Maximum principle for viscous fluids flowing through regions of pure fluid and mixtures

A variational formulation for the problem of the flow of an incompressible Newtonian fluid through two, or more, regions is presented. Some of these regions are occupied only by the fluid and others by a saturated mixture of the fluid with a homogeneous rigid porous medium. The boundary between a region of pure fluid and another of mixture is formed by an impermeable interface that allows exchange of momentum but not of mass. The maximum of the functional is the solution of the problem.