The Brinkman Model for Natural Convection in a Shallow Porous Cavity With Uniform Heat Flux

The Brinkman model is used to study steady-stale natural convection in a shallow porous cavity with uniform heating and cooling through opposite walls. The horizontal boundaries are considered rigid-rigid, rigid-free, or free-free, and heating is through the bottom or the side wall. An approximate solution is obtained by assuming parallel flow in the core region of the cavity and a numerical solution by solving the complete governing equations. The flow and heat transfer variables are obtained in terms of the Darcy-Rayleigh number R and the Darcy number Da. The critical Darcy-Raleigh number for the onset of convection in a bottom-heated cavity is predicted. The results for a viscous fluid (Da → ∞) and the Darcy porous medium (Da → 0) emerge from the present analysis as limiting cases.