The Stability of a Developing Thermal Front in a Porous Medium. I Linear Theory

In this paper, we analyze the stability of the developing thermal boundary layer that is induced by suddenly raising the temperature of the lower horizontal boundary of a uniformly cold semi-infinite porous domain. A full linear stability analysis is developed, and it is shown that disturbances are governed by a parabolic system of equations. Numerical solutions of this system are compared with the neutral stability curve obtained by approximating the system as an ordinary differential eigenvalue problem. Different criteria are used to mark the onset of convection of an evolving disturbance, namely, the maximum disturbance temperature, the surface rate of heat transfer, and the disturbance energy. It is found that these different measures yield different neutral curves. We also show that the disturbances have a favoured evolutionary path in the sense that disturbances introduced at different times or with different initial profiles eventually tend toward that common path.