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On thin evaporating drops: When is the $d^{2}$-law valid?

Published online by Cambridge University Press:  29 February 2016

M. A. Saxton ,
J. P. Whiteley ,
D. Vella  and
J. M. Oliver
M. A. Saxton
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
J. P. Whiteley
Affiliation:
Department of Computer Science, University of Oxford, Parks Road, Oxford OX1 3QD, UK
D. Vella
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
J. M. Oliver*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
*
†Email address for correspondence: [email protected]
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Abstract

We study the evolution of a thin, axisymmetric, partially wetting drop as it evaporates. The effects of viscous dissipation, capillarity, slip and diffusion-dominated vapour transport are taken into account. A matched asymptotic analysis in the limit of small slip is used to derive a generalization of Tanner’s law that takes account of the effect of mass transfer. We find a criterion for when the contact-set radius close to extinction evolves as the square root of the time remaining until extinction – the famous $d^{2}$-law. However, for a sufficiently large rate of evaporation, our analysis predicts that a (slightly different) ‘$d^{13/7}$-law’ is more appropriate. Our asymptotic results are validated by comparison with numerical simulations.

JFM classification

Phase change: Condensation/evaporation Interfacial Flows (free surface): Contact lines Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 792 , 10 April 2016 , pp. 134 - 167
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Copyright
© 2016 Cambridge University Press 

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