Sigma-convergence for thin heterogeneous domains and application to the upscaling of Darcy-Lapwood-Brinkmann flow
The sigma-convergence concept has been up to now used to derive macroscopic models in full space dimensions. In this work, we generalize it to thin heterogeneous domains given rise to phenomena in lower space dimensions. More precisely, we provide a new approach of the sigma-convergence method that is suitable for the study of phenomena occurring in thin heterogeneous media. This is made through a systematic study of the sigma-convergence method for thin heterogeneous domains. Assuming that the thin heterogeneous layer is made of microstructures that are distributed inside in a deterministic way including as special cases the periodic and the almost periodic distributions, we make use of the concept of algebras with mean value to state and prove the main compactness results. As an illustration, we upscale a Darcy-Lapwood-Brinkmann micro-model for thin flow. We prove that, according to the magnitude of the permeability of the porous domain, we obtain as effective models, the Darcy law in lower dimensions. The effective models are derived through the solvability of either the local Darcy-Brinkmann problems or the local Hele-Shaw problems.