A progressive asymptotic approach procedure for simulating steady-state natural convective problems in fluid-saturated porous media

In a fluid-saturated porous medium, a change in medium temperature may lead to a change in the density of pore-fluid within the medium. This change can be considered as a buoyancy force term in the momentum equation to determine pore-fluid flow in the porous medium using the Oberbeck-Boussinesq approximation model. The momentum equation used to describe pore-fluid flow in a porous medium is usually established using Darcy's law or its extensions. If a fluid-saturated porous medium has the geometry of a horizontal layer, and is heated uniformly from the bottom of the layer, then there exists a temperature difference between the top and bottom boundaries of the layer. Since the positive direction of the temperature gradient due to this temperature difference is opposite to that of the gravity acceleration, there is no natural convection for a small temperature gradient in the porous medium. In this case, heat energy is solely transferred from the high temperature region (the bottom of the horizontal layer) to the low temperature region (the top of the horizontal layer) by thermal conduction. However, if the temperature difference is large enough, it may trigger natural convection in the fluid-saturated porous medium. This problem was first treated analytically by Horton and Rogers (1945) as well as Lapwood (1948), and is often called the Horton-Rogers-Lapwood problem.