Homogenization of a micropolar fluid past a porous media with non-zero spin boundary condition

We consider a micropolar fluid flow in a media perforated by periodically distributed obstacles of size $\epsilon$. A non-homogeneous boundary condition for microrotation is considered: the microrotation is assumed to be proportional to the rotation rate of the velocity on the boundary of the obstacles. The existence and uniqueness of solution is analyzed. Moreover, passing to the limit when $\epsilon$ tends to zero, an analogue of the classical micropolar Darcy law in the theory of porous media is derived.