The asymptotic analysis of a Darcy-Stokes system coupled through a curved interface

The asymptotic analysis of a Darcy-Stokes system modeling the fluid exchange between a narrow channel (Stokes) and a porous medium (Darcy) coupled through a $C^{2}$ curved interface, is presented. The channel is a cylindrical domain between the interface ($\Gamma$) and a parallel translation of it ($\Gamma + \epsilon \, \boldsymbol{\widehat{e}}_{N}$). The introduction of a change variable to fix the domain's geometry and the introduction of two systems of coordinates: the Cartesian and a local one (consistent with the geometry of the surface), permit to find a Darcy-Brinkman lower dimensional coupled system as the limiting form, when the width of the channel tends to zero ($\epsilon \rightarrow 0$).