Scalings in homogenisation of reaction, diffusion and interfacial exchange in a two-phase medium
Malte A. Peter, Michael Böhm · Czech digital mathematics library · 2007
Abstract. We consider the homogenisation of a coupled system of parabolic partial differential equations in a heterogeneous two-phase medium and study various choices of scaling of the material parameters with powers of the homogenisation parameter. The system may be regarded as modelling a reaction–diffusion problem, the Stokes problem of single-phase flow of a slightly compressible fluid or as a heat conduction problem (with or without interfacial resistance). A proper nondimensionalisation shows that, depending on the ratio of the characteristic diffusion times of the different species, different scalings of the diffusivities and the interfacial-exchange coefficient with the scale parameter ε appear reasonable. It is shown that, starting with the same type of 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 in the limit. In a unified approach, the limit problems are classified for a whole range of scaling parameters. New limit problems arise and well-known results from the literature are recovered as special cases for certain scalings such as the models of Barenblatt et. al and Arbogast et. al for single-phase flow. Key words. Homogenisation, micro-macro, reaction–diffusion AMS subject classifications. 35B27, 35B30, 35K50, 35K57, 80A20, 80M40.