# A thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflow

Sullivan, J. M. and Paterson, C. and Wilson, S. K. and Duffy, B. R.
(2012)
*A thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflow.*
Physics of Fluids, 24 (8).
ISSN 1070-6631

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## Abstract

We use the lubrication approximation to analyse three closely related problems involving a thin rivulet or ridge (i.e. a two-dimensional droplet) of fluid subject to a prescribed uniform transverse shear stress at its free surface due to an external airflow, namely a rivulet draining under gravity down a vertical substrate, a rivulet driven by a longitudinal shear stress at its free surface, and a ridge on a horizontal substrate, and find qualitatively similar behaviour for all three problems. We show that, in agreement with previous numerical studies, the free surface profile of an equilibrium rivulet/ridge with pinned contact lines is skewed as the shear stress is increased from zero, and that there is a maximum value of the shear stress beyond which no solution with prescribed semi-width is possible. In practice, one or both of the contact lines will de-pin before this maximum value of the shear stress is reached, and so we consider situations in which the rivulet/ridge de-pins at one or both contact lines. In the case of de-pinning only at the advancing contact line, the rivulet/ridge is flattened and widened as the shear stress is increased from its critical value, and there is a second maximum value of the shear stress beyond which no solution with a prescribed advancing contact angle is possible. In contrast, in the case of de-pinning only at the receding contact line, the rivulet/ridge is thickened and narrowed as the shear stress is increased from its critical value, and there is a solution with a prescribed receding contact angle for all values of the shear stress. In general, in the case of de-pinning at both contact lines there is a critical “yield” value of the shear stress beyond which no equilibrium solution is possible and the rivulet/ridge will evolve unsteadily. In an Appendix we show that an equilibrium rivulet/ridge with prescribed flux/area is quasi-statically stable to two-dimensional perturbations.

Item type: | Article |
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ID code: | 41282 |

Notes: | (c) American Institute of Physics |

Keywords: | aerodynamics, computational fluid dynamics, contact angle, drops, external flows, perturbation theory, free surface, fluid drops, lubrication, shear flows, numerical solutions, Physics, Physics and Astronomy(all) |

Subjects: | Science > Physics |

Department: | Faculty of Science > Mathematics and Statistics > Mathematics Faculty of Science > Mathematics and Statistics |

Depositing user: | Pure Administrator |

Date deposited: | 28 Sep 2012 14:27 |

Last modified: | 03 Apr 2017 11:53 |

Related URLs: | |

URI: | http://strathprints.strath.ac.uk/id/eprint/41282 |

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