Effective velocity boundary condition at a mixed slip surface

This paper studies the nature of the effective velocity boundary condition for liquid flow over a plane boundary on which small free-slip islands are randomly distributed. It is found that art effective Navier partial-slip condition for the velocity emerges from a statistical analysis valid for arbitrary fractional area coverage beta. As an example, the general theory is applied to the low-beta limit and this result is extended heuristically to finite beta with a resulting slip length proportional to alpha beta/(1-beta), where a is a characteristic size of the islands. A specification of the nature of the free-slip islands is not required in the analysis. They could be nano-bubbles, as suggested by recent experiments, or hydrophobic surface patches. The results are also relevant for ultra-hydrophobic, surfaces exploiting the so-called 'lotus effect'.