Numerical homogenization in the Rayleigh–Taylor problem of filtering two immiscible incompressible liquids

The present work is devoted to the plane motion of two immiscible incompressible fluids with different densities, separated by a surface of contact discontinuity in the elastic pore space. We discuss results of numerical calculations of the exact models on the microscopic level with a free boundary for the absolutely rigid skeleton and for the elastic skeleton. There is a small parameter $\varepsilon$ in the model, equal to the ratio average pore size to the size of the hole domain. If we decrease parameter $\varepsilon$, then solutions of the model on the microscopic level describe averaged picture of two immiscible liquids movement. In this case, the movement of fluids in the elastic skeleton preserve the surface of a contact discontinuity, while for the motion of fluids in an absolutely rigid skeleton instead of without free boundary occurs some mixed zone (mushy region), occupied by a mixture of two fluids.