From Stokes to Darcy in infinite cylinders: do limits commute? (English) Zbl 1299.35044

Donato, Patrizia; Mardare, Sorin; Vernescu, Bogdan

Differ. Integral Equ. 26, No. 9-10, 949-974 (2013).

Summary: The Darcy flow problem in a porous medium in an infinite cylinder is looked at as a two-parameter limit problem, in terms of the characteristic pore size and the cylinder length. As the characteristic pore size tends to zero, the Stokes problem on the finite cylinder converges to a Darcy problem, and the Darcy problem in the infinite cylinder is obtained as its limit when the length of the cylinder goes to infinity. But one could do this in the opposite order: first consider the limit of the Stokes problem in an infinite cylinder and then consider the homogenized limit to obtain Darcy in an infinite cylinder. Would these two procedures yield the same result? In other words do the limits commute? The answer is shown to be affirmative.

MSC:

35B40	Asymptotic behavior of solutions to PDEs
35B27	Homogenization in context of PDEs; PDEs in media with periodic structure
76D07	Stokes and related (Oseen, etc.) flows
76S05	Flows in porous media; filtration; seepage

Keywords:

Darcy problem; Stokes problem; flows in porous media; asymptotic behavior