## 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 φ. 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 N = 1, 2, 4, 8, .... We conjecture that it is exact for all N.

Original language | English (US) |
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Article number | 880 |

Pages (from-to) | R4 |

Journal | Journal of Fluid Mechanics |

Volume | 880 |

DOIs | |

State | Published - Dec 10 2019 |

## All Science Journal Classification (ASJC) codes

- Condensed Matter Physics
- Mechanics of Materials
- Mechanical Engineering
- Applied Mathematics

## Keywords

- drops and bubbles
- low-Reynolds-number flows