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Time-periodic convection in porous media: the evolution of Hopf bifurcations with aspect ratio

Published online by Cambridge University Press:  26 April 2006

D. S. Riley  and
D K. H. Winters
D. S. Riley
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
D K. H. Winters
Affiliation:
Theoretical Studies Department, Harwell Laboratory, Didcot, Oxon OX11 ORA, UK
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Abstract

Techniques of bifurcation theory are used to study the porous-medium analogue of the classical Rayleigh-Bénard problem, Lapwood convection in a two-dimensional saturated porous cavity heated from below. The focus of the study concerns the destabilization, through symmetry-preserving Hopf bifurcations, of the various stable convective flow patterns that can form in a rectangular cavity. We show how the limits of stability of steady convection in a porous medium can be determined by bifurcation techniques that locate Hopf bifurcations, and we predict a surprisingly complex evolution of the Hopf bifurcation along the unicellular branch as the aspect ratio varies. The continuation methods that we adopt reveal interactions of Hopf bifurcations with limit points that signal complicated dynamical behaviour for certain container sizes. The study demonstrates the role of Hopf bifurcation in destabilizing completely the unicellular flow at aspect ratios greater than 2.691. A simple relationship between symmetry-preserving Hopf bifurcations from the alternative steady flows is also derived, and used to define upper limits on the stability of the alternative steady flows and the thresholds for oscillatory convection as a function of aspect ratio.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 223 , February 1991 , pp. 457 - 474
Copyright
© 1991 Cambridge University Press

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