Resonant penetrative convection in porous media with an internal heat source/sink effect

We study the model of penetrative convection in a porous layer which involves a heat source which varies linearly with vertical height across the layer. This allows us to obtain very strong resonance between sub-layers. The mathematical analysis involves a linear instability technique which yields a definite instability boundary coupled with a global nonlinear energy stability analysis which yields a definite stability threshold. In addition to a linearized instability analysis, the global unconditional nonlinear stability thresholds are derived. Then, the accuracy of the linear instability thresholds are tested using a three dimensional simulation. The results support the assertion that the linear theory , in general, is accurate in predicting the onset of convective motion, and thus, regions of stability.