Homogenized limits of Stokes flow and advective transport in thin perforated domains

Abstract

We deal with the rigorous homogenization and dimension reduction of flow and transport problems posed in thin $\varepsilon$-periodic perforated layers with thickness of order $\varepsilon^{\alpha}$ with $\alpha \in (0,1)$ and therefore the thickness of the layer is large compared its porosity. The aim is the derivation of effective models for $\varepsilon\to 0 $, when the thickness of the layer tends to zero. For the flow problem we consider incompressible Stokes equations with a pressure boundary condition on the top/bottom of the layer, and the transport problem is given by reaction-diffusion-advection problem with advective flow governed from the fluid velocity from the Stokes model and different scalings for the diffusion coefficient modelling low and fast diffusion in the horizontal direction. In the limit, a Darcy-type law is obtained for the Stokes flow with Darcy-velocity depending only on the derivative of the Darcy-pressure in the vertical direction. The effective equation for the transport problem is again of diffusion-advection-type including homogenized coefficients, and with advective flow given by the Darcy-velocity and only taking place in the vertical direction. In the case of slow diffusion in the vertical direction, effective diffusion only takes place in the vertical direction, where in the case of high diffusion in horizontal direction, we obtain effective diffusion in all space directions. To pass to the limit we use the method of two-scale convergence adapted to our microscopic geometry, which is based on uniform a priori estimates. Critical parts in the derivation of the macro-models are the control of the fluid pressure, for which we construct a Bogovskii-operator for thin perforated domains, as well as the strong two-scale convergence for the microscopic solution of the transport equation, necessary to pass to the limit in the advective term.