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A nonlinear stability analysis of the Bénard–Marangoni problem

Published online by Cambridge University Press:  20 April 2006

A. Cloot  and
G. Lebon
A. Cloot
Affiliation:
Department of Mechanics, University of Liège, B5, Sart Tilman, 4000 Liège, Belgium
G. Lebon
Affiliation:
Department of Mechanics, University of Liège, B5, Sart Tilman, 4000 Liège, Belgium
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Abstract

A nonlinear analysis of Bénard–Marangoni convection in a horizontal fluid layer of infinite extent is proposed. The nonlinear equations describing the fields of temperature and velocity are solved by using the Gorkov–Malkus–Veronis technique, which consists of developing the steady solution in terms of a small parameter measuring the deviation from the marginal state. This work generalizes an earlier paper by Schlüter, Lortz & Busse wherein only buoyancy-driven instabilities were handled. In the present work both buoyancy and temperature-dependent surface-tension effects are considered. The band of allowed steady states of convection near the onset of convection is determined as a function of the Marangoni number and the wavenumber. The influence of various dimensionless quantities like Rayleigh, Prandtl and Biot numbers is examined. Supercritical as well as subcritical zones of instability are displayed. It is found that hexagons are allowable flow patterns.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 145 , August 1984 , pp. 447 - 469
Copyright
© 1984 Cambridge University Press

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