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The transition from roll to square-cell solutions in Rayleigh–Bénard convection

Published online by Cambridge University Press:  20 April 2006

D. R. Jenkins  and
M. R. E. Proctor
D. R. Jenkins
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW, England
M. R. E. Proctor
Affiliation:
Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW, England
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Abstract

We consider three-dimensional finite-amplitude thermal convection in a fluid layer with boundaries of finite conductivity. Busse & Riahi (1980) and Proctor (1981) showed that the preferred planform of convection in such a system is a square-cell tesselation provided that the boundaries are much poorer conductors than the fluid, in contrast to the roll solutions which are obtained for perfectly conducting boundaries. We determine here the conductivity of the boundaries at which the preferred planform changes from roll to square-cell type. We show that, for low-Prandtl-number fluids (e.g. mercury), square-cell solutions are realized only when the boundaries are almost insulating; while, for high-Prandtl-number fluids (e.g. silicone oils), square-cell solutions are stable when the boundaries have conductivity comparable to that of the fluid.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 139 , February 1984 , pp. 461 - 471
Copyright
© 1984 Cambridge University Press

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