Engineering Sciences Report, Kyushu University
Vol.28, No.1 pp.1–8, JUN. 2006

九州大学大学院総合理工学報告第 28 巻 第 1 号 1-8 頁 平成 18 年 6 月

Thermocapillary convection in a shallow
annular pool of silicone oil
Wanyuan Shi*1,2 and Nobuyuki Imaishi*3,E-mail of corresponding author: imaishi@cm.kyushu-u.ac.jp(Received April 25, 2006)

Thermocapillary flow in a shallow annular pool (Ri=20 mm, Ro=40 mm and depth d=1.0 mm) of silicone oil (0.65 cSt,
Pr=6.7), heated from the outer wall and cooled at the inner wall, is investigated by numerical simulation. Numerical results
clarified details of pattern formation and oscillatory behavior of hydrothermal waves (HTW) as well as the criticalconditions for their incipience. In non-rotating pool, the critical Marangoni number Mac for the incipience of the HTW is8.396×103(∆Tc=5.03K). The critical azimuthal wave number mc is 27. At slightly super critical conditions, a single group of
HTW propagating in the azimuthal direction is dominant after a long calculation time. Further increase in Ma causes
coexistence of several groups of HTW with different wave numbers and propagation directions. Effect of a slow rotation of
the pool around its central axis destabilizes the basic steady axisymmetric flow against HTW. At Ta=0.322 (corresponding toa rotation rate of 2 r.p.m.), the Mac was determined as 8.096×103 (∆Tc=4.85K) with mc=30. Over a range of Ma from
8.76×103 to 2.0×104, numerical simulations indicate that the HTW propagates azimuthally opposite to the direction of thepool rotation in a rotating coordinate. This phenomenon, i.e. a selection of propagation direction, is caused by the azimuthal
velocity component in the basic flow field induced by the Coriolis force. At Ma=1.34×104, the azimuthal wave number mincreases up to 54 accompanied by an appearance of finger-shaped patterns. At Ma=2.0×104, two groups of HTW withgreatly different wave numbers (m=48 and m=5) coexist and propagate in the opposite azimuthal directions.
Keywords: Hydrothermal wave, thermocapillary convection, rotation, numerical simulation, silicone oil, pattern formation

1. Introduction
In past few decades, thermocapillary convection had
been increasingly interested in many fields, such as
subcooled nucleate pool boiling 1-3), thin-film coating4), and melt growth of single crystals 5-6). In
single-crystal growth processes, such as the
Czochralski (Cz) methods, thermocapillary force
gives significant effects on the stability of melt flow
7-9)
. Spatio-temporal changes of the unstable melt
flow cause inhomogeneous distribution of dopant andpoint defects in the grown crystals. In Cz melt pools,
convection is driven by buoyancy force, thermocapillary force, centrifugal force and Coriolis force.
Combination of these driving forces sometime leadsvery much complex and unstable flow patterns
accompanied by time-dependent three dimensional
oscillations of temperature and concentration fields
10). Spoke patterns on oxide melt surfaces are
explained by the Marangoni effect caused by atemperature gradient perpendicular to the liquidsurface (Marangoni instability of Pearson type) 7-8).
However, Azami et al. suggested the hydrothermal
wave type instability (caused by a horizontal
temperature gradient on the surface) is a possible
mechanism of pattern formation in shallow pool ofsilicon melt in a Cz furnace based on their*1

Interdisciplinary Graduate School of Engineering Sciences,
Kyushu University, Japan*2
College of Power Engineering, Chongqing University, China*3
Institute for Materials Chemistry and Engineering, Kyushu
University, Japan

experiment 10).
The hydrothermal wave (HTW) type flow
instability was first predicted by Smith and Davis(hereafter SD) 11). They investigated stability ofthermocapillary flows in rectangular shallow pool ofliquid induced by a horizontal temperature gradienton the surface and found a new type of oscillatory
instability on the basis of three-dimensional,
time-dependent linear stability analysis (LSA).
In rectangular pools, this type of flow instability
had been confirmed by experiments using various
silicone oils with different Prandtl numbers 12-16). To
eliminate the effects of buoyancy, experiments wereconducted mostly in shallow liquid layers. Riley and
Neitzel 13) and Burguete et al. 14) obtained a stability
limit diagram for thermocapillary flow as a functionof liquid depth. Their results suggested that thecritical Marangoni number (Ma*=γT∆Τd2/µαL)
increases with liquid depth d, where γT is thetemperature coefficient of surface tension 13).
Thermocapillary flow instability in annular liquid
pools had been studied by several researchers.
Kamotani et al. 17-19) conducted a large set of
microgravity experiments on oscillatory thermocapillary flow of silicone oil in open cylindricalcontainers with aspect ratios (As=(ro-ri)/d) close to 1
or 2 (with cylindrical diameters of 12, 20 and 30 mm)
where liquid was heated by a hot solid cylinder
located at the pool axis or a laser beam. Theyobserved two or three lobed surface temperaturepatterns.

2

Thermocapillary convection in a shallow annular pool of silicone oil

Mukolobwiez et al. 20) observed HTW traveling in
the azimuthal direction in a shallow annular channelof silicone oil (Pr=10) (1.7 mm in depth, 10mm inwidth with a mean radius 80 mm) heated from theinner wall. The influence of liquid depth on the
wavelength of the HTW was investigated by
Schwabe et al. 21) through their experiments onthermocapillary flow in annular liquid pools of
ethanol (Pr=17) with thickness ranging from 0.6 mm
to 3.6 mm (inner radius 20mm, outer radius 77mm),
heated at the inner rod. They observed shortwavelength temperature patterns with curved arms
(part of spirals) in shallow liquid pools (d<1.4 mm)
and long-wavelength temperature patterns in deeper
pools (d>1.4 mm). Hoyas et al. 22, 23) conducted linear
stability analysis and obtained a stability diagram forthe incipience of oscillatory flow in annular liquid
pools heated at the inner wall (aspect ratio 2.5 ≤ As≤ 10), taking the surface heat transfer to the ambient
air into account.
For annular pools heated at the outer wall andcooled at the inner wall, Garnier and Chiffaudel 24)
clearly observed HTWs with spiral-like arms in anannular container of silicone oil (Pr=10, 135mm indiameter with an inner cold rod of 8 mm in diameter,
and with a depth of 1.2 mm or 1.9 mm; As=52.9 or
33.4). They observed pulsating, target-like wave
patterns (i.e., coaxial circles traveling outward in theradial direction) dominant only near the cold innerwall, as well as the curved arms of HTW dominant in
the whole area of the liquid pool. Later, Schwabe etal. 25, 26) found that gravity significantly stabilizes thebasic steady radial thermocapillary flow of siliconeoil (Pr=6.8), by comparing their results of on-groundexperiments and microgravity experiments on the
FOTON-12 satellite.
Since it is difficult to understand the details ofthermocapillary convection through experiments,
numerical simulations had been carried out by someresearchers. Xu and Zebib 27) conducted a set ofnumerical simulations of thermocapillary convection
in a rectangular pool and obtained a stability diagram
and wave numbers. Thermocapillary flow in anannular pool of silicone oil was worked out by Sim etal. 28) for the same geometry as that of Schwabe’s
experiment on FOTON-12. On the other hand, Li andcoworkers 29) also conducted numerical simulationsof thermocapillary convection in annular pools withthe same geometry as those of the FOTON-12
experiments. Their stability diagram was comparedwith those of FOTON-12 microgravity experiments.
Recently, Shi and Imaishi 30) conducted a set ofnumerical simulations for silicone oil (Pr=6.7) in anannular container with the same geometry as that of
Schwabe’s experiment 26) but much thinner depth(d=1mm, As=20) and evaluated the effect of gravity
on the critical Marangoni number and wave patterns.
Effects of the Coriolis force on the criticalcondition for the incipience of hydrothermal waves
were analyzed in order to evaluate the effect ofsystem rotation (order of the rotation rate Ω is about
10-3 rad/s) during the microgravity experiments inspace vehicles orbiting around the earth with a periodof 2 hours or so. Zebib 31) conducted a linear stability
analysis of thermocapillary convection in an
infinitely extended rectangular pool. His results
indicate the effect of system rotation on the
incipience of three dimensional secondary flows overa wide range of the Taylor number (Ta=Ωd2/ν)between –50 to 100. For rotation around an axis
perpendicular to the liquid surface, Zebib’s LSA
predicts that the basic flow is slightly stabilizedagainst HTW but destabilized against stationary three
dimensional disturbances if Pr number is smaller
than a critical value, Prc, which depends on Ta, suchas Prc=1 for Ta=3 and Prc=7.1 for Ta=5, respectively.
At Ta=8, HTW disappears and the basic flow
becomes unstable against stationary 3D disturbances
at much smaller Mac than those of SD. Further
increase in Ta increases the critical Marangoni
number. At Ta ≥ 50, the basic flow is stabilizedcompared with SD in the whole range of Pr against
stationary disturbances. Bauer and Eidel 32) analyzed
the axisymmetric thermocapillary flow (the basicflow) in deep cylindrical containers similar to themicrogravity experimental apparatus used by
Kamotani, heated at the outer wall under gravity androtating about its center axis over a range ofTa*=Ωa2/ν = 1 - 400 (here, a is the radius of the outercylinder). The Coriolis force induces an azimuthal
velocity in a rotating annular pool. The azimuthal
velocity near the free surface is large and directs inthe same direction as that of the pool rotation. Far
below the free surface, the azimuthal velocity is small
and directs opposite to that of the pool rotation. Sim
and Zebib 33) numerically investigated the effect oforbiting motion of the space vehicle on the transition
to an oscillatory flow from the basic thermocapillary
convection in an open deep cylindrical annulus
heated from the inner wall with As=1.0 and Pr=30,
exactly corresponding to Kamotani’s microgravity
experiments. They found that Mac decreases withincreasing Taylor number in a range of Ta= 0 - 10.
However, there are few numerical reports on the
HTW in very thin annular pools of silicone oil, suchas 1.0 mm. In this work, we numerically investigate
the thermocapillary convection and the effects of thepool rotation (Ta) on the basic thermocapillary flow
and the critical conditions for the incipience ofhydrothermal waves in a shallow annular pool ofsilicone oil heated from the outer wall. The value of
Taylor number in this work Ta=0.322 (rotating rate of
0.21 rad/s) is much smaller than those in theliteratures 31-33).

平成 18 年

九州大学大学院総合理工学報告第 28 巻第1号

3

2. Model and methods
2.1 Physical and mathematical models
As shown in Fig.1, the model system is an annularpool with an open top free surface and a solid bottom,
with an inner solid wall (radius Ri=20 mm), an outerwall (radius Ro=40 mm) and depth d=1.0 mm. These
geometries are same as those of Schwabe’s
experimental apparatus 26) except for the liquid depth.
The pool can rotate around the vertical central axis
with constant rotating rate. The fluid is silicone oil.
Its physical properties are listed in Table 1.
The outer wall is maintained at a constanttemperature Th while the inner wall at a lower
temperature Tc (Th>Tc and ∆T=Th-Tc). Top andbottom surfaces are considered to be adiabatic. The
silicone oil is regarded as an incompressible
Newtonian fluid with constant properties except for
the temperature dependence of surface tension. The
Marangoni effect (thermocapillary force) acts on thesurface. The flow is considered to be laminar. The
liquid top surface is assumed to be flat andnondeformable.
Under the above assumptions, a mathematicalmodel is expressed by the following nondimensional
equations in a coordinate system co-rotating with thepool around the z axis. In the cases of axisymmetricsteady simulations, we assume that all gradients inthe azimuthal direction are zero.
∇ ⋅V = 0(1)∂V2+ V ⋅ ∇V = −∇ P + ∇ V − 2Ta e z × V
(2)
∂τ
∂Θ
1 2
+ V ⋅ ∇Θ =
∇ Θ
(3)
∂τ
Pr
Here, ez is a unit vector in z direction. The followings
are used for the boundary conditions.
At the top surface:∂u
Ma ∂Θ ∂v
Ma ∂Θ
,
,
=−
=−
∂z
Pr ∂r
∂z
Pr r∂θ

∂Θ= 0.
∂z
At the bottom:
w=(4a, b)(4c,d)∂Θ=0.
(5a, b, c, d)∂z
At the inner wall:u = v = w = 0, Θ = 0.
(6a, b, c, d)
At the outer wall:u = v = w = 0, Θ =1.
(7a, b, c, d)Initial conditions are as follows.
For axisymmetric steady simulations,ln (r ri ).
(8a, b)V = 0, Θ =ln (ro ri )For 3D simulations, either Eqs. 8a, b, or a 3D
results obtained at a smaller Ma was used as an initialcondition. While for the simulations to
u = v = w = 0,

Fig.1 Configuration of the model system.
Table 1 Physical properties of silicone oil
Prandtl number, Pr=6.7
Density, ρ=760 kg/m3
Thermal diffusivity, α=9.7×10-8 m2/s
Kinematic viscosity, ν=6.5×10-7 m2/s
-3
Thermal expansion coefficient, ρT=1.34×10 1/K
Temperature coefficient of surface tension, γT=8.0×10-5 N/m K

determine Mac, axisymmetric steady fields obtainedby axisymmetric simulation under the same Ma wereused.
Here, u, v and w are velocities in r, θ and z
directions, respectively. The length, time τ, pressure
P, velocity and phase (angular) velocity ω of HTW
are scaled by d, d2/ν, νµ/d2, ν/d and ν/d2,
respectively. Non-dimensional temperature is definedas Θ=(T-Tc)/∆T, here ∆T=Th-Tc. The Marangoni,
Prandtl and Taylor numbers are defined as
Ma=γTd∆T/µα, Pr=ν/α, Ta=Ωd2/ν, respectively. Here,µ is the viscosity, γT=-∂γ/∂T the temperature
coefficient of surface tension, γ the surface tension,and α the thermal diffusivity of silicone oil. Ω is theangular velocity of the pool rotation. The stream
1 ∂ψ
1 ∂ψ
function ψ is defined as u = −
.
, w=
r ∂z
r ∂r
2.2 Numerical methods
The equations (1)-(8) were discretized by the control
volume method in the staggered grid system. The
non-uniform grids are constructed to achieve finer
meshes in the regions near the free surface andbottom and also near the sidewalls where theboundary layer develops. The space intervals
between two neighbor grids were changed in asinusoidal function within a half cycle to provide
finer grids near the boundaries. For example, in amesh system with grid points of 202r×603θ×21z, theminimum space intervals in the radius and z direction
were 0.015 and 0.013, respectively. The azimuthal
direction has uniform staggered grids in all cases.
The central difference approximation was introduced
for the diffusion terms. Convective terms weretreated by the second-order upwind scheme excepted
for the azimuthal direction where the QUICK scheme
was adopted. The equations were discretized by afully implicit method in the time marching. The

4

Thermocapillary convection in a shallow annular pool of silicone oil

SIMPLEC algorithm 34) was used to handle thepressure coupling. The preconditioned Bi-CGStab
algorithm 35) was applied to solve the discretized
equations. Convergence at each time step was
assumed when both of the following conditions weresatisfied:| R i |max ≤ 10-Nand |φ i +1 − φ i|max ≤ 10 −4 ,φ i +1

where | R i |max is the absolute maximum residual of
the nondimensional continuity equation among allcontrol volumes in ith iterating step. φ indicates any
one of the variables u, v, w and Θ. N was varied
between 12 and 10 depending on Ma i.e., N=12 wasused for a small Ma and N=10 for a larger Ma. The
numerical simulations were conducted on one PE of
Fujitsu VPP5000/64 supercomputer at the Computing
and Communications Center of Kyushu University.
The grid convergence was carefully checked
through numerical simulations using different grid
systems30). These results confirm the gridconvergences. In order to save computation time,
most of the following results were obtained using agrid 202r×363θ×16z for non-rotation pools. However,the critical Marangoni number and the criticalfrequency were determined on the basis of the results
using a grid system of 202r×603θ×16z. In the rotating
pool, due to larger azimuthal wave number, a finermesh 202r×603θ×21z was used to provide higher
spatial resolutions. The grid convergence in a rotatingcase is shown in Table 2.
Table 2 Mesh dependency of the oscillatory parameters under
Ma=1.0×104(∆T=6.0K), Ta=0.322.
Mesh sizes (r×θ×z)

Wave number mFrequency f (Hz)202×603×16
202×603×21
202×723×21

27
30
30

0.547
0.547
0.547

Fig.2 Surface temperature distributions as a function of radius R.
Dashed line: ∆T=2.0 K. Solid line: ∆T=4.0 K.

(a)

(b)
Fig. 3 Axisymmetric steady flow. (a). The radial velocity
component profiles versus radius r on the surface. (b) The radial
velocity component profiles versus axis z at r=30. Dashed line:
Ma=3.34×103 (∆T=2.0 K). Solid line: Ma=6.68×103 (∆T=4.0 K).

3. Results in a non-rotating pool
3.1 The basic axisymmetric steady flow
In this system, a horizontal temperature gradient is
imposed on the fluid in the radial direction, thus thethermocapillary flow arises at any finite value of Ma.
When Ma is small, the thermocapillary flow is 2D
steady and axisymmetric, called as the “basic flow”.
Numerical simulation results with small values of Ma
are shown in Figs.2-4. Fig.2 shows the radial
distribution of the surface temperature at two
different ∆T. Thermal boundary layers appear near
the inner and outer walls. With increase of Ma (∆T),
the surface temperature gradient (∂T/∂R) in the mid
area increases. Fig.3-a shows the radial velocity u on

Fig.4 Streamlines and isotherms of the basic flow.
(a). Ma=3.34×103, ψmin=-26.47, ψmax=0.013.
(b). Ma=6.68×103, ψmin=-49.63, ψmax=0.029.

the surface as a function of r. The radial velocity
increases with increase in Ma due to the larger radial
temperature gradient on the surface. The radial
velocity u at r=30 is shown in Fig. 3-b as a functionof z. These profiles are similar to those ofthermocapillary flow in a rectangular pool11). Fig. 4
shows the stream function and isotherms. It should be
noted that there appears a strong roll cell near the hotwall driven by the large surface temperature drop
there. In contrast, a very large temperature drop near

平成 18 年

九州大学大学院総合理工学報告

the cold inner wall causes a large surface velocity
peak near the cold wall, followed by a steep
deceleration down to zero at the cold wall as shownin Fig. 3-a.
As Ma increases, a second roll cell is induced next
to the strong roll cell near the hot wall. However, forthe steady basic flow in this pool the maximum
number of the roll cells is two. If we conduct 2-D
simulations under much larger Ma values, 2-D flow
field becomes oscillatory and 7 or 8 roll cells arepropagating in the radial direction from the cold wall
to the hot wall.
3.2 Critical conditions for the incipience of HTW
As Ma exceeds a certain threshold value, the flow
becomes unstable and oscillations start. The flow is
3D oscillatory. In the early period of the simulations,
local radial velocities and local temperatures indicate
temporal oscillations with very small amplitudes.
These oscillations increase their amplitude
exponentially with time following Eq. (9), as
described in our previous paper 30).
(9)X ( τ ) = X 0 exp [( β + i 2 π f ∗ ) τ ]第 28 巻

第1号

Number of HTW groups decreases with the progressof computation. However, two groups of HTW, withthe same wave number but opposite azimuthal
propagating directions, coexist for a considerably
long time but gradually one of them becomes weaker.
After a long time of calculation, at t=1140 s, only one
group of HTW is dominant, as shown in Fig. 5a. A
spatiotemporal diagram (STD: a plot of δΘ3 ( r=25, θ,
z=1,τ ) over a time span of 10 seconds) indicates the
propagation speed and counterclockwise traveling
direction of these patterns as shown in Fig. 5b. Thesesurface temperature patterns with curved arms areapproximately expressed as A(r)sin[m(ωτ+θ+ϕ(r))].
Here, A is the amplitude of the temperaturefluctuation, m the azimuthal wave number,
ω = 2πf ∗ m the phase velocity of the wave
propagation in the azimuthal direction and ϕ thephase shift angle. A and ϕ are function of r. In caseswith small Ma, the HTW patterns are clearlyobservable only in the inner part of the pool, being
faded in the outer region. As Ma increases, HTW
becomes dominant over the whole area of the liquidsurface.

The nondimensional frequency of oscillations f ∗ isdefined as f ∗ = f d 2 ν , where f [Hz] is oscillation
frequency of local values.
For each value of Ma, we can determine thegrowth rate constant β as a slope of a plot of thelogarithm of a local value of surface velocity as afunction of τ. A plot of β vs. Ma determines thecritical Marangoni number for the incipience of 3D
oscillatory flow as the Ma at which β becomes zero,
since the state β=0 corresponds to a marginal stability
limit. The wave number and phase velocity obtainedby a simulation conducted at a Ma close to Mac are
regarded as the critical wave number and criticalphase velocity. Thus determined critical conditions
are: Mac=8.396×10 3 (∆Tc=5.03K), mc=27 and fc*=
0.93. These are consistent within 2% of errors withthe results by LSA 36).
3.3 HTW patterns
In order to express spatial distributions of oscillatory
variables, we introduce the “fluctuation of X” for the
3D case, δX3, where X stands for local temperatures,
velocities and other parameters. δX3 is defined as adeviation of a local value from its average value(averaged over the azimuthal direction), given byδX 3 (r , θ , z , τ ) = X (r , θ , z , τ ) −

1
2π

2π

∫ 0 X (r, θ , z, τ )dθ . (10)Thus defined fluctuation of surface temperatureclearly shows that surface temperature pattern (HTW),
as shown Fig.5a. Generally, in the early periods ofcalculation, several groups of HTW originate atseveral locations and propagate in different directions.

Fig. 5 Snapshot of the surface temperature deviation pattern when
Ma=1.0×104 (∆T=6.0K). Azimuthal wave number m=26.
(a). Surface temperature fluctuation (HTW).
(b). STD of δΘ at r=25 on the surface.

Fig.6 Surface pattern of the temperature deviation (a) and its STD(b) at r=30, Ma=2.0×104(∆T=12.0 K).

Simulation with a larger Ma, such as
Ma=2.0×10 4(∆T=12.0 K), indicates several groups of
HTW with different wave numbers coexisting in thepool with different traveling directions throughout thesimulation (t ≤ 1000s), as shown in Fig.6. Typicalvalues of the wave number are m=28 for the HTW
dominating near the inner wall and m=46 for the
HTW dominating in the outer region. The
coexistence of different groups of HTW is also5

6Thermocapillary convection in a shallow annular pool of silicone oilreported by Garnier et al. 24) although his poolgeometry is different from that of the present system.
However, we did not observe “targetlike (coaxial)
waves” (m=0) propagating outward in the radial
direction, which were reported by Garnier and
Chiffaudel 24) near the inner wall of his annular poolwhich has much larger outer wall and smaller innerwall radiuses (67.5 mm×4 mm). The larger inner wall
radius in the present system could produce smaller
surface temperature gradient near the inner wall thanits critical value for the incipience of the target-likewaves.
A set of simulations with different values of Ma,
ranging from 1.0×104 to 3.0×104, reveals that a single
mode HTW occurs and their wave number slightly
increases with Ma in a slightly super critical region.
However, multi-mode patterns, such as shown in Fig.
6, appear at higher Ma range.

4. Effects of pool rotation
4.1 Basic flow in a rotating pool
If Ma is less than a certain threshold value, thethermocapillary flow in a rotating pool with Ta=0.322(Ω=0.21rad/s in the anticlockwise direction) is steady
and axisymmetric. Numerical results indicated thatsuch a slow pool rotation gives practically no
influence on the basic temperature and radial velocity
distributions. However, there appears an azimuthal
velocity component v in the pool. Fig. 7 shows aradial distribution of v on the surface and also a v
profile in z direction at r=30. Since both the depth

(a)

(b)
Fig.7 Azimuthal velocity distributions for Ta=0.322, Ma=8.35×103(∆T=5.0K). (a) radial distribution of v on the surface. (b) profile ofv in z direction at r=30.

and rotation rate are small, the induced azimuthal
velocity is positive in almost whole part of the liquid
pool, except in limited areas near the bottom close to
the outside wall, not shown in the figure.
4.2 Critical conditions for the incipience of HTW
If Ma exceeds a certain threshold value, the basic
flow can’t keep its steady state against HTW. A set ofnumerical simulations with Ta=0.322 determined thecritical conditions for the incipience of the HTW as
Mac=8.096×10 3 (∆Tc=4.85K), mc=30 and ωc=- 0.176.
These are consistent with the result of LSA 36) withinerrors of less than 1%. It should be noted that thenegative value of ωc indicates the HTW propagates in
the clockwise direction, i.e., in the opposite direction
to the pool rotation. The propagation phase velocity
in a co-rotating coordinate system is ω=-0.176 andthe pool rotation angular velocity is Ta=0.322. Thus,
if observed from a fixed point, the HTW pattern isrotating in the anticlockwise direction with a phasevelocity of 0.146.
It can be seen that in a rotating pool, the Mac is
3.6% smaller than that of non-rotating case. Thisimplies that the pool rotation destabilizes the steadyaxisymmetric basic thermocapillary flow. This trend
is similar to the numerical results of Sim and Zebib
33)
although the heating direction is reversed.
4.3 HTW in a rotation poolIn the non-rotating pool, according to the analysis by
SD 11) and our previous numerical simulations 30),
propagation direction of HTW is either in theclockwise or anticlockwise. Both cases should appear
in equal probability. However, the present numericalsimulations in a rotating pool over a whole range of
Ma values ranging between 8.76×10 3 and 2.0×10 4
indicate that the HTW propagate in the opposite
direction to the pool rotation, as shown in Fig.8-a and8-b. This feature is confirmed by some simulations. A
simulation was conducted using an artificial HTW
patterns propagating in the same direction as the poolrotation. After some calculation time, the propagation
direction was reversed. If the pool rotation direction
is reversed, the HTW also changes its azimuthal
propagation direction. As mentioned in the previous
section, pool rotation induces an azimuthal velocity
component. Due to this additional azimuthal velocity
component of the basic flow, the steady axisymmetricbasic thermocapillary flow becomes less stableagainst a HTW-type disturbances propagating in the
opposite to the pool rotation, i.e., HTW propagating
opposing to the azimuthal velocity.
The result under a condition of Ma=1.34×10 4(∆T=8.0 K) and Ta=0.322 shows a sudden increase of
the azimuthal wave number m up to 54 from m=30
for Ma≤1.0×104, as shown in Fig.9. In this case, thickheavy short arm-like patterns appear near the inner平成 18 年

九州大学大学院総合理工学報告

wall and they tend to break up into two thinnercurved arms. Let us name them as finger-like arms.
Break up occurs at almost 2/3 of the thick wave
patterns. The total wave number of the thicker
patterns near the inner wall is m=32 on the other hand,
m=54 in the outer region. However, the frequenciesof the local temperature oscillation measured at those
two regions show no difference. Then the phasevelocity of the inner waves is larger than that of theouter waves by a factor of ωinner/ ωouter=54/32=1.69.
Due to the difference in the phase velocity between
the inner and outer waves, the finger-like breakingsof the thicker arms are not steady but propagate in theazimuthal direction repeating alternate break-up andrecombination as seen in Fig.9a. After one fingershape wave separates into two thin long waves, thepreceding one quickly catches up and gradually links
to the preceding thick arm. After a very short time oftraveling, a reassembled finger-shape wave comesout again. At the same time, Fig. 9-b (STD) indicates
there is a third group of HTW with m=18 propagating
along anticlockwise direction.
For a condition of Ma=2.0×10 4 (∆T=12.0 K) and
Ta=0.322, the azimuthal wave number m of HTW
dominated in the pool slightly decrease to 48(Fig.10a). STD (Fig.10b) indicates another HTWgroup of m= 5 traveling in the anticlockwise direction,although indistinguishable in Fig.10a.
The azimuthal wave number, propagation phasevelocity of the HTW and their critical conditions in
the rotating and non-rotating pools are summarized in
Fig.11. In most cases, the azimuthal wave number
increases and the phase velocity |ω| decrease by the

第 28 巻

第1号

Fig. 10 Snapshot of the HTW (a) and its STD at r=25 on thesurface (b) when Ma=2.0×104 (∆T=12K), Ta=0.322.

Fig. 11 Ma dependencies of wave number m and phase velocity of
the HTW in rotating and non-rotating pools. The half filled marks
correspond to another group of the HTW. The solid marks
correspond the critical conditions.

pool rotation when Ma ranged from 8.76×10 3 to
3.0×104.

5. Conclusions

Fig.8 Snapshot of the HTW (a) and its STD at r=25 on the surface(b) when Ma=1.0×104(∆T=6.0K) and Ta=0.322. m=30.

Fig.9 Finger-shape HTW when Ma=1.34×104 and Ta=0.322. (a)Snapshot of HTW, m=54 for dominant group, m=32 for innergroup and m=18 for fading group traveling along anticlockwise
direction. (b) STD on a circle with radius r=30 on the surface.

Details of the three dimensional oscillatory thermocapillary flow in a shallow annular pool (Ri=20 mm,
Ro=40 mm and depth d=1.0 mm) of silicone oil (0.65
cSt, Pr=6.7) heated from the outer wall and cooled at
the inner wall, were investigated by numericalsimulation. In non-rotating annular pool, the criticalconditions for the incipience of HTW was determined
to be Mac= 8.396×10 3, mc=27 and ωc=0.217. At
slightly super critical conditions, a single group of
HTW propagating in the azimuthal direction becomesdominant after very long calculation time. At larger
Ma, several groups of HTW with different wave
numbers and different propagation directions coexist
in the pool.
In a rotating pool at a rate of Ta=0.322, the basic
2-D steady flow becomes less stable against 3-D
oscillatory disturbances propagating in the opposite

7

8

Thermocapillary convection in a shallow annular pool of silicone oil

direction to the pool rotation. The critical conditions
are determined as Mac=8.096×10 3, mc=30 and ωc=-0.176. With Ma from 8.76×103 to 2.0×104, numericalsimulations indicate that the HTW always propagatesin the direction opposite to the pool rotation. This iscaused by the azimuthal component of the basic flow
induced by the Coriolis force. The result at
Ma=1.34×104 showed significant increase of wave
number up to m=54 in the outer region. The
finger-shaped patterns appear near the inner wall, thisis caused by the instability of wave pattern near the
inner wall. The result at Ma=2.0×104 showed twogroups of HTW with different wave numbers, i.e.,
m=48 propagating in the clockwise direction and m=5
propagating in the clockwise direction.
Numerical results indicate that the pool rotation
destabilizes the basic steady axisymmetric thermocapillary flow. At Ta=0.322 the critical temperature
difference is decreased about 3.6% from that for
non-rotating case.

23)

References

24)

R. Marek, J. Straub, Int. J. Heat Mass and Transfer, 44,
pp.619-632, 2001.
2) C. Reynard, M. Barthes, R. Santini and L. Tadrist, Experimental
Thermal and Fluid Science, 29, pp.783-793, 2005
3) C. Buffone, K. Sefiane and W. Easson, Physical Review E,
71, pp.1-8, 2005
4) P. Carles, A. Cazabat, J. Colloid and Interface Science, 157,
pp.196-201, 1993.
5) C. E. Chang, W. R. Wilcox, Int. J. Heat and Mass Transfer,19, pp. 355-366, 1976.
6) D. Scwabe, A. Scharmann, F. Pleisser, R. Oeder, J. Crystal
Growth, 43, pp. 305-312, 1978.
7) D. C. Miller and T. L. Pernell, J. Crystal Growth, 58,
pp.253-260, 1982.
8) K. -W. Yi, K. Kakimoto, M. Eguchi, M. Watanabe, T. Shyoand T. Hibiya, J. Crystal Growth, 144, pp. 20-28, 1994.
9) K. Kakimoto, H. Ozoe, Computational Materials Science,
10, pp.127-133, 1998.
10) T. Azami, S. Nakamura, M. Eguchi, T. Hibiya, J. Crystal
Growth, 233, pp. 99-107,2001.

11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)

1)

25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)

M. K. Smith, S. H. Davis, J. Fluid Mech., 132, pp.119-144, 1983.
F. Daviaud, J. M. Vince, Phys. Rev. E, 48, pp.4432-4436, 1993.
R. J. Riley, G. P. Neitzel, J. Fluid Mech., 359, pp.143-164, 1998.
J. Burguete, N. Mukolobwiez, F. Daviaud, N. Garnier, A.
Chiffaudel, Physics of Fluids, 13, pp.2773-2787, 2001.
M. A. Pelacho, J. Burguete, Phys. Rev. E, 59, pp.835-840, 1999.
B. Ezersky, A. Garcimartin, H. L. Mancini, and C.
Perez-Garcia, Phys. Rev. E, 48, pp.4414-4422, 1993.
Y. Kamotani, S. Ostrach and A. Pline, Physics of Fluids, 6, pp.
3601-3609,1994.
Y. Kamotani, S. Ostrach and J. Masud, Int. J. Heat and Mass
Transfer, 42, pp.555-564, 1998.
Y. Kamotani, Adv. Space Res., 24, pp. 1357-1366, 1999.
N. Mukolobwiez, A. Chiffaudel, F. Daviaud, Phys. Rev.
Letters, 80, pp.4661-4664, 1998.
D. Schwabe, U. Moller, J. Schneider, A. Scharmann, Phys.
Fluids A, 4, pp.2368-2381, 1992.
S. Hoyas, H. Herrero, A. M. Mancho, Phys. Rev. E, 66,
pp.057301-1-4, 2002.
S. Hoyas, H. Herrero, A. M. Mancho, J. Phys. A: Math. Gen.,35, pp.4067-4083, 2002.
N. Garnier, A. Chiffaudel, The European Phys. J. B, 19,
pp.87-95, 2001.
D. Schwabe, S. Benz, Adv. Space Rec. 29, pp. 629-638, 2002.
D. Schwabe, A. Zebib, B-C. Sim, J. Fluid Mech., 491, pp.
239-258, 2003.
J. Xu and A. Zebib, J. Fluid Mech., 364, pp. 187-209, 1998.
B. C. Sim, A. Zebib, D. Schwabe, J. Fluid Mech., 491, pp.
259-274, 2003.
Y. R. Li, L. Peng, Y. Akiyama, N. Imaishi, J. Crystal Growth,259, pp.374-387, 2003.
W. Y. Shi, N. Imaishi, J. Crystal Growth, 290, pp.280-291, 2006.
A. Zebib, Physics of Fluids, 8, pp.3209-3211, 1996.
H. F. Bauer, W. Eidel, Heat and Mass Transfer, 34, pp.79-90, 1998.
B. C. Sim, A. Zebib, Physics of Fluids, 14, pp.225-231, 2002.
J. P. Van Doormaal, G. D. Raithby, Numerical Heat Transfer,7, pp.147-163, 1984.
H. A. Van Der Vorst, SIAM J. Sci. Stat. Comput., 13, pp.
631-644, 1992.
W. Y. Shi, M. K. Ermakov, N. Imaishi, J. Crystal Growth, inpress.