Flow through porous media due to high pressure gradients

While Darcy’s equations are adequate for studying a large class of flows through porous media, there are several situations wherein it would be inappropriate to use Darcy’s equations. One such example is a flow wherein the range of pressures involved is very large and high pressures and pressure gradients are at play. Here, after developing an approximation for the flow through a porous solid, that is a generalization of an equation developed by Brinkman, we study a simple boundary value problem that clearly delineates the difference between the solution to these equations and those due to the equations that are referred to as “Darcy Law”. We find that the solutions for the equations under consideration exhibit markedly different characteristics from the counterpart for the Brinkman equations or Darcy’s equations (or the Navier–Stokes equation if one neglects the porosity) in that the solutions for the velocity as well as the vorticity lack symmetry and one finds the maximum value of the vorticity occurs at the boundary near which the fluid is less viscous in virtue of the pressure being lower. We also find that for a certain range of values for the non-dimensional parameters describing the flow, boundary layers develop in that the vorticity is confined next to the boundary adjacent to which the viscosity is lower, such boundary layers being absent in the other classical cases.