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On the roughness-induced effective boundary conditions for an incompressible viscous flow. (English) Zbl 1009.76017
The authors study the steady flow of a viscous incompressible fluid through a channel having roughly a square cross-section. The real boundary is obtained by perturbing a flat surface with periodic irregularities whose characteristic length and amplitude are given by some small parameter \(\varepsilon\). The flow is driven by a constant pressure gradient. The problem with no-slip boundary condition is considered first with the aim of getting information on the limit \(\varepsilon \to 0\). Through a series of estimates derived under the restriction that the pressure gradient is sufficiently small, the authors obtain a precise description of how the tangential velocity and its normal derivative become related as \(\varepsilon\) approaches zero. Their result provides a rigorous justification of the so-called Navier’s slip condition.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
76M50 Homogenization applied to problems in fluid mechanics
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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