A thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflow
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 d...
| Main Authors: | , , , |
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| Format: | Journal article |
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2012
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| _version_ | 1797067007112249344 |
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| author | Sullivan, J Paterson, C Wilson, S Duffy, B |
| author_facet | Sullivan, J Paterson, C Wilson, S Duffy, B |
| author_sort | Sullivan, J |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-06T21:50:06Z |
| format | Journal article |
| id | oxford-uuid:4af5b209-045c-4f6f-a540-8346a0a88f2f |
| institution | University of Oxford |
| last_indexed | 2024-03-06T21:50:06Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:4af5b209-045c-4f6f-a540-8346a0a88f2f2022-03-26T15:40:42ZA thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflowJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4af5b209-045c-4f6f-a540-8346a0a88f2fMathematical Institute - ePrints2012Sullivan, JPaterson, CWilson, SDuffy, BWe 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. |
| spellingShingle | Sullivan, J Paterson, C Wilson, S Duffy, B A thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflow |
| title | A thin rivulet or ridge subject to a uniform transverse
shear stress at its free surface due to an external airflow |
| title_full | A thin rivulet or ridge subject to a uniform transverse
shear stress at its free surface due to an external airflow |
| title_fullStr | A thin rivulet or ridge subject to a uniform transverse
shear stress at its free surface due to an external airflow |
| title_full_unstemmed | A thin rivulet or ridge subject to a uniform transverse
shear stress at its free surface due to an external airflow |
| title_short | A thin rivulet or ridge subject to a uniform transverse
shear stress at its free surface due to an external airflow |
| title_sort | thin rivulet or ridge subject to a uniform transverse shear stress at its free surface due to an external airflow |
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