Effect of anisotropy and boundary conditions on Darcy and Brinkman porous penetrative convection

We investigate the effects of anisotropic permeability and changing boundary conditions upon the onset of penetrative convection in a porous medium of Darcy type and of Brinkman type. Attention is focussed on the critical eigenfunctions which show how many convection cells will be found in the porous layer. The number of cells is shown to depend critically upon the ratio of vertical to horizontal permeability, upon the Brinkman coefficient, and upon the upper boundary condition for the velocity which may be of Dirichlet type or constant pressure. The critical Rayleigh numbers and wave numbers are determined, and it is shown how an unconditional threshold for nonlinear stability may be derived