Penetrative convection in a superposed porous-medium–fluid layer via internal heating

Abstract

Penetrative convection in a two-layer system in which a layer of fluid overlies and saturates a porous medium is simulated via internal heating. The motion in the porous medium is described via Darcy's law and in the fluid layer by the Navier–Stokes equations with a Boussinesq approximation. The lower porous surface is held fixed at a temperature $T_L$, while the upper fluid surface is stress free and held at $T_U\,{>}\,T_L$. Internal heating takes place in both layers and allows the model to describe penetrative convection. The strength of heating has a dramatic effect on both the onset of convection and the nature of the ensuing convection cells. It is found that a heat source/sink $Q$ in the fluid layer has a destabilizing effect on the porous layer whereas one in the porous medium $Q_m$ has a stabilizing influence on the fluid. The effect of $Q$ and $Q_m$ on their respective layers, however, depends strongly upon the temperature difference $T_U\,{-}\,T_L$, and the strength and type of heating in the opposite layer. When $Q$ and $Q_m$ are varied, a range of streamlines are presented that exhibit novel behaviour. The model is compared with an alternative in which the density is assumed to have quadratic temperature dependence and there is no internal heating. When the two models are mathematically adjoint they are shown to yield the same critical instability threshold but different eigenfunctions. It is also shown that the initiating cell is not necessarily the strongest one. This curious behaviour is explained and illustrated with a range of streamlines for variable permeability.