Natural convection in a shallow porous cavity—the Brinkman model

Abstract

The natural convection in a shallow porous rectangular cavity with differentially heated sidewalls is examined using the Brinkman model. The heat transfer rate through the cavity is determined in terms of a Nusselt number, in the limit of vanishingly small aspect ratio. Two types of boundary conditions are considered. Case I deals with a cavity with all rigid boundaries so that the no-slip boundary conditions can be imposed. In case II, the cavity has a free upper surface. The present analysis shows that the Brinkman model and Darcy's law give virtually the same result for the heat transfer rate when the Darcy number, based on the depth of the cavity, is less than the order of 10 −4. We also find that the presence of a free surface can significantly increase the heat transfer rate through the cavity, especially when the permeability of the medium is high.