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

J. M. Sullivan, C. Paterson, S. K. Wilson, B. R. Duffy

Research output: Contribution to journalArticle

8 Citations (Scopus)
58 Downloads (Pure)

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 languageEnglish
Article number082109
Number of pages19
JournalPhysics of Fluids
Volume24
Issue number8
DOIs
Publication statusPublished - 24 Aug 2012

Keywords

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

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