Bounds for heat transport in a porous layer

Abstract

Bounds on the heat transport in a porous layer are derived using the variational method of Howard (1963) and Busse (1969b). The relatively simple structure of the variational problem in the case of porous convection allows one to formulate the theory more simply and to investigate some of the mathematical questions posed by the earlier work. A precise characterization of the solution with N wavenumbers is given. The variational problem is solved exactly among functions with a single overall wavenumber and this solution is in good agreement with a nonlinear perturbation solution of the governing equations and with experiments. An N-wavenumber solution is constructed for large Nusselt numbers by boundary-layer methods. The asymptotic solution is compared with a numerical solution of the problem for N = 2. The comparison supports the boundary-layer assumptions introduced in the asymptotic analysis.