Boundary Asymptotic Analysis for an Incompressible Viscous Flow: Navier Wall Laws

TL;DR

This paper develops a new asymptotic analysis method for incompressible viscous flows near boundaries, deriving generalized Navier wall laws involving measure-valued matrices, with applications to special cases and optimal control.

Contribution

It introduces a novel approach using $ ext{Gamma}$-convergence to derive generalized Navier wall laws with measure-valued matrices, extending boundary condition modeling.

Findings

Derived a general Navier law involving measure-valued matrices.

Characterized the measure matrix in two special cases.

Applied the framework to an optimal control problem.

Abstract

We consider a new way of establishing Navier wall laws. Considering a bounded domain $\Omega$ of R N , N=2,3, surrounded by a thin layer $\Sigma \epsilon$, along a part $\Gamma$2 of its boundary $\partial \Omega$, we consider a Navier-Stokes flow in $\Omega \cup \partial \Omega \cup \Sigma \epsilon$ with Reynolds' number of order 1/$\epsilon$ in $\Sigma \epsilon$. Using $\Gamma$-convergence arguments, we describe the asymptotic behaviour of the solution of this problem and get a general Navier law involving a matrix of Borel measures having the same support contained in the interface $\Gamma$2. We then consider two special cases where we characterize this matrix of measures. As a further application, we consider an optimal control problem within this context.