Effective velocity boundary condition at a mixed slip surface

This paper studies the nature of the effective velocity boundary conditions for liquid flow over a plane boundary on which small free-slip islands are randomly distributed. It is found that, to lowest order in the area fraction $\beta$ covered by free-slip regions with characteristic size $a$, a macroscopic Navier-type slip condition emerges with a slip length of the order of $a\beta$. The study is motivated by recent experiments which suggest that gas nano-bubbles may form on solid walls and may be responsible for the appearance of a partial slip boundary conditions for liquid flow. The results are also relevant for ultra-hydrophobic surfaces exploiting the so-called ``lotus effect''.