A nonlinear analysis of convection in a porous vertical slab

In 1969 A. E. Gill showed that convection in a porous vertical slab is not possible when the temperature difference is maintained across the layer. He attributed this to the lack of fluid inertia in Darcy's law and based his arguments on two-dimensional linear stability theory. We readdress the Gill problem but by using integral inequality estimates on the full three-dimensional nonlinear theory. We again find there is always stability, but now provided the initial data are less than a threshold which is proportional to the inverse of the Rayleigh number. Of course, this is in complete agreement with the result of Gill, although we believe it adds insight into the physical situation.