Darcy’s law for porous media with multiple microstructures. (English) Zbl 1517.35184
Summary: In this paper we study the homogenization of the Dirichlet problem for the Stokes equations in a perforated domain with multiple microstructures. First, under the assumption that the interface between subdomains is a union of Lipschitz surfaces, we show that the effective velocity and pressure are governed by a Darcy law, where the permeability matrix is piecewise constant. The key step is to prove that the effective pressure is continuous across the interface, using Tartar’s method of test functions. Secondly, we establish the sharp error estimates for the convergence of the velocity and pressure, assuming the interface satisfies certain smoothness and geometric conditions. This is achieved by constructing two correctors. One of them is used to correct the discontinuity of the two-scale approximation on the interface, while the other is used to correct the discrepancy between boundary values of the solution and its approximation.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 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 |
| 76M50 | Homogenization applied to problems in fluid mechanics |
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