7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Thermocapillary Migration of Deformable Drops at Moderate to Large Marangoni Number in Microgravity
JianFu ZHAO, ZhenDong LI
Key Laboratory of Microgravity (National Microgravity Laboratory), Institute of Mechanics, Chinese Academy of Sciences.
Beijing 100190, China
Keywords: thermocapillary migration, drop, levelset method
Abstract
Using the levelset method and the continuum interface model, the axisymmetric thermocapillary migration of liquid drops in
an immiscible bulk liquid with a temperature gradient at moderate to large Marangoni number is simulated numerically.
Constant material properties of the two phases are assumed. Steady state of the motion can always be reached. The terminal
migration velocity decreases monotonously with the increase of the Marangoni number due to the wrapping of isotherms
around the front surface of the drop. Good agreements with space experimental data and some previous theoretical and
numerical studies in the literature are evident. Slight deformation of drop is observed, but no distinct influence on the motion
occurs. It is also found that the influence of the convective transport of heat inside drops cannot be neglected at finite
Marangoni number, while the influence of the convective transport of momentum inside drops may be actually negligible.
Introduction
A fluid particle (gas bubble or liquid drop) floating in an
immiscible bulk fluid with a temperature gradient can be
moved by the nonuniform surface tension at the particle
interface. This motion is well known as the thermocapillary
or Marangoni migration. It plays an important role in many
natural physical processes as well as a host of industrial
activities, particularly in space material processing and
many other scientific and engineering applications in
microgravity. Thus, it attracts much more interests of
researchers all over the world along with the progress of
human space activities. However, most of the work on this
subject is relatively recent, as summarized in the
monograph by Subramanian and Balasubramaniam (2001).
Focusing upon the present topic, only studies relating to the
thermocapillary migration of liquid drop are briefly
reviewed here.
Thermocapillary migration was first analyzed by Young et
al. (1959) in the case of infinitesimal Reynolds and
Marangoni numbers, in which convective transport of
momentum and heat can be neglected comparing to
molecular transport of these quantities and the governing
equations can then be linearized. They derived the named
YGB theory predicting the following steady migration
velocity
VYGB 2U (1)
(2+3 2/,1)(2+k2/k,)
where U = c VTR/,p1 is the named thermocapillary
velocity, R is the drop radius, p is the dynamic viscosity, k
is the thermal conductivity, oy is the rate of change of
interfacial tension with temperature, V T is the
temperature gradient imposed in the continuous bulk fluid.
The subscripts 1 and 2 denote the material properties of the
continuous bulk fluid and the fluid particle, respectively.
The Reynolds and Marangoni numbers are defined as
Re = UR/v1 and Ma =UR/1 Here, v denotes the
kinematic viscosity, A denotes the thermal diffusivity. If the
Prandtl number is defined as Pr = v1 / = Ma / Re .
The analysis of Young et al. (1959) was extended by many
others to include convective influence. For example, using
asymptotic expansion technique, the migration velocity of a
nondeformable gas bubble for small but nonzero
convective heat transfer in the limit of zero Reynolds
number was obtained by Subramanian (1981). 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, Subramanian (1983) later extended
this work to liquid drops, in these paper, they concluded
that the heat convection may reduce the scaled migration
velocity of the drop for small Ma. Balasubramaniam and
Subramanian(2000) studied the effect of the convection
inside the drop and found the terminal velocity of a single
drop first decreases with increasing Marangoni number,
attains a minimum and then increases with a further
increase in the Marangoni number.
On the other hand, HajHariri et al. (1997) calculated
numerically the threedimensional thermocapillary motion
of deformable drops at finite Reynolds and Marangoni
numbers and found that the strong heat convection may
retard the thermocapillary motion of the drop. Ma et al.
(1999) analyzed the thermocapillary motion of a
nondeformable single drop, and concluded that the scaled
migration velocity decreases with Ma, reaches a minimum,
and then increases with Ma when Ma is large enough. Nas
(1995, 2003) adopted the fronttracking method to calculate
the thermocapillary interaction of two drops, Yin et al.
(2008) adopted the same method numerically investigate
the thermocapillary migration phenomena of a single
nondeformable spherical drop.
In addition to theoretical and numerical developments, there
are some results from experiments in earthbased
laboratories, as well as in reduced gravity conditions.
Because of the nonlinearity of the problem, the
thermocapillary motion with finite values of the Reynolds
and the Marangoni numbers can be observed
experimentally only in microgravity environment in order
to avoid the buoyant convection. Hadland et al. (1999) and
Xie et al. (2005) reported some results on the
thermocapillary migration of FC75 droplets in silicone oil
aboard a NASA space shuttle and the ShenZhou4
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
spaceship of China, respectively. Both two experiments
cover a wide range of the Reynolds and Marangoni
numbers.
In the present paper, a numerical study on the
thermocapillary motion of deformable drops at moderate to
large Marangoni number is reported. The results of the
thermocapillary migration velocity will be compared with
experimental data of Hadland et al. (1999) and Xie et al.
(2005), as well as some asymptotic predictions and previous
numerical simulations. The structures of the flow and
temperature fields will be presented to help understanding
the characteristic of this phenomenon.
Mathematical formulation and numerical method
The thermocapillary migration of a single deformable drop
in an immiscible bulk liquid is considered here with the
following assumptions: 1) the fluids in both phases are
Newtonian, viscous and incompressible; 2) the material
properties are constant and not influenced by the
temperature; 3) the surface tension depends linearly upon
the temperature; 4) the motion in both phases is
axisymmetric and laminar.
To capture the interface of the drop, the levelset method
(Osher et al., 1988) is used. The levelset function is
denoted as 9 which is positive in the continuous bulk fluid
and negative inside the drop. So the interface between two
phases is the zero levelset of q9, which can be advanced by
the following convection equation:
C+u V( = 0 (2)
at
where u denotes the velocity vector. A thirdorder
RungeKutta method in time and a fifthorder WENO
method in space are used to solve the equation.
In the numerical simulation, however, 9 will no longer be a
distance function as the interface evolves. In order to keep p
approximately equal to signed distance, the reinitialization
equation (Sussman et al., 1994)
= sgn( )(1 IVl) (3)
Or
needs to be solved to steady state periodically during
simulation with the following initial condition
P(x,0)= 9(x) (4)
where r is the virtual time, qo is the levelset function at the
time t and sgn() is the smoothed sign function.
Furthermore, to guarantee the mass conversation of the drop
the following HamiltonJacobi equation is also solved to
steady state periodically during simulation (Son et al., 2007)
+(V, V)(a+bc)Vpl = 0 (5)
Or
where Vo is the initial total volume of the drop and V is the
total mass corresponding to the levelset function Op(r), c is
the surface curvature. The parameters a, b and c are chosen
in the present study as 1, 0 and 0, respectively.
Thus, based on the levelset function and the continuum
interface model, the dimensionless governing equations can
be written as
V.u = (6)
p'v v.u+v'u )
O .Vu Re (7)
at p' Fca(TrT) (I n).VT i
We Re
'cP', +uVT) v2T (8)
cSt Ma
in which R, U, and IVTR are used as the characteristic
length, velocity and temperature, respectively. To is a
dimensionless reference temperature which is set as 0, or
the value at the initial position of the center of the drop in
the present study. The other dimensionless parameters
appeared in the above equations are defined as
I P k=l i CP ki /LU
'= = c k'= Ca= and
Pi /1 cp k, oo
We= ReCa. The last two dimensionless parameters are the
capillary and Weber numbers, which determine the
deformation of the drop.
The unit surface normal, the surface curvature and the delta
function appeared in the above equations are also defined as
follows
n = V p/V'p
K = Vn
1+cos i;)/(3h) pI <1.5h)
0 (pH 1.5h)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
g = g2 + (g 2)H (12)
where g represents p, p, and so on. The Heaviside function
His defined as
1
H= 1+ +sin 2 /2;T
2 3h 3h
0
( >1.5h)
p< 1.5h)
(q < 1.5h)
The projection method is used to solve the above governing
equations with uniform staggered grid of equal spacing at
the following initial conditions
u= = 0, T=z (t=0) (14)
and boundary conditions:
u = o = 0,
u=o=0,
Or Or
u = = 0,
T =A
T=B
T=z
where z = A, z = B, and r = C denote the bottom, top,
and the outer boundaries, respectively. The center of the
drop locates at the point (0, 0) at the beginning, and A is
always set as 3.
Resolution test and validation
In order to estimate the accuracy of the numerical algorithm,
the method described above is first applied to several test
cases. The dimensionless numbers except Re and Ma are
chosen as Ca =0.2, P P2/P = 1/1 p=2//p =1/1,
l= k2/k=l/1,and 2=c2/cPl =1/1.
o
4
.,
U,
u,
E
'Ei
(9)
(10)
(11)
where h is the grid spacing.
Furthermore, in order to avoid numerical instability caused
by their jumps across the interface, the material properties
of the fluids are smoothed by the Heaviside function
YGB
0.12
0.09
0.06
0.03
0.00
0.00 0.01 0.02 0.03 0.0
dimensionless time
Figurel. Comparison between the numerical result and the
YGB theory.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Fig. 1 shows the comparison between the numerical result
for a test case of Re=0.01 and Ma=0.01 and the analytical
prediction of YGB theory. A computational domain of 15 x5
and a grid of 300x100 are used. A good agreement is
evident since the effects of convective transport of
momentum and heat are negligible in this case.
Before the above validation, the influence of the grid
spacing is firstly studied at Re=l and Ma=100 with a
computational domain of 15x5, namely A=3, B=12, and
C=5. It is found that the difference of the terminal migration
velocity is no more than 1% between two grids of 300 x 100
and 450x 150. The influence of the computational domain is
also studied with the same grid spacing corresponding to
the first one of the above cases. It is also found that
differences of the terminal migration velocity are no more
than 1% among three computational domains of 15x5, 20x5,
and 15 x7. Thus, all results reported in the present paper are
obtained with the computational domain of 15x5 and the
gridof300x100.
Influences of dimensionless parameters
Fig. 2 presents evolutions of drop migration velocity for
different Marangoni number at a fixed Reynolds number of
1. The other dimensionless parameters are the same as those
in Fig. 1. Steady state of the thermocapillary migration can
be reached. It is evident that the migration velocity is a
monotonically decreasing function of the Marangoni
number, which is consistent with the previous theoretical
and numerical studies for the case of nondeformable drops.
YGB
10
10
Ma=100
u.uu  i  i 
0 5 10 15 20
dimensionless time
Figure 2. Influence of Marangoni number on
drop migration velocity.
A) Ma=l
B) Ma=10
C) Ma=100
Figure 3. The temperature fields at different Marangoni
number.
The dependence of the migration velocity on the Marangoni
number can easily be explained by the isotherms
surrounding the drop, which are shown in Fig. 3
equallyspaced with dimensionless temperature increments
25 30 of 0.4. Obviously, the enhanced convective transport of
momentum and heat with the increase of the Marangoni
number results in the wrapping of the isotherms around the
evolution of front of the drop, leading to a substantial reduction in the
surface temperature gradient and diminishing the driving
force for the motion of the drop over much of the front
surface.
Figs. 4 and 5 show respectively the influences of the
viscosity ratio a and the thermal conductivity ratio / on the
terminal migration velocity at Re= and Ma= 100. The other
parameters are the same as those in Fig. 1. In these figures,
the terminal migration velocities V of drops are scaled using
the corresponding values predicted by the YGB theory.
Thus, according to Eq. (1), the scaled velocities are equal to
1 for any values of a and / in the absence of convective
transport of momentum and heat. On the contrary, the
results shown in Figs. 4 and 5 indicate a different trend in
which the scaled velocity decreases with the decrease of a
or p. Thus, the influence of the convective transport of heat
and the convective transport of momentum inside drops
cannot be neglected.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
5. Compare with space experimental data
The thermocapillary motion of deformable FC75 drops in
silicone oil is studied numerically using the above algorithm
at the same conditions as those in the space experiments of
Hadland et al. (1999) and Xie et al. (2005). Constant
material properties, however, are assumed to be
independent of temperature. The following values are
adopted for the dimensionless parameters in the problem,
namely j=1.89, a=0.14, f=0.47, p=0.69, and Pr=83.3.
Furthermore, a much larger value for the capillary number,
i.e. Ca=0.2 which is still much less than 1 to guarantee no
distinct deformation of the drop, is used here than those in
the space experiments, which are of the order of 101 or less,
in order to prevent the virtual flow caused by the strong
jump of the normal stress across the interface of drops.
U..J
Figure 4. The
viscosity ratio.
U.. I
scaled migration velocity as a function of
0 1
Figure 5. The scaled mil
thermal conductivity ratio.
ration velocity as a function of
10 20 30 40 50
dimensionless time
Figure 6. Evolutions of drop migration velocity at different
Marangoni number.
Fig.6 shows evolutions of drop migration velocities at
different Marangoni number. A steady state can always be
reached for the thermocapillary migration of FC75 drops.
The predicted dimensionless terminal velocity is a
monotonically decreasing function of the Marangoni
number. In Fig. 7, the terminal migration velocities are also
scaled using the corresponding values predicted by the
YGB theory, which are compared with the experimental
data of Hadland et al. (1999) and Xie et al. (2005), labeled
respectively as HBWS1999 and XHZLH2005 for the
brevity of the figure. Generally, good agreements are
evident. Numerical results of HajHariri et al. (1997), Ma
(1998) and Yin et al. (2008), labeled respectively as
HHSB1997, Ma1998 and YGHC2008.
1.2 e
S1Experimental Data
1.0 eG I e HBWS1999
0.8 j E B XHZLH2005
m 0.6 .
0.4 Numerical Data a ft
0Present e
 YGHC2008 A
Ma 1998
 HHSB1997 "
0.2
101 102 103 104
Ma
Figure 7. Comparison of the predicted thermocapillary
migration velocity with the experimental data.
Except the prediction of Ma (1998), the other three
numerical results are all show that the predicted
dimensionless terminal velocity is a monotonically
decreasing function of the Marangoni number. The
differences between the present results and those of
HajHariri et al. (1997) or Yin et al. (2008) may caused by
the different values of parameters used in the calculations.
HajHariri et al. (1997) used =a=fl=Z=0.5, and Ca=0.1, or
Yin et al. (2008) used =a=P==l, and nondeformable drop.
According to the findings by HajHariri et al. (1997), Yin et
al. (2008) and this paper, the terminal migration velocities
can be affected by the change of physical parameters, thus,
the differences are reasonable and may be diminished if
these influences are taken into account.
Table 1. The locations of the center of the vortex inside
drops in the local reference frame attached to the center of
drops (I) and in the laboratory reference frame (II). The
origins of the two frames are located at the center of drops.
Maa (r, z)I (r, Z)II
2 (0.68, 0.0510) (0.84, 0.0110)
5 (0.68, 0.0754) (0.84, 0.0354)
10 (0.68, 0.0930) (0.88, 0.0423)
20 (0.68, 0.1314) (0.88, 0.0514)
50 (0.68, 0.1353) (0.88, 0.0553)
100 (0.68, 0.1453) (0.92, 0.0548)
200 (0.68, 0.1525) (0.92, 0.0653)
500 (0.68, 0.1568) (0.92, 0.0668)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Isotherms in the terminal steady state of the thermocapillary
migration for several typical cases are shown as dashed
lines in Fig. 8 in a meridian plane. Streamlines are also
plotted as solid lines in the same figure, in which the left
part is those in the local reference frame attached to the
center of the drop and the right part is those in the
laboratory reference frame. As mentioned above, the
enhanced convective transport of momentum and heat with
the increase of the Marangoni number leads to a substantial
reduction in the surface temperature gradient and
diminishes the driving force for the motion of the drop over
much of the front surface. That is the reason of the fact that
the migration velocity decreases monotonically with the
increase of the Marangoni number. An annular vortex can
be observed inside the drop. The center of the vortex locates
near the interface of the drop. As shown in Table 1, the
transverse position of the vortex in the local reference frame
is not changed with the Marangoni number, however, in the
laboratory reference frame it moves outward with the
Marangoni number; the longitudinal positions in both two
reference frames move downstream with the increase of the
Marangoni number.
The reason for this fact should be found in the distribution
of temperature. According to Fig. 8, the temperature
gradient is nearly uniform along the drop interface at small
Marangoni number, while a much uneven distribution of
temperature will be observed at high Marangoni number.
Although the surface temperature gradient over whole
surface decreases at high Marangoni number, it becomes
larger along the rear surface than that along the front
surface. Larger temperature gradient means larger driving
force and faster motion, so the center of the vortex inside
drops moves with the increase of the Marangoni number. In
Fig.8, we can also see that because of the enhanced
convective transport, the isotherms in the terminal steady
state are more similar to the streamline structure with the
increase of the Marangoni number, and even at Ma=500,
there are annular isotherms inside the drop just like the
vortex.
Fig.9 shows the aspect ratio between the longitudinal and
transverse lengths of deformed drops in the terminal steady
state of the thermocapillary migration. It decreases quickly
with the increase of the Marangoni number at Ma<100,
while the decreasing rate lowers when the Marangoni
number exceeds 100. The biggest variation of aspect ratio,
however, is no more than 1% in the present study, because
both the Reynolds and capillary numbers are very small. It
ought to be pointed out here that the actual deformation
may be much smaller than the predictions since the
capillary number in space experiments has much smaller
values than that used here. Therefore, no distinct influence
of the deformation of drops can occur in the
thermocapillary migration of FC75 drops, at least within
the present range of the Marangoni number.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
has been studied numerically at moderate to large
Marangoni number. The axisymmetric governing equations
are solved using the projection method with a uniform
staggered grid of equal spacing at appropriate initial and
boundary conditions. The levelset method and the
continuum interface model are used to account for finite
drop deformation.
0.998
0.997
0.996
0.995
0.994
0.993
0.992
0.99'
A) Ma=2
B) Ma=50
C) Ma=500
Figure 8. Isotherms (dashed line) and streamlines (solid
line) at different Marangoni number.
Conclusions
The thermocapillary migration of deformable FC75 drops
'0.
'cQ
.0
1 
102
Ma
Figure 9. Aspect ratio of the deformable FC75 drop at
different Marangoni number.
It is found from the numerical results that steady state of the
thermocapillary migration of deformable FC75 drops can
always be reached. The terminal migration velocity of drops
scaled by the named YGB velocity decreases with the
decrease of the ratios of the dynamic viscosity and thermal
conductivity between the drop and continuous bulk liquid
phases. The scaled velocity may keep a monotone decrease
with the decrease of the ratio of the thermal conductivity
and the dynamic viscosity throughout its possible range. It
is suggested that the influence of the convective transport of
heat and momentum inside drops cannot be neglected.
The numerical results of the simulations under the
conditions corresponding to the space experiments of
Hadland et al. (1999) and Xie et al. (2005) are compared
with experimental data. Good agreement can be observed.
The structures of the flow and temperature fields are also
presented to reveal the influences of convective transport on
the phenomenon. It is verified by the present results that no
distinct influence of the deformation of drops can occur in
the thermocapillary migration of FC75 drops, at least
within the present range of the Marangoni number.
Acknowledgments
The present work is supported financially by the National
Natural Science Foundation of China under the grant of
10972225.
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