## Abstract

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 of grooves and the solid fraction. Using matched asymptotic expansions we analyse the large- limit, seeking the mobility enhancement from the respective homogeneous-cylinder mobility value. We thus find the two-term approximation, 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. We conjecture that it is exact for all.

Original language | English (US) |
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Pages (from-to) | R41-R413 |

Journal | Journal of Fluid Mechanics |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

## All Science Journal Classification (ASJC) codes

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering

## Keywords

- drops and bubbles
- low-Reynolds-number flows