A Two-Parameter Family of Basic State in Porous Media Leading to Darcy–Bénard Convection

The current research undertakes the onset and formation of Darcy–Bénard convection within a channel of finite depth filled with fluid-saturated porous medium. The Darcy model of porosity is adopted, and a new family of solutions controlled by two parameters is identified from the governing equations representing the flow and thermal fields. These solutions cover most of the basic states studied so far associated with the Darcy law equations. Overall, they exhibit the relaxed impermeable wall constraints resulting in a linear vertical velocity distribution and with non-uniform lateral velocities. The corresponding thermal fields under different thermal boundary conditions are also evaluated analytically in terms of the error function. Despite the fact of the presence of non-uniform streamwise or spanwise velocities, it is shown that a linear stability analysis is still accessible through successively orienting the vortex perturbation, and hence, the conditions setting the convective instability onset are sought. In the two-parameter space, the pairs of most unstable Rayleigh wavenumber are determined for each thermal cases studied. Their contributions to triggering/delaying of emergence of the Darcy–Bénard cells are illuminated. The impacts of further physical mechanisms on the disclosed family in the present paper are open to new researches.