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

• Published in: Mathematical methods in the applied sciences Vol. 42; no. 1; pp. 403 - 413
• Main Authors: Gómez, Delfina, Lobo, Miguel, Pérez‐Martínez, María‐Eugenia
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 15-01-2019
• Subjects: Adsorption; asymptotic expansions; Barriers; boundary homogenization; Computational fluid dynamics; critical parameters; Domains; evolution problems; Fluid flow; Incompressible flow; Incompressible fluids; Mathematical models; nonlinear problems; Parameters; Partial differential equations; Time dependence
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.5323

We consider a time‐dependent model for the diffusion of a substance through an incompressible fluid in a perforated domain Ωϵ,
Ωϵ⊂Ω⊂Rn with n = 3,4. The fluid flows in a domain containing a periodical set of “obstacles” (Ω\Ωϵ) 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.