at the same height while the smaller droplet rises further, overtakes the larger droplet, and attains its final height much later. The final arrangement confirms, as already shown by Delitzsch, Eckelmann, and Wuest,I4-16 that a Marangoni force acts on the droplets. In Fig. 5 a horizontal migration of the droplets can also be ob­served. This is probably due to a very sman horizontal tem­perature gradient present in the test cell which leads to a horizontal component of the Marangoni force. Since there is no restituting force in this direction the droplet will then move horizontally. This effect will always occur when the test cell is not perfectly axisymmetric. In the present case, a deviation from the axisymmetry is due to the two opposite windows that are necessary for observation and to the tem­perature sensors necessary for the determination of the verti­cal temperature distribution.

IV. EXPERIMENTAL RESULTS

Two independent sets of measurements, distinguished by different temperature measuring devices and by different temperature gradients, with a total of seven runs have been performed. The evaluation of the particular experiments was carried out with the help of the trajectories of which Fig. 6 shows an example. As expected, the smaller droplets always attain higher final positions than the larger ones. The time that the droplets stay in the field of view is generally suffi­cient to determine their final position. Only those droplets that stay in the field of view long enough are evaluated. The slight overshoot of their final position by the larger droplets is caused by inertia during the ascent phase.

The droplet velocities as a function of the height z or correspondingly of the temperature T in the test cell have been calculated by differentiating the trajectories (Fig. 7). The velocities for corresponding droplet diameters are also calculated by Eq. (1) and depicted in Fig, 8. Both experi­ment and calculation show that the velocity decreases stronger for larger droplets than for smaller ones; the experi­mental values are, however. much smaller than those result-

FIG. 6. Trajectories of four different droplets. The numbers given in the figure are the droplet diameters in millimeters. Here dT Idz = 2.07 ·C/em.

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FIG. i. Droplet velocities calculated from the trajectories ill Fig. 6.

ing from Eg. (1). The main reason for this difference is due to the assumption of steady state, which is not given during the ascent. Since the rising droplet does not warm up instan­taneously, its temperature is always lower than that of the environment at equal height. Therefore its density is slightly increased, resulting in a smaller density difference between the droplet and the continuous phase, and thus in a reduced buoyancy. This leads to a smaller migration velocity; an ef~ fect that increases with increasing droplet diameter since the warm-up time of the droplet increases with the diameter.

The Peelet number

Pe = woD/k

for the droplet liquid based on the velocities measured at the location of equal density is in the range of O.03<Pe<O.5. When based onto the matrix liquid it becomes about double that value.

V. COMPARISON WITH THE THEORY

Only few droplets left the field of view; in most of the cases it was possible to determine their final position from the trajectories. According to Eq. (5), the reciprocal diame­ter of the droplets should be a linear function of the deviation from the height of equal density and independent of the tem­perature gradient. As shown in Figs. 9(a) and 9(b), the values at the various temperature gradients can be represent­ed by straight lines, but they do not collapse onto a single line. The heights of equal density, which correspond to the

1.5 ~- 1 [m~~'~ '\ . ,-_.

0.5 ~~ \ J 0.7~~~~

0,0 O.~'~ i'~ <::;;-=r--.30.8 32,0 .33,2 .34.4 .35.6

T [Oc]

FIG. 8. Droplet velocities calculated by Eq. (I) for the same parameters as ill Figs. 6 and 7.

1464 PhyS. Fluids A, Vol. 1, No.9, September 1989

2.8 ~----r----~----,---.,

O}8 (b)

[mm-'] 2.1

°30 2,5 5.0 7.5 10,0 <:" [mm]

FIG. 9. (a) Reciprocal droplet diameter as a functioll of the deflection from the height of equal density. The first set of measurements with covered PT-100 temperature sensors. Here, X: 0.83 'Clem; 0: 1.36 'C/cm; 0: 2.0i 'C/ cm. (b) Same as ill (a). The second set of measurements with uncovered PT-IOO temperature sensors. Here, +: 0.48 'Clem; \7: 1.05 'C/em; 0: 1.83 'C/em; fl: 3.01 'Clem.

position where droplets of infinite diameter should be locat­ed, were obtained from regression lines. The correlation co­efficients of the linear regression varies between 0.998 for the highest and 0.965 for the lowest temperature gradient, The uncertainty of the measured values is less than the symbol size used in the figures. For comparison, in Figs. 9(a) and 9 (b) a line is plotted (theory) that is calculated with the help ofEq. (2) by using the material data of the butyl benzo­ate/water system. To obtain the necessary material data, the density/temperature and the dynamic viscosity/tempera­ture functions of butyl benzoate were measured directly while the corresponding data of water and the temperature­independent data of both liquids were obtained from the li­terature. The error of the measured densities is less than 0.01 %; the dynamic viscosities are accurate to 1 %. The per-

TABLE 1. Physical properties of the two liquids used for the experiments.

Thermal cOllductivity A (J/msec 'C) Thermal diffusivity k (mz/sec) Prandtl number (30 'C) DYllamic viscosity (30 'C) J.l (kg/msec) Density p (kg/m3)

at 31 'C at 32 'c at 33 'C

Butyl benzoate Water

1.18 X 10-2

6.63X 10-8

38 2.49X 10.- 3

995.87 995.02 994.16

6.40X 1O~ 2

1.50x 10-7

5,3 7.57X 10-4

995.34 995.03 994.71

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tinent physical properties of the two liquids are collected in Table I.

The temperature coefficient of the interfacial tension, dr/dT, can only be determined directly by exploiting the thermocapiUary forces [e.g., Eq. (5)]; or obtained indirect~ ly by differentiating the interfacial tension/temperature function. For our liquid two-component system with ap­proximately equal density the rule of Antonoff22 is well suit­ed to determine the interfacial tensions. This rule states that the interfacial tension YI2 between two liquids is equal to the difference of the surface tensions of the mutually saturated liquids (0"1(2) ,0"2(1) ) against a common gas

rl2 = 10"1(2) - 0"2(1) I· (6)

This formula was confirmed by Donahue and BarteU23 for organic liquids that spread on water like the butyl benzoate used for our measurements. The interfacial tension/tem­perature function was determined here first by measuring the surface tension of both mutually saturated liquids at dif­ferent temperatures and then substracting the surface ten­sion/temperature functions from each other. The slope of the resulting line represents the temperature coefficient of the interfacial tension; its value is - 7.4 X 1O~5 N/m ·C. It should be noted that the method used here to determine dr/ dTis only good for an estimate of the order of magnitude; the obtained value is not very accurate because it is a difference of two large quantities which are themselves subject to error.

As can be seen from Figs. 9 (a) and 9 (b), the line repre­senting the theory is a lower limit value for the experimental data. The slopes of the various straight lines denoted by A approach that of the theory line with decreasing temperature gradient.

VI. DISCUSSION

Equation (5) can lead to the suggestion that the depen­dence ofthe slopes A on the temperature gradients is due to a decrease of the temperature coefficient of the interfacial ten­sion dr/dT with increasing temperature gradient But this cannot be true since the temperature coefficient is a material constant and thus should not depend on the temperature gradient. The systematic deviation of the experimental re­sults from the theory can be explained by the missing con­vection term in the energy equation as used by Young, Gold­stein, and Block4 for the derivation of Eq. (1). This term, however, cannot be neglected in the present case since the Peclet numbers reach the order or unity. Convection inside and outside of the droplet increases the heat transport con­siderably and thus leads to a reduction of the temperature gradient acting at the interface. This results in a reduced Marangoni force, Le., the droplet experiences a smaller de­flection from the height of equal density.

A theory that also considers the convection term in the energy equation has not been given yet. Therefore only an empirical fit of Eq. (5) to the experimental data can be pre­sented. The ansatz here is that the temperature gradient in Eq. (3) is replaced by the effective temperature gradient

- =- 1 +a - ,a=const. dTI dT[ (dT)2] ~ I dz eff dz dz

(7)

1465 Phys. Fluids A, Vol. 1, No.9, September 1989

1 <2 ~-~---,----,---,----, A

[mm-2J 0<9

0<6

4 6 8 10

(dT / dz)2 [OC 2 / cm 2}

FIG. 10. Slopes ofthelines of Figs. 9(a) and 9(b) asafunction ofthe square of the temperature gradient. The line given in the figure was obtained by linear regression.

A reason for the quadratic appearance of the gradient in the additional term of Eq. (7) is that the effect produced by convection should not depend on the direction of the tern· perature gradient. Replacing in Eq. (2) dT /dzbydT /dzl etr

and going through the same algebra that leads to Eq. (5), yields

D -I = [1 + a( ~~Y](:;r- !G~. (8)

Provided that Eq. (8) represents the measured data correct~ ly, the slopes A in Figs, 9 (a) and 9 (b) should be described as

[ (dT)2](dY) -!

A = 1 + a dz dT G, (9)

hence. as a linear function of (dT / dz) 2. This is confirmed by Fig. 10. As already mentioned. the value of the theory of Young, Goldstein, and Block is also approached with de~ creasing temperature gradient. The unknown constants a and dy I dT were determined by the regression line also shown in this figure. The coefficient of the interfacial tension dr/dT calculated from the ordinate intercept amounts to - 4.4 X 1O~5 N/m "C. This value is obtained from many

single measurements and is thus more accurate than that obtained from the rule of Antonoff ( - i.4 X 10-5 N/m °C). Further support for the smaller value is the fact that all mea­sured reciprocal diameters, when divided by A and plotted

10<0 ,-----,...---,.-----,----:>1

(DA)-!

[mmJ 7<5

+ 5<0 ~

5<0 7.5 10<0 t; [mmJ

FIG. 11. Normalized reciprocal droplet diameter as afunction of the deflec­tion from the height of equal density for both sets of measurements given in Figs. 9(a) and 9(b).

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against the deflection from the height of equal density S, collapse on one single line with unity slope (Fig. 11).

With a given a, and thus known dT Idzletf, it is possible to calculate the droplet velocities w [Eg. (1) 1 and compare these values with the observed ones. Droplet velocities deter­mined for a temperature gradient of2.07 ·C/cm are shown in Fig. 7. At the location of equal density, where in Eg. (1) the first term vanishes, the droplet velocity Wo is about the same for all diameters, whereas Eg. (1) demands that it be propor­tional to the diameter. This is another hint that relaxation processes are involved during the ascent of the droplets.

Since the problem of calculating the convective heat transport can probably not be solved analytically, numerical procedures would have to be developed to verify the experi­mental results. Similar procedures have already been pre­sented by Szymczyk24 and Shankar and Subramanian25 for the thermocapillary migration of bubbles under no gravity conditions.

ACKNOWLEDGMENTS

The authors are grateful to Dr. W. Mohring and Dr. W. Poppe for helpful discussions; to Dr. D. Buss from the Anor­ganic Chemical Institute of the Georg-August-Universitat Gottingen for his advice in finding an appropriate two-com­ponent system as well as for his help with the gas chromato­graphic analysis; to Dr. G. Wozniak from the Universitat­GH-Essen and Mr. D. Haase from the Max-Planck-Institut fUr Biophysikalische Chemie Gottingen for their help with the measurements of material data; and to Mrs. R. Kretschmer, K. H. Nortemann, and H. J. Schafer for techni­cal assistance.

This research was supported by the Bundesminister fUr

1466 Phys. Fluids A, Vol. 1, No.9, September i 989

Forschung und Technologie under Contract No. 01 QV 494A.

'c. G. M. Marangolli, Ann. Phys. (Poggendor/f) 143,337 (lB71). 2c. G. M. Marangoni, Nuovo Cimento, Ser. 25/6,239 (1878). 'C. G. M. Marangoni, Ann. Phys. (Poggelldorlf) 3, 842 (1879). 'No O. Young, J. S. Goldstein, and M. J. Block, J. Fluid Mech. 6, 350 ( 1959).

SR. S. Subramanian, AIChEJ. 27, 646 (1981). "R. S. Subramanian, in Advances in Space Research, edited by Y. Malmejec

(Pergamon, London, 1983), Yol. 3, No.5, p. 145. 7R. S. Subramanian, J. Fluid Mech. 153, 389 (1985). "J. M. Papazian and W. R. Wilcox, AIAA J. 16,447 (1978). 9S. C. Hardy, J. Colloid Interface Sci. 69, IS7 (1979). '"R. L. Thompson, Ph.D. thesis, University of Toledo, 1979. "R. Nahle, D. Neuhaus, J. Siekmann, G. Wozniak, and J. Sruijes, Z. F1ug­

wiss. Weltraumforsch. 11, 211 (1987). '2E. Chifu, I. Stan, Z. Finta, and E. Gavrila, J. Colloid Interface Sci. 93, 140

(1983). uD. Langbein and W. Heide, Z. Flugwiss. Weltraumforsch. 8, 192 (1984). I.y. Delitzsch, H. Ecke1mann, and W. Wuest, in ResultsoJSpacelab-l (Eu­

ropean Space Agency, Paris, 1984), ESA SP-222, p. 245 . • 5y. Delitzsch, H. Eckelmann, and W. Wuest, in Optical Methods in Dy­

namics oj Fluids and Solids, rUT AM Symposium, Liblice, Czechoslova­kia, 1984, edited by M. Pichal (Springer, Berlin, 1985), p. 373.

lby. Delitzsch, H. Eckelmann, and W. Wuest, Max-Planck-Institut fiir Striimungsforschung, Giittingen, Report No. 103/1985, 1985.

nG. Wozniak, Ph.D. thesis, Universitiit-GH-Essen, 1986. '"V. Delitzsch and H. Eckelmann, Max-Planck-Institut fUr Stromungsfors­

chung, Gottingen, Report No. 106/1987,1987. ,gM. Hahne!, Max-Planck-Institut flir Stromungsforschung, Gottingen,

Report No. 14/1988, 1988. 20J. Hadamard, C. R. Acad. Sci. (Paris) 152, 1735 (1911). 2'W. Rybczynski, Bull. lot. Acad. Sci. Cracovie, Vol. 1,40 (1911 l. 22G. N. Antonoff, J. Chim. Phys. 5, 372 (1907). 23D. J. Donahue and F. E. Bartell, J. Phys. Chern. 56, 480 (1952). 24J. A. Szymczyk, Ph.D. thesis, Universitiit-GH-Essen, 1985. 25N. Shankar and R. S. Subramanian, J. Colloid Interface Sci. 123, 512

(1988).

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