Effect of Brinkman boundary layer on the onset of Marangoni convection in a fluid-saturated porous layer

The effect of Brinkmann boundary layer on the onset of convection driven by surface tension gradients, called Marangoni convection, in a thin horizontal fluid-saturated sparsely packed porous layer bounded by adiabatic free boundaries is studied analytically by means of linear stability analysis. The single-term Galerkin expansion technique is shown to be convenient and instructive to establish eigenvalues. The convergence of the results obtained by this approximate technique is checked by comparing the results with those of exact solutions obtained using a regular perturbation technique. By comparing the results of the two techniques, we found that the single-term Galerkin expansion is accurate only for small values of the porous parameter sigma(< 10). The effect of large values of sigma on the onset of Marangoni convection is determined using a method of matched asymptotic expansions. The effect of a boundary layer that exists for large values of sigma is shown to increase the critical Marangoni number by an amount of 2 sigma compared to that for small values of sigma. This uniformly valid solution permits a unified treatment of Marangoni convection and provides the means for a deeper explanation of the physical phenomena. The results obtained are valid for highly porous materials of current practical importance.