Derivation of the Forchheimer law via homogenization

In this paper we derive the Forchheimer law via the theory of homogenizatio n. In particular, we study the nonlinear correction to Darcy's law due to i nertial effects on the flow of a Newtonian fluid in rigid porous media. A g eneral formula for this correction term is derived directly from the Navier -Stokes equation via homogenization. Unlike other studies based on the same approach that concluded for the nonlinear correction to be cubic in veloci ty for isotropic media, the present work shows that the nonlinear correctio n is quadratic. An example is constructed to illustrate our theory. In this example, the analytic solution to the Navier-Stokes equation is obtained a nd is utilized to show the validity of the quadratic correction. Both incom pressible and compressible fluids are considered.