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Bounds for heat transport in a porous layer

Published online by Cambridge University Press:  29 March 2006

V. P. Gupta  and
D. D. Joseph
V. P. Gupta
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota Present address: The Mitre Corporation, McLean, Virginia 22101, U.S.A.
D. D. Joseph
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota
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Abstract

Strongly nonlinear heat transport across a porous layer is studied using Howard's (1963) variational method. The analysis explores a bifurcation property of Busse's (1969) multi-a solution of this variational problem and complements the 1972 study of Busse & Joseph by further restricting the fields which are allowed to compete for the maximum heat transported a t a given temperature difference. The restriction arises, as in the case of infinite Prandtl number convection studied by Chan (1971), from letting a parameter tend to infinity from the outset; here, however, the parameter which is assumed infinitely large (the Prandtl-Darcy number) is actually seldom smaller than O(107).

The theoretical bounding heat-transport curve is computed numerically. The maximizing Nusselt number (Nu) curve is given a t first by a functional of the single-a solution; then this solution bifurcates and the Nusselt number functional is maximized for an interval of Rayleigh numbers (R) by the two-a solution. The agreement between the numerical analysis and recent experiments is striking. The theoretical heat-transport curve is found to be continuously differentiable but has piecewise discontinuous second derivatives.

The results of an asymptotic (R → ∞) analysis following Chan (1971) are in qualitative agreement with the results of numerical analysis and give the asymptotic law Nu = 0.016R. This law is consistent with the result of the porous version of the well-known dimensional argument leading to the one-third power law for regular convection. The asymptotic results, however, do not appear to be in good quantitabive agreement with the numerical results.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 57 , Issue 3 , 20 February 1973 , pp. 491 - 514
Copyright
© 1973 Cambridge University Press

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