Modeling the Lid Driven Flow: Theory and Computation

Abstract

Motivated by the study of the corner singularities in the so-called cavity flow, we establish in the first part of this article, the existence and uniqueness of solutions in $L^2(Ω)^2$ for the Stokes problem in a domain $Ω$, when $Ω$ is a smooth domain or a convex polygon. This result is based on a new trace theorem and we show that the trace of $u$ can be arbitrary in $L^2(∂Ω)^2$ except for a standard compatibility condition recalled below. The results are also extended to the linear evolution Stokes problem. Then in the second part, using a finite element discretization, we present some numerical simulations of the Stokes equations in a square modeling thus the well known lid-driven flow. The numerical solution of the lid driven cavity flow is facilitated by a regularization of the boundary data, as in other related equations with corner singularities ([9], [10], [45], [24]). The regularization of the boundary data is justified by the trace theorem in the first part.