Dynamics of Perturbations under Diffusion in a Porous Medium

Abstract

We consider the dynamics of finite perturbations of a plane phase transition surface in the problem of evaporation of a fluid inside a low-permeability layer of a porous medium. In the case of a nonwettable porous medium, the problem has two stationary solutions, each containing a discontinuity. These discontinuities correspond to plane stationary phase transition surfaces located inside the low-permeability porous layer. One of these surfaces is unstable with respect to long-wavelength perturbations, while the other is stable. We study the evolution of perturbations of the stable plane phase transition surface. It is known that when two phase transition surfaces are located close enough to each other, the dynamics of a weakly nonlinear and weakly unstable wave packet is described by the Kolmogorov–Petrovskii–Piskunov (KPP) diffusion equation. As traveling wave solutions, this equation has heteroclinic solutions with either oscillating or monotonic structure of the front. The boundary value problem in the full statement, which should be considered if the distance between the stable and unstable plane phase transition surfaces is not small, also has similar solutions. We formulate a sufficient condition for the decrease of finite perturbations of the stable plane phase transition surface. This condition depends on their position with respect to the standing wave type and traveling front type solutions of the model equations in the model description when the KPP equation holds.