Surface-Tension-Driven Convection in a Fluid Overlying a Porous Layer

An accurate numerical analysis for the onset of thermal convection in a two-layer system is presented. The system comprises a saturated porous layer over which lies a layer of the same fluid. The layered system is heated from below, the upper (fluid) surface is free to the atmosphere, and convection driven by surface tension is allowed for. The eigenvalues and eigenfunctions for the instability problem are derived by utilizing a D2 Chebyshev tau method (J. J. Dongarra, B. Straughan, and D. W. Walker, 1996, Appl. Numer. Math.22, 399–435). This allows us to obtain highly accurate eigenvalues and eigenfunctions in a very efficient manner. The onset of convection is seen to have a bimodal nature in which convection may be dominated by the porous medium or by the fluid, depending on the depths of the relative layers and the strength of the tension in the fluid surface. The effect of surface tension is investigated in detail and it is found that for the parameter d̂ (=depth of fluid layer/depth of porous layer) very small, the surface tension has a strong effect on convection dominated by the porous medium, whereas for d̂ larger the surface tension effect is observed only with the fluid mode.