Asymptotics for models of non‐stationary diffusion in domains with a surface distribution of obstacles

Abstract

We consider a time-dependent model for the diffusion of a substance through an incompressible fluid in a perforated domain Ω , Ω ⊂ Ω ⊂ R n with n = 3, 4. The fluid flows in a domain containing a periodical set of "obstacles" (Ω∖Ω ) placed along an inner (n − 1)-dimensional manifold Σ ⊂ Ω. The size of the obstacles is much smaller than the size of the characteristic period . An advection term appears in the partial differential equation linking the fluid velocity with the concentration, while we assume a nonlinear adsorption law on the boundary of the obstacles. This law involves a monotone nonlinear function of the concentration and a large adsorption parameter. The "critical adsorption parameter" depends on the size of the obstacles , and, for different sizes, we derive the time-dependent homogenized models. These models contain a "strange term" in the transmission conditions on Σ, which is a nonlinear function and inherits the properties of . The case in which the fluid velocity and the concentration do not interact is also considered for n ≥ 3. CorrespondenceMaría-Eugenia Pérez-Martínez, ETSI Caminos, Canales y Puertos. Av. de los Castros s/n.,

Boundary homogenization with large reaction terms on a strainer-type wall

“…The technique of matched asymptotic expansions, which follows from that in [9,10] and [12], with the suitable modifications, leads us to the homogenized problems listed below:…”

“…Let us refer to [5,6] and references therein for rapidly alternating Dirichlet-Steklov boundary conditions and [11,18,28] for further references and possible applications in the framework of Geophysics and Winkler beds (foundations). See [9][10][11][12][13][14][15] and [32] for an extensive and updated bibliography on different boundary homogenization problems with Robin-type boundary conditions. Finally, we also mention the first works [16] and [19] where different strange terms in the homogenization of volume perforated media with nonlinear-Robin boundary conditions have been introduced.…”