The derivation of a nonlinear filtration law including the inertia effects via homogenization
The authors develop a nonlinear relationship between the pressure gradient and the filtration velocity starting from the stationary incompressible Navier-Stokes equations in a periodic porous medium. Considering that the local Reynolds number is of order 1 (in this case the inertial terms are not negligible) and performing the formal asymptotic expansion with respect to a small parameter representing the ratio between the microscopic length scale and the macroscopic characteristic length, they replace the non-dimensional incompressible Navier-Stokes system with a homogenized problem for which they give existence and uniqueness results. According to the constitutive law found out by the authors, the filtration velocity is a nonlinear function depending on the external force and pressure gradient. After establishing the analyticity of the nonlinear function in a ball around zero, the authors prove that for some values of the Reynolds number, it is possible to approximate the nonlinear filtration law by a quadratic law. Finally, the proof of the convergence of the homogeneization process justifies rigorously the nonlinear filtration law.
scientific article; zbMATH DE number 1491641