Local thermal non-equilibrium effects in the Horton-Rogers-Lapwood problem with a free surface

The onset of thermoconvective instability in a modified Horton-Rogers-Lapwood problem is here investigated. Since the local thermal non equilibrium model is employed, two temperatures, one for the solid phase and one for the fluid phase, are considered. The porous layer is saturated by a Newtonian fluid and the lower plate is impermeable. A horizontal free surface is assumed as top boundary. The free surface is subject to a uniform pressure condition and a third kind boundary condition rules the heat transfer with the external environment. The lower boundary is subject to a uniform heat flux modelled by means of Model A proposed by Amiri et al. [1]. The linear stability of the basic state is investigated by means of normal modes method. An eigenvalue problem characterised by ordinary differential equations is obtained. This eigenvalue problem is governed by a number of parameters. This feature gives the chance of investigating different limiting cases. Some of these cases are solved analytically. These analytical solutions are employed as benchmark and as guess values for the numerical solver employed to solve the general case: a fourth order Runge-Kutta method coupled with the shooting method. The critical values of the stability parameter for the onset of convective instability are obtained for a number of cases.