The asymptotic analysis of a Darcy-Stokessystem coupled through a curved interface

We present the asymptotic analysis of a Darcy-Stokes coupledsystem, modeling the fluid exchange between a narrow channel(Stokes flow)and a porous medium (Darcy flow), coupled through aC2curved interface.The channel is a cylindrical domain between the interface (Γ) and a paralleltranslation of itself (Γ +ǫbeN, ǫ >0). The introduction of a change variable(to fix the domain geometry) and the introduction of two systems of coor-dinates: the Cartesian and a local one (consistent with the geometry of thesurface), permit to find the limiting form of the system when the width of thechannel tends to zero (ǫ→0). The limit problem is a coupled system withDarcy flow in the porous medium and Brinkman flow on the curved interface(Γ).