Effective permeability of regular arrays and wavy channels

Various crossovers of the effective permeability of certain analytically treatable models of the Darcy flow in porous media are studied. They account for the critical behavior as well for the regimes with low concentrations of obstacles. Transverse permeability of spatially periodic arrays of impenetrable cylinders is found in an analytical form and accounts for various asymptotic regimes. Longitudinal permeability for a square array of cylinders is found as well. Transverse flows past hexagonal and square arrays of cylinders are also considered based on expansions for small concentrations and lubrication approximation for high concentrations of cylinders. Three-dimensional periodic arrays of spherical obstacles are considered as well. Formulas for the drag force exerted by various lattices of obstacles are derived from low-concentration expansions. The Stokes flow through two-dimensional and three-dimensional channels enclosed by two wavy walls is studied by means of expansions for small waviness amplitudes. Compact formulas for permeability are derived in the form of factor approximants for arbitrary values of waviness. Various power laws are accounted for in the regime of large waviness parameters, as well as the existing expansions at small amplitudes.