Time-Evolving to Space-Evolving Thermal Instability of a Porous Medium Flow

The Rayleigh-B\'enard instability of the stationary throughflow in a horizontal porous layer, also known as Prats' problem, is here analysed in a fresh new perspective. In fact, the classical analysis of the linear instability, carried out by employing space-periodic Fourier modes, is here reconsidered by focussing on the effects of time-periodic and space-evolving modes. The basic stationary flow is assumed to be perturbed by a localised source of perturbation which is steady-periodic in time. Then, the streamwise development of such perturbations is monitored in order to detect their possible amplification or decay in the direction of propagation. Accordingly, the spatial stability/instability threshold is determined. The study is carried out by employing a Fourier transform formalism, where the transformed variable is time.