Can porous vertical penetrative convection become linearly unstable?

This study offers a renewed perspective on the stability of buoyancy-driven parallel flow in a vertical porous slab by extending classical theory to encompass two pivotal generalizations: the presence of a density maximum and the incorporation of partially permeable boundaries through Robin-type velocity boundary conditions. The latter provide a seamless interpolation between the impermeable and fully permeable limits. Gill's original proof of stability is shown to hold strictly for impermeable boundaries; once this assumption is relaxed, the stability of the base flow must be assessed numerically by solving the resulting eigenvalue problem. The analysis reveals that easing the impermeability conditions leads to the onset of instability, with the critical Darcy–Rayleigh number asymptotically approaching a minimum threshold of 908.9492 as the boundaries become fully permeable. A comparison between linear and quadratic density–temperature laws (QDTLs) further reveals the delicate yet decisive role of the density maximum: although not a direct trigger of instability, it exerts a profound influence on the stability of the base flow once instability has set in.