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Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I: Abstract framework, a volume distribution of holes. (English) Zbl 0724.76020

The homogenization of the Stokes (or Navier-Stokes) equations with a no- slip (Dirichlet) boundary condition in open sets perforated with tiny holes (obstacles) is considered for flows in porous media as well as in mixing grids. For flows in a porous medium, modeled as a periodic repetition of an elementary cell of size \(\epsilon\), the cirtical size of a solid obstacle (hole) is defined as \(\exp (-1/\epsilon^ 2)\) in the two-dimensional case and \(\epsilon^{N/(N-2)}\) in the N-dimensional case (N\(\geq 3)\). The homogenization of the Stokes equations converges to the Brinkman law for the critical hole size, to the Darcy law if the hole size is asymptotically larger than the critical size, and to the Stokes law if it is asymptotically smaller than the critical size. For flows in mixing grids, the homogenization of the Stokes equations converges to the Darcy law if the holes have the same size as the period, and to a particular form of the Brinkman law in a mixing grid represented by its vanes of size \(\epsilon^ 2\) in the three-dimensional case periodically distributed at the nodes of a regular mesh of size \(\epsilon\). This part of the article deals with the abstract formulation for these two kinds of flows. The Brinkman law is derived in this part, and it is proved that in the two-dimensional case the limiting Brinkman-type law is independent of the shape of the holes. Effective equations associated with holes are established on a hypersurface rather than throughout the medium. Optimal \(L^ 2\)-estimates of the pressure, leading to a proof of the convergence of the homogenization process and some new results, including correctors and error estimates, are derived.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations

Citations:

Zbl 0724.76021
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References:

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