Using PHF format - Clarkson University

PHYSICS OF FLUIDS

VOLUME 12, NUMBER 4

APRIL 2000

ARTICLES

The migration of a drop in a uniform temperature gradient at large Marangoni numbers R. Balasubramaniama) National Center for Microgravity Research on Fluids and Combustion, NASA Glenn Research Center, Mail Stop 110-3, Cleveland, Ohio 44135

R. Shankar Subramanian Department of Chemical Engineering, Clarkson University, Potsdam, New York 13699-5705

共Received 24 September 1999; accepted 20 December 1999兲 The steady thermocapillary motion of a spherical drop in a uniform temperature gradient is treated in the situation where convective transport of energy is predominant in the drop phase as well as in the continuous phase, i.e., when the Marangoni numbers are large. It is assumed that the Reynolds numbers in both phases are large as well; to leading order, the velocity fields are given by a potential flow field in the continuous phase and Hill’s vortex inside the drop. The migration velocity of the drop is obtained by equating the rate at which work is done by the thermocapillary stress to the rate of viscous dissipation of energy. The analysis deals with an asymptotic situation wherein convective transport of energy dominates with conduction playing a role only where essential. This leads to thin thermal boundary layers both outside and within the drop. The method of matched asymptotic expansions is employed to solve the conjugate heat transfer problem in the two phases. It is shown that the demand for energy within the drop, necessary to increase its temperature at a steady rate as it moves into warmer surroundings, results in a large temperature difference between the surface of the drop and its interior. The variation of temperature over the drop surface is large as well, and leads to a linear increase of the migration velocity of the drop with increasing Marangoni number. This result is strikingly different from that for the limiting case when the viscosity and thermal conductivity inside the drop become negligible compared to the corresponding properties in the continuous phase. This limit, which holds for a gas bubble, is recovered correctly from the analysis. © 2000 American Institute of Physics. 关S1070-6631共00兲01504-X兴

I. INTRODUCTION

vides a measure of the relative importance of convective transport of energy when compared to conduction. Definitions of these groups are given in the next section. For the purposes of the present discussion, it is only necessary to appreciate the physical significance of these groups. We analyzed the limiting case of a gas bubble in Ref. 4 共Balasubramaniam and Subramanian, hereafter abbreviated as Ba-S兲. In the gas bubble limit, the thermal conductivity of the fluid within the drop is considered negligible, as is its viscosity. This decouples the energy and momentum equations for the fluid within the drop from those for the continuous phase. One needs to solve only the transport problems in the continuous phase. In Ba-S we showed that the migration speed of the bubble, when scaled by a suitable reference, approaches a constant value as Ma→⬁, both when Re→0 and when Re→⬁. The constants are slightly different in the two limiting cases. In light of this, the natural expectation is that a similarly scaled migration velocity in the more general case, where the transport properties within the drop cannot be considered negligible, would also approach a constant which depends upon some parameters involving physical properties. This proves not to be the case. The asymptotic

When a gas bubble or an immiscible liquid drop is present in a liquid in which a temperature gradient is imposed, it is propelled toward warm liquid so that the surface energy is minimized. This motion, called thermocapillary migration, is important in the reduced gravity environment of orbiting spacecraft for the processing of materials and other applications. Reviews of the literature can be found in Refs. 1–3. Our goal in this paper is to analyze the problem of steady migration of a drop in a continuous phase when subjected to a temperature gradient under conditions such that inertial terms in the momentum equation and convective transport terms in the energy equation dominate over the corresponding molecular transport terms. A suitably defined Reynolds number 共Re兲 characterizes the relative importance of inertia when compared to viscous forces and the Marangoni number 共Ma兲, which serves as a Pećlet number, proa兲

Author to whom correspondence must be addressed. Electronic mail: [email protected]

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© 2000 American Institute of Physics

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Phys. Fluids, Vol. 12, No. 4, April 2000

analysis we present here reveals that the scaled migration velocity of the drop should be proportional to the Marangoni number as Ma→⬁. In physical terms, we find that the velocity of a drop is proportional to the square of the temperature gradient and the cube of the radius of the drop in this limiting situation. This observation is in contrast to that in the gas bubble limit, in which the linear dependence upon the radius and the applied temperature gradient, which occurs when convective heat transport effects are negligible, is preserved in the opposite situation when convective heat transport dominates. We show how this remarkable feature arises because of the demand for heat made by a drop as it moves into warmer surroundings. We also show how the gas bubble limit is indeed properly recovered as the thermal conductivity of the drop phase becomes negligibly small. We restrict the present analysis to the case of Re→⬁, which is tractable. Before embarking on the analysis, it is appropriate to briefly review the literature on thermocapillary migration when convective transport effects are included. The very first analysis of this problem is due to Young, Goldstein, and Block,5 who neglected such effects altogether. Subramanian6 considered the motion of a gas bubble including the effect of convective transport of energy as a small perturbation. He showed that a regular perturbation expansion in Ma for the temperature field in the continuous phase fails to satisfy the boundary condition far from the bubble, and therefore analyzed the problem using the method of matched asymptotic expansions. He found the migration speed of a gas bubble is reduced by the inclusion of the effect of convective transport of energy when Ma is small. In a later article, Subramanian7 extended the analysis to the case of a fluid drop, accounting for the transport problems in both phases. He showed that the migration speed of a drop can be reduced or enhanced for small Ma depending on the values of the various parameters. We already have mentioned our asymptotic analysis 共Ba-S兲 for Ma→⬁. A development similar to ours in many respects has also been presented by Crespo and Jimeńez-Fernańdez.8,9 Numerical solutions for the migration of bubbles and drops have been obtained by several investigators. Szymczyk, Wozniak and Siekmann10 and Szymczyk and Siekmann11 calculated the steady migration speed of a gas bubble for Re and Ma up to 100 and 1000, respectively. Balasubramaniam and Lavery12 extended the calculations of Szymczyk et al. to Reynolds numbers up to 2000. They found that the migration speed is influenced more by the Marangoni number than by the Reynolds number. Similar computational results for a gas bubble have also been obtained by Treuner et al.13 Shankar and Subramanian14 used an analytical solution for the velocity field in the limit of negligible Re in the energy equation, which was then solved numerically for Ma up to 200. Ehmann, Wozniak and Siekmann15 calculated the migration speed of drops of paraffin oil in aqueous ethanol and drops of aqueous ethanol in paraffin oil. In both cases, the predictions agree with the result of Young et al. in the limit of negligible convective transport. When aqueous ethanol is the drop phase, they found that the migration speed first increases, and subsequently decreases, when convective transport is present. The initial increase of the drop speed above the result of Young

R. Balasubramaniam and R. S. Subramanian

et al. is qualitatively consistent with the results of Subramanian.7 Comprehensive numerical simulations of the thermocapillary motion of deformable drops including transport of momentum and energy within the drop and unsteady effects have been performed by Nas16 and Haj-Hariri, Shi, and Borhan.17 Nas has performed some three-dimensional calculations with as many as nine drops interacting with each other. Many of the results presented by Nas for a single drop appear to be for a two-dimensional case. Haj-Hariri et al. have performed fully three-dimensional calculations for an isolated drop. The results of Nas and Haj-Hariri et al. are consistent with the predictions of Young et al. for negligibly small values of Re and Ma. They also show that a small deformation of the drop has a significant impact on the migration velocity. Haj-Hariri et al. report quantitative agreement with the results of Balasubramaniam and Lavery for an undeformed gas bubble with Re⫽1 and Ma up to 100. There are some qualitative differences between the results of Nas and Haj-Hariri et al. which are noted in Ma, Balasubramaniam, and Subramanian.18 In that study we reported numerical results for the steady migration velocity of an undeformed drop in axisymmetric motion when convective transport of energy is predominant in both the continuous phase and within the drop. The scaled speed of the drop is predicted to first decrease as Ma increases from zero, then attain a minimum, and finally increase with the Marangoni number when it is large. This increase with Ma is consistent with the asymptotic prediction made here. The paper is organized as follows. The bulk of the material contains the formulation and analysis of the theoretical problem for the thermocapillary motion of a drop in the limit as Ma→⬁. This is divided into several subsections. First, we state the problem, write the governing equations and boundary conditions, and obtain the leading order outer temperature fields in the continuous phase and within the drop. This is followed by analysis of the leading order inner fields in the continuous phase and within the drop near its surface, including the details of the solution of an integral equation that is necessary to complete this portion of the analysis. The fruit of this labor is a leading order result for the migration speed of the drop which increases linearly with increasing values of Ma. We then show how the present analysis is reconciled with that of Ba-S in the gas bubble limit by noting that when the ratio of the thermal conductivity of the drop to that of the continuous phase approaches 0, the coefficient of the Ma term in the result for the migration velocity also approaches zero. A higher order analysis is briefly outlined which shows how to extract the result in our earlier work for the gas bubble. Finally, we provide a few remarks placing the prediction from this work in context in light of the computational results of Ma et al. Even though we have reported results from thermocapillary migration experiments at relatively large values of the Marangoni number in Hadland, Balasubramaniam, Wozniak, and Subramanian,19 it is not possible to compare the present prediction against those results. We comment on this issue in Sec. III, concluding with suitable remarks.


The migration of a drop in a uniform temperature . . .

II. ANALYSIS A. Formulation of the problem

Consider the steady migration of a spherical drop of radius R 0 in a fluid of infinite extent with density ␳ , viscosity ␮ , thermal conductivity k and thermal diffusivity ␬ . The drop has a density ␥ ␳ , dynamic viscosity ␣␮ , thermal conductivity ␤ k and thermal diffusivity ␭ ␬ . The rate of change of the interfacial tension between the drop and the continuous phase fluid with temperature is denoted by ␴ T and is assumed to be a negative constant, consistent with the movement of the drop toward the warm portion of the continuous phase. All the physical properties of the fluids are assumed to be constant. A constant temperature gradient with magnitude G is assumed to be imposed in the continuous phase and gravitational effects are neglected. A suitable reference velocity for the motion in the fluids may be obtained from the tangential stress balance at the interface of the drop, where the jump in the tangential stress across the interface equals the thermocapillary stress at the interface. This velocity scale is v 0⫽

共 ⫺ ␴ T 兲 GR 0

␮

.

共1兲

The speed of the drop is scaled by v 0 , and the nondimensional steady drop speed is denoted by v ⬁ . In addition to the property ratios ␣ , ␤ , ␥ and ␭, the dimensionless parameters that are important are the Reynolds and Marangoni numbers. Definitions of these parameters which are based on the above velocity scale and the properties of the continuous phase fluid are given below: Re⫽

Ma⫽

␳ v 0R 0 , ␮ v 0R 0

␬

.

共2兲共3兲

When the Reynolds number is large, the Weber number Wb⫽ ␳ v 20 R 0 / ␴ , where ␴ denotes the interfacial tension, is a parameter that influences the shape of the drop. We assume WbⰆ1, which permits us to neglect deformation from the spherical shape. A spherical polar coordinate system is used, with the origin located at the center of the drop. The radial coordinate, scaled by the radius of the drop, is r and the polar coordinate measured from the direction of the temperature gradient is ␪ . The azimuthal coordinate is labeled ␾ . The temperature in the fluids is scaled by subtracting the temperature in the undisturbed continuous phase at the location of the drop and dividing by the quantity GR 0 : T⫽

¯T ⫺G v 0 v ⬁ ␶ , GR 0

共4兲

where ¯T is the physical temperature and ␶ denotes time. The problem is analyzed in a reference frame attached to the moving drop. It is assumed that the Reynolds and Marangoni numbers in both fluids are large. In a large Re asymptotic expansion, the flow field to O(Re⫺1/2) is given

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by the potential flow field in the continuous phase and Hill’s spherical vortex within the drop 共see Harper and Moore20兲. The steady migration speed of the drop is determined by an energy balance as shown by Levich,21 and specifically for the case of the thermocapillary migration of a gas bubble by Crespo and Manuel.22 The rate of decrease of interfacial energy that equals the rate at which work is done by the thermocapillary stress must equal the rate of viscous dissipation of kinetic energy. From this balance, an expression for v ⬁ may be written as follows: v ⬁ ⫽⫺

⫽

冕

1 2 共 2⫹3 ␣ 兲

1 共 2⫹3 ␣ 兲

冕

␲

0

␲

0

sin2 ␪

⳵T 共 1,␪ 兲 d ␪ ⳵␪

sin␪ cos␪ T 共 1,␪ 兲 d ␪ .

共5兲

Thus to determine v ⬁ , the temperature field on the surface of the drop must be known. This requires the solution of the energy equation. In general, the problem for v ⬁ is nonlinear because the motion of the drop influences the temperature field in the fluids. The energy equations for the temperature fields in the continuous phase and within the drop are written as follows: 1⫹u

⳵T v ⳵T ⫽ ⑀ 2 ⵜ 2 T, ⫹ ⳵r r ⳵␪

共6兲

⳵T⬘ v⬘ ⳵T⬘ ⫽␭ ⑀ 2 ⵜ 2 T ⬘ . ⫹ ⳵r r ⳵␪

共7兲

1⫹u ⬘

In the above, the symbols u and v stand for the radial and angular velocity components in the continuous phase, scaled by the velocity of the drop, and primes are used to designate quantities within the drop phase. The small parameter ⑀ is defined as

⑀⫽

1

冑Mav ⬁

共8兲

,

and the scaled velocity components are written as follows:

冉冊冉冊

u⫽⫺cos␪ 1⫺

v ⫽sin␪ 1⫹

1

r3

1

,

共9兲

,

共10兲

u ⬘ ⫽ 23 cos␪ 共 1⫺r 2 兲 ,

共11兲

v ⬘ ⫽3sin␪ 共 r 2 ⫺ 21 兲 .

共12兲

2r 3

The boundary conditions are that the temperature field depends linearly on distance far away from the drop, the temperature is continuous across its interface, and the jump in the heat flux at the interface is related to the stretching of interfacial area elements: T→rcos␪ as r→⬁,

共13兲

T 共 1,␪ 兲 ⫽T ⬘ 共 1,␪ 兲 ,

共14兲

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⳵T ⳵T⬘ E sv ⬁ ⳵ 共 1,␪ 兲 ⫺ ␤ 共 1,␪ 兲 ⫽ 关v共 1,␪ 兲 sin␪ 兴 . ⳵r ⳵r sin␪ ⳵ ␪


共15兲

We also require that the temperature within the drop remain bounded. On the right-hand side of Eq. 共15兲, E s ⫽( ␴ ⫺e s ) ␴ T /( ␮ k), where e s stands for the internal energy per unit area of the interface. This term arises from energy changes associated with the stretching and shrinkage of interfacial area elements. This effect was first considered by Harper, Moore, and Pearson,23 and most recently by Torres and Herbolzheimer.24 Its contribution is negligible when E s Ⰶ1 when ReⰆ1 and MaⰆ1. When both Re and Ma are large, as is the case in the present problem, it can be shown from the asymptotic scalings for the interfacial temperature field, the thermal boundary layer thickness, and the velocity of the drop, presented later in this work, that the effect of stretching and shrinkage of interfacial area elements is negligible when E s ⰆMa. Since E s Ⰶ1 for most common fluids at room temperature, we neglect the right-hand side of Eq. 共15兲 from here onward. Therefore, the heat flux is continuous at the surface of the drop. From an overall energy balance on the drop, the following equation can be written:

冕⳵

␲⳵T

0

r

共 1,␪ 兲 sin␪ d ␪ ⫽

冕 ␤ ⳵⳵ ⬘ ␲

0

2␤ 1 T . 共 1,␪ 兲 sin␪ d ␪ ⫽ r 3␭ ⑀ 2 共16兲

Equation 共8兲 represents a significant departure from our earlier work in Ba-S. There we defined ⑀ ⫽ 1/冑Ma with the implicit assumption that v ⬁ ⬃O(1) when ⑀ Ⰶ1. Here, the expansion parameter ⑀ includes v ⬁ in its definition. While the dependence of v ⬁ on the Marangoni number is unknown, we assume that ⑀ is a small quantity when Ma is large, subject to verification a posteriori. When ⑀ is small, boundary layers are expected to occur near the drop surface in both phases. The temperature fields outside the boundary layers, i.e., the outer solutions, will be denoted by T and T ⬘ in the continuous phase and the drop, respectively. Similarly, the inner temperature fields will be denoted by t and t ⬘ . The outer variables are (r, ␪ ) while the inner variables in the two phases are (x, ␪ ) and (x ⬘ , ␪ ), where x⫽

r⫺1 , ⑀

x ⬘⫽

1⫺r

⑀ 冑␭

r⭓1,

,

r⭐1.

共17兲

1 t⫽ t 0 ⫹¯, ⑀

共20兲

1 t ⬘ ⫽ t 0⬘ ⫹¯. ⑀

共21兲

To determine the scaling of the outer temperature field within the drop, let us examine the consequence of substituting a straightforward expansion in ⑀ of the form T ⬘ ⫽T ⬘0 ⫹o(1) into Eq. 共7兲. This yields a problem for the leading order temperature field, in which convective transport of energy balances the sink. Thus, as a fluid element goes around a closed circulation loop in the Hill’s vortex within the drop, it would lose energy. This is clearly not acceptable, since the steady temperature field cannot be multi-valued within the drop. Therefore, we must begin the expansion at an order other than O(1). We can rule out the possibility that T ⬘ is of o(1), since it will not be possible to balance the sink term using such an expansion. Therefore, we must conclude that T ⬘ is of an order which is lower than O(1). A solution that is of the form T ⬘ ⫽ (1/⑀ )T 0⬘ ⫹o(1/⑀ ) leads to the same difficulty as that encountered before. In this case, there is no problem with the leading order outer temperature field, which is constant along each closed streamline. However, the next higher order field, which must be of O(1) in order to balance the sink term, violates the requirement that the temperature field be single-valued within the drop. Therefore, we are led to the following outer expansion for the scaled temperature field within the drop:

共18兲 T ⬘⫽

Next, we determine the asymptotic scalings for the leading order temperature fields in the outer and inner regions. The outer temperature field in the continuous phase is of O(1), because this scaling is controlled by the temperature field far away from the drop: T⫽T 1 ⫹¯.

FIG. 1. Asymptotic scalings of the inner and outer temperature fields near the surface of the drop.

共19兲

The reason for the choice of the subscript 1 on the leading order outer field in Eq. 共19兲 will become clear as we pursue the analysis. The scaling for the inner fields can be obtained from Eq. 共16兲, rewritten in terms of the inner variables:

1

⑀2

T 0⬘ ⫹T 1⬘ ⫹¯.

共22兲

Using this expansion in Eq. 共7兲, it can be seen that the sink is balanced against convective transport of energy in the problem for T 1⬘ , but, in addition, there is a nonhomogeneous term in the governing equation for T 1⬘ of the form ␭ⵜ 2 T 0⬘ , which balances the sink when integrated around a closed streamline. This leads to a self-consistent formulation, wherein the leading order temperature field, which is constant along each streamline, is determined by conduction across the streamlines balancing the sink term. Higher order fields satisfy the condition that they be single-valued within the drop. Note



that we have not introduced a term at O(1/⑀ ) in Eq. 共22兲. Formally, such a term should be included. However, the governing first order partial differential equation for the field is homogeneous at this order. Matching the outer and inner fields leads to the conclusion that the only contribution to the outer solution at O(1/⑀ ) is a constant. Therefore, we have labeled the next contribution at O(1) as T ⬘1 , for convenience in notation. Figure 1 shows a sketch of the various regions near the interface of the drop and the order of magnitude of the temperature in these regions. The order of the scaled temperature in Eq. 共22兲 implies that the magnitude of the temperature within the drop is very large, compared to that in the continuous phase. Since heat is being transferred to the drop, its interior must be very cold, relative to the surrounding fluid. The extent of the temperature difference is characterized by a factor O(1/⑀ 2 ). The reason for these very cold conditions is the need to sustain a sensible heat flux into the drop in the limit when convective transport dominates over conduction. Physically, one can regard this as the limit when the thermal conductivities are small, so that it takes a large difference in temperature to drive the requisite heat flux at the interface. One might wonder about the mechanism by which the interior of the drop can become so cold compared to the exterior fluid. We can explain it by considering the initial period, during which the drop achieves its quasi-steady physical velocity V ⬁ . In the following discussion, we assume that this is the order of magnitude of the velocity in the transient period. The physical temperature of the interior of the drop can change only when the changes at the surface are communicated to it. Since the streamlines in the Hill’s vortex within the drop are closed, and convective transport is rapid compared to conduction, the temperature changes at the surface are communicated to the stagnation ring of the Hill’s vortex in the interior by conduction across the streamlines. The time scale for this process is R 2 /(␭ ␬ ). During this period, the drop moves a distance of order V ⬁ R 2 /(␭ ␬ )⫽(R/␭)Mav ⬁ . Therefore, this distance is of the order R/ ⑀ 2 . The physical temperature of the continuous phase at the location of the drop is then of the order O(GR/ ⑀ 2 ), while the temperature within the drop is adjusting to the changes at the surface. Thus the interior of the drop, near the stagnation ring, is very cold relative to the continuous phase, when conduction effects are small. Substituting Eq. 共21兲 into Eq. 共5兲, the asymptotic scaling for v ⬁ can be obtained as follows: v ⬁ ⫽Mav ⬁ 0 ⫹¯.

共23兲

Thus the leading order scaled migration speed of the drop is proportional to the Marangoni number. From Eq. 共23兲 and the definition of ⑀ , we find that, at this order, ⑀ ⫽1/(Ma冑v ⬁ 0 ). Therefore, the assumption that ⑀ →0 as Ma→⬁ is justified. At higher orders, the dependence of ⑀ on Ma is complicated and is unknown a priori because of the appearance of v ⬁ in its definition. The thicknesses of the thermal boundary layers on both sides of the interface scale as ⑀ and hence are proportional to 1/Ma. This is in contrast to the gas bubble limit in Ba-S, in which we found the thick-

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ness of the thermal boundary layer in the continuous phase to be proportional to 1/冑Ma. Since Ba-S found in the gas bubble limit that v ⬁ ⬃O(1) as Ma→⬁, their result is consistent with the scaling given here. We shall show in Sec. II E that v ⬁ 0 →0 as ␤ →0. B. Outer temperature field in the continuous phase

The equation for the outer temperature field at leading order can be obtained from Eqs. 共6兲 and 共19兲 to be 1⫹u

⳵T1 v ⳵T1 ⫹ ⫽0 ⳵r r ⳵␪

共24兲

with the boundary condition T 1 →rcos␪ as r→⬁.

共25兲

The leading order outer temperature field is the same as that in the case of a gas bubble. Transforming from (r, ␪ ) to (r, ␺ ) coordinates where ␺ ⫽ 21 sin2␪ (r2⫺1/r) is the streamfunction, Ba-S have obtained the following solution: T 1 ⫽rcos␪ ⫹

冕

⬁

r

˜ 兲 ⫺1 兲共 3 ␺ / 共˜r 2 ⫺1/r ˜. dr ˜ 兲兲 1/2 共˜r 3 ⫺1 兲共 1⫺2 ␺ / 共˜r 2 ⫺1/r 1

共26兲

Near r⫽1, the result can be expressed as follows: T 1 ⫽1⫹

␲

1 1 2 ⫺ ln48⫹ ln共 r⫺1 兲 ⫹ ln共 1⫹cos␪ 兲 . 3 3 6 冑3 6 共27兲

C. Outer temperature field within the drop

From Eqs. 共7兲 and 共22兲, the equation for T ⬘0 within the drop can be written as follows: u⬘

⳵ T 0⬘ v ⬘ ⳵ T 0⬘ ⫹ ⫽0. ⳵r r ⳵␪

共28兲

The solution is T 0⬘ ⫽F 共 ␺ ⬘ 兲 ,

共29兲

where ␺ ⬘ is the streamfunction within the drop, defined as

␺ ⬘ ⫽ 43 sin2 ␪ 共 r 4 ⫺r 2 兲 ,

共30兲

and F( ␺ ⬘ ) is an unknown function yet to be determined. The physical implication of Eqs. 共29兲 and 共30兲 is that T 0⬘ is constant along each recirculating streamline within the drop. We show below that F( ␺ ⬘ ) can be obtained by considering the equation for the next higher order temperature field T 1⬘ , 1⫹u ⬘

⳵ T 1⬘ v ⬘ ⳵ T 1⬘ ⫽␭ⵜ 2 F. ⫹ ⳵r r ⳵␪

共31兲

To solve Eq. 共31兲, we follow the procedure used by Kronig and Brink25 and Brignell26 and transform to coordinates along a streamline and orthogonal to it in a meridian plane. The transformed coordinates are (m,q) which are related to (r, ␪ ) as shown below: m⫽⫺

16 3

␺ ⬘,

共32兲

738


q⫽

r 4 cos4 ␪


共33兲

.

2r 2 ⫺1

for the leading order outer temperature field within the drop may finally be written in a form that is applicable near the surface of the drop as follows:

The metrical coefficients for the transformation are 1 , h m⫽ 8r⌬sin␪

h q⫽

共 2r 2 ⫺1 兲 2

4r 3 ⌬cos3 ␪

h ␾ ⫽r sin ␪ ,

,

共34兲

T 0⬘ ⫽

⫺

where ⌬ 2 ⫽ 共 1⫺2r 2 兲 2 sin2 ␪ ⫹ 共 1⫺r 2 兲 2 cos2 ␪ .

共35兲

Equation 共31兲 can be written in (m,q) coordinates as follows: 1⫹

冉

冊

⳵ h q h ␾ dF ␭ vq ⳵T1 . ⫽ hq ⳵q h m h q h ␾ ⳵ m h m dm

共36兲

Here v q is the velocity along a streamline. It can be shown that v q h m h ␾ is a constant. Multiplying both sides of Eq. 共36兲 by h m h q h ␾ and integrating with respect to q for one circuit along a streamline yields the following equation for F(m): H 共 m 兲 ⫽␭

冉

冊

dF d J共 m 兲 , dm dm

共37兲

where H共 m 兲⫽ ⫽

⫽ J共 m 兲⫽

冖

h m h q h ␾ dq

冑2 K 8 共 1⫹ 冑1⫺m 兲 1/2 1 4 共 1⫹ 冑m 兲

冖

1/2

K

冉

2 冑1⫺m 1⫹ 冑1⫺m

冉冊 1⫺ 冑m

1⫹ 冑m

冊共38兲

,

h qh ␾ 2 冑2 dq⫽ 共 1⫹ 冑1⫺m 兲 1/2共 4⫺3m 兲 hm 3

⫻E

冉

2 冑1⫺m 1⫹ 冑1⫺m

冋

冊

⫺

16 mH 共 m 兲 3

冉冊冉冊册 1⫺ 冑m

1⫹ 冑m

,

共39兲

and K(x) and E(x) are the complete elliptic integrals of the first and second kinds, respectively 共see Abramowitz and Stegun27兲. Kronig and Brink obtained the expressions for H(m) and J(m) and we have rewritten them in the notation used by Abramowitz and Stegun. The solution for F is given below: F⫽

冋冕冕

1 B ⬘⫹ ␭ 0

m

0

dx J共 x 兲

x

1

册

H 共 s 兲 ds ,

冉

冊

册

3 2 m ln m ⫹O 共 m 3 ln m 兲 . 512

共41兲

The second term inside the bracket on the right-hand side of Eq. 共41兲 reveals that ⳵ T 0⬘ / ⳵ r is positive as r→1. Thus energy is being provided to the drop at a constant rate by the continuous phase. This is necessary for the physical temperature of the drop to increase at a constant rate as it moves into the warmer surroundings. However, at O(1/⑀ 2 ), the outer temperature field in the continuous phase is zero, implying that the fluid is isothermal. Since the applied temperature gradient far away from the drop is responsible for its migration, one might wonder how it enters into the problem at leading order. The answer is that the applied temperature gradient appears in the leading order problem indirectly, via the demand for energy within the drop. This is represented by the sink in Eq. 共36兲, which becomes evident when the equation is rewritten in terms of physical quantities. Since the temperature fields in the two fluids, in the absence of conduction, are disparate in order, inner 共boundary兲 layers must exist near the surface of the drop in both phases that provide energy to the drop while maintaining continuity of temperature and heat flux across the interface. Matching with the inner field, to be obtained shortly, leads to the result that B ⬘0 ⫽0. However, it will become necessary to introduce a constant temperature field, labeled B ⬘ , at O( ⑀ ) to satisfy matching requirements at leading order. Therefore, the physical picture is as follows. The interior of the drop displays scaled temperature variations of O(1/⑀ 2 ), overlaid on a constant scaled temperature that is of O(1/⑀ ). We now proceed to the analysis of the inner temperature fields. D. Leading order inner temperature fields

4 1⫺ 冑m ⫽ 共 1⫹ 冑m 兲 1/2 共 4⫺3m 兲 E 3 1⫹ 冑m ⫺ 共 4 冑m⫺3m 兲 K

冋

1 1 3 3 7 B ⬘ ⫺ m⫹ m2 ln2⫺ ␭ 0 16 16 16 64

Using the definitions of the inner variables 关Eqs. 共17兲 and 共18兲兴 and the inner expansions 关Eqs. 共20兲 and 共21兲兴 in the energy equation 关Eqs. 共6兲 and 共7兲兴, the governing equation for the leading order inner temperature fields can be written as follows: ⫺3xcos␪

⳵t0 3 ⳵ t 0 ⳵ 2t 0 ⫽ , ⫹ sin␪ ⳵x 2 ⳵␪ ⳵x2

⫺3x ⬘ cos␪

⳵ t 0⬘

⳵ t 0⬘ ⳵ 2 t 0⬘ 3 ⫹ sin␪ ⫽ . ⳵ ␪ ⳵ x ⬘2 ⳵x⬘ 2

共42兲

共43兲

The boundary conditions are given below: 共40兲

where B 0⬘ is an unknown constant. It can be shown that 兰 10 H(s)ds⫽ 31 and J and H can be expanded near m ⫽0 as J(m)⫽ 163 ⫺5m⫹O(m 2 ln m), H(m)⫽ 83 ln 2⫺ 161 ln m ⫹O(mln m), while J(1)⫽0, H(1)⫽ ␲ /(8冑2). The solution

t 0 共 0,␪ 兲 ⫽t 0⬘ 共 0,␪ 兲 ,

␦

⳵ t ⬘0 ⳵t0 共 0,␪ 兲 ⫽⫺ 共 0,␪ 兲 , ⳵x ⳵x

t 0 共 x→⬁, ␪ 兲 →0,

共44兲共45兲共46兲


t ⬘0 共 x ⬘ →⬁, ␪ 兲 →B ⬘ ⫺

1 2 冑␭


x ⬘ sin2 ␪ ,

共47兲

where ␦ ⫽ 冑␭/ ␤ ⫽ 冑k ␳ C p /(k ⬘ ␳ ⬘ C ⬘p ). Equations 共46兲 and 共47兲 represent matching conditions between the inner and outer temperature fields. When we rewrite the leading order outer field in Eq. 共41兲 in the inner variables (x ⬘ , ␪ ), and expand for small ⑀ , it is seen that only the terms up to O(m) need to be matched at leading order in the inner field. To complete the specification of the boundary conditions, ‘‘starting’’ conditions must be provided at ␪ ⫽0. The starting condition for t 0 may simply be written as t 0 ⫽0 at ␪ ⫽0. In other words, near the front stagnation point, the temperature field in the continuous phase is isothermal at the edge of the boundary layer for all streamlines. Of course the outer temperature field demands variations of O(1), but at O(1/⑀ ) there is no change in the temperature with distance along the forward stagnation streamline and in a thin bundle of streamlines surrounding it. The starting condition for t ⬘0 is complicated because the temperature distribution near the front stagnation point within the drop is influenced by the inner thermal wake that convects energy from the rear stagnation region to the forward stagnation region. A similar situation was encountered by Harper and Moore in their analysis of the momentum boundary layer within a drop for large Reynolds numbers. Harper and Moore showed that to leading order one may neglect diffusion of vorticity in the internal wake and assume that the vorticity distributions near the front and rear stagnation points are identical. We use the same idea here—the temperature distributions at leading order near the forward and rear stagnation points are identical, and the internal thermal wake within the drop merely convects the temperature field passively along streamlines. A simple argument, originally given by Brignell26 can be used to justify this assumption. The thermal boundary layer at the drop surface has a thickness of O( ⑀ ) and the temperature variation across it is of O(1/⑀ ). The internal wake comprises the boundary layer fluid that turns around near the rear stagnation region. Therefore, the temperature change across the internal wake is O(1/⑀ ) as well. The thickness of the thermal wake within the drop is O( 冑⑀ ). This is established by using the result that the mass flow rate of fluid entering the wake is the mass flow rate leaving the thermal boundary layer. The balance of convection along streamlines in the wake and conduction across the wake yields the result that the temperature change along the streamlines is O(1) and can be neglected at leading order. The expressions for the starting conditions are provided in Eqs. 共58兲 and 共59兲, after transforming to new variables. We transform the independent variables from ((x,x ⬘ ), ␪ ) to (( ␩ , ␩ ⬘ ), ␰ ) and the dependent variables from (t 0 ,t ⬘0 ) to ( f , f ⬘ ): 共 ␩,␩⬘兲⫽

␰⫽

3 16

冕

␪

0

冉

3 4 冑2

sin3˜␪ d˜␪ ⫽

3

冊

x ⬘ sin2 ␪ ,

共48兲

1 共 2⫹cos␪ 兲共 1⫺cos␪ 兲 2 , 16

共49兲

xsin2 ␪ ,

4 冑2

f ⫽ 冑␭t 0 ,

739

共50兲

f ⬘ ⫽ 冑␭ 共 t 0⬘ ⫺B ⬘ 兲 ⫹

2 冑2 ␩ ⬘. 3

共51兲

The equations for f , f ⬘ and the associated boundary conditions are written below.

⳵ f ⳵2 f ⫽ , ⳵␰ ⳵ ␩ 2

共52兲

⳵ f ⬘ ⳵2 f ⬘ ⫽ , ⳵␰ ⳵ ␩ ⬘ 2

共53兲

f 共 0,␰ 兲 ⫽ f ⬘ 共 0,␰ 兲 ⫹

␦

B

共54兲

,

␦

2 冑2 ⳵ f ⬘ ⳵f ⫺ 共 0,␰ 兲 ⫽ 共 0,␰ 兲 , ⳵␩ 3 ⳵␩⬘

共55兲

f 共 ␩ →⬁, ␰ 兲 →0,

共56兲

f ⬘ 共 ␩ ⬘ →⬁, ␰ 兲 →0,

共57兲

f 共 ␩ ,0兲 ⫽0,

共58兲

f ⬘ 共 ␩ ⬘ ,0兲 ⫽ f ⬘ 共 ␩ ⬘ , ␰ 共 ␲ 兲兲 ⬅g 共 ␩ ⬘ 兲 ,

共59兲

g 共 ␩ ⬘ →⬁ 兲 →0.

共60兲

The constant B in Eq. 共54兲 is B⫽(␭/ ␤ )B ⬘ . The solution to Eqs. 共52兲–共57兲 can be obtained in a straightforward manner using methods suggested by Harper and Moore: f 共 ␩,␰ 兲⫽

冉冊冋冉冊冑冉冊册冉冊冋冉冊冑冉冊册

2 冑2 1 B R 共 ␩ , ␰ 兲 ⫺R ⫺ 共 ␩ , ␰ 兲 ⫹ ␩⫹ 1⫹ ␦ ⫹ 3 ␦ ⫻erfc

f ⬘共 ␩ ⬘, ␰ 兲 ⫽

␩

2 冑␰

2␰ ␩2 exp ⫺ ␲ 4␰

4 3

⫺

共61兲

,

2 冑2 1 R ⫹共 ␩ ⬘, ␰ 兲 ⫹ ␦ R ⫺共 ␩ ⬘, ␰ 兲 ⫹ ␩ ⬘ ⫺B 1⫹ ␦ 3 ⫻erfc

␩⬘

⫺

2 冑␰

4 3

2␰ ␩ ⬘2 exp ⫺ ␲ 4␰

,

共62兲

where R ⫾ 共 p,q 兲 ⫽

1

冕冋冉冉冊册

2 冑␲ q

⬁

0

⫾exp ⫺

g 共 ˜p 兲 exp ⫺

共 p⫹ ˜p 兲 2 4q

共 p⫺ ˜p 兲 2 4q

d ˜p .

冊共63兲

The starting condition given in Eq. 共59兲 yields a Fredholm integral equation of the second kind for g( ␩ ⬘ ):

740


1 g共 ␩⬘兲⫽ 1⫹ ␦ ⫹

冋冉

1

冑␲

冊

2 冑2 2 ␩ ⬘ ⫺B erfc␩ ⬘ ⫺ 3 3

冕

⬁

0

冉

˜ 2 g 共 ˜␩ 兲 e ⫺( ␩ ⬘ ⫺ ␩ ) ⫹

冑


2 ⫺␩ 2 e ⬘ ␲

册

TABLE I. Numerically calculated values of h( ␦ ) and B( ␦ ).

冊

1⫺ ␦ ⫺( ␩ ⫹ ˜␩ ) 2 e ⬘ d ˜␩ . 1⫹ ␦ 共64兲

The unknown constant B is determined by the condition given in Eq. 共60兲. For a given value of ␦ , g( ␩ ⬘ ) and B are determined numerically. It is convenient to define G 共 ␩ ⬘ 兲 ⫽ 共 1⫹ ␦ 兲 g 共 ␩ ⬘ 兲 .

冉冑

冊

2 2 2 ␩ ⬘ ⫺B erfc␩ ⬘ ⫺ 3 3

⫹

冕冑␲ 1

⬁

0

冑

冉

2 ⫺␩ 2 e ⬘ ␲

˜ 2 G 共 ˜␩ 兲 e ⫺( ␩ ⬘ ⫺ ␩ ) ⫹

冊

1⫺ ␦ ⫺( ␩ ⫹ ˜␩ ) 2 e ⬘ d ˜␩ . 1⫹ ␦

The inner temperature field on the surface of the drop is now used in Eq. 共5兲 to determine the migration speed of the drop at leading order:

␭ 共2⫹3 ␣ 兲共1⫹␦兲 2

⫹

1 共 1⫹ ␦ 兲冑␲␰

冋冕冉 ␲

Ma 2

冕

⬁

0

0

⫺

冑2 共 2⫹cos␪ 兲 1/2共 1⫺cos␪ 兲 3 冑␲

冊

册

2

˜2 G 共 ˜␩ 兲 e ⫺ ␩ /4 ␰ d ˜␩ sin␪ cos␪ d ␪ .

共67兲 The above result for the drop speed can be written as v ⬁⫽

4h 共 ␦ 兲 Ma ␭ 共 2⫹3 ␣ 兲 2 共 1⫹ ␦ 兲 2

共68兲

.

For a chosen value of ␦ , the numerical calculations to determine h( ␦ ) were performed as follows. The integral on the right-hand side of Eq. 共66兲 was approximated in the following manner:

冕

⬁

0

G 共 ˜␩ 兲 H 共 ␩ ⬘ , ˜␩ 兲 d ˜␩ ⫽

B( ␦ )

0 0.25 0.5 1 5 10 ⬁

0.005 68 0.006 11 0.006 42 0.006 83 0.007 75 0.007 98 0.008 26

⫺0.418 ⫺0.407 ⫺0.399 ⫺0.389 ⫺0.366 ⫺0.361 ¯

G i ⫽L i ⫹

共66兲

v ⬁⫽

h( ␦ )

共65兲

Equation 共64兲 can be recast for G( ␩ ⬘ ) as follows: G共 ␩⬘兲⫽

␦

冕

Yl

0

G 共 ˜␩ 兲 H 共 ␩ ⬘ , ˜␩ 兲 d ˜␩

⫹G 共 Y l 兲

冕

⬁

Yl

H 共 ␩ ⬘ , ˜␩ 兲 d ˜␩ .

兺j G j H i j

or

兺j G j 共 ␦ i j ⫺H i j 兲 ⫽L i ,

共70兲

where L i represents the first two terms on the right-hand side of Eq. 共66兲. For a guessed value of the unknown constant B, the values of G i were determined by solving the above linear system of equations. The value of B was obtained by trial and error such that G(Y l )→0. An interpolation function was then constructed for G( ␩ ⬘ ) which was equal to G i for ␩ ⬘ ⭐Y l and zero otherwise. This interpolation function was used in Eq. 共67兲 to numerically determine v ⬁ . The procedure given above for numerical solution of the integral equation is essentially that given by Hammersley.28 The numerical algorithm was checked by reproducing the solution to Eq. 4.9 of Hammersley’s paper 共with oscillatory convergence as ␩ ⬘ →⬁, see Stewartson29 in this regard兲 as well as an example integral equation where the kernel H( ␩ ⬘ , ˜␩ ) is a separable product of two exponential functions, for which an analytical solution can be readily obtained 关i.e., G( ␩ ⬘ )⫽1 ˜ ⫹ 兰 ⬁0 G( ˜␩ )e ⫺ ␩ ⬘ e ⫺ ␩ d ␩ ⬘ ⇒G( ␩ ⬘ )⫽1⫹2e ⫺ ␩ ⬘ ]. The values for h( ␦ ) and B( ␦ ) obtained from the numerical calculations are given in Table I. It is seen that h( ␦ ) is a monotonically increasing function. The ratio h(⬁)/h(0) is approximately 1.45. The value for h(⬁) is 1/(18␲ ) (16/7 ⫺12冑3/7) 2 ⫽0.008 261 8 and is obtained by analytically evaluating the integral on the right-hand side of Eq. 共67兲 in the limit ␦ →⬁. The function G( ␩ ⬘ ) is related to the inner temperature field in the forward stagnation region within the drop. Figure 2 shows a plot of G( ␩ ⬘ ) for ␦ ⫽0 and 10. It is seen that G( ␩ ⬘ ) is quite insensitive to the value of ␦ and exhibits

共69兲

A typical value for Y l was 2.5 and Y l ⫽4 was also tested in a few selected cases. The first term on the right-hand side of Eq. 共69兲 was approximated by using the trapezoidal rule for numerical integration on N⫺1 equally spaced intervals 共typically N⫽51). The integral in the second term on the right-hand side of Eq. 共69兲 was also evaluated numerically after truncating the infinite domain by trial and error to assure that the value of the integral was accurate to six significant figures. Writing G i for the N unknown values of G( ␩ ⬘ ) in the interval 0⭐ ␩ ⬘ ⭐Y l , Eq. 共66兲 can be written in the following discrete form where ␦ i j is the Kronecker delta:

FIG. 2. Results for G( ␩ ⬘ ) vs ␩ ⬘ from the numerical solution of Eq. 共66兲.



741

Also, t 1 must match with the O(1) outer temperature field given in Eq. 共27兲. Since t 0 →0 as ␦ →⬁, the nonhomogeneous terms on the right-hand side of Eq. 共71兲 vanish. The governing equation and boundary conditions for t 1 are identical to those obtained by Balasubramaniam and Subramanian.4 Thus, their analysis for the migration of a gas bubble at O(1) is recovered. III. DISCUSSION AND CONCLUDING REMARKS

FIG. 3. The distribution of the scaled temperature on the surface of the drop.

oscillatory convergence with small amplitudes as ␩ ⬘ →⬁. The function (1⫹ ␦ ) f (0,␪ )⫺ B/ ␦ ⫽(1⫹ ␦ ) 冑␭t 0 (0,␪ ) ⫺ B/ ␦ is plotted against ␪ in Fig. 3. Note that the entity being plotted as the ordinate is relatively insensitive to change in the value of ␦ . We can infer from this figure that as the value of ␦ increases, the average scaled surface temperature of the drop becomes less negative.

The central result in this paper is the expression for the steady migration speed of a drop 关Eq. 共68兲兴 at leading order that is valid for large values of the Reynolds and Marangoni numbers. In this limit, the scaled drop speed is proportional to the Marangoni number. In contrast, for the case of a gas bubble, the migration speed is independent of the Marangoni number when it is large. The physical migration speed of the drop is written as V ⬁⫽

4 兩 ␴ T 兩 2 G 2 R 30 h 共 ␦ 兲

␮ 2 ␭ ␬ 共 2⫹3 ␣ 兲 2 共 1⫹ ␦ 兲 2

1⫺3xcos␪

⳵t1 3 ⳵ t 1 ⳵ 2t 1 ⫹ sin␪ ⫺ ⳵x 2 ⳵␪ ⳵x2

⫽ 共 2⫺6x 2 cos␪ 兲

⳵t0 ⳵t0 . ⫹3xsin␪ ⳵x ⳵␪

共71兲

The boundary conditions are

⳵t1 共 0,␪ 兲 ⫽0, ⳵x

共72兲

⳵t1 共 x,0兲 ⫽0. ⳵␪

共73兲

共74兲

In contrast, the physical migration speed of the drop in the limit Re, Ma→0 is

E. The limit of a gas bubble

The case of a gas bubble is obtained when the viscosity and the thermal conductivity of the drop are small compared to the corresponding properties in the continuous phase, i.e., in the limit ␣ , ␤ →0. When the Reynolds number is large, the velocity field around a gas bubble is given by the potential flow solution given in Eqs. 共9兲 and 共10兲. Equation 共5兲 then relates the migration speed of the bubble to the temperature distribution on its surface. From the definition of ␦ , we note that ␦ →⬁ when ␤ →0. In the limit ␦ →⬁, we see from the results of Sec. II D 关Eqs. 共61兲 and 共68兲兴 that the inner temperature field at O(1/ ⑀ ) in the continuous phase and the migration speed at O(Ma) for the gas bubble vanish. This is not surprising because the inner temperature fields considered in Sec. II D are driven by the demand for energy within the drop and in the limit of a gas bubble, to leading order in ␤ , no heat enters the bubble. The leading order inner temperature field in the continuous phase must therefore be of O(1) to match with the outer temperature field given by Eq. 共26兲. The governing equation for this temperature field t 1 (x, ␪ ) can be written as follows:

.

V YGB⫽

2 兩 ␴ T 兩 GR 0 . ␮ 共 2⫹3 ␣ 兲共 2⫹ ␤ 兲

共75兲

Except for a numerical factor that depends weakly on Re, Eq. 共75兲 also holds for large Ma in the limit ␣ , ␤ →0 for a gas bubble. This regime of motion of the drop occurs because of the demand for energy within the drop necessary to increase its temperature at a constant rate as it moves into warmer fluid in the continuous phase. In a reference frame moving with the drop, except for a thin boundary layer at the surface, the temperature within the drop is constant along a streamline, with heat being transferred across the streamlines by conduction. Since conduction is weak compared to convection when the Marangoni number is large, a substantial temperature difference is present in the drop, and the fluid near its internal stagnation point is very cold. In contrast, a large change in the temperature in the continuous phase occurs only near the drop and is necessary to conduct energy to it. Outside this thermal boundary layer, the gradient in the fluid temperature is modest and commensurate with the imposed gradient far away from the drop. We have recently performed numerical calculations for the motion of a drop of fluorinert FC-75 in a continuous phase of silicone oil of kinematic viscosity 50 centistokes.18 For Ma⫽400, we found that the isotherms and the streamlines within the drop are virtually identical outside the thermal boundary layer, in accord with the results in Sec. II C. The temperature difference between the stagnation points on the surface of the drop, as well as its migration speed, were found to increase with Ma for Ma⬎100, almost linearly. As anticipated, the drop is quite cold compared to the undisturbed continuous phase fluid. The drop migration speed was also computed for Marangoni numbers in the range 0⭐Ma ⭐1000 for Prandtl numbers of both fluids equal to 1 and 5.

742



The speed of the drop was not very sensitive to the Prandtl number in this range. The computed drop speeds at large Ma were qualitatively in agreement with Eq. 共68兲. We regret that due to a typographical error in an earlier version of the present analysis, we used a result for the asymptotic drop speed which is lower than that given here in Eq. 共68兲 by a factor of 4 in computing the result shown in Figure 8 of Ref. 18. When the above correction is made, the asymptotic result from the present work is quite removed from the numerical solution obtained by Ma et al. In order to understand the reason for the discrepancy, we explored the nature of the higher order corrections to the result given here in Eq. 共68兲. When the result for T 0⬘ from Eq. 共41兲 is rewritten in inner variables, the first correction beyond the leading order is O(ln⑀). Therefore, the inner expansions in Eqs. 共20兲 and 共21兲 must have contributions at O(ln⑀). When an inner expansion of the form (1/⑀ ) t 0 ⫹ln⑀ tl1 is substituted into Eq. 共5兲, since ⑀ ⫽1/冑Mav ⬁ , we obtain the expansion for v ⬁ to be of the form v ⬁ ⫽a 0 Ma⫹a l 1 ln Ma⫹a 1 ⫹a l 2

⫹¯.

ln Ma 共 ln Ma兲 2 ⫹b l 2 Ma Ma 共76兲

The result for the constant a 0 is already given in Eq. 共68兲. Since the other constants depend on t l 1 (0,␪ ), which is unknown, a direct comparison cannot be made with the numerical results. However, we empirically fitted the results used in Figure 8 of Ref. 18 to the form shown in Eq. 共76兲, wherein the coefficient a 0 is obtained from the best fit. For Pr⫽1, ␣ ⫽␭⫽ ␦ ⫽1, and 200⭐Ma⭐1000, this fit is 共 ln Ma兲 2 v ⬁ ⫽0.002 36 Ma⫺3.383ln Ma⫹24.29⫺88.34 Ma

ln Ma ⫹232.0 . Ma

共77兲

We found the fit to be excellent, with a difference between the computed values and that given by the fit to be less than 0.2%. This remarkably good fit suggests that the form used in Eq. 共76兲 is indeed correct. Unfortunately, the labor involved in analytically obtaining the constants in Eq. 共76兲 is prohibitively large. Therefore, we have not undertaken that task here. We note that the coefficient of Ma in Eq. 共68兲 for this set of parameters is 0.002 05. This value is close to the result for the fitted coefficient a 0 given in Eq. 共77兲, with a difference of around 15%. Since the higher order terms carry substantial weight, the leading order result for the drop speed differs significantly from the numerically obtained value even at Ma⫽1000. Experimental results are not available for large values of the Reynolds and Marangoni numbers to compare with the prediction in Eq. 共74兲. We have recently reported results from reduced gravity experiments for the motion of fluorinert FC-75 drops in silicone oils in Ref. 19. The maximum Marangoni number in these experiments is around 400 and the maximum Reynolds number is approximately 4.7. The drop speed from Eq. 共74兲 is compared with experimental

results in Ref. 19. Unfortunately, the same typographical error in our earlier version of the analysis, mentioned in the context of Figure 8 of Ref. 18, was present in Eq. 共8兲 of Ref. 19. Therefore, the values shown for the asymptotic prediction in Figure 5 of Ref. 19 should have been larger by a factor of 4. The correct asymptotic prediction lies substantially above the observed speeds. However, the Reynolds numbers were so small in the experiments that the comparison with the present asymptotic prediction is inappropriate. It is worth noting that the variation of viscosity with temperature makes the interpretation of the experimental data difficult. The migration speed of a drop increases along its trajectory because of this effect. We have pointed out in Ref. 19 that a simple quasi-steady interpretation of the data is inadequate, and a fully transient analysis is necessary in order to properly describe the behavior of the drops in the experiments. Nevertheless, it is interesting to note that when the scaled migration speed of a drop along its trajectory is plotted against the Marangoni number for that drop, this speed indeed increases with increasing values of Ma. However, data for different drops clearly show an opposite trend with increasing values of the Marangoni number, for reasons mentioned above. Suitable experiments that need to be performed to test the asymptotic prediction require the time scales for diffusion of momentum and energy within the drop, the convective residence time of the fluid in the continuous phase, and the inertial time scale for acceleration of the drop from rest to be sufficiently small. This is to ensure that steady velocity and temperature fields can be comfortably attained in a fraction of the time for the traverse of the drop. The experiments require a reduced gravity environment. Since the duration of reduced gravity in available facilities on the Earth is a few seconds, the requirement for a small thermal diffusion time can only be met in experiments with drops of a liquid metal that have a diameter of a few millimeters. The temperature distribution in a stratified liquid layer, in which a drop moves solely due to a body force such as gravity, can readily be obtained from the analysis given in this paper when the Reynolds and Pećlet numbers associated with the motion are large. In such a case, when the thermocapillary effect is neglected, along with that of buoyant convection, the reference velocity v 0 is simply redefined to be the steady speed of the drop, and the leading order temperature fields in the two phases can be obtained from the results in Sec. II. ACKNOWLEDGMENT

This work was supported in part by NASA’s Microgravity Sciences and Applications Division via Grant No. NAG31122 to Clarkson University. 1

G. Wozniak, J. Siekmann, and J. Srulijes, ‘‘Thermocapillary bubble and drop dynamics under reduced gravity – survey and prospects,’’ Z. Flugwiss. Weltraumforsch. 12, 137 共1988兲. 2 R. S. Subramanian, ‘‘The motion of bubbles and drops in reduced gravity,’’ in Transport Processes in Bubbles, Drops and Particles, edited by R. P. Chhabra and D. DeKee 共Hemisphere, New York, 1992兲, pp. 1–42. 3 R. S. Subramanian, R. Balasubramaniam, and G. Wozniak, ‘‘Fluid mechanics of bubbles and drops,’’ in Physics of Fluids in Microgravity, ed-

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The migration of a drop in a uniform temperature . . . 16

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