The effects of imposed nonlinear oscillatory shear upon the onset of Marangoni-Bernard convection, as predicted by linear theory, in a layer of liquid with a deformable free surface were reported upon by Or and Kelly for small amplitude oscillations. Depending on the operating conditions, either stabilization or destabilization might occur. The aim of the current paper is to report the results for finite amplitude imposed oscillations so that the actual amount of stabilization or destabilization can be determined for prescribed operating conditions

Kelly, Robert E.

Or, Arthur C.

English

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STUDIES IN THERMOCAPILLARY CONVECTION OF THE MARANGONI-
BENARD TYPE
R. E. Kelly
A. C. Or
Mechanical and Aerospace Engineering Department
University of California
Los Angeles, CA 90095-1597
ABSTRACT G
The effects of an imposed nonplanar, oscillatory shear upon the onset of Marangoni-
Benard convection, as predicted by linear theory, in a layer of liquid with a deformable free
surface were reported upon by Or and Kelly (ref. 1) for small amplitude oscillations.
Depending upon the operating, conditions, either stabilization or destabilization might occur.
The aim of the current paper ts to report results for finite amplitude imposed oscillations so
that the actual amount of stabilization or destabilization can be determined for prescribed
operating conditions.
INTRODUCTION
The small amplitude analysis of Or and Kelly (ref. 1) predicts that the finite
wavelength mode of thermocapillary instability, which tends to be the critical mode on earth,
can be stabilized by imposing an oscillatory, nonplanar shear on the fluid layer. However, the
same shear has a destabilizing influence upon the long wavelength mode associated with
surface deformation, which tends to be the most unstable mode under microgravity
conditions. Although the small amplitude analysis is useful for an initial approach to the
problem in view of the many nondimensional parameters involved, a fully numerical
approach is necessary in order to predict how much stabilization or destabilization is obtained
for finite amplitude oscillations. Therefore, an expansion of the solution was made in terms of
a Fourier representation in a plane parallel to the bounding wall and Chebyshev functions in
the direction normal to it (using the Tau method). This procedure yields a set of ordinary
differential equations with time-periodic coefficients that is then investigated by use of
Floquet theory in order to determine the stability boundaries. Only one figure showing some
preliminary results was shown in 1994 (Fig. 5 of ref. 2); much more extensive results have
since been obtained and are now reported.
PROBLEM DESCRIPTION
A layer of Boussinesq fluid with mean thickness h rests above a solid horizontal plate
that can perform simple harmonic motion along each of the axes defining the plane of the
wall. A difference in phase (8) between the two oscillations creates an oscillatory nonplanar
shear field, as defined for the basic state by the well-known solution of Stokes. The layer is
heated at the isothermal wall and, since viscous dissipation is neglected, the temperature of the
basic state is governed by the steady conduction equation and so varies linearly with distance
normal to the wall. The surface of the layer, in general, is allowed to deform in accordance
with the free surface boundary conditions, and heat transfer there is characterized by a Biot
number (Bi).
The formulation of the problem has been given by Or and Kelly (ref. 1) for arbitrary
amplitude oscillations and so is not repeated here. The numerical problem consists of solving
eqs. (22) and (23) of ref. 1 subject to the boundary conditions given there as eqs. (24) - (28).
For lack of space, neither these equations nor details of the numerical solution are given here.
The governing nondimensional parameters will be described below.
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RESULTS
Besides the phase angle 8, the other nondimensional parameters governing the basic
state's velocity are the nondimensional frequency fl - ((.oh 2 / 2 v), where o_ is the dimensional
frequency and v is the fluid's kinematic viscosity, a Reynolds number (Re) that involves the
magnitude of the wall's velocity in one direction, and a parameter _. that is a ratio of the
magnitude of the two components of the wall's motion. For _ - ;r/2 , the effects of the
nonplanar aspect of the fluid motion are the greatest, and _. is a parameter that governs the
pattern selection, which will not be discussed here.
Additional nondimensional physical parameters entering into the equations are the
Prandfl number Pr = v  r(l¢ being the thermal diffusivity), the Rayleigh number
Ra = agATh 3 / vr (a being the coefficient of thermal expansion, g gravity, and ATthe
temperature difference across the layer in the basic state), the Marangoni number
M= 7ATh/pvr (7 being proportional to the rate of change of surface tension with
temperature), the Bond number Bo = pgh2/o (o being the surface tension) and the Crispation
number C = ,ovK" / oh. We shall assume that Ra << 1 so that the effects of buoyancy can be
neglected.
As fl --_ 0, the shear in the basic flow vanishes whereas, for 13>> 1, the thickness of
the Stokes layer is much less than h. In either case, no change occurs in, say, the critical value
of M c relative to the case Re = 0. Hence, an optimal value of 13 exists which has a value
between one and two, as shown in Fig. 1.
Some typical values for a case typical of conditions on earth (Bo = 0.15) are shown in
Figure 2. For the given conditions, the finite wavenumber mode is critical when Re = 0. Fig. 1
indicates that this mode can be stabilized substantially as Re increases. However, the nearly
vertical solid line near Re = 190 indicates that a limit to the stabilization exists. This line is the
stability boundary for long wavelength disturbances which are much more stable at low values
of Re. We therefore conclude that stabilization of the system is limited by a pronounced
destabilization of the long wavelength mode at sufficiently large values of Re. This
destabilization is due to the action of the unsteady shear at the free surface and has been
discussed for the isothermal case by Yih (ref. 3) and Or and Kelly (ref. 4).
As the value of g diminishes relative to the value on earth (go), the nearly vertical line
in Fig. 2 moves to the left and both the maximum amount of stabilization and the Reynolds
number at which this is achieved diminish. These cut-off values are shown in Fig. 3 as a
function of g/go" In particular, as g/go -_ 0 stabilization becomes impossible and, instead,
destabilization occurs for all Re > 0. A sequence of neutral stability boundaries for Bo =
0.01 is given in Fig. 4 as Re increases. In Figs. (4a) and (4b), Mc increases with Re because
the finite wavenumber mode is critical and tends to be stabilized by the oscillations. However,
the long wavelength mode (k _ 0) is also becoming less stable. When Re = 70, the long
wavelength mode is already critical and destabilization of the layer occurs. For Re = 100, the
layer is unstable even when the wall is cooled relative to the ambient.
Because the destabilization of the long wavelength mode is associated with the
interfacial instability discussed first by Yih (ref. 3), the isothermal case was explored to a
greater extent than done by Yih. As Fig. 5 indicates, more than one unstable, long wavelength
region of instability occurs as _ increases, but the regions are separated from each other.
However, when finite wavelength disturbances are also considered, the corridors of stability
296
are eliminated. Current research is aimed at clarifying the connection between these new finite
wavenumber modes existing for the isothermal case with the finite wavenumber modes
already established for the thermocapillary case.
REFERENCES
1. Or, A. C., and Kelly, R. E., Int. J. Heat Mass Transfer 38 (1995), 2269-2279.
2. Kelly, R. E., and Or, A. C., NASA CP 3276 (1994), 15-20.
3. Yih, C. S., J. Fluid Mech. 31 (1968), 737-751.
4. Or, A. C., and Kelly, R. E., Bull. APS 40 (1995), 1949 (abstract only).
700
600
400
_t
/// \\
100 _m
0.5 1 1.6 2 _6 $ a.6 4 4.6 6
m
Figure !: Critical Marangoni number versus nondimensional frequency for various Re;
Pr= 7, Cr= 2 x 10"6, Bo = 0.15, Bi = 0.1. Insert shows variation of critical
wavenumber with frequency for one Reynolds number.
297
Figure2:
91X)
6OO
500
_l 40O
)00
i
L A
50 100
1
!
I
i
150 _00
Critical Marangoni number as a function of Reynolds number for 13= 1.1
(solid), 1.4 (short dashes) and 1.7 (long dashes); Pr = 7, Cr = 2 x 10 -6,
Bo = 0.15, Bi = 0.1.
Figure 3:
800
w • i i • • • • •
.oo........_ ....i ........i .......i .......i ......i ....i .........i ......
i ! i i ! i • ' : :
_,°° ...... I
_oo .......i ........._A..,._,.-....... .........i ........ -......... i.........i........._........
OO'B O_
0 ' i i , , , , , , ,
0 0.1 02 0.8 0A 0.1J 0.8 0.7 0.0 0.0
The cut-off values of Reynolds number (lower curve) and Marangoni number
(upper curve) for maximum stabilization at 13= 1.1 as a function of g/go;
Pr=7, Cr=2x 10.6 ,Bo=0.15atg=go, Bi=0.1,13= 1.1.
298
Figure 4:
4OOO
8000
• 0000
1000
0
O
(8) Re-tO
i
OJI '1 1Ji
.-4000
80OO
118000
1000
(b) Re.SO
i
........... . ............. i .............
0.1 1 1.1
4000
8000
•SO00
1000
I
0
(e) Roo?O
i :
O,S 1
k
(d) Ro-100
.o.t .......+ ..........!.............
,OOOl|V..!.. .........+.............
.,ooo,"°°°V...........!! .........+i.............
0 OJ 1 IJ
k
Critical Marangoni number as a function of wavenumber for various
Reynolds numbers for 13 = 1.1 (solid), 1.4 (dashes), 1.7 (dash-dot); Pr = 7,
Cr = 2 x 10 "6, Bo = 0.01, Bi = 0.1.
10u
lO'
10 +
Figure 5:
I
10o0 1 2 I 4 8 O 7 O O 10
Scaled critical Reynolds number as a function of frequency for the isothermal
case with Bo = 0.1; _ = gh3/2v 2. Black dots mark points at which finite k
neutral curves bifurcate from the k --o 0 neutral curves shown as loops.
299



Studies in Thermocapillary Convection of the Marangoni-Benard Type

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