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The elastic Landau–Levich problem on a slope

Published online by Cambridge University Press:  26 November 2019

Katarzyna L. P. Warburton Open the ORCID record for Katarzyna L. P. Warburton [Opens in a new window] ,
Duncan R. Hewitt Open the ORCID record for Duncan R. Hewitt [Opens in a new window]  and
Jerome A. Neufeld
Katarzyna L. P. Warburton*
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CambridgeCB3 0WA, UK
Duncan R. Hewitt
Affiliation:
Department of Mathematics, University College London, 25 Gordon Street, LondonWC1H 0AY, UK
Jerome A. Neufeld
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, CambridgeCB3 0WA, UK BP Institute, University of Cambridge, Madingley Rise, CambridgeCB3 0EZ, UK Department of Earth Sciences, Bullard Laboratories, University of Cambridge, Madingley Rise, CambridgeCB3 0EZ, UK
*
†Email address for correspondence: klpw3@cam.ac.uk
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Abstract

The elastic analogue of the Landau–Levich dip-coating problem, in which a plate is withdrawn from a bath of fluid on whose surface lies a thin elastic sheet, is analysed for angle of withdrawal $\unicode[STIX]{x1D703}$ to the horizontal. The flow is controlled by the elasticity number, $El$, which is a measure of the relative importance of viscous and bending stresses, and $\unicode[STIX]{x1D703}$. The leading-order solution for small $El$ is a steady profile in which the thickness of the film on the plate is found to vary as $El^{3/4}/(1-\cos \unicode[STIX]{x1D703})^{5/8}$. This prediction is confirmed in the limit $\unicode[STIX]{x1D703}\ll 1$ by comparison with numerical simulation. Finally, the circumstances under which the assumption of a steady solution is no longer valid are discussed, and the time-dependent solution is described.

JFM classification

Interfacial Flows (free surface): Thin films Low-Reynolds-number flows: Lubrication theory Materials Processing Flows: Coating
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A40
Copyright
© 2019 Cambridge University Press

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