Displacement of oil by water in a single elastic capillary

In this paper we consider the evolution of a free boundary separating two immiscible viscous fluids with different constant densities. The joint motion of liquids in the solid skeleton is described by Stokes equations coupled with Lamé equations, driven by the input pressure and the force of gravity. We prove the existence and uniqueness of classical solutions global in time, and we emphasize the study of the properties of the moving boundary separating the two fluids.