Influence of the Reynolds number on non-Newtonian flow in thin porous media

We study the effect of the Reynolds number on the flow of a generalized Newtonian fluid through a thin porous medium in R3. This medium is a domain of thickness ε≪1, which is perforated by periodically distributed solid cylinders of size ε. We consider the nonlinear stationary Navier-Stokes system with viscosity following the Carreau law with flow index r, with 1<r<+∞. The Reynolds number is assumed to scale as ε−γ, where γ is a real constant. The methodology combines homogenization theory with dimension reduction techniques appropriate for thin domains. We perform a scaling in the vertical direction and introduce an appropriate extension operator to handle the perforated geometry. The treatment of the inertial terms, which represents a major analytical challenge, relies on a combination of refined interpolation techniques and sharp estimates for the extended pressure fields. Our main theoretical result identifies the existence of a critical Reynolds number of order 1/ε (i.e. γ=1), independent of the flow index r, that separates regimes where inertial terms vanish or persist in the homogenized limit. Specifically, for Reynolds number of order smaller than or equal to 1/ε, the inertial term becomes negligible in the homogenized limit (giving linear Darcy's law for γ<1 and nonlinear Darcy's law for γ=1), whereas for Reynolds numbers greater than the critical value (γ>1), inertial effects are expected, and the classical Darcy's law approximation fails. Finally, we propose a new numerical method for solving the nonlinear Darcy's law describing the effective flow in the critical case. We present a mesh convergence analysis for the 3D cell describing the periodic structure of the material, and study the influence of rheological and geometrical parameters of the flow on the resulting numerical solutions.