7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Coupled Marangoni convection and thermocapillary convection
in liquid layers
K. Li', W Liao' Z.M. Tang', W R. Hu'
Key Laboratory of Microgravity, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China
and National Microgavity Laboratory, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China
2 The Graduate School of Chinese Academy of Sciences, Beijing 100190, China
likai(iaimech.ac.cn
liaowenpinimech.ac.cn
zmtang(inimech.ac.cn
wrhu(aimech.ac.cn
Key words: liquid layer, Marangoni convection, thermocapillary convection, instability
Abstract
In the present paper, the coupled thermocapillary convection and Marangoni convection in a liquid layer of finite
extension with curved free surface in microgravity was studied by a twodimensional numerical simulation. For different
volume ratios and Prandtl numbers of the liquid layers, the cell structure and its evolution of the coupled convection were
investigated. For a liquid layer with fixed Prandtl number, the main influence is from the configuration of the liquid layer. The
corresponding marginal stability boundary of the coupled flow consists two branches separated by the volume ratio (V/V ), i.e.
V/Vo > 1 and V/Vo < 1 (see Fig. 1). On the other hand, the coupled convection is dependent of the Prandtl number of the liquid
layer. The numerical study also revealed some characteristics of the coupled convection quite different from the corresponding
linear stability analysis results.
Introduction
Pearson's theoretical study, by means of linear stability
analysis, indicated that Marangoni convection arose in a
liquid layer of infinite extension heated from the liquid/solid
boundary at the bottom when the applied temperature
difference exceeds a threshold (Pearson 1958). This
phenomenon suggested a new instability mechanism in the
fluid system, which is different from Rayleigh instability
(Rayleigh 1916). Many works have been devoted to the
investigation of Marangoni convection in liquid layers.
Most of them assumed that the liquid/gas interface usually
noted as free surface (see Fig. 1) is planar and
nondeformable, and focused on the multicellular structures
of Marangoni convection depending on the aspect ratio of
liquid layers (e.g., see refs. Rosenblat et al. 1982a, 1982b).
It is worth noting that periodically oscillatory Marangoni
convection arises under certain conditions. Dijkstra et al.
(1992, 1995) investigated the multicellular structures of the
Marangoni convection in twodimensional and
threedimensional containers. The preferred convection
modes were summarized in the x and y aspect ratio plane.
Moreover, effects of the boundary conditions on the
preferred convection modes were also studied. On the other
hand, the investigation of thermocapillary convection under
an inclined temperature gradient was conducted by Oleg &
Alexander (2"114). The results brought to light the
importance of the horizontal temperature gradient on the
formation and development of convective regimes in a plane
liquid layer heated from liquid/solid boundary at the bottom.
The geometric simplification of a planar free surface
adopted in the aforementioned investigations is not always
real in practice. The free surface of a liquid layer in the
microgravity conditions is generally curved due to the much
y
Fr~^
AT4
I rk
AT IT 2
Fig. 1. A sketch of liquid layer with curved free surface
weaker static pressure compared to the case in the terrestrial
conditions. Therefore, even if the applied temperature
gradient is perpendicular strictly to the liquid layer, there is
a temperature gradient along the curved free surface, which
results in the coexistence and interaction of the Marangoni
convection and thermocapillary convection in the
microgravity conditions. In our previous study (Tang et al.
2008), the coupled Marangoni convection and
thermocapillary convection in a liquid layer of finite
extension (Pr = 105.6) with a curved free surface in the
microgravity conditions was numerically investigated using
a twodimensional liquid layer model. The study revealed
the significant influence of volume ratio of the liquid layer
on the evolution of the multicellular convection structure
and the corresponding critical Marangoni number profile in
the microgravity conditions.
In present study, twodimensional numerical
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
simulations are conducted on the coupled Marangoni
convection and thermocapillary convection in liquid layers
of finite extension (Pr = 11.6) with a curved free surface in
the microgravity conditions. Influence of the Prandtl
number of the liquid layer on the multicellular structure of
the coupled convection and the corresponding marginal
stability boundary are studied in detail. The physical model
and mathematical formulations of the problem are given in
the next section. Section 3 describes the numerical method
and the code validation. The computed results are presented
in section 4. The last section is the discussion.
2. Physical model and mathematical formulation
Fig. 1 shows a twodimensional liquid layer of 1 cSt
silicon oil (Pr = 11.6). The liquid layer is surrounded by an
ambient gas of temperature Te (25C). The heights of the
liquid layer at both sidewalls are ho = 3mm and the
extension of the liquid layer is 1 = 55.5mm. By heating from
the bottom up to a given value Tb, a vertical temperature
gradient is set up in the liquid layer. Heat is transferred from
the liquid to the ambient gas, and the heat transfer
coefficient at the free surface is q. Both sidewalls are
assumed to be thermally isolated and impermeable but
slippery. The liquid is considered as incompressible with
constant viscosity and thermal diffusivity, and Boussinesq
approximation is adopted. The surface tension at the free
surface is a linear function of the temperature T, and c/ DT
is assumed to be constant. The free surface is described as y
= h (x). The temperature difference AT = TbTe, velocity U,
= I d /o AT/p and pressure puUjho are used as the
characteristic scales for the length, temperature, velocity and
pressure, where p is dynamic viscosity.
Introducing the nondimensional vorticity
vector = (o co, co) and stream function vector
S= (v/', q/, /; ), which are defined as
Vxq=V (1)
VxVx = o (2)
The nondimensional equations and boundary conditions
can be expressed in terms of vorticity a and stream
function W. The mass conservation equation is satisfied
automatically. The momentum conservation equation and
energy conservation equation can be expressed
respectively as equations (3) and (4) as follows.
+ V V.V w, VV= 1 V2a, (3)
St Re
d0 1
+V.VO = V20. (4)
Qt Ma
AT
where 0 is nondimensional temperature.
For the twodimensional problem, the boundary
conditions are as follows:
a>
x = andx= /h h :i = 0, co = x2
O:o
y = 0: = 0, 0 = T = T0,,
5^
0 T
0, 0
9 x
0,
(5)
At the free surface y = ;h \ ii..
V = 0,
(1+ h'2) T
(1 h' 2) S
(6)
4h' 2// 1 q 2 2
[_ +2
(1 h' ) OxCy 2y
(7)
aT
= LT,
an (8)
where L = qh/2, A notes thermal conductivity of the liquid
and q is the rate of heat loss per unit area from the upper
surface. The vorticity at the free surface is determined
from equilibrium of tangential stress (7), in which S points
the tangential direction. Eq. (8) means that the rate of heat
supply to the surface from the liquid must equal to the rate
of heat loss from the free surface to the environment. L
depends on the efficiency of the process for transferring
heat from the surface. The value of q depends on the
circumstances, and is retained as a parameter in the
subsequent calculation.
The free surface shape is calculated by equilibrium of
normal stress,
LW aU
h + a
9x 9y
h"
Ca (I+h'2) /
where U and W are respectively the nondimensional
velocities in the directions x and y, and the dynamic
deformation of the free surface during heating process is
negligible. The free surface at both the left and right
boundaries locates at y = 1, which means the method of
controlling the height of the liquid layer at the interface of
solid and liquid is adopted. The initial condition may be
obtained from the static case without applied temperature
difference and convection.
3. Numerical method and code validation
A hybrid finite element method of fractional steps is
adopted in the present study to solve the governing
equations due to its stability, simplicity and flexibility in
handling complex geometry. In this method, to account the
nonlinear convection effect, the characteristic procedure is
used for convection operator, and the finite element
method is used for diffusion calculation. The hybrid
method of fractional steps was first suggested by Vanenko
(1971). The hybrid finite element method of fractional
steps was presented by Wu (1985) for numerical modeling
of the pollution in aquatic environment. The detailed
stability analysis and a twodimensional sample for
checking the divergence and precision were also report in
(Wu 1985). Irregular triangular grids are adopted in the
present study. The numbers of the cell are 121x21 in the
directions x, y respectively, and the calculated domain is
divided in to 4800 triangular elements with 2541 nodes.
The shape of the free surface is determined according to
the volume ratio by using eq. (9) with U = 0, and W = 0.
The detailed validations of the present code can be found
( 2 [h'2 aU
1+h'2) L x
aw
ay
in (Tang et al. 2008) which are ignored here.
4. Numerical results
The liquid layer with nondeformed free surface, i.e.,
the volume ratio V/Vo = 1, is firstly studied, where V is the
practical volume of liquid layer and Vo is the volume of
liquid layer with planar free surface. Only the Marangoni
convection is involved in this case. Different from the case
of infinite extension, a pair of convective roll emerges in
the vicinity of the sidewalls initially even with the slippery
boundary condition which means zero shear stress. The
convective roll gradually fills up the liquid layer with the
increasing temperature difference (Tang et al. 2008).
Therefore, the critical state in the present study is defined
as the moment the convective rolls reach the central line
(x= 1/2ho) of the liquid layer, and the related temperature
difference is defined as the critical temperature difference
(ATc). Figure 2 shows the maximum dimensionless vertical
velocity We at central line, where W = Woh/v, Wo is the
dimensional vertical velocity and v the kinematic viscosity.
We is zero when the applied temperature difference is
lower than ATc, otherwise it increases sharply. In the
present study, the critical Marangoni number for the liquid
layer of Pr = 11.6 is Mac = 336.6 (AT,= 0.132 C). The
numerical results for the liquid layer of Pr = 105.6 (Tang et
al. 2008) is also shown in Fig. 2. The corresponding
critical Marangoni number is Mac = 392.0 (ATc.= 1.93 C).
It can be seen that with the same configuration, the Mac for
liquid layer of lower Prandtl number is much smaller.
0.15 , , ,
0.1 Ld
0.05
0
0.05 
300 400 500 600
Maa
Fig. 2. The maximum dimensionless vertical velocity We
and related Marangoni number at the central line of the
planar liquid layer.
Before the further investigation of the influence of
volume ratio on the coupled Marangoni convection and
thermocapillary convection, it is necessary to describe the
determination of the Mac for the onset of Marangoni
convection in the liquid layer of finite extension with a
curved free surface. It is known that the thermocapillary
convection sets in under any temperature difference. The
vertical velocity We at the central line of the liquid layer
changes between positive value and negative value
alternatively although it is nearly zero in the subcritical
state. When W changes from negative (positive) to positive
(negative) and then increases sharply, the applied
temperature difference is defined as ATc. Figure 3 shows
the volumeratio dependent Mac profile determined
through the above criteria. The curved free surface
introduces the lateral temperature gradient along the free
surface, and then the thermocapillary convection which
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
influences the multicellular structure of the coupled
convection and the corresponding Mac compared to the
case of planar free surface. Similar to the liquid layer of Pr
= 105.6 (Tang et al. 2008), in the cases of concave free
surface, the minimum depth and the maximum temperature
at the free surface are at the central line of the liquid layer.
The thermocapillary convection is from the central line to
the sidewalls at the free surface. The initial
thermocapillary convection is constrained to the vicinity of
the sidewall. With the increasing temperature difference,
new convective roll pairs generate and extend towards the
central line (not shown here). Figure 4 shows the
multicellular structures of the coupled convection in
liquid layers of different volume ratios at the critical states.
0.8, 0.19C 0.8, 0.19C
0., 0.14C 0.9, 0.145*C
1.0, 0.132*C 1.0, 0.132C
1.1, 0.09C
1.2, 0.075aC
1.3, 0.073C
1.1, 0.09C
12, 0 .075
1.3, 0.073*C
Fig. 4. Streamlines (left) and isotherms (right) for liquid
layers of different volume ratios at the critical states.
On the other hand, the maximum depth and the minimum
temperature at the free surface are at the central line of the
liquid layer in the cases of convex free surface. The
thermocapillary convection is from the sidewalls to the
central line at the free surface and nearly occupies the
whole liquid layer. New convective roll pairs gradually
grow and split from the initial roll pair until the formation
of the multicellular structure of the coupled convection.
However, the convective roll pairs at the critical states in
the cases of convex free surface are much less compared to
the cases of concave free surface (see Fig. 4). Similar to
the numerical results for the liquid layer of Pr = 105.6
(Tang et al. 2008), the marginal instability boundary for
liquid layer of Pr = 11.6 consists two branches, i.e., one
corresponds to V/Vo < 1 and the other corresponds to V/Vo
> 1. However, with the same configuration, Mac for the
concave liquid layer of Pr = 11.6 are smaller than those of
liquid layers of Pr = 105.6 while Mac for the convex liquid
layer of Pr = 11.6 are generally larger than those of liquid
layers of Pr = 105.6. The different cellular evolutions of
the coupled convection and critical temperatures for liquid
layers of different volume ratios are due to two factors, the
configuration of the liquid layer and the thermocapillary
convection. However, the main influence is from the
former.
An interesting phenomenon is also revealed in the
present study on liquid layer of Pr = 11.6, which was not
reported in the previous study (Tang et al. 2008). When the
critical Marangoni number is exceeded, oscillatory
multicellular coupled convection may occur in concave
liquid layers of small volume ratios (see Fig. 5).
0 . . I I . . I . . 1 . . 1
0.4 0.6 0.8 1 1.2 1.4
Fig. 5. The marginal instability boundary for liquid layers
of Pr = 11.6 with different volume ratios.
Particularly, in the narrow range of 0.8 < V /o < 0.85,
the coupled convection experiences the regimes of steady
multicellular convection, oscillatory multicellular
convection and steady multicellular convection in
sequence with the increasing Marangoni number. As an
example, figure 6 shows the time evolution of the
oscillatory multicellular coupled convection in liquid
layer of VVo = 0.8. It can be seen that the maximum
dimensionless vertical velocity W, at the central line
oscillates periodically accompanied with the alternative
change of the roll number between 8 pairs and 7 pairs.
During this process, the propagation of the velocity
disturbance U' and temperature disturbance T' of the
oscillatory coupled convection at the free surface of the
liquid layer are shown in Fig. 7. It is worthy noting that the
traveling wave originates from the central of the free
surface, i.e., the hot spot, and propagates to the sidewalls
of the liquid layer, i.e., the cold spot. This phenomenon
contradicts to the propagation direction of the
hydrothermal wave which is usually adopted to explain the
oscillation mechanism of the surface tension driven
convection of large Pr fluid.
5. Conclusions
In the present paper, effects of the volume ratio of a
=12800s W13800s
1t13400s 114500s
(a)
0.04
0.02
0.02
0.04 ....
12000 13000 14000 15000 16000
t(s)
(b)
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Fig. 6. Time evolution of (a) multicellular structure and
(b) maximum dimensionless vertical velocity W, at the
central line of the oscillatory coupled convection in liquid
layer of VVo = 0.8.
16000
14000
12000
u'(mmisec)
0 191837
0 142B57
0 093B77B
0 044898
0 00408163
0 0530612
0 102041
0 15102
02
16000
14000
12000
T'(C)
000767347
000571429
00037551
000179592
0 000163265
000212245
000408163
000604082
0008
(b)
Fig. 7. The propagation of the (a) velocity disturbance U'
and (b) temperature disturbance T' of the oscillatory
coupled convection at the free surface of the liquid layer of
V/Vo = 0.8.
liquid layer of Pr = 11.6 on the coupled Marangoni
convection and thermocapillary convection and its stability
properties are numerically studied based on a
twodimensional liquid layer model with finite extension.
Due to the free surface configuration and the introducing
of the thermocapillary convection, the marginal instability
boundary consists two branches separated by V/Vo = 1, i.e.
V/Vo < 1 and V/Vo > 1. Moreover, the numerical results
reveal that a pair of rolls originates in the whole region or
in the vicinity of sidewalls depending on the liquid layer
configuration, and new pairs of rolls gradually generate to
cover the whole region with the increasing temperature
difference. These phenomena are similar to the case of Pr
=105.6 (Tang et al. 2008) but different from the linear
stability analysis results on liquid layer of infinite
extension, in which a series of rolls generate
simultaneously at the critical temperature difference.
In the linear stability analyses, the marginal curves
can be obtained by solving an eigenvalue problem
involving the parameters of Ma, L, and the critical
conditions are independent of Pr. However, the present
study reveals that for the same configuration, the critical
Marangoni numbers for the concave liquid layer of Pr =
11.6 are smaller than that those of liquid layers of Pr =
105.6 while the critical Marangoni numbers for the convex
liquid layer are generally larger than those of liquid layers
of Pr = 105.6. it is worthy noting that for liquid layers of Pr
= 11.6, when the critical Marangoni number is exceeded,
 Multiple
Oscillation flow M
SMultiple
steady flow
steady flow
oscillatory multicellular coupled convection may occur in
concave liquid layers of small volume ratios. Particularly,
in the narrow range of 0.8 < V/Vo < 0.85, the coupled
convection experiences the regimes of steady
multicellular convection, oscillatory multicellular
convection and steady multicellular convection in
sequence with the increasing Marangoni number. Moreover,
the propagation of the corresponding velocity disturbance
U' and temperature disturbance T' of the oscillatory
coupled convection at the free surface of the liquid layer
propagates in the reverse direction compared to
propagation of the hydrothermal wave which is usually
adopted to explain the oscillation mechanism of the surface
tension driven convection of large Pr fluid. These
phenomena were not reported in the previous study on
liquid layer of Pr =105.6 (Tang et al. 2008) and require
further insights.
Acknowledgment
This research is supported by the National Natural
Science Foundation of China (Grant No. 10872202) and
the Knowledge Innovation Project of the Chinese
Academy of Sciences (Grant No. KJCX2YWL08).
References
J.R.A. Pearson, On convection cells induced by surface
tension. J. Fluid Mech. 4, 489500 (1958).
L. Rayleigh, Scientific Papers 6, 432, Cambridge
University Press (1916).
S. Rosenblat, S.H. Davis, GM. Homsy, Nonlinear
Marangoni convection in bounded layers. Part 1. Circular
cylindrical containers. J. Fluid Mech. 120, 91122
(1982a).
S. Rosenblat, GM. Homsy, S.H.
Marangoni convection in bounded
Rectangular cylindrical containers. J.
123138 (1982b).
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
J.H. Wu, A hybrid method of fractional steps with
Loostability for numerical modeling of aquatic
environments, Advances in Hydrodynamics 1 (3) 2736 (in
Chinese) (1985).
Davis, Nonlinear
layers. Part 2.
Fluid Mech. 120,
H.A. Dijkstra, On the structure of cellular patterns in
RayleighBenardMarangoni flows in two dimensional
containers with rigid sidewalls. J. Fluid Mech. 243, 73102
(1992).
H.A. Dijkstra, Surface tension driven cellular patterns in
threedimensional boxesPart II: A bifurcation study.
Microgravity Sci. Technol. VIII/2, 7076 (1995).
E.S. Oleg, A.N. Alexander, Thermocapillary flows under
an inclined temperature gradient. J. Fluid Mech. 504,
99132 (2"'14).
Z.M. Tang, K. Li, W.R. Hu, Influence of free surface
curvature of a liquid layer on the critical Marangoni
convection, Int. J. Heat Mass Trans. 51, 51025107 (2008).
N.N. Vanenko, The method of fractional steps.,
SpringerVerlag, Berlin, Heidellerg, New York (1971).
