THE ONSET OF CONVECTION IN THE WOODING PROBLEM WITH ANISOTROPIC PERMEABILITY

We consider the Wooding problem, namely the onset of convection in a semi-infinite saturated porous medium with uniform downward suction into a horizontal and uniformly hot bounding surface. In particular we shall begin to examine the stability properties of convection for the case of a mechanically anisotropic porous medium. A linearized stability analysis is performed and the partial differential system of governing equations is transformed into an ordinary differential eigenvalue problem for the critical Darcy–Rayleigh number, Ra, as a function of wavenumber, k, and the anisotropy ratio, ξ. The eigenvalue problem is solved numerically through the use of the MATLAB routine BVP4C. Neutral curves are presented and the critical parameters are found as a function of ξ. It is found that both the critical Darcy–Rayleigh number and wavenumber decrease with increasing values of ξ. An asymptotic analysis is also presented for ξ ≫ 1 where we find that Rac ∼ 3.67049 + O(ξ⁻¹/²) and kc ∼ 0.96565ξ⁻¹/4 .