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

Mustapha El Jarroudi (1) , Alain Brillard (2)

1 FST Tanger, Département de Mathématiques - Facullté des Sciences et Techniques Tanger, Département de Mathématiques
2 GRE - Laboratoire de Gestion des Risques et Environnement

We consider a new way of establishing Navier wall laws. Considering a bounded domain Ω of R N , N=2,3, surrounded by a thin layer Σ ε , along a part Γ2 of its boundary ∂Ω, we consider a Navier-Stokes flow in Ω∪∂Ω∪Σ ε with Reynolds' number of order 1/ε in Σ ε . Using Γ-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 Γ2. 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.

Mots clés: Navier law; Navier-Stokes flow; Γ-convergence; Asymptotic behaviour; Optimal control problem

Domaines: Equations aux dérivées partielles [math.AP]

Mustapha El Jarroudi, Alain Brillard. Boundary Asymptotic Analysis for an Incompressible Viscous Flow: Navier Wall Laws. Applied Mathematics and Optimization, 2008, 57 (3), pp.371-400. ⟨10.1007/s00245-007-9026-5⟩. ⟨hal-00539904⟩