Marangoni instability in a finite container with a deformable interface in the absence of gravity has been investigated. It is shown that the critical Marangoni number Macr is a non-monotonic function of the length of the container. Two different physical mechanisms driving convection are indicated. The advection of heat is essential for the first, advective (“classical”) mechanism that gives rise to short wavelength modes. The interface deformation is essential for the second mechanism that gives rise to long wavelength modes. If the container is sufficiently long, the second mechanism leads to an unconditional instability. The available results suggest that the unconditional instability leads to segmentation of the interface.

Issue Section:
Natural and Mixed Convection
Keywords:
convection, surface tension, flow instability, confined flow, Heat Transfer, Instability, Natural Convection, Surface Tension, Thermocapillary
Topics:
Cavities, Containers, Convection, Deformation, Flow (Dynamics), Surface tension, Wavelength, Temperature, Stability
1.
Be´nard
,
H.
,
1900
, “
Les Tourbillons Cellulaires dans une Nappe Liquide
,”
Rev. Gen. Sci. Pures Appl.
,
11
, pp.
1261
–
1271
;
2.
11
, pp.
1309
–
1328
.
1.
Block
,
M. J.
,
1956
, “
Surface Tension as the Cause of Benard Cells and Surface Deformation in a Liquid Film
,”
Nature (London)
,
178
, pp.
650
–
651
.
2.
Pearson
,
J. R. A.
,
1958
, “
On Convection Cells Induced by Surface Tension
,”
J. Fluid Mech.
,
4
, pp.
489
–
500
.
3.
Scriven
,
L. E.
, and
Sterling
,
C. V.
,
1964
, “
On Cellular Convection Induced by Surface Tension Gradients: Effect of Mean Surface Tension and Viscosity
,”
J. Fluid Mech.
,
19
, pp.
321
–
340
.
4.
Smith
,
M. K.
,
1966
, “
On Convective Instability Induced by Surface Tension Gradients
,”
J. Fluid Mech.
,
24
, pp.
401
–
414
.
5.
Davis
,
S. H.
, and
Homsy
,
G. M.
,
1980
, “
Energy Stability Theory for Free Surface Problems: Buoyancy-Thermocapillary Layers
,”
J. Fluid Mech.
,
98
, pp.
527
–
553
.
6.
Davis, S. H., 1983, “Rupture of Thin Liquid Films,” in Waves on Fluid Interfaces, R. E. Meyer, ed., Academic Press, New York, pp. 291–302.
7.
Davis
,
S. H.
,
1987
, “
Thermocapillary Instabilities
,”
Annu. Rev. Fluid Mech.
,
19
, pp.
403
–
435
.
8.
Goussis
,
D. A.
, and
Kelly
,
R. E.
,
1990
, “
On the Thermocapillary Instabilities in a Liquid Layer Heated from Below
,”
Int. J. Heat Mass Transf.
,
33
, pp.
2239
–
2245
.
9.
Krishnammoorthy
,
S.
,
Ramaswamy
,
B.
, and
Joo
,
S. W.
,
1995
, “
Spontaneous Rupture of Thin Liquid Films due to Thermocapillary: A Full Scale Numerical Simulation
,”
Phys. Fluids
,
7
, pp.
2291
–
2293
.
10.
VanHook
,
S. J.
,
Schatz
,
M. F.
,
Swift
,
J. B.
,
McCormick
,
W. D.
, and
Swinney
,
H. L.
,
1997
, “
Long-Wavelength Surface-Tension-Driven Benard Convection: Experiment and Theory
,”
J. Fluid Mech.
,
345
, pp.
45
–
78
.
11.
Winters
,
K. H.
,
Plesser
,
Th.
, and
Cliffe
,
K. A.
,
1988
, “
The Onset of Convection in a Container due to Surface Tension and Buoyancy
,”
Physica D
,
29
, pp.
387
–
401
.
12.
Dijkstra
,
H. A.
,
1992
, “
On the Structure of Cellular Solution in Rayleigh-Benard-Marangoni Flows in Small-Aspect-Ration Containers
,”
J. Fluid Mech.
,
243
, pp.
73
–
102
.
13.
Dijkstra
,
H. A.
,
1995
, “
Surface Tension Driven Cellular Patterns in Three-Dimensional Boxes—Linear Stability
,”
Microgravity Sci. Technol.
,
7
, pp.
307
–
312
.
14.
Dijkstra
,
H. A.
,
1995
, “
Surface Tension Driven Cellular Patterns in Three-Dimensional Boxes—A Bifurcation Study
,”
Microgravity Sci. Technol.
,
7
, pp.
307
–
312
.
15.
Dauby
,
P. D.
, and
Lebon
,
G.
,
1996
, “
Marangoni-Benard Instability in Rigid Rectangular Containers
,”
J. Fluid Mech.
,
329
, pp.
25
–
64
.
16.
Perez-Garcia
,
C.
,
Echebarria
,
B.
, and
Bestehorn
,
M.
,
1998
, “
Thermal Properties in Surface-Tension-Driven Convection
,”
Phys. Rev. E
,
57
, pp.
475
–
481
.
17.
Hamed
,
M.
, and
Floryan
,
J. M.
,
2000
, “
Marangoni Convection: Part 1—Cavity with Differentially Heated Side Walls
,”
J. Fluid Mech.
,
405
, pp.
79
–
110
.
18.
Hamed
,
M.
, and
Floryan
,
J. M.
,
1998
, “
Numerical Simulation of Unsteady Nonisothermal Capillary Interfaces
,”
J. Comput. Phys.
,
145
, pp.
110
–
140
.
19.
Chen
,
C.
, and
Floryan
,
J. M.
,
1994
, “
Numerical Simulation of Non-isothermal Capillary Interfaces
,”
J. Comput. Phys.
,
111
, pp.
183
–
193
.
20.
Koschmieder
,
E. L.
, and
Prahl
,
S. A.
,
1990
, “
Surface-Tension-Driven Benard Convection in Small Containers
,”
J. Fluid Mech.
,
215
, pp.
571
–
583
.
21.
Floryan
,
J. M.
, and
Chen
,
C.
,
1994
, “
Thermocapillary Convection and Existence of Continuous Layer in the Absence of Gravity
,”
J. Fluid Mech.
,
277
, pp.
303
–
329
.
Copyright © 2001
 by ASME
You do not currently have access to this content.