The onset of Darcy-Bénard instability in a horizontal porous channel with a free surface using a thermal nonequilibrium model

The intent beyond this present work is to investigate the onset of instability in homogeneous horizontal porous channel saturated with Newtonian fluid bounded by free surface from the top and impermeable wall from the bottom. The upper free surface is cooled by a third kind boundary condition, while the lower one is heated by two different ways. The wall is modelled with uniform temperature in the first case and with a perfect heat flux (Model A) in the second case. Relaxing the assumption of local thermal equilibrium model can provide in general two different Biot numbers Bs and Bf as well as two temperature profiles at the basic state, one for each phase. The linear stability of the basic motionless state is studied by using the normal modes analysis. The neutral stability curves are obtained numerically by applying a sixth order Runge-Kutta method in combination with a shooting method. The critical values of R at which the instability initiates in the porous medium with isothermal wall or isoflux wall are calculated.