THE ONSET OF CONVECTION IN THE WOODING PROBLEM WITH ANISOTROPIC PERMEABILITY

Abstract

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, <i>k</i>, and the anisotropy ratio, <i>&xi;</i>. 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 <i>&xi;</i>. It is found that both the critical Darcy-Rayleigh number and wavenumber decrease with increasing values of <i>&xi;</i>. An asymptotic analysis is also presented for <i>&xi;</i> &#62;&#62; 1 where we find that Ra_c &#126; 3.67049 &#43; <i>O</i> (<i>&xi;</i>^-1/2) and <i>k_c</i> &#126; 0.96565<i>&xi;</i>^-1/4.