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On the structure of cellular solutions in Rayleigh–Bénard–Marangoni flows in small-aspect-ratio containers

Published online by Cambridge University Press:  26 April 2006

Henk A. Dijkstra
Henk A. Dijkstra
Affiliation:
Mathematical Sciences Institute, Cornell University, Ithaca, NY 14853, USA Present address: Institute of Meteorology and Oceanography, University of Utrecht, Princetonplein 5, 3584 CC Utrecht, The Netherlands.
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Abstract

Multiple steady flow patterns occur in surface-tension/buoyancy-driven convection in a liquid layer heated from below (Rayleigh–Bénard–Marangoni flows). Techniques of numerical bifurcation theory are used to study the multiplicity and stability of two-dimensional steady flow patterns (rolls) in rectangular small-aspect-ratio containers as the aspect ratio is varied. For pure Marangoni flows at moderate Biot and Prandtl number, the transitions occurring when paths of codimension 1 singularities intersect determine to a large extent the multiplicity of stable patterns. These transitions also lead, for example, to Hopf bifurcations and stable periodic flows for a small range in aspect ratio. The influence of the type of lateral walls on the multiplicity of steady states is considered. ‘No-slip’ lateral walls lead to hysteresis effects and typically restrict the number of stable flow patterns (with respect to ‘slippery’ sidewalls) through the occurrence of saddle node bifurcations. In this way ‘no-slip’ sidewalls induce a selection of certain patterns, which typically have the largest Nusselt number, through secondary bifurcation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 243 , October 1992 , pp. 73 - 102
Copyright
© 1992 Cambridge University Press

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