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arXiv · 0711.3610

The Navier wall law at a boundary with random roughness

David Gerard-Varet
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Abstract

We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size $\eps \ll 1$. In a parent paper, we derived a homogenized boundary condition of Navier type as $\eps \to 0$. We show here that for a large class of boundaries, this Navier condition provides a $O(\eps^{3/2} |\ln \eps|^{1/2})$ approximation in $L^2$, instead of $O(\eps^{3/2})$ for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables.

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David Gerard-Varet. 2007-11-22. The Navier wall law at a boundary with random roughness. https://doi.org/10.1007/s00220-008-0597-z

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