Homogenization for thermoconductivity in a porous medium with periods of different orders in the different directions

Abstract

We study here the of heat transfer in a porous medium consisting of a homogeneous isotropic material. We assume that the period of the medium is not the same in the two directions and that the ratio of these two periods tends to zero. We find the explicit representation of the coefficients of the averaged equations and prove strong convergence to the solution of the averaged problem. 1. Statement of the and simplest estimates By averaging processes in periodic structures appears the to solve the so-called cell problem (Sanchez-Palencia [7, 8], Bakhvalov [1], Bensoussan et al. [3]) in heterogeneous media. Except in the one-dimensional case this cannot be solved in exact form and for solving it we need to use numerical methods. The same arises in a porous medium with periodic holes (Cioranescu and Saint Jean Paulin [4]). In the case when the magnitude of the period is not the same in all the directions, some results are already known. For a heterogeneous material, the averaged equations in explicit form are obtained by Dubinskaya [6] who gives a whole asymptotic expansion without boundary conditons, when the coefficients are smooth. In a porous tall structure with rectangular holes, the limit equation and convergence results are obtained by Cioranescu and Saint Jean Paulin [5] using variational methods. In the present paper, we consider the thermoconductivity case in a two-dimensional isotropic porous medium with symmetric holes. The period is El in the Xl direction and E2 in the X2 direction, with E2 « El « 1. The perforations in the porous material are obtained from a model perforation KO 'The work of N.S. Bakhvalov was partially supported by the Russian Foundation for basic research, 93-01-01729, and most of it was done while he enjoyed the hospitality of the Mathematics Department of the University of Metz. 0921-7134/96/$8.00 © 1996 Elsevier Science B.Y. All rights reserved 254 N.S. Bakhvalov and J. Saint Jean Paulin / Homogenization for thermoconductivity ~Yl=1/2 (0,1) f----~' ___ ----,