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
.
(Submitted)

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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 twodimensional 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 semiwidth is possible. In practice, one or both of the contact lines will depin before this maximum value of the shear stress is reached, and so we consider situations in which the rivulet/ridge depins at one or both contact lines. In the case of depinning 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 depinning 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 depinning 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 quasistatically stable to twodimensional perturbations.
Item Type:  Article 

Subjects:  D  G > General 
Research Groups:  Oxford Centre for Collaborative Applied Mathematics 
ID Code:  1554 
Deposited By:  Peter Hudston 
Deposited On:  05 Jul 2012 07:49 
Last Modified:  29 May 2015 19:15 
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