Abstract
This paper treats the homogenization of the Stokes or Navier-Stokes equations with a Dirichlet boundary condition in a domain containing many tiny solid obstacles, periodically distributed in each direction of the axes. (For example, in the three-dimensional case, the obstacles have a size of ε3 and are located at the nodes of a regular mesh of size ε.) A suitable extension of the pressure is used to prove the convergence of the homogenization process to a Brinkman-type law (in which a linear zero-order term for the velocity is added to a Stokes or Navier-Stokes equation).
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Allaire, G. Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes I. Abstract framework, a volume distribution of holes. Arch. Rational Mech. Anal. 113, 209–259 (1991). https://doi.org/10.1007/BF00375065
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DOI: https://doi.org/10.1007/BF00375065