Effective models of the Navier–Stokes flow in porous media with a thin fissure

We consider the Navier–Stokes equations in a bounded domain consisting of two porous media with a thin fissure {0⩽xn⩽η} of width η between them. We suppose the porous medium having a periodic structure with a period ε and consider the limit when ε and η tend to zero. The effective behaviors in porous media and the fissure are determined, which depend on the comparison between ε and η.