6 Fluid mechanics of bubbles and drops

R. Shankar Subramanian, R. Balasubramaniam

and GuÈnter Wozniak

The literature on the motion of bubbles and drops, due to the action of interfacial
tension gradients which arise from temperature variation along the interface, is
reviewed, highlighting recent work. First, theoretical developments are considered.
These include asymptotic analyses of the motion of an isolated drop or bubble when
convective transport effects are either very small or very large as well as a variety of
numerical solutions of the problem. Also, theoretical work on drops interacting with
other drops or with neighboring surfaces is mentioned very briefly. This is followed by a
discussion of results from ground-based and reduced gravity experiments. Experiments
on thermocapillary migration on the ground have been restricted to conditions wherein
convective transport effects play only a small role. The only ground-based experiments
in which such effects are large involve drops that are held stationary under the combined
influence of gravity and thermocapillarity. Experiments were initiated under reduced
gravity conditions in the late seventies, and gathered momentum in the last two decades
of the twentieth century due to the availability of long periods of reduced gravity
conditions aboard the United States space shuttle. We conclude this review with com-
ments about future prospects for this field, pointing out the importance of conducting
experiments under low gravity conditions.

6.1 INTRODUCTION

Bubbles and drops are encountered in our day-to-day life. They also appear in a
variety of engineering and materials processing applications. The word bubble repre-
sents an object that contains a gas or vapour and the word drop is used to designate an
object that contains liquid. In this review, we shall frequently use the term drop to refer
to both types of objects. In some cases, it is desirable to have a collection of bubbles in
a material as in polymer foams. In other situations, such as when growing a crystal, we
wish to produce a material that is free of such inclusions. Stable suspensions of drops
are necessary for producing a composite material by cooling a liquid containing more
than one component through a miscibility gap into a solid. On the other hand, after
contacting one liquid dispersed as drops with another to exchange some component
between them, it is necessary to settle out the drops and separate the two phases in
mass transfer applications.
On Earth, gravity is always present and when the density of a dispersed phase differs
from that of the continuous phase, the dispersed phase material will sink if it is more
dense than the continuous phase or rise if it is less dense. We depend on this motion
induced by gravity for a variety of applications. In experiments carried out aboard a
150 R. S. Subramanian et al.
space vehicle such as an orbiting space laboratory or an outer space probe, gravita-
tional effects on particles, bubbles, or drops suspended in a fluid are so small as to be
virtually unimportant in this respect. This is the motivation for studying the motion of
these objects due to forces other than gravity. One can think of a few mechanisms
independent of gravity that will cause a drop or a rigid particle suspended in a liquid to
move. For instance, under suitable conditions, an electric or magnetic field can be used
to induce the motion of an object. The most common mechanism that has been used is,
however, the application of a temperature gradient in the continuous phase. This is
because, it is easy to produce temperature variations in a liquid and such gradients also
occur naturally in many materials processing applications because of the use of heating
or cooling as an integral part of the process. The subject of this review is the motion of
drops in a temperature gradient. The occurrence of this motion depends critically on
the presence of a mobile interface between the drop and the surrounding fluid.
Therefore the same mechanism does not operate on rigid particles, which also will
move in a temperature gradient due to thermophoresis. The motion of rigid particles in
reduced gravity is an important subject in its own right; however, we shall not consider
that topic in the present review. Also, we shall not discuss the effects of electric and
magnetic fields, and the motion within a stationary suspended drop that is caused
perhaps by oscillations or rotation of the drop. Some experiments in reduced gravity
have indeed been performed on rotating and/or oscillating drops and a brief discussion
can be found in Wang (1992).
In this chapter, the continuous phase always is considered to be a liquid. There
will be no sharp interface between two gases in the absence of an intervening liquid
film. With such a film, a bubble containing gas can be present in another gas. A
familiar example is a soap bubble. Little fundamental research has been carried out
on the motion of bubbles of this type or that of liquid drops in a gas in reduced
gravity. Therefore, we confine our review to the motion of a drop or bubble in a
liquid.
A drop moves in a liquid under the action of a temperature gradient because of the
variation of its interfacial tension  with temperature T. It is typical of most fluids that
the interfacial tension decreases with increase in temperature so that the coefficient T
is negative. Its magnitude lies usually in the range 0:01 0:1mN/…m  K †. The change in
interfacial tension over the drop surface causes a shear stress to be exerted on the
neighboring fluids on either side of the interface, typically from the warm pole toward
the cold pole. This is called a thermocapillary stress, because, it arises from a tem-
perature gradient at the interface. The resulting fluid motion induces a hydrodynamic
force on the drop whose sense is to cause the drop to move toward warmer fluid, in the
direction of the applied temperature gradient. Once the drop starts moving, it
encounters resistance to this motion from the fluid, and it is possible to imagine steady
motion when the hydrodynamic force on the drop is exactly zero. In practice, drops
rarely move at steady state because the physical properties of the fluids vary with
temperature. The most important property in this respect is the viscosity. One can
usually expect the drop to continue to accelerate as it moves into warmer fluid because
of the reduction in viscosity with temperature. In the less common instance where the
interfacial tension increases with increase in temperature, a drop, when released from a
state of rest, would initially accelerate toward cooler fluid but will ultimately decel-
erate as it encounters more viscous liquid. This type of situation does not arise in pure
liquids, but is encountered sometimes in the case of mixtures.
 Fluid mechanics of bubbles and drops 151
The literature on thermocapillary migration up to about 1989 is reviewed in Woz-
niak et al. (1988) and Subramanian (1992). Also, a summary of research carried out in
one of our laboratories (RSS) has been provided by Subramanian (1995). Therefore,
our principal goal in this contribution is to highlight recent work which has not been
discussed in detail in the above reviews and present our thoughts regarding the direction
of future research in this field. After briefly discussing the relevant dimensionless
groups, we consider theoretical developments in a historical context, and then the
experiments. The reader interested in the mathematical aspects of the problems, such
as the detailed governing equations, boundary conditions, and techniques used for
solution, may wish to consult the monograph by Subramanian and Balasubramaniam
(2001).

6.2 IMPORTANT DIMENSIONLESS GROUPS

The most important entity that is measured in thermocapillary migration experiments

is the trajectory of the drop. This permits one to infer its velocity at a given instant in
time. Of course, from visual records of the migration process on video or motion
picture film, the size and shape of the drop can be obtained. Other items of interest are
the velocity and temperature fields in the two phases. Theoretical models attempt to
provide predictions of some or all of these quantities, subject to idealized assumptions.
When the governing equations and boundary conditions are non-dimensionalized,
three important dimensionless groups emerge. They are the Reynolds, Marangoni, and
Capillary numbers, represented by the symbols Re, Ma, and Ca respectively. Defin-
itions based on the properties of the continuous phase follow:

Rv0
Re ˆ (6:1)

Rv0
Ma ˆ (6:2)

v0
Ca ˆ (6:3)


Here, R is the radius of the drop,  is the kinematic viscosity of the continuous phase,
 is its thermal diffusivity, and  is its dynamic viscosity. The symbol  refers to the
interfacial tension between the drop and the continuous phase. The reference velocity,
v0 , obtained by balancing the thermocapillary stress at the interface with a typical
viscous stress, is defined as:

j T jj rT1 j R
v0 ˆ (6:4)


where, T is the rate of change of interfacial tension with temperature, and rT1 is the
temperature gradient imposed in the continuous phase fluid. The ratios of the dynamic
viscosities, thermal conductivities, thermal diffusivities, and the densities of the two
phases are additional parameters in the problem that one must include in modelling
the process. Therefore, in the general problem, there are seven independent dimen-
sionless parameters. When inertial effects are important, the Reynolds number and the
152 R. S. Subramanian et al.
Weber number, We ˆ CaRe ˆ Rv20 /, influence deformation. Here,  is a reference
density usually taken to be that of the continuous phase. The Marangoni number plays
the role of a PeÂclet number, reflecting the relative importance of convective transport
of energy when compared with conduction. It is the product of the Reynolds number
and the Prandtl number (Pr ˆ /).
Clearly, one can define similar groups in the drop phase and it is possible to consider
situations wherein convective transport can be important in one phase and not the
other. Usually, the physical property ratios involved are such that large values of either
the Reynolds number or the Marangoni number in one phase imply similarly large
values of the corresponding group in the other phase. Exceptions arise when one
considers fluid pairs with a large contrast in kinematic viscosity or thermal diffusivity,
but the above statement is correct in most common systems.

6.3 THEORETICAL DEVELOPMENTS

The first study that focussed specifically on thermocapillary movement of bubbles was
that of Young et al. (1959) who performed experiments on air bubbles in a column of
silicone oil suspended between the anvils of a micrometer. The authors used a down-
ward temperature gradient to cause the bubbles to almost come to a standstill, and
measured the radius of a bubble that is nearly stationary in a given temperature gra-
dient. Young et al. also constructed the first theoretical description of the motion of a
drop under the combined influence of gravity and thermocapillarity. They solved the
governing conservation equations in the linear limit when convective transport of
energy and momentum are negligible, namely when Re ! 0, and Ma ! 0. In this
limit, the contribution to the steady velocity of a drop from gravity, predicted by
Hadamard (1911) and RybczynÂski (1911), and that from thermocapillarity are addi-
tive. We use the symbol VYGB to designate the magnitude of this thermocapillary
contribution to the quasi-steady velocity of a drop. The velocity can be considered
only quasi-steady since physical properties such as viscosity and density depend upon
temperature. Since the drop continues to move into warmer fluid, these properties will
change continuously and true steady state cannot be obtained. Therefore, one assumes
that the time taken for the fields to achieve steady representations corresponding to
the local conditions around the drop is small compared to the time taken for the drop
to move into a region in which the properties are appreciably different. Note that the
problem does not arise in isothermal situations involving purely gravitational settling,
where steady motion can be conveniently realized. The result for VYGB is

2v0
VYGB ˆ (6:5)
…2 ‡ 3†…2 ‡ †

where,  ˆ 0 / is the ratio of the dynamic viscosity of the drop phase to that of the
continuous phase and  ˆ k0 /k is a similar ratio of thermal conductivities. In the limit
of negligible convective transport, the scaled velocity is independent of the density
ratio and the thermal diffusivity ratio mentioned earlier in the list of parameters. It can
be established that when the solution of Young et al. is applicable, the drop takes
on a spherical shape regardless of the value of the Capillary number. We note that
this result for the shape in the limit Re ! 0 and Ma ! 0 holds only when the
 Fluid mechanics of bubbles and drops 153
applied temperature gradient is uniform. If a spherical shape is assumed, the migration
velocity can be calculated from equation (6.5) even when the applied temperature
gradient is not uniform. In doing so, the value of the undisturbed temperature gradient
evaluated at the location of the centre of the drop must be used for calculating v0 from
equation (6.4). A similar result can be written for the steady migration velocity of a
drop due to axisymmetric absorption of incident radiation as shown by Oliver and
DeWitt (1988). In this case, one replacesR the applied temperature gradient in the
1 
definition of the reference velocity with 4k 0 q() sin 2d, where,  is the polar angle
measured from the forward stagnation streamline, and q() stands for the distribution
of the radiant heat flux that is absorbed.
Bratukhin (1975) used an asymptotic expansion in the Reynolds number to obtain a
result that he presumed to be valid for small non-zero values of Re. Bratukhin did not
specify the shape of the drop except to require departures from the spherical shape to
be small. He found that the correction to the result for the migration velocity given in
equation (6.5) was zero at O(Re), but was able to use his solution to calculate small
deformations from the spherical shape. Later, Thompson et al. (1980) extended
Bratukhin's solution to the next higher order in Re; however, the solution for the
temperature field at the second order fails to satisfy the boundary condition at infinity.
This is a well-known problem in fluid mechanics dating back to Whitehead's attempt
to improve Stokes's solution for flow past a rigid sphere. The difficulty with a simple
perturbation scheme is that the terms representing convective transport of momentum
(and energy in the present problem) are presumed uniformly small everywhere in space
when compared to the molecular transport terms. In the unbounded domain asso-
ciated with the problem, as recognized by Oseen (1910), the convective terms become
comparable in importance to the molecular transport terms at a distance that is of the
order of the ratio R/Re. Proudman and Pearson (1957) showed how the method of
matched asymptotic expansions can be used to overcome the difficulty. The method
consists of constructing a second asymptotic expansion in the Reynolds number while
keeping a new radial coordinate  ˆ Re Rr fixed. The expansion that is valid near the
sphere, called the inner expansion, is then matched with that valid far from the sphere,
called the outer expansion, in an overlap domain since both expansions represent the
same function. Subsequently, Acrivos and Taylor (1962) solved the heat transfer
analog of this problem for a rigid sphere. By using this method of matched asymptotic
expansions, it is possible to overcome the difficulty encountered in the solution pro-
cedure used by Bratukhin and Thompson et al. Subramanian (1981; 1983) pointed this
out and developed an analytical solution in the case of highly viscous fluids, setting the
Reynolds number to zero, and writing a perturbation expansion in the Marangoni
number. He found that the correction to the result given in equation (6.5) at O(Ma) is
zero, and that the first non-zero correction appears at O(Ma2 ). This result was sub-
sequently extended by Merritt (1988) to O(Ma4 ). Crespo et al. (1998) performed a
similar perturbation expansion in the Marangoni number for the case of a gas bubble
when the Reynolds number is large. These authors found that the first correction to
the result in equation (6.5) in that problem also occurs at O(Ma2 ). To date, the cor-
responding asymptotic problem including inertial effects and deformation has not
been solved.
Crespo and Manuel (1983) and, independently, Balasubramaniam and Chai (1987),
discovered that the solution for the velocity field in purely thermocapillary motion
given by Young et al. which is a potential flow in the continuous phase and Hill's
154 R. S. Subramanian et al.
spherical vortex in the drop phase, is an exact solution at all values of the Reynolds
number, provided the temperature field is that given by the solution of Laplace's
equation. The latter implies that convective transport of energy must be ignored so
that the solution applies for fluids of relatively small values of the Prandtl number, Pr.
The shape is no longer spherical, however, and Balasubramaniam and Chai obtained a
result for small inertial corrections to the spherical shape. Subsequently, Haj-Hariri
et al. (1990) constructed the solution of this problem in invariant form and calculated
the correction to the migration velocity caused by the shape deformation.
In the limit when convective transport of momentum and energy are negligible, the
unsteady motion of a drop can be analysed using Laplace transforms. This was done
by Dill and Balasubramaniam (1992) who provided predictions including asymptotic
trends in limiting situations. A similar analysis was published by Galindo et al. (1994)
who also included a gravitational contribution to the motion in their development for
completeness.
Perturbation solutions for accommodating small amounts of convective momentum
or energy transport are clearly not sufficient because they only permit predictions to be
made for relatively small values of Re and Ma, respectively. Therefore, efforts began in
the eighties to obtain solutions by numerical means. In the limit of a gas bubble, only
the transport problems in the continuous phase need to be solved and therefore this
was the first type of problem to be addressed. Shankar and Subramanian (1988)
considered the Stokes motion of a gas bubble for 0 < Ma4200 and obtained a solu-
tion of the energy equation by the method of finite differences. For the velocity field,
they used an analytical solution of Stokes's equation. The authors identified an
interesting flow structure that arises in thermocapillary migration problems which
involves a separated reverse flow wake behind the moving bubble, and discussed the
physical reason for the appearance of this structure. Subsequently, Merritt and Sub-
ramanian (1992) used the same finite difference technique to obtain a numerical
solution in the case when a bubble moves under the combined action of buoyancy and
thermocapillarity. The solution is reported only for Marangoni number in the range
0±5, but results are given for a wide range of values of a group that characterizes the
relative importance of buoyancy when compared with thermocapillarity. It is the only
reported solution as of this writing, in which the role of the gravitational force is
accommodated and convective transport effects are considered. Returning now to
purely thermocapillary motion, Szymczyk and Siekmann (1988) and Balasu-
bramaniam and Lavery (1989) solved both the momentum and the energy equations in
the gas bubble case using finite differences. Szymczyk and Siekmann reported
results for 0 < Re4100 for Prandtl numbers in the range 0.01±10 and for 0 < Re450
for Pr ˆ 100. Balasubramaniam and Lavery permitted the Reynolds number to vary
from 0.1 to 2 000 and the Prandtl number to vary from 0.01 to 1 000; however, pairs of
values of these groups were chosen such that the product of the two, namely, the
Marangoni number, was constrained to the range 0±1 000. In all these studies, the
bubble was assumed to be spherical and therefore the normal stress balance, which
could not be satisfied, was not used. The authors scaled the migration velocity of the
bubble with v0 so that the scaled velocity v1 would be 1/2 in the limit Re ! 0 and
Ma ! 0. They found that this scaled velocity decreased with increasing values of Ma
for fixed Re and that it increased gently with increasing Re for fixed Ma. Balasu-
bramaniam and Lavery (1989) also noted the appearance of a separated reverse flow
wake behind the bubble when fluid inertial effects were included, similar to that found
 Fluid mechanics of bubbles and drops 155
by Shankar and Subramanian (1988) who neglected inertia. More recently, Balasu-
bramaniam (1995) and Crespo et al. (1998) have reported numerical solutions of the
thermocapillary migration problem for a gas bubble, assuming potential flow to pre-
vail in the continuous phase. Results from these calculations are in agreement with
those from Balasubramaniam and Lavery when Ma ! 0 and an asymptotic result in
the limit Ma ! 1 that is mentioned later. We note that Treuner et al. (1996) per-
formed calculations in the gas bubble limit, similar to those of Balasubramaniam and
Lavery (1989), up to a Marangoni number of 2 500. Chen and Lee (1992) permitted the
bubble to deform from the spherical shape, and showed that a small deformation can
lead to a large decrease in the quasi-steady migration velocity. Unsteady thermo-
capillary motion of a bubble was first treated by Oliver and DeWitt (1994) who solved
the problem assuming negligible inertia. They showed that steady state can be expected
when a bubble has moved a distance approximately equal to 1±5 radii, at values of the
Marangoni number up to 200. This is logical because this is the order of the time
required to reach steady state in the thermal boundary layer at the bubble surface.
Later, Welch (1998) numerically solved the transient problem, accommodating both
inertia and deformation of the bubble shape. Welch demonstrated that a true steady state
solution is not possible because of the continuous increase in the deformation of a bubble
as it moves into warmer fluid, caused by the corresponding decrease in surface tension.
In recent years, numerical work has been performed in the case of migrating liquid
drops. In some cases deformation of the shape from a sphere has been accommodated,
which makes the computations more involved. The list of articles includes Ehmann
et al. (1992), Nas (1995), Haj-Hariri et al. (1997), and most recently, Ma et al. (1999).
Ehmann et al. were the first to make computations for spherical liquid drops. Ma et al.
solved this problem numerically by extending the code developed by Balasu-
bramaniam and Lavery (1989) for a gas bubble. The most important observation
made by Ma et al. is that the scaled velocity of a liquid drop initially decreases in
magnitude with increasing values of the Marangoni number, but that beyond a value
of Ma that lies somewhere in the range 50±200 depending upon the parameters, the
scaled velocity increases with further increase in Ma. We shall see shortly that this
remarkable result is consistent with an asymptotic prediction by Balasubramaniam
and Subramanian (2000). Nas and Haj-Hariri et al. accounted for the possibility of
shape deformation. As demonstrated by Chen and Lee (1992) and Welch (1998) in the
case of a gas bubble, the authors found that a slight deformation of shape has a
substantial effect on the migration velocity of a liquid drop. Most of Nas's computa-
tions are for 2D drops which are deformed infinite cylinders moving with a velocity
perpendicular to their axes. There are some qualitative differences in the predictions
between the 2D case considered by Nas and the 3D case treated by Haj-Hariri et al. for
reasons which are not entirely clear at this time.
Numerical solutions cannot easily handle the asymptotic situation where the par-
ameters become very large. If the Reynolds number in the continuous phase is large, it
can be established that the flow in the continuous phase is described by potential flow
almost everywhere. There is a momentum boundary layer next to the surface of the
bubble in which the tangential stress adjusts to the correct value at the interface, but
the velocity change due to the presence of this boundary layer is small. If the Mara-
ngoni number in the continuous phase is large, the situation is more complicated. This
problem for a gas bubble was analysed in detail by Balasubramaniam and Sub-
ramanian (1996). Over most of the continuous phase, heat transfer is dominated by
156 R. S. Subramanian et al.
convection, with conduction being of negligible consequence. The solution obtained
by ignoring conduction is, however, unable to satisfy the boundary condition of neg-
ligible heat flux at the bubble surface and a thermal boundary layer is formed over the
bubble surface in which conduction is as important as convective transport. Further-
more, the temperature itself undergoes substantial change in this boundary layer
because the field obtained ignoring conduction is singular on the bubble surface, being
proportional to log(r ± R) where r is a radial coordinate measured from the centre of
the bubble. There are additional complications because the temperature field both
outside and within the thermal boundary layer becomes singular as the rear stagnation
point is approached, once again in a logarithmic dependence on the angle measured
from the rear stagnation streamline. This wake singularity is relieved by a solution
that accommodates conduction in the angular direction at leading order. Balasu-
bramaniam and Subramanian obtained solutions in the limit Ma ! 1 in two cases.
For Re ! 1 they confirmed the earlier result of Crespo and Jimenez-Fernandez
(1992a), namely, v1 ˆ 13 18 log 3  0:1960. But for Re ! 0, they found v1 ˆ 0:1538 in
contrast to the value obtained by Crespo and Jimenez-Fernandez (1992b). The authors
went on to provide an improved result in the case Re ! 1, namely:

1 1 1
v1 ˆ log 3 ‡ p ‰0:06845 log…Ma† ‡ 0:6578Š (6:6)
3 8 Ma

Balasubramaniam and Subramanian (2000) analysed the motion of a drop in the case
when both the Reynolds number and the Marangoni number are large. The flow in the
continuous phase is described by a potential flow velocity field while that within the
drop is Hill's spherical vortex. The transport of thermal energy is dominated by con-
vection in both fluids over most of the domain. Conduction is important only in thin
thermal boundary layers in each fluid adjacent to the interface. These boundary layers
ensure continuity of temperature and heat flux at the interface. In this asymptotic
limit, the authors found that the migration velocity, scaled by v0 , is proportional to the
Marangoni number. As noted earlier, this is consistent with the numerical finding of
Ma et al. (1999). This remarkable result, which implies that the physical velocity is
proportional to j T j2 j rT1 j2 R3 , is very different from the corresponding asymp-
totic result obtained by Balasubramaniam and Subramanian (1996) in the gas bubble
limit. As noted earlier, the asymptotic migration velocity of a gas bubble is propor-
tional to j T jj rT1 j R. The striking contrast between the two results arises from the
fact that in the gas bubble problem, the thermal conductivity of a gas bubble is
assumed to be zero. The drop makes a demand for energy to increase its temperature
at a constant rate as it moves into warmer fluid at a constant velocity. This energy
must be supplied from the continuous phase to the contents of the drop. Within the
drop, heat flows by conduction across recirculating streamlines when viewed from the
perspective of the drop. Since conduction is weak, the temperature gradient required
to supply this heat is relatively large, leading to relatively low temperatures in the
vicinity of the drop and within it. The scaled temperature is predicted to be smaller by
a factor of Ma in the thermal boundary layers near the drop surface when compared
with the values far from the drop. It is smaller by another factor of Ma in the core of
the drop when compared with the values that prevail in the thermal boundary layers.
Recently, Balasubramaniam (1998) has analysed the motion of a gas bubble in the
limit Ma ! 1 in a liquid in which the viscosity varies linearly with temperature. The
 Fluid mechanics of bubbles and drops 157
analysis applies for large values of the Reynolds number, so that potential flow can
be assumed in the liquid. Balasubramaniam permitted a buoyant contribution to the
motion as well as a thermocapillary contribution. He concluded that in purely thermo-
capillary migration, the decrease in viscosity with temperature leads to a reduction in
the scaled velocity of the bubble when compared with the value that would be cal-
culated using an average viscosity at the temperature corresponding to the current
location of the bubble. The physical reason is that in any given plane perpendicular to
the direction of motion, the liquid near the bubble is cooler than that far away from it.
Thus, if one uses an average temperature based on the temperature field far from the
bubble, one under-estimates the viscosity of the liquid.
It is known that surfactants, which adsorb on the interface between the drop and the
continuous phase, can affect the motion of a drop. In gravitational settling, the effect
is to reduce the mobility of the interface and it is possible for sufficiently small drops to
move as though their surface is rigid, as noted by Bond (1927) and Bond and Newton
(1928). Surfactants have a larger impact on thermocapillary migration than on gravi-
tational settling because the entire driving force for thermocapillary motion resides at
the interface. When a drop moves, surfactant adsorbed at the interface is swept to the
rear of the drop. This leads to a concentration gradient of surfactant which causes the
interfacial tension to decrease as one moves from the front to the rear of the drop. In
thermocapillary migration, this opposes the interfacial tension gradient that arises
from the temperature difference. Theoretical developments for low Reynolds number
motion were developed by Kim and Subramanian (1989a,b) who used an idealized
linear equation of state for the surfactant. When the presence of an insoluble surfac-
tant is accommodated, a new dimensionless group, which can be termed the Elasticity
number E, is introduced in the analysis. It is defined as E ˆ RT0 /(v0 ), where, R is
the gas constant, T0 represents a reference temperature at the interface, and stands
for a reference concentration of the surfactant on the interface. The Elasticity number
provides a measure of the relative role of the surfactant in lowering the interfacial
tension, when compared with that of the varying temperature in altering the interfacial
tension. Also, a dimensionless gas constant  ˆ R /( T ) multiplies the product of the
scaled interfacial temperature and the scaled surfactant concentration on the interface
in an additive contribution to the scaled interfacial tension. When the surfactant
concentration at the interface is relatively small, the contribution from this non-linear
term can be neglected. Kim and Subramanian considered the case when surfactant
forms a stagnant cap (1989a) and a more general situation where surfactant is present
over the entire surface (1989b). Using a perturbation scheme, Nadim and Borhan
(1989) obtained the solution permitting the shape of the drop to be slightly deformed
from a sphere in the situation when the dependence of the interfacial tension on the
surfactant concentration is relatively weak. Shortly thereafter, Nadim et al. (1990)
calculated the correction to the migration velocity due to the shape distortion, using a
technique similar to that employed by Haj-Hariri et al. (1990) for calculating a similar
correction due to small inertial effects. Recently, Chen and Stebe (1997) have analysed
the thermocapillary motion of a drop in the presence of surfactants using more real-
istic models of surfactant adsorption permitting the possibility of monolayer satur-
ation and non-ideal surfactant interactions. They find that the Langmuir framework,
which accommodates monolayer saturation effects, leads to less retardation of drop
motion by a surfactant, when compared with the predictions from a linear model. This
is because the linear model permits adsorbed surfactant concentration to increase
158 R. S. Subramanian et al.
without limit, whereas the Langmuir framework places a maximum limit on the con-
centration. The authors also use the Frumkin framework, which takes interactions
among surfactant molecules into account in addition, to show that cohesion of the
molecules increases surface concentration gradients, leading to strong retardation
effects from the surfactant. Repulsion among the surfactant molecules has the oppos-
ite effect. Chen and Stebe point out that when the adsorption and desorption occur
rapidly when compared with surface convective transport, the interface can
be remobilized. Physically, this means that surfactant can be distributed nearly uni-
formly over the surface so as to make the interfacial tension gradient arising from the
surfactant concentration gradient negligible. This is accomplished by using elevated
bulk concentrations of the surfactant or by the addition of appropriate amounts of
a remobilizing surfactant. We note that reverse flow structures in the wake similar to
those found by Shankar and Subramanian (1988) are reported by Chen and Stebe
(1997), and they occur due to the same reason. Other interesting flow structures arise
when a drop moves under the combined action of gravity and thermocapillarity.
Details can be found in Merritt et al. (1993).
It is evident that considerable progress has been made in developing theoretical
predictions in the case of an isolated drop. Theoretical advances also have been made
in the context of interactions between a pair of drops or a drop and a boundary and
more recently on the thermocapillary motion of a collection of drops. Virtually all of
the theoretical work has been restricted to the limit Re ! 0 and Ma ! 0 so that a
linear problem can be posed and solved. Most of the available results are restricted to
spherical drops. This is implied in the discussion of the literature here, and we expli-
citly point out when shape deformation is accommodated by the authors. In the case
of two drops or a drop and a plane boundary, axisymmetric solutions of Laplace's and
Stokes's equation in bispherical coordinates obtained by Jeffery (1912) and Stimson
and Jeffery (1926), respectively, were specialized in the gas bubble limit by Meyyappan
et al. (1981; 1983). A more general solution applicable to asymmetric situations, pro-
vided by O'Neill (1964), was used by Meyyappan and Subramanian (1987) in ana-
lysing the case of a gas bubble moving in an arbitrary direction with respect to a plane
surface. A solution for two drops obtained using the method of reflections is given by
Anderson (1985) who also calculated a result for a suspension of equal-sized drops.
Results for a liquid drop moving normal to a plane surface are given by Barton and
Subramanian (1990) and Chen and Keh (1990); Barton and Subramanian (1991)
subsequently conducted experiments which confirmed the correctness of their predic-
tions for the migration velocity. Recently, Chen (1999a,b) has provided approximate
solutions of problems involving the thermocapillary motion of a drop near a surface
by the method of reflections. The case of a deformable liquid drop moving normal to a
plane surface was solved numerically by Ascoli and Leal (1990). Morton et al. (1990)
analysed the thermocapillary motion of a compound drop which implies a droplet
present within a drop. The drop migrates in the applied temperature gradient, and
the droplet moves within the drop. The authors considered both the concentric con-
figuration and a more general eccentric configuration, but restricted the analysis to
axisymmetric fields. Borhan et al. (1992) analysed the axisymmetric motion of a
concentric compound drop accommodating the influence of surfactants, and also
obtained corrections to the spherical shapes of both the drop and the droplet, for small
values of the Capillary number. Keh and Chen (1990) obtained results for interactions
between a pair of drops immersed in a continuous phase. The case of three bubbles or
 Fluid mechanics of bubbles and drops 159
drops aligned in a chain was solved by Keh and Chen (1992; 1993) and Wei and
Subramanian (1993) who used the solution for a single sphere given in Lamb (1932)
and satisfied the boundary conditions on selected collocation rings. Nas (1995) solved
model problems involving a collection of several drops using a full numerical solution
of the governing equations, and accommodating shape deformation. The motion of
a collection of bubbles of identical size was analysed by Acrivos et al. (1990); subse-
quently the treatment was extended to a bidisperse collection by Wang et al. (1994).
The collision and subsequent coalescence of drops undergoing thermocapillary
migration leads to an evolution of the size distribution with time. This subject is dis-
cussed in Satrape (1992), Zhang and Davis (1992), and Zhang et al. (1993). When
a drop and a surface are very close, or when two drops are very close to each other,
lubrication theory can be used to make predictions as shown by Loewenberg and
Davis (1993a,b). In a variation on the theme, Golovin (1995) analysed the motion of
a rigid sphere and a gas bubble, which arises because of a temperature difference
between the rigid sphere, assumed to be at a fixed temperature, and the continuous
phase, which is isothermal in the undisturbed state. The resulting temperature gradient
in the fluid causes thermocapillary migration of the bubble, and a drift of the rigid
sphere in the resulting flow. A similar analysis is presented by Leshansky et al. (1997)
for the case of a rigid sphere placed near a fluid interface. In a recent article, Lav-
renteva et al. (1999) have treated the influence of unsteady and convective transport
effects on the Stokes motion of a pair of drops that is driven by interfacial tension
gradients. These gradients arise as a result of non-uniform transport of a surfactant
between the drops and the continuous phase, in which the concentration of surfactant
is uniform in the undisturbed state. This is the only article at the time of this writing in
which both the unsteady and convective transport effects are treated in a problem
involving interacting drops. We also note that Balasubramaniam and Subramanian
(1999) have analysed the axisymmetric motion of two bubbles in a temperature gra-
dient when convective transport effects are asymptotically large, both in the momen-
tum transport and in the energy transport problem. Subject to the assumptions made
by the authors, the leading bubble is unaffected, whereas the trailing bubble moves less
rapidly than it otherwise would. While this particular analysis was motivated by our
experiments in reduced gravity, it will be seen from subsequent sections that experi-
ments have generally lagged behind theory in the area of interactions among drops.

6.4 EXPERIMENTS ON THE GROUND

The first experiments on the thermocapillary motion of bubbles were performed by

Young et al. (1959) as mentioned in the introduction to the previous section. Young
et al. held bubbles nearly stationary in a downward temperature gradient. From
their theory, they predicted that the temperature gradient needed to hold a bubble still
would be proportional to its radius, and independent of the viscosity of the liquid. In
spite of the uncertainties associated with their measurements, they indeed verified
these predictions. Interestingly, it can be shown that it would be difficult to hold a gas
bubble still in a liquid by applying a downward temperature gradient. While a bubble,
which neither dissolves nor grows, can be placed at a certain location where the forces
on it are balanced, slight departures from that location can cause it to move away.
Whether the bubble actually does so will depend upon the physical properties of the
160 R. S. Subramanian et al.
liquid and the magnitude of the applied temperature gradient. Dissolution or growth that
invariably occurs because the liquid can be saturated with the gas at only one tempera-
ture, also would make it impossible to hold a bubble still. In the experiments of Young
et al. the bubbles moved so slowly that the authors considered them practically
stationary. Hardy (1979) refined the experiment, using a single bubble at a time, and
found the bubbles to move up and down in the cell in a cyclic fashion. From the turning
points where the bubble changed direction, assuming the prediction made by Young
et al. to be correct, he evaluated the coefficient T for a silicone oil and found it to be
in good agreement with the value he measured using the pendant drop technique. The
linear scaling of the migration velocity with the radius was verified in the case of gas
bubbles by Merritt and Subramanian (1988) who performed experiments on air bubbles
moving under the combined influence of buoyancy and a downward temperature gra-
dient in silicone oils. Later, the linear dependence of the migration velocity upon the
radius as well as that on the applied temperature gradient was confirmed by Barton and
Subramanian (1989) who observed the motion of ethylsalicylate drops in diethylene
glycol in an upward temperature gradient. These authors found, however, that the values
of the coefficient, T , evaluated from the migration experiments, were lower than the
values obtained from static Du NouÈy Ring measurements made by others. We also note
that Morick and Woermann (1993) reported experimental results on the motion of gas
bubbles in a vertical temperature gradient which are consistent with the earlier obser-
vations made by Hardy and Merritt and Subramanian.
Very little has been done experimentally on the role of surfactants in influencing
thermocapillary migration. Young et al. (1959), in their original experiments, demon-
strated the important role of surfactants by showing that they could not make bubbles
move downward in n-hexadecane by using a downward temperature gradient when
trace amounts of a silicone oil, which is a surfactant in that liquid, were added. Fol-
lowing their example, Barton and Subramanian (1989) demonstrated the role of
thermocapillarity in their experiments by deliberately adding a surfactant, Triton
X-100, to both phases to suppress the thermocapillary contribution. Nallani and
Subramanian (1993) reported that methanol drops moved in a silicone oil at velocities
lower than those predicted by assuming an uncontaminated interface, and fitted their
data to the stagnant surfactant cap model given in Kim and Subramanian (1989a).
Work on the ground on moving drops has been mostly confined to the linear limit
when Re and Ma are both negligible. This has been for two reasons. One is that the
superposition principle permits us to extract the purely thermocapillary contribution
to the velocity of a drop under these conditions. As an example, Merritt and Sub-
ramanian (1989) inferred that, in Stokes flow, the thermocapillary interaction with a
boundary is much weaker than the interaction that occurs when motion is driven by
gravity. They were able to decouple the two contributions because the experiments
were performed in the linear limit. The other reason for using small values of Re and
Ma is to minimize interference from gravitational effects. To obtain Reynolds and
Marangoni numbers that are not small, one must use relatively large bubbles and
drops and apply a sufficiently large temperature gradient. The relative importance of
the gravitational force on a drop increases with the size of the drop. When a downward
temperature gradient is used, due to the resulting unstable density stratification, the
maximum temperature gradient that can be employed is limited by the onset of cellular
convection in the continuous phase. Even when an upward temperature gradient is
employed, which leads to stable stratification, increasing the temperature gradient
 Fluid mechanics of bubbles and drops 161
leads to problems. This is because of the resulting increased temperature contrast
between the experimental test cell and the surroundings. This increases horizontal
temperature gradients, and therefore, horizontal density variations, leading to more
buoyant convection in the test cell.
Much of the work on the ground on relatively large drops has been conducted on
liquid pairs of nearly matched densities in which a drop can be held motionless under
the action of a temperature gradient. Therefore, the gravitational and thermocapillary
effects on the drop are balanced. It is possible to achieve substantial values of the
Reynolds and Marangoni numbers on the ground in this case. In the experiments,
an upward temperature gradient is applied and a drop is inserted into the continuous
phase. The liquids are chosen such that the coefficient of thermal expansion of the
continuous phase is larger than that of the drop phase. At some temperature, the drop
phase has exactly the same density as that of the continuous phase. At larger tem-
peratures, the drop phase is more dense. Drops migrate above the level of matched
densities and find a stationary location. At this location, the net downward force due
to gravity, which is the weight of the drop minus the buoyant force, is precisely
balanced by the upward hydrodynamic force exerted on the drop due to the motion
induced by the interfacial tension gradient. This equilibrium is stable to small vertical
disturbances. While the drop itself is stationary, there is considerable motion, espe-
cially in the vicinity of the interface. These types of experiments have been performed
by Wozniak (1986), HaÈhnel et al. (1989), Rashidnia and Balasubramaniam (1991),
Chen et al. (1997), and most recently Ma (1998), using a variety of immiscible pairs of
fluids. All the studies report information on the ultimate stationary location of the
drop relative to the matched density level, for drops of various sizes and for different
applied temperature gradients. In addition, Wozniak obtained interferometry images
which permitted him to describe the temperature field around the drop. Rashidnia and
Balasubramaniam used tracer particles to follow the motion. Using this technique,
they inferred the magnitudes of the velocity at the interface and clearly demonstrated
the role of thermocapillarity in driving the observed motion in both fluids. Chen et al.
used the same system as HaÈhnel et al. but studied the role of an added surface-active
chemical which reduced the distance at which the drop was located above the neutral
density point, depending upon its concentration. As discussed in Section 6.3, a gradient
in surfactant concentration on the interface, which arises from interfacial motion,
leads to an interfacial tension gradient which opposes that caused by temperature
variation. This is the reason for the observations reported by Chen et al. Ma 1998
made a comparison of his observations on two pairs of fluids with a solution of the
governing conservation equations and boundary conditions obtained by numerical
means, and presented the results in a suitable dimensionless form. He reported reason-
able agreement between his observations and predictions.
In the only available study on interactions between a pair of bubbles moving under
the combined action of gravity and thermocapillarity, Wei and Subramanian (1994)
demonstrated that the predictions from the method of reflections were sufficient to
describe their observed results. It was possible to superpose predictions for motion
driven by a body force with those for purely thermocapillary motion because the
Reynolds and Marangoni numbers were negligibly small, making the theoretical
problems linear. The authors also commented that a large bubble moving downward
in a vertical temperature gradient can cause a neighbouring small bubble to move
upward. If the large bubble were not present, the small bubble would move downward
162 R. S. Subramanian et al.
in the same temperature gradient. This interesting reversal of direction of the small
bubble is due to the upward thermocapillary flow generated by the large bubble in its
vicinity. Wei and Subramanian (1995) subsequently provided a detailed discussion of
flow structures that arise when two bubbles move under the combined action of
buoyancy and thermocapillarity.
Finally, we note that when a drop immersed in a liquid or surrounded by a gas is
allowed to approach a solid or fluid surface, the application of a temperature differ-
ence between the drop and the surface can prevent the drop from spreading over the
surface. This is a consequence of the presence of a thin film of fluid between the drop
and the surface, which is entrained by thermocapillary flow in the drop. This topic is
considered in detail by Castagnolo and Monti (2001) in Chapter 4. The authors discuss
both experimental work and theoretical models.

6.5 EXPERIMENTS IN REDUCED GRAVITY

After the appearance of the initial article by Young et al. very little was published on
thermocapillary migration for approximately 15 years. The literature on the subject
began to grow in the seventies and eighties mainly because of the impetus provided by
the space programme. Research was conducted on the ground as much as possible, but
suffered from interference caused by gravitational effects. Therefore, experiments in
reduced gravity were initiated beginning in the late seventies in the NASA sounding
rocket program named SPAR which provided about 5 min of reduced gravity condi-
tions. The first experiments were performed by Papazian and Wilcox (1978) who
correctly surmised that thermocapillary motion can be an important mechanism for
managing bubbles in crystal growth in reduced gravity. They proceeded to study the
behaviour of bubbles at a solidification interface in carbon tetrabromide aboard a
sounding rocket flight. Unfortunately, the bubbles did not move in the temperature
gradient and the authors advanced some possible explanations. The most likely cause
would have been the presence of some surface-active contaminant which adsorbed on
the interface. In later experiments conducted aboard similar sounding rocket flights,
Wilcox and coworkers (Smith et al. 1982; Meyyappan et al. 1982) indeed observed the
motion of gas bubbles in the direction of a temperature gradient in a sodium borate
melt.
Thompson (1979) conducted a systematic study of the motion of bubbles of nitrogen
in different liquids in the drop tower at NASA Lewis Research Centre (now Glenn
Research Centre), and the results are reported in Thompson et al. (1980). The reduced
gravity experiment time was approximately 5 s. Thermocapillary migration could not
be observed in water, likely because of surfactant contamination effects. Migration
velocities in ethylene glycol were consistent with the predictions of Young et al. given
in equation (6.5), even up to a Reynolds number of 5.66 and Marangoni numbers as
large as 713. This is a puzzling observation. However, bubbles in ethanol and a silicone
oil were observed to move at smaller velocities than those given in equation (6.5).
Given the conditions of Thompson's experiments, one must conclude that the non-
linearity of the temperature profile, coupled with the initial transients, must have had
significant impact on the measured velocities of the bubbles.
Langbein and Heide (1984) performed experiments on sounding rockets using
a binary liquid mixture of cyclohexane and methanol in which drops were observed to
 Fluid mechanics of bubbles and drops 163
move in the direction of the temperature gradient. Shortly thereafter, experiments were
conducted aboard the D-1 mission of the NASA space shuttle by Siekmann and col-
leagues, and the results are reported in NaÈhle et al. (1987) and Szymczyk et al. (1987).
The authors injected a collection of air bubbles into a silicone oil in one cell, and a
collection of water drops into the same silicone oil in a second cell. A temperature
gradient was applied to the silicone oil and it evolved with time during the experiment.
The water drops did not move while the air bubbles moved in the direction of the
applied temperature gradient. The experimental data on the velocity of air bubbles up
to a value of Ma ˆ 288 were presented in Szymczyk et al. and shown to be consistent
with predictions from a numerical solution of the governing equations.
Aboard the same D-1 mission of the space shuttle, another experiment on thermo-
capillary migration was carried out by Neuhaus and Feuerbacher (1987). These
authors used three similar silicone oils as the continuous phase. Air bubbles in AK100
silicone oil moved at velocities consistent with the predictions of Young et al. whereas,
bubbles moved at velocities smaller than those predicted in silicone oil AS100, and did
not move at all in silicone oil AP100. No information is available on the Reynolds or
Marangoni numbers corresponding to these experiments. Based on results from addi-
tional ground-based experiments on gravitational rise, the authors suggest that one
must account for dissipation processes at the interface through an interfacial dilational
viscosity, but it is not clear why this particular hypothesis is put forward.
Wozniak (1991) conducted experiments aboard sounding rockets on drops of par-
affin oil moving in a solution of ethanol and water, the same pair on which he had
conducted studies on the ground earlier. Approximately 7 min of reduced gravity time
was available. The drops ranged in radius from 0.7 to 2.4 mm and a temperature
gradient of 0.8±0.9 K/mm was used. The maximum Reynolds and Marangoni numbers
in the continuous phase were approximately 25 and 588, respectively. Predictions were
made from a numerical solution of the governing equations. Wozniak found that the
measured velocities were only a fraction of the predicted values, and ranged from
approximately 3.6±31.7% of the predicted values with the worst agreement being dis-
played for the smallest drop. He attributes the discrepancy to possible interactions
with the boundaries and to the influence of surface-active contaminants even though
every effort was made to work with very pure liquids. Braun et al. (1993) performed
experiments aboard a sounding rocket that provided approximately 5.5 min of
experiment time in reduced gravity conditions using a mixture of 2-butoxyethanol and
water, which exhibits an inverted miscibility gap. Drops of a mean diameter of 11 mm,
rich in 2-butoxyethanol, were observed to move toward cooler fluid in a temperature
gradient. This direction of movement was consistent with the sign of the coefficient,
T , which is reported to be positive for this system; that is, the interfacial tension
increases with increasing temperature. The Marangoni numbers ranged from 10 5 to
10 6 and the Reynolds numbers were even smaller. The migration velocities were
found to be consistent with predictions from equation (6.5). Experiments in another
system, which exhibits a positive value of T over a certain range of temperatures, were
carried out aboard the space shuttle in 1994 and 1996 by Viviani and Golia (1998). The
authors used a solution of n-heptanol in water which exhibits a surface tension min-
imum around 40  C, according to measurements made by PeÂtre et al. (1983). Air bub-
bles injected into this solution were observed to move from a region near the hot wall
toward cooler regions in the test cell. The bubbles came to rest at a location where the
temperature was estimated to be approximately 8±10  C, near the cold wall which was
164 R. S. Subramanian et al.
held at 5  C. The temperature estimate was based on an assumed linear temperature
distribution between the hot and cold walls. As noted, the static measurements reveal a
minimum in surface tension to occur in the neighbourhood of 40  C in this aqueous
solution. This implies a positive value of T at temperatures larger than approximately
40  C, and a negative value of T at lower temperatures. Therefore, the bubbles should
have moved from the hot wall toward cooler regions until this temperature was
reached and should have stopped at that location. They continued to move, however,
into cooler regions in a direction opposite to that which would have been predicted
from theory for an isolated bubble. The observation remains unexplained. Since the
temperatures in the liquid were not measured directly, but inferred from an assumed
linear distribution, there is some question about the actual temperature prevailing in
the region where the bubbles came to rest, but the discrepancy is unlikely to be so large
as to explain the observation.
Recently, Treuner et al. (1996) conducted thermocapillary migration experiments in
the drop tower in Bremen, Germany, which provides approximately 4.7 seconds of low
gravity conditions. Bubbles of air, varying from 0.5 mm to 2 mm in diameter, were
found to move in a temperature gradient in three liquids, n-octane, n-decane, and
n-tetradecane. Much of the data are in the transient regime and Treuner et al. provide
several plots of data on the velocities of the bubbles as well as an interferometry image
that displays refractive index distributions around the moving bubbles. The observed
velocities were consistent with those predicted from a numerical solution. As a bubble
moved into warmer fluid, however, the Marangoni number for the bubble changed
with time, as did the scaled velocity of the bubble. This variation of the scaled velocity
of an individual bubble with the Marangoni number was quite often in a direction
opposite to that predicted by a quasi-steady numerical solution.
We conclude this section with a brief discussion of our own results from reduced
gravity experiments conducted aboard the NASA space shuttle Columbia in two ser-
ies, the first in summer 1994, and the second in summer 1996. The experiments in 1994
were included in the International Microgravity Laboratory-2 (IML-2) mission and
those in 1996 were part of the Life and Microgravity Spacelab (LMS) mission. We
have described the apparatus, procedure, and results in detail in Balasubramaniam
et al. (1996) and Hadland et al. (1999), and only a summary is provided here.
The apparatus, known as the Bubble, Drop, Particle Unit (BDPU), was designed and
built under the auspices of the European Space Agency (ESA), and made available for
our use through a cooperative arrangement with NASA. The BDPU consisted of
a facility that provided power, optical diagnostics and illumination, imaging facilities
including a video camera and a motion picture camera, and other support services such
as heating and cooling. Different test cells could be inserted into the BDPU facility by the
astronauts when needed. We used two identical test cells that were filled with a silicone
oil. One test cell permitted the injection of air bubbles, while the second test cell was
equipped to inject drops of Fluorinert FC-75. The cells were of rectangular cross-
section, 60 mm long and 45  45 mm square in cross-section. The continuous phase
used in the IML-2 mission was a DC-200 silicone oil of nominal viscosity 50 cs,
whereas, that used in the LMS mission was a silicone oil of nominal viscosity 10 cs.
During injection or extraction of a bubble or drop, the test cell was connected to
mechanical systems that ensured the compensation of the volume of the bubble or
drop. In each experiment, a steady and uniform temperature gradient was established
at first over a period of 2 h. Then bubbles and drops of the desired size were injected,
 Fluid mechanics of bubbles and drops 165
and their traverses were captured on videotape. Images in selected runs were recorded
on motion picture film. The bubbles and drops ranged in radius from approximately
0.5 mm to 8 mm. It was found that the bubbles and drops traversed the cell sufficiently
rapidly for size change to be negligible. In evaluating the velocity of an object from the
video or cine film, data on position vs. time near the middle of the cell, away from the
end walls, were used in order to minimize the effects of interaction with these boundaries.
In our flight experiments, the Marangoni number in the continuous phase ranged
from 5.4 to 5 780 in the case of air bubbles, and from 2.5 to 3 700 for the migration of
Fluorinert drops. The corresponding Reynolds number ranges were 0.0096±87.2 for
bubble runs, and 0.0045±49.1 in the case of drops. The Prandtl number for the silicone
oil used in the IML-2 experiments ranged from 371±567 while the range was 59.4±92.9
in the LMS experiments. The Reynolds and Marangoni numbers were not varied
independently because, the Prandtl numbers of the fluids did not change by much from
one run to another. Therefore, the Reynolds and Marangoni numbers virtually
tracked each other. In Figure 6.1, the scaled velocity of air bubbles, from the IML-2
and the LMS experiments, is displayed against the Marangoni number. The velocity is
scaled with the value it would have if convective transport of momentum and energy
played a negligible role, namely, VYGB . In calculating the values of VYGB and v0 , the
local value of the viscosity at the instantaneous location of the bubble was used.

1.2
LMS: ∇ Τ ∞ = 0.33 K/mm
1.0 LMS: ∇ Τ ∞ = 1.0 K/mm
Scaled velocity v/vYGB

IML-2
0.8 Numerical Solution
Asymptote
0.6 Ma → ∞, Re → ∞

0.4

0.2
Asymptote
Ma → ∞, Re → 0
0.0
1 10 100 1000 104
Marangoni number Ma
Figure 6.1 Scaled velocity of air bubbles migrating in a DC-200 silicone oil, plotted
against the Marangoni number; data from the LMS and IML-2 experiments
aboard the space shuttle are shown, along with predictions from a numerical
solution and two asymptotic solutions; reproduced from Hadland et al. (1999).
Thermocapillary migration of bubbles and drops at moderate to large
Marangoni number and moderate Reynolds number in reduced gravity.
Experiments in Fluids 26, 240±248, with permission from Springer-Verlag
GmbH & Co. KG.
166 R. S. Subramanian et al.
During the course of an experiment, as a bubble moved into warmer liquid, it accel-
erated. Therefore, the value of the scaled velocity, as well as that of the Marangoni
number, changed during the traverse. In the LMS runs, we have displayed this vari-
ation by using a set of data points for each bubble. These data were obtained for
approximately 10 mm of traverse in the middle region of the cell between the hot and
cold walls. In the IML-2 experiments, only a single data point was obtained for each
bubble. In Figure 6.1, a numerical solution of the governing equations obtained by Ma
(1998) is included for comparison, along with the asymptotic results in the limit
Ma ! 1 for Re ! 0 and for Re ! 1. The former is just a constant equal to 0.3076
and the latter is calculated from equation (6.6). It is evident from the figure that the
results from the flight experiments are generally consistent with the numerical pre-
dictions. Also, even though the Prandtl number of the silicone oil in the IML-2
experiments was approximately six times larger than that in the LMS experiments, the
scaled velocities are not very sensitive to the variation in the Prandtl number. This is
because the Prandtl number is large in both series of experiments. The results appear
to be approaching the asymptotic prediction for small Reynolds number as the Maran-
goni number becomes large. It is interesting that the prediction for large Reynolds
number does remarkably well when compared with the experimentally observed

1.4
LMS: ∇ Τ ∞ = 0.25 K/mm
1.2 LMS: ∇ Τ ∞ = 1.0 K/mm
Scaled velocity v/vYGB

IML-2
1.0 Numerical Solution

0.8

0.6

0.4

0.2

0.0
1 10 100 1000
Marangoni number Ma
Figure 6.2 Scaled velocity of FC-75 Fluorinert drops migrating in a DC-200 silicone oil,
plotted against the Marangoni number; data from the LMS and IML-2
experiments aboard the space shuttle are shown, along with predictions from
a numerical solution; reproduced from Hadland et al. (1999). Thermocapillary
migration of bubbles and drops at moderate to large Marangoni number and
moderate Reynolds number in reduced gravity. Experiments in Fluids 26,
240±248, with permission from Springer-Verlag GmbH & Co. KG.
 Fluid mechanics of bubbles and drops 167
velocities. We suspect that if the next term in the expansion for large Ma can be cal-
culated in the limit Re ! 0, that prediction would do well when compared with the
data. Finally, it is worth noting the trend of the scaled velocity vs. Marangoni number
for a given bubble during its traverse, which is opposite to that expected from the
quasi-steady predictions. This is consistent with the observations reported by Treuner
et al. (1996) from their drop tower experiments.
The results for Fluorinert FC-75 drops smaller than approximately 2.5 mm in radius
are plotted in Figure 6.2 along with a numerical solution obtained by Ma (1998).
Again, data are displayed from both the IML-2 and the LMS experiments. The LMS
data for each drop consist of a set of points corresponding to a traverse of approxi-
mately 10 mm in the middle of the cell, just as in the case of air bubbles. In the IML-2
runs, only a single data point was obtained for each drop. The data are consistent with
the numerical prediction up to about Ma ˆ 90. Beyond that value of Ma, the scaled
velocities continue to decrease with increasing values of the Marangoni number, while
the theoretical prediction is for them to increase. The experimental data for relatively
large drops, not included in Figure 6.2, fell in the range of values of Ma from 1 300 to
3 700. They showed a lack of sensitivity of the scaled migration velocity to change in
the value of Ma in this range. It was concluded, however, that these data corresponded
to a regime where the velocity of the drop was quite far from steady state. So, no
meaningful comparison with the quasi-steady solution could be attempted for the data
on these large drops. We also found that even the largest drops displayed no measurable
deformation from a spherical shape. The largest bubbles, of radius larger than 6 mm, for
which the Weber number exceeded unity, were slightly deformed, becoming oblate
spheroids. These conclusions are based on a view of the objects from a single direction.

6.6 FUTURE PROSPECTS

In this section, we highlight some of the gaps in our knowledge about thermocapillary
migration identifying potential experiments in reduced gravity as well as areas of
advancement in theoretical research. The most interesting experimental problems are
those involving two or more bubbles or drops. Very little has been done on this subject
in reduced gravity. In our IML-2 flight experiments, we had an opportunity to study
the axisymmetric interaction between a pair of drops. Several pairs of drops were
formed in which the leading drop was smaller than the trailing drop. We noted that the
leading small drop moves virtually as though it is isolated, but that the trailing large
drop is drastically slowed by the interaction with the leading drop. We attributed this
to the effect of the thermal wake left by the leading drop in which the temperature
gradient is weakened. This fluid wraps around the interface of the trailing drop and
therefore reduces the driving force for its motion. As noted earlier, Balasubramaniam
and Subramanian (1999) have recently analysed the motion of two gas bubbles of
negligible viscosity and thermal conductivity, in the limit Re ! 1 and Ma ! 1, and
predicted precisely such an effect. However, the analysis is not directly applicable to
the experiments because, the Reynolds number in the experiments was relatively small,
and they were conducted using liquid drops whose conductivity cannot be considered
negligible. In the LMS flight, we observed the same phenomenon in some runs, but
noticed even more interesting physical behaviour in others. Often, we found that a
large trailing bubble or drop would move away from the axis of the cell. Its velocity
168 R. S. Subramanian et al.
toward the hot wall would increase, and in some cases, the trailing drop would pass the
smaller leading drop. A likely cause of this is an instability of the motion of the trailing
drop in the wake left by the leading drop. If a natural disturbance leads to a slight
displacement of the trailing drop from the centreline of the cell along which it was
injected, this instability would result in its moving further away from the centreline.
When the trailing drop has moved a sufficient distance away from the axis, the
interaction with the thermal wake of the leading drop would be weak. The hydro-
dynamic interaction with the leading drop is likely to be weak as well, so long as the
separation distance is at least three times the radius of the leading drop. Being unin-
fluenced by the leading drop, the trailing drop would move more rapidly, as though it
were isolated. Since the velocity of isolated drops increases with their size, the trailing
drop can pass the leading drop when there is sufficient time during the traverse to do it.
Unfortunately, this phenomenon could not be explored systematically in the LMS
experiments, and it will need to be studied in detail in a future flight experiment.
Another interesting observation we made in a very small number of runs involved a
chain of drops. These were typically introduced into the cell by accident, usually
during the first attempt to inject a drop. These drops executed a 3D trajectory through
the cell. Unfortunately, only one view could be captured on video. In this view, the
trajectory of each drop across the cell appears as an undulating pattern, not quite
sinusoidal, but qualitatively similar. Also, these spatial oscillations in position take the
drop away from the axis. Interferometry images recorded from an orthogonal direc-
tion reveal a similar pattern, even though making precise measurements from this view
is not possible. Therefore, we are led to conjecture that the drops did not proceed in
a straight path along the axis of the cell, but rather followed a spiral path. This
phenomenon remains unexplained and unexplored at this time.
We note that no experiments have been carried out on a well-characterized collec-
tion of drops moving in a continuous phase under the influence of a temperature
gradient. Predictions have been made for monodisperse and bidisperse collections of
this type by Acrivos and co-workers, but these remain to be tested. Also, predictions
have been made by Zhang et al. (1993) regarding the evolution of the size distribution
of a cloud of drops undergoing thermocapillary migration. These authors consider
pairwise interactions in a dilute suspension and predict how the size distribution
changes with time as the drops grow by coalescence. It would be interesting to perform
controlled experiments in which the size distribution is measured experimentally in
situ, perhaps by optical techniques.
The ground-based experiments of Merritt and Subramanian (1989), and later Bar-
ton and Subramanian (1991), have established the correctness of the predictions from
the low Reynolds number theory for the motion of a drop normal to a plane surface.
No experimental data have been reported for motion parallel to a plane surface, or in
the more general case of motion in an arbitrary direction with respect to the surface. It
would be useful to perform experiments on drops moving near boundaries to gauge
the effects of the boundaries on thermocapillary migration when convective transport
of momentum and energy are not negligible. In this case, the problem is inherently
unsteady. Therefore, theoretical predictions, while straightforward to make in prin-
ciple, are difficult to obtain because of the need for numerical solution of the governing
equations. Of course, the experiments would have to be done in reduced gravity
because, the problems are non-linear and the gravitational contribution in ground-
based experiments cannot be simply decoupled.
 Fluid mechanics of bubbles and drops 169
Eventhough a number of experiments have been performed in reduced gravity on
isolated bubbles and drops, much work remains to be done in this area. Recall that the
prediction for large values of Ma in the case of drops is that the scaled velocity should
increase with increasing values of the Marangoni number. In physical terms, for a
fixed set of property ratios, this implies that the physical velocity must be proportional
to the cube of the radius of the drop and the square of the applied temperature gra-
dient. In contrast, for a gas bubble of negligible thermal conductivity, the prediction
at large values of the Marangoni number is for the velocity to depend on the first
power of the radius and the applied temperature gradient, the same as the prediction
at negligible values of Ma. The theoretical result for drops is so remarkable that
experiments should be performed to verify it. The main difficulty here is in achieving
quasi-steady conditions within the drop. One needs to use a drop of sufficiently large
thermal diffusivity that conduction across streamlines within the drop can be expected
to occur rapidly. This was not possible in the IML-2 and LMS flight experiments
because the thermal diffusivity of Fluorinert FC-75 was too small. A liquid metal is a
good choice for the drop phase in such experiments.
In certain applications, such as in metallurgy, drops will change size rapidly during
the course of their migration. Dill (1991) constructed a theoretical description of the
motion of an isolated drop that changes size in the case when the Reynolds and
Marangoni numbers are negligibly small. He defined a dimensionless parameter that
contrasts the rate of size change with the typical velocities resulting from thermo-
capillary motion. For small values of this parameter, he predicted that the size change
should have no effect on the migration velocity because it merely produces an additive
potential flow that is radially directed. This is consistent with the experimental
observations of Merritt and Subramanian (1988). No predictions are available, how-
ever, for situations where the assumptions of Dill's analysis are relaxed. It would be
useful to perform thermocapillary migration experiments on drops and bubbles that
dissolve or grow at rapid rates under reduced gravity conditions. Such observations
would be extremely useful in extending the present theoretical descriptions to accom-
modate mass transfer situations.
Earlier, we noted the limited nature of the available experimental results involving
the role of added surfactants on thermocapillary migration. The fluids used for
studying thermocapillary migration were carefully selected to avoid contamination by
trace impurities that act as surface-active agents. This is the primary reason for the
choice of silicone oils as the continuous phase in numerous studies since contaminants
in these liquids appear to stay in bulk solution instead of adsorbing at the air±silicone
oil interface. The second reason is that silicone oils in a given family are available in a
wide range of viscosities, while the other physical properties remain virtually constant.
If thermocapillary migration is to be used as a practical tool in applications in the
reduced gravity environment, a variety of liquids must be used as the continuous and
dispersed phases and some contamination by surfactants is inevitable. Therefore,
studies of the role of surfactants in influencing thermocapillary migration must be
performed in reduced gravity. For this purpose, a well-characterized surfactant should
be used so that the physical parameters associated with surfactant adsorption, desorp-
tion, and transport in the bulk as well as on the interface, would be known. Then,
thermocapillary migration experiments can be performed, in which the concentration
of this surfactant in each phase is varied systematically, and the results used to validate
and improve the theoretical descriptions.
170 R. S. Subramanian et al.
Temperature gradients naturally arise at the surface of a translating drop in an
otherwise isothermal fluid. The physical basis of this phenomenon, first analysed by
Harper et al. (1967), is as follows. Elements of area of the interface are growing in the
forward half of the drop surface as they move toward the equatorial region. The
internal energy that is needed is provided by the neighbouring fluid which is cooled as
a consequence. The reverse occurs in the rear half. The resulting interfacial tension
gradient leads to a thermocapillary stress that opposes the motion of the drop. Harper
et al. were interested in establishing whether this phenomenon could explain the fact
that small drops and bubbles translating through a second fluid, encounter more
resistance than that predicted by theory. They concluded that the effect was too small
in common fluids, and that the explanation provided by Frumkin and Levich (1947),
based on the adsorption of surface active contaminants, is the correct one. The phe-
nomenon analysed by Harper et al. is indeed important in certain fluids. This was
recognized by Torres and Herbolzheimer (1993), who pointed out that this effect can
lead to macroscopic temperature gradients when a swarm of bubbles rises in an
otherwise isothermal liquid. The same phenomenon will reduce the thermocapillary
migration velocity of a drop, as noted by these authors. Its relative importance is
characterized by a dimensionless group Es ˆ (es )T /(k), where es stands for the
interfacial internal energy per unit area. When this group is of the order unity or larger,
the effect is predicted to be important. Examples of fluids in which Es is sufficiently
large for the effect to be pronounced, cited by Torres and Herbolzheimer (1993), are
cyclopentanone, dimethylphenyl carbinol, abietic acid, and even water at elevated
temperatures, as well as liquid nitric oxide, methane, and carbon dioxide. No results
from thermocapillary migration experiments in these fluids have been reported that
confirm the predictions of even the linear theory, which applies when convective
transport effects are negligible. Experiments should be performed on gas bubbles
migrating in these fluids under the action of a temperature gradient, both on the
ground, and in reduced gravity in the long run.
It is possible for a drop, initially stationary in isothermal surroundings, to move due
to the action of a uniform source of heat within the drop, or uniform generation of
heat at the surface of the drop due to chemical reaction, as first pointed out by Rya-
zantsev (1985). Normally, the source of energy would lead to a uniform temperature
on the surface of the drop, which can be different from that of the undisturbed con-
tinuous phase, resulting in no motion. However, a disturbance that causes a slight
movement of the drop will lead to a temperature variation on the surface of the drop,
because of non-uniform heat transport between the drop and the surrounding fluid
arising from the action of the convective transport terms. Under the right conditions,
the drop will continue to move in the same direction due to the action of the thermo-
capillary stress, sustaining the motion indefinitely. Since the appearance of the original
work by Ryazantsev (1985), a variety of situations have been analysed. Examples can
be found in Golovin et al. (1986), and Rednikov et al. (1994a). A review is provided by
Rednikov et al. (1994b). No experiments have been reported in which these predictions
have been tested. It would be appropriate to perform such experiments under low
gravity conditions so that interference from gravitational effects can be minimized.
We conclude by noting that the achievement of true quasi-steady conditions may
very well be impossible for large values of the Reynolds and Marangoni numbers if the
viscosity (or any other relevant physical property, for that matter) varies strongly
with temperature. Therefore, experimental measurements must be evaluated against
 Fluid mechanics of bubbles and drops 171
predictions made from a consideration of the fully transient problem accommodating
the dependence of physical properties on temperature. Such predictions have yet to be
made. This is the most important theoretical problem that needs to be addressed for an
isolated drop. Of course, a variety of problems involving size change, interaction of
drops with each other and with boundaries, and the behaviour of a collection of drops,
remain to be solved under conditions when the Reynolds and Marangoni numbers
assume that which are not negligible.

ACKNOWLEDGEMENTS

The research efforts of R.S. Subramanian and R. Balasubramaniam on thermocapillary

migration, including work on two flight experiments aboard the space shuttle, was
supported by NASA's Microgravity Sciences and Application Division (now, Physical
Sciences Research Division). Also, G. Wozniak gratefully acknowledges support of his
work by the European Space Agency (ESA) and the German Space Agency (DARA).

NOMENCLATURE

Ca Capillary number
es internal energy of the interface per unit area
E Elasticity number
Es dimensionless group representing the relative importance of temperature
gradients naturally arising on the interface due to drop motion
k thermal conductivity of the continuous phase
k0 thermal conductivity of the drop phase
Ma Marangoni number
Pr Prandtl number
q radiant heat flux absorbed by a drop surface
r radial coordinate
R radius of the drop
R universal gas constant
Re Reynolds number
T temperature
T0 reference temperature on the interface
v0 reference velocity
v1 physical migration velocity scaled by the reference velocity
VYGB velocity predicted by Young et al.
We 0 Weber number
 ˆ  ratio of the dynamic viscosity of the drop phase to that of the continuous
phase
0
 ˆ kk ratio of the thermal conductivity of the drop phase to that of the continuous
phase
reference concentration of surfactant on the interface
 polar angle measured from the forward stagnation streamline
 thermal diffusivity of the continuous phase
 dimensionless gas constant
172 R. S. Subramanian et al.
 dynamic viscosity of the continuous phase
0 dynamic viscosity of the drop phase
 kinematic viscosity of the continuous phase
 density of the continuous phase
 interfacial tension between the drop phase and the continuous phase
T rate of change of interfacial tension with temperature
 scaled radial coordinate in an outer expansion of the velocity field
rT1 applied temperature gradient

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