Pressure-driven flows in thin and porous domains

The present thesis is devoted to the derivation of Darcy's law for incompressible viscous fluid flows in perforated and thin domains by means of homogenization techniques. The problem of describing asymptotic flows in porous/thin domains occurs in the study of various physical phenomena such as filtration in sandy soils, blood circulation in capillaries, lubrication and heationg/cooling processes. In all such cases flow characteristics are obviously dependent of microstructure of the fluid domains. However, in the most of practical applications the significant role is played by average (or integral) quantities, such as permeability and macroscopic pressure. In order to obtain them there exist several mathematical approaches collectively referred to as homogenisation theory. This thesis consists of five papers. Papers I and V represent the general case of thin porous domains where both parameters ε - the period of perforation, and δ - the thickness of the domain, are involved. We assume that the flow is governed by the Stokes equation and driven by an external pressure, i.e. the normal stress is prescribed on a part of the boundary and no-slip is assumed on the rest of the boundary. Let us note that from the physical point of view such mixed boundary condition is natural whereas in mathematical context it appears quite seldom and raises therefore some essential difficulties in analytical theory. Depending on the limit value λ of mutual δ / ε -ratio, a form of Darcy's law appears as both δ and ε tend to zero. The three principal cases namely are very thin porous medium (λ =0), proportionally thin porous medium (0< λ<∞) and homogeneously thin porous medium (λ =∞). The results are obtained first by using the formal method of multiple scale asymptotic expansions (Paper I) and then rigorously justified in Paper V. Various aspects of such justification (a priori estimates, two-scale and strong convergence results) are done separately for porous media (Paper II) and thin domains (Paper III). The vast part of Papers II and III is devoted to the adaptation of already existing results for systems that satisfy to no-slip condition everywhere on the boundary to the case of mixed boundary condition mentioned above. Alternative justification approach (asymptotic expansion method accomplished by error estimates) is presented in Paper IV for flows in thin rough pipes.