Abstract:

Our goal is to present the rigorous derivation of new interface conditions in porous medium flow using homogenization and boundary-layer theory. In the first part of the talk, we consider the fluid flow in a domain with porous boundary. We start from the Stokes system in a domain with an array of small holes on the boundary and on each hole we impose the value of the normal stress corresponding to the exterior conditions. Letting the period of the porous boundary tend to zero, we propose the interface condition in the form of the generalized Darcy law. If no further assumptions are made concerning the isotropy of the geometry of the porous boundary, the obtained result generalizes the classical Beavers-Joseph condition. In the second part of the talk, we generalize the setting and propose the effective condition on the interface separating the porous from the non-porous (free flow) part of the domain. Using higher-order asymptotics, we derive the interface condition acknowledging the leaking from the free part to the porous part. This phenomenon naturally occurs in case of the blood flow through an artery since the wall of the artery is a live tissue which is fed by that same blood. This is a joint work with Eduard Marusic-Paloka (University of Zagreb).