A note on interface dynamics for convection in porous media

We studied the fluid interface problem through porous media given by two incompressible 2-D fluids of different densities. This problem is mathematically analogous to the dynamics interface for convection in porous media, where the free boundary evolves between fluids with different temperatures. We found a new formula for the evolution equation of the free boundary parameterized as a function in the periodic case. In this formula there is not a principal value in the non-local integral operator involved in the equation, giving a simpler system. Using this formulation, we perform numerical simulations in the stable case (denser fluid below) which shows a strong regularity effect in the periodic interface.