Rotation of a superhydrophobic cylinder in a viscous liquid

The hydrodynamic quantification of superhydrophobic slipperiness has traditionally employed two canonical problems - namely, shear flow about a single surface and pressure-driven channel flow. We here advocate the use of a new class of canonical problems, defined by the motion of a superhydrophobic particle through an otherwise quiescent liquid. In these problems the superhydrophobic effect is naturally measured by the enhancement of the Stokes mobility relative to the corresponding mobility of a homogeneous particle. We focus upon what may be the simplest problem in that class - 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 (velocity-to-torque ratio). The associated two-dimensional flow problem is defined by two geometric parameters - namely, the number N of grooves and the solid fraction phi. Using matched asymptotic expansions we analyse the large-N limit, seeking the mobility enhancement from the respective homogeneous-cylinder mobility value. We thus find the two-term approximation, 1 + 2/N ln csc pi phi/2, for the ratio of the enhanced mobility to the homogeneous-cylinder mobility. Making use of conformal-mapping techniques and inductive arguments we prove that the preceding approximation is actually exact for N = 1, 2, 4, 8, .... We conjecture that it is exact for all N.