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On finite-amplitude patterns of convection in a rectangular-planform container

Published online by Cambridge University Press:  26 April 2006

P. G. Daniels  and
M. Weinstein
P. G. Daniels
Affiliation:
Department of Mathematics, City University, Northampton Square, London EC1V OHB, UK
M. Weinstein
Affiliation:
Raphael, PO Box 2250, Haifa 31021, Israel
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Abstract

This paper considers the development of finite-amplitude patterns of convection in rectangular-planform containers. The horizontal dimensions of the container are assumed to be large compared with the critical wavelength of the motion. An interaction between rolls parallel and perpendicular to the lateral boundaries is modelled by a coupled pair of nonlinear amplitude equations together with appropriate conditions on the four lateral boundaries. At Rayleigh numbers above a critical value a steady-state solution is established with rolls parallel to the shorter lateral sides, consistent with the predictions of linear theory. At a second critical value this solution becomes unstable to cross-rolls near the shorter sides and a new steady state evolves. This consists of the primary roll pattern together with regions near the shorter sides where there is a combination of rolls parallel and perpendicular to the boundary.

Analytical and numerical methods are used to describe both the evolution and steady-state structure of the solution, and a comparison is made with the results of full numerical simulations and experiments.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 317 , 25 June 1996 , pp. 111 - 127
Copyright
© 1996 Cambridge University Press

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