The Transition Between the Navier–Stokes Equations to the Darcy Equation in a Thin Porous Medium

Abstract

Listen

Translate article

We consider a Newtonian flow in a thin porous medium $$\Omega _{\varepsilon }$$ of thickness $$\varepsilon $$ which is perforated by periodically distributed solid cylinders of size $$a_\varepsilon $$ . Generalizing (Anguiano and Suarez-Grau, ZAMP J Appl Math Phys 68:45, 2017), the fluid is described by the 3D incompressible Navier–Stokes system where the external force takes values in the space $$H^{-1}$$ , and the porous medium considered has one of the most commonly used distribution of cylinders: hexagonal distribution. By an adaptation of the unfolding method, three different Darcy’s laws are rigorously derived from this model depending on the magnitude $$a_{\varepsilon }$$ with respect to $$\varepsilon $$ .