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Natural convection in porous media: effect of weak dispersion on bifurcation

Published online by Cambridge University Press:  26 April 2006

Xiaowei S. He  and
John G. Georgiadis
Xiaowei S. He
Affiliation:
Department of Mechanical Engineering and Materials Science. Duke University, Durham, NC 27706, USA
John G. Georgiadis
Affiliation:
Department of Mechanical Engineering and Materials Science. Duke University, Durham, NC 27706, USA
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Abstract

We use weakly nonlinear analysis via a two-parameter expansion to study bifurcation of conduction into cellular convection of an internally heated fluid in a porous medium that forms a horizontal layer between two isothermal walls. The Darcy–Boussinesq model of convection is enhanced by including two nonlinear terms: (i) quadratic (Forchheimer) drag; and (ii) hydrodynamic dispersion enhancement of the thermal conductivity described by a weak linear relationship between effective conductivity and local amplitude of filtration velocity. The impact of the second term on the shape of the bifurcation curve for two-dimensional rolls is profound in the presence of uniform volumetric heating. The resulting bifurcation structure is unlike any pitchfork bifurcations typical of the classical Bénard problem. Although direct experimental validation of the novel bifurcation is not available, we would like to register it as an alternative or a supplement to models of small imperfections, and as an attempt to account for the scatter of observed critical values for the first bifurcation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 216 , July 1990 , pp. 285 - 298
Copyright
© 1990 Cambridge University Press

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