Nonlinear corrections to Darcy's law at low Reynolds numbers

Under fairly general assumptions, this paper shows that for periodic porous media, whose period is of the same order as that of the inclusion, the nonlinear correction to Darcy's law is quadratic in terms of the Reynolds number, i.e. cubic with respect to the seepage velocity. This claim is substantiated by reinspection of well-known experimental results, a mathematical proof (restricted to periodic porous media), and numerical calculations.