Diffusion in a highly rarefied binary structure of general periodic shape

Abstract

We study the homogenization of a diffusion process that takes place in a binary structure formed by an ambiental connected phase surrounding a suspension of very small particles of general form distributed in an ε-periodic network. The asymptotic distribution of the concentration is determined for both phases, as ε → 0, assuming that the suspension has mass of unity order and vanishing volume. Three cases are distinguished according to the values of a certain rarefaction number. When it is positive and finite, the macroscopic system involves a two-concentration system, coupled through a term accounting for the non-local effects. In the other two cases, where the rarefaction number is either infinite or going to zero, although the form of the system is much simpler, some peculiar effects still account for the presence of the suspension.

Keywords:

• diffusion
• homogenization
• fine-scale substructure

2000 Mathematics Subject Classifications:

• 35B27
• 35K57
• 76R50

Acknowledgements

This work was done during the visit of F. Bentalha and D. Polişevschi at the I.R.M.A.R.'s Department of Mechanics (University of Rennes 1), whose support is gratefully acknowledged. It corresponds to a part of D. Polişevschi's work for the CERES Research Program 4-187/2004.