Unsteady micropolar fluid flow in a thin domain with Tresca fluid–solid interface law

Abstract

We consider a micropolar fluid flow in a two-dimensional domain. We assume that the velocity field satisfies a non-linear slip boundary condition of friction type on a part of the boundary while the micro-rotation field satisfies non-homogeneous Dirichlet boundary conditions. We prove the existence and uniqueness of a solution. Then motivated by lubrication problems we assume that the thickness and the roughness of the domain are of order 0<ε<<1 and we study the asymptotic behaviour of the flow as ε tends to zero. By using the two-scale convergence technique we derive the limit problem which is totally decoupled for the limit velocity and pressure (v0,p0) on one hand and the limit micro-rotation Z0 on the other hand. Moreover we prove that v0, p0 and Z0 are uniquely determined via auxiliary well-posed problems.