Asymptotic behavior of a two fluid flow in a thin domain: From Stokes equations to Buckley–Leverett equation and Reynolds law

We consider the flow of two Newtonian, incompressible and nonmiscible fluids in a 2D thin domain. Starting from the Stokes equations, we derive a generalized Buckley–Leverett equation for the first fluid saturation. We study the asymptotic behavior of the flow when the thickness of the gap tends to zero. Assuming that the fluids interface, which is a free boundary, is described by curves of uniformly bounded variation, we prove that the limit problem obeys a generalized Reynolds law. Moreover, when the two sides of the gap are fixed, the saturation of the limit problem is solution of the classical Buckley–Leverett equation.