The Brinkman model for natural convection in a porous layer: Effects of nonuniform thermal gradient

Abstract

The effects of nonlinear temperature distribution on stability and natural convection in a horizontal porous layer, with heating from below, are investigated using the Brinkman model. The horizontal boundaries are either rigid/rigid, rigid/stress-free, or stress-free/stress-free. Constant-flux thermal boundary conditions are considered for which the onset of convection is known to correspond to a vanishingly small wavenumber. An analytical solution for the flow and heat transfer variables, based on a parallel flow assumption, is obtained in terms of the Darcy-Rayleigh number, R, and the Darcy number, Da. The critical Rayleigh number for the onset of convection arising from sudden heating or cooling at the boundaries is also predicted. Various basic temperature profiles are considered. Closed form solutions are obtained from which results for a viscous fluid ( Da→ ∞) and the Darcy porous medium ( Da → 0) emerge from the present analysis as limiting cases.