Asymptotics for models of non‐stationary diffusion in domains with a surface distribution of obstacles. (23rd October 2018)

• Record Type: Journal Article
• Authors: Gómez, Delfina; Lobo, Miguel; Pérez‐Martínez, María‐Eugenia
• Is Part Of: Mathematical methods in the applied sciences. Volume 42:Number 1(2019)
• Page Start: 403
• Page End: 413
• Publication Date: 2018-10-23
• Subjects: asymptotic expansions -- boundary homogenization -- critical parameters -- evolution problems -- nonlinear problems; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.5323
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : We consider a time‐dependent model for the diffusion of a substance through an incompressible fluid in a perforated domain Ω ϵ, Ω ϵ ⊂ Ω ⊂ R n with n = 3, 4. The fluid flows in a domain containing a periodical set of "obstacles" (Ω\Ω ϵ ) 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 ϵ . 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 σ 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 σ . The case in which the fluid velocity and the concentration do not interact is also considered for n ≥ 3.