Asymptotic behaviour of solutions of lubrication problem in a thin domain with a rough boundary and Tresca fluid-solid interface law

Abstract

We study the asymptotic behavior of the solution of a Stokes flow in a thin domain, with a thickness of order ε \varepsilon , and a rough surface. The roughness is defined by a quasi-periodic function with period ε \varepsilon . We suppose that the flow is subject to a Tresca fluid-solid interface condition. We prove a new result on the lower-semicontinuity for the two-scale convergence, which allows us to obtain rigorously the limit problem and to establish the uniqueness of its solution.