Rotation of a superhydrophobic cylinder in a viscous liquid

Motivated by recent experiments, we address the motion of a superhydrophobic particle through an otherwise quiescent liquid. In these problems the superhydrophobic effect is naturally quantified by the enhancement of the Stokes mobility. We focus upon what may be the simplest problem in that class, namely the rotation of an infinite circular cylinder whose boundary is periodically decorated by a finite number of infinite grooves, with the goal of calculating the rotational mobility. The associated two-dimensional flow problem is defined by two geometric parameters, namely the number N of grooves and the solid fraction. Using matched asymptotic expansions we analyze the large-N limit (and arbitrary solid fraction), obtaining an approximation for the hydrodynamic mobility. Making use of conformal mapping techniques we show that the preceding approximation actually holds for all N, and therefore constitutes an exact result.