Recent Developments in Multiscale Problems Coming from Fluid Mechanics

Two topics will be discussed: (i) Homogenization of Flow Problems in the Presence of Rough Boundaries and Interfaces We consider tangential viscous flows over rough surfaces and obtain the effective boundary condition. It is the Navier’s slip condition, used in the computations of viscous flows in complex geometries. The effective coefficients, the Navier’s matrix, is determined by upscaling. It is given by solving an appropriate boundary layer problem. Then we address application to the drag reduction. Finally, we consider viscous flows through domains containing two or more sub-domains, separated by interfaces. Sub-domains are supposed to be geometrically different. A typical example is a viscous flow over a porous bed. It will be shown that in the upscaled problem, it is enough to consider the free fluid part and impose the law of Beavers and Joseph at the interface. Another class of problems is a flow through 2 different porous media. The homogenization, coupled with the boundary layers, gives the effective law at the interface. In this chapter we’ll explain how those results are obtained, give precise references for technical details and present open problems. (ii) Interactions Flow-Structures We consider viscous and inviscid fluid flows through a deformable porous medium. The solid skeleton is supposed to be elastic. We present the modeling of the problem and a derivation of Biot’s type equations by homogenization.