Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium

Abstract

SUMMARYSeveral choices of scaling are investigated for a coupled system of parabolic partial differential equations in a two-phase medium at the microscopic scale. This system may be regarded as modelling a reactiondiffusion problem, the Stokes problem of single-phase flow of a slightly compressible fluid or as a heat conduction problem (with or without interfacial resistance), for example. It is shown that, starting with the same problem on the microscopic scale, different choices of scaling of the diffusion coefficients (resp. permeability or conductivity) and the interfacial-exchange coefficient lead to different types of macroscopic systems of equations. The characterization of the limit problems in terms of the scaling parameters constitutes a modelling tool because it allows to determine the right type of limit problem. New macroscopic models, not previously dealt with, arise and, for some scalings, classical macroscopic models are recovered. Using the method of two-scale convergence, a unified approach yielding rigorous proofs is given covering a very broad class of different scalings.

Mathematical justification of macroscopic models for diffusion MRI through the periodic unfolding method

“…We show that there are only five possible limit models, including of course the model of [7]. These limit models are similar to those that have already appeared in the literature when homogenizing elliptic problems with jump condition (see [14]- [15]) using two-scale convergence. Such problems have also been addressed through the periodic unfolding method in [9].…”

Homogenization of a nonlinear multiscale model of calcium dynamics in biological cells

“…The value of the scaling number m is related to the speed of the interfacial exchange, cf. [10,11] for details.…”

“…e.g. [9,10,11]. On the other hand, the notions of convergence in periodic homogenization are of weak type, which implies that they are not compatible with nonlinear terms a priori.…”

Homogenization Approach to the Dispersion Theory for Reactive Transport through Porous Media

“…Scaling in homogenization is an important issue (see e.g. [31], [33]). The characteristic length L R coincides in fact with the "observation distance".…”