Abstract
We homogenize stationary incompressible Stokes flow in a periodic porous medium. The fluid is assumed to satisfy a no-slip condition on the boundary of solid inclusions and a normal stress (traction) condition on the global boundary. Under these assumptions, the homogenized equation becomes the classical Darcy law with a Dirichlet condition for the pressure.
Highlights
Various physical phenomena can be described in terms of fluid flow in porous media
There exist several mathematical approaches collectively referred to as homogenization theory as well as heuristic methods based on phase averaging (Whitaker, 1986)
We consider the flow of an incompressible fluid in a perforated domain which depends on a small parameter
Summary
Various physical phenomena can be described in terms of fluid flow in porous media. It occurs e.g. in the study of filtration in sandy soils or blood circulation in capillaries, see Bear (1975) and Hornung (1997) for more examples and motivation. Our objective is to homogenize the Stokes system under the mixed boundary condition, i.e. we want to find the effective (or macroscopic) set of equations that govern pressure and velocity in the limit as the parameter tends to zero This includes (1) to define a suitable notion of convergence for solutions defined on a sequence of perforated domains Ω as → 0; (2) to deduce the limit (homogenized) boundary value problem in the unperforated domain Ω; and (3) to rigorously derive the corresponding Darcy law in terms of homogenized problem solution. All previous mathematical studies concern the Dirichlet problem for the velocity field This leads to Darcy’s equation with a Neumann condition, which implies that the homogenized pressure distribution can only be determined up to a constant.
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