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

Handle URI:
http://hdl.handle.net/10754/597426
Title:
A thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflow
Authors:
Sullivan, J. M.; Paterson, C.; Wilson, S. K.; Duffy, B. R.
Abstract:
We use the lubrication approximation to analyze 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 the Appendix, we show that an equilibrium rivulet/ridge with prescribed flux/area is quasi-statically stable to two-dimensional perturbations. © 2012 American Institute of Physics.
Citation:
Sullivan JM, Paterson C, Wilson SK, Duffy BR (2012) A thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflow. Phys Fluids 24: 082109. Available: http://dx.doi.org/10.1063/1.4744980.
Publisher:
AIP Publishing
Journal:
Physics of Fluids
KAUST Grant Number:
KUK-C1-013-04
Issue Date:
2012
DOI:
10.1063/1.4744980
Type:
Article
ISSN:
1070-6631
Sponsors:
J.M.S. and C.P. gratefully acknowledge the financial support of the United Kingdom Engineering and Physical Sciences Research Council (EPSRC) via a Doctoral Training Account (DTA) studentship and of the University of Strathclyde via a Postgraduate Research Scholarship, respectively. This work was completed while S.K.W. was a Visiting Fellow in the Department of Mechanical and Aerospace Engineering, School of Engineering and Applied Science, Princeton University, USA, and a Visiting Fellow in the Oxford Centre for Collaborative Applied Mathematics (OCCAM), Mathematical Institute, University of Oxford, United Kingdom. This publication was based on work supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).
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Full metadata record

DC FieldValue Language
dc.contributor.authorSullivan, J. M.en
dc.contributor.authorPaterson, C.en
dc.contributor.authorWilson, S. K.en
dc.contributor.authorDuffy, B. R.en
dc.date.accessioned2016-02-25T12:33:00Zen
dc.date.available2016-02-25T12:33:00Zen
dc.date.issued2012en
dc.identifier.citationSullivan JM, Paterson C, Wilson SK, Duffy BR (2012) A thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflow. Phys Fluids 24: 082109. Available: http://dx.doi.org/10.1063/1.4744980.en
dc.identifier.issn1070-6631en
dc.identifier.doi10.1063/1.4744980en
dc.identifier.urihttp://hdl.handle.net/10754/597426en
dc.description.abstractWe use the lubrication approximation to analyze 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 the Appendix, we show that an equilibrium rivulet/ridge with prescribed flux/area is quasi-statically stable to two-dimensional perturbations. © 2012 American Institute of Physics.en
dc.description.sponsorshipJ.M.S. and C.P. gratefully acknowledge the financial support of the United Kingdom Engineering and Physical Sciences Research Council (EPSRC) via a Doctoral Training Account (DTA) studentship and of the University of Strathclyde via a Postgraduate Research Scholarship, respectively. This work was completed while S.K.W. was a Visiting Fellow in the Department of Mechanical and Aerospace Engineering, School of Engineering and Applied Science, Princeton University, USA, and a Visiting Fellow in the Oxford Centre for Collaborative Applied Mathematics (OCCAM), Mathematical Institute, University of Oxford, United Kingdom. This publication was based on work supported in part by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherAIP Publishingen
dc.titleA thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflowen
dc.typeArticleen
dc.identifier.journalPhysics of Fluidsen
dc.contributor.institutionUniversity of Strathclyde, Glasgow, United Kingdomen
kaust.grant.numberKUK-C1-013-04en
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