From Stokes to Darcy in infinite cylinders: do limits commute?

Abstract

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.