Derivation of a contact law between a free fluid and thin porous layers via asymptotic analysis methods

Abstract

We describe the asymptotic behaviour of an incompressible viscous free fluid in contact with a porous layer flow through the porous layer surface. This porous layer has a small thickness and consists of thin channels periodically distributed. Two scales are present in this porous medium, one associated to the periodicity of the distribution of the channels and the other to the size of these channels. Proving estimates on the solution of this Stokes problem, we establish a critical link between these two scales. We prove that the limit problem is a Stokes flow in the free domain with further boundary conditions on the basis of the domain which involve an extra velocity, an extra pressure and two second-order tensors. This limit problem is obtained using Γ-convergence methods. We finally consider the case of a Navier–Stokes flow within this context.

Keywords:

• free fluid flow
• Stokes flow
• Navier–Stokes flow
• thin porous layer
• contact surface
• homogenization
• Γ-convergence

AMS Subject Classifications::

• 35B27
• 76D07
• 76D05

Acknowledgements

This work is supported by the Comité Mixte Franco–Marocain under the program PHC MA/08/183. We also thank the referees for the useful comments which improve a first version of this article.