Homogenization of the inviscid incompressible fluid flow through a 2D porous medium
Andro Mikelić, Lætitia Paoli · Proceedings of the American Mathematical Society · 1999
We consider the non-stationary incompressible Euler equations in a 2D porous medium. We suppose a periodic porous medium, with the period proportional to the characteristic pore size ε \varepsilon and with connected fluid part. The flow is subject to an external force, corresponding to an inflow. We start from an initial irrotational velocity and prove that the effective filtration velocity satisfies a transient filtration law. It has similarities with Darcy’s law, but it now connects the time derivative of the filtration velocity with the pressure gradient. The viscosity does not appear in the filtration law any more and the permeability tensor is determined through auxiliary problems of decomposition type. Using the limit problem, we construct the correction for the fluid velocity and prove that C 1 ( [ 0 , T ] ; L 2 ( Ω ) 2 ) C^1 ( [0,T]; L^2(\Omega )^2 ) -norm of the error is of order ε \varepsilon . Similarly, we estimate the difference between the fluid pressure and its correction in C ( [ 0 , T ] ; L 1 ( Ω ) ) C ( [0,T]; L^1(\Omega ) ) as C ε C \varepsilon .