On the Roughness-Induced Effective Boundary Conditions for an Incompressible Viscous Flow

AbstractWe consider the laminar viscous channel flow with the lateral surface of the channel containing surface irregularities. It is supposed that a uniform pressure gradient is maintained in the longitudinal direction of the channel. After studying the corresponding boundary layers, we obtain rigorously the Navier friction condition. It is valid when the size and amplitude of the imperfections tend to zero. Furthermore, the coefficient in the law is determined through an auxiliary boundary-layer type problem, and the tangential drag force and the effective mass flow are determined up to order O(ε3/2). The value of the effective coefficient is shown to be independent with respect to the position of the mean surface in the range of O(ε)

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.Comment: 32 page

Approximation of homogenized coefficients in deterministic homogenization and convergence rates in the asymptotic almost periodic setting

For a homogenization problem associated to a linear elliptic operator, we prove the existence of a distributional corrector and we find an approximation scheme for the homogenized coefficients. We also study the convergence rates in the asymptotic almost periodic setting, and we show that the rates of convergence for the zero order approximation, are near optimal. The results obtained constitute a step towards the numerical implementation of results from the deterministic homogenization theory beyond the periodic setting. To illustrate this, numerical simulations based on finite volume method are provided to sustain our theoretical results.Comment: 49 pages, 10 figure

About loss of regularity and "blow up" of solutions for quasilinear parabolic systems.

Starting from sufficient conditions for regularity of weak solutions to quasilinear parabolic systems, uecessary conditions for loss of regularity are formulated. It is shown numerically that in some situations loss of regularity ("blow up") really happens accordingly to these conditions