Some new results in homogenization of flow in porous media with mixed boundary condition

Sammanfattning: The present thesis is devoted to derivation of Darcy’s Law for incompressible Newtonian fluid in perforated domains by means of homogenization techniques.The problem of describing flow in porous media occurs in the study of various physical phenomena such as filtration in sandy soils, blood circulation in capillaries etc. In all such cases physical quantities (e.g. velocity, pressure) are dependent of the characteristic size ε 1 of the microstructure of the fluid domain. However in most practical applications the significant role is played by averaged characteristics, such as permeability, average velocity etc., which do not depend on the microstructure of the domain. In order to obtain such quantities there exist several mathematical techniques collectively referred to as homogenization theory.This thesis consists of two papers (A and B) and complementary appendices. We assume that the flow is governed by the Stokes equation and that global normal stress boundary condition and local no-slip boundary condition are satisfied. Such mixed boundary condition is natural for many applications and here we develop the rigorous mathematical theory connected to it. The assumption of mixed boundary condition affects on corresponding forms of Darcy’s law in both papers and raises some essential difficulties in analysis in Paper A.In both papers the perforated domain is supposed to have periodical structure and the fluid to be incompressible and Newtonian. In Paper A the situation described above is considered in a framework of rigorous functional analysis, more precisely the theorem concerning the existence and uniqueness of weak solutions for the Stokes equation is proved and Darcy’s law is obtained by using two-scale convergence procedure. As it was mentioned, vast part of this paper is devoted to adaptation of classical results of functional analysis to the case of mixed boundary condition.In Paper B the Navier–Stokes system with mixed boundary condition is studied in thin perforated domain. In such cases it is natural to introduce another small parameter δ which corresponds to the thickness of the domain (in addition to the perforation parameter ε). For the case of thin porous medium the asymptotic behavior as both the film thickness δ and the perforation period ε tend to zero at different rates is investigated. The results are obtained by using the formal method of asymptotic expansions. Depending on how fast the two small parameters δ and ε go to zero relative to each other, different forms of Darcy’s law are obtained in all three limit cases — very thin porous medium (δ ε), proportionallythin porous medium (δ ∼ λε, λ ∈ (0,∞)) and homogeneously thin porous medium (δ ε).