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

Article type: Research Article

Authors: Paoli, Laetitia

Affiliations: Équipe d'Analyse Numérique de Saint Etienne et UMR 5585 MAPLY, Université Jean Monnet, 23, rue du Docteur Paul Michelon, 42023 Saint Étienne cedex 2, France E‐mail: paoli@anum.univ‐st‐etienne.fr

Abstract: 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.

Keywords: two fluid Stokes equation, 2D thin domain, asymptotic behavior, Buckley–Leverett equation, generalized Reynolds law

Journal: Asymptotic Analysis, vol. 34, no. 2, pp. 93-120, 2003