Asymptotic analysis of Oseen type equations in a channel at small viscosity

Abstract

Our object in this article is to study the boundary layer appearing at large Reynolds number (small viscosity ε) in Oseen type equations in space dimension two in a channel. These are Navier-Stokes equations linearized around a fixed velocity flow: we study the convergence as ε → 0 to the inviscid type equations, we define the correctors needed to resolve the boundary layer and obtain convergence results valid up to the boundary; we study also the behavior of the boundary layer when simultaneously time and the Reynolds number tend to infinity in which case the boundary layer tends to pervade the whole domain. To avoid the difficulties related to complicated geometries we restrict ourselves to the flow in a channel. This article extends previous results [22] concerning the Stokes equations, i.e. the Navier-Stokes equations linearized around rest. An important fact which appears here and which did not appear in [22], is the mixing of the layers in the tangential direction which is due to the transport term.