### 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.

Original language | English |
---|---|

Article number | 082109 |

Number of pages | 19 |

Journal | Physics of Fluids |

Volume | 24 |

Issue number | 8 |

DOIs | |

Publication status | Published - 24 Aug 2012 |

### Keywords

- aerodynamics
- computational fluid dynamics
- contact angle
- drops
- external flows
- perturbation theory
- free surface
- fluid drops
- lubrication
- shear flows
- numerical solutions

## Fingerprint Dive into the research topics of 'A thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflow'. Together they form a unique fingerprint.

## Cite this

Sullivan, J. M., Paterson, C., Wilson, S. K., & 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), [082109]. https://doi.org/10.1063/1.4744980