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The interaction between a fluid and a wall is described with a certainboundary condition for the fluid velocity at the wall. To understand how fluidsbehave near a rough wall, the fluid velocity at every point of the roughsurface may be provided. This approach requires detailed knowledge of, andlikely depends strongly on the roughness. Another approach of modeling theboundary conditions of a rough wall is to coarse grain and extract apenetration depth over which on average the fluid penetrates into theroughness. In this work we show that for a broad range of periodic roughnesspatterns and relative flow velocities, a universal penetration depth functioncan be obtained. We obtain these results with experiments and complementarynumerical simulations. Our results show that wall roughness boundary conditionscan be captured with an average ``slip length'' and so indicate that surfacepatterning yields extensive control over wall slip.

Quantitative coarse graining of laminar fluid flow penetration in rough boundaries