Determination of the macroscopic (partial) slip boundary condition for a viscous flow over a randomly rough surface with a perfect slip microscopic boundary condition

Abstract

Consider a viscous fluid, at zero Reynolds number, moving over a solid surface flat except for a random array of microscopic defects having a small area fraction c. Assuming a microscopic boundary condition of perfect slip, the macroscopic boundary condition is determined from first principles. The asymptotic structure of the solution for a random surface with finite slope is quite different from those of earlier studies in the limit of an ‘‘almost flat’’ surface. The results of this study show that very small amounts of roughness can well approximate a no-slip boundary condition macroscopically, for example, one defect of the order of 10−9 m per (10−7 m)2 gives a slip length of only 10−5 m.