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 ⩽ x n ⩽ 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.