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The Stokes equations with Fourier boundary conditions on a wall with asperities. (English) Zbl 1007.35058

The authors study the effect of wall rugosity on the solution of the Stokes system with Fourier boundary conditions, which include the stress tensor and a friction coefficient. For small periodic asperities, the authors prove that the velocity field, pressure and drag, respectively, converge to the velocity field, pressure and drag of a homogenized Stokes problem with smooth walls, where a homogenized friction coefficient appears. This shows that, contrary to Dirichlet boundary conditions, rugosity is dominant for Fourier boundary conditions. In the particular case of a plate, the limit drag is larger than the drag of smooth wall.

MSC:

35Q30 Navier-Stokes equations
76D07 Stokes and related (Oseen, etc.) flows
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
76M50 Homogenization applied to problems in fluid mechanics
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References:

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