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On the distinguished limits of the Navier slip model of the moving contact line problem

Published online by Cambridge University Press:  28 April 2015

Weiqing Ren ,
Philippe H. Trinh Open the ORCID record for Philippe H. Trinh [Opens in a new window]  and
Weinan E
Weiqing Ren
Affiliation:
Department of Mathematics, National University of Singapore, Singapore 119076, Singapore Institute of High Performance Computing, A*STAR, Singapore 138632, Singapore
Philippe H. Trinh*
Affiliation:
Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
Weinan E
Affiliation:
Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544-1000, USA School of Mathematics, Peking University, Beijing 100871, PR China
*
†Email address for correspondence: trinh@maths.ox.ac.uk
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Abstract

When a droplet spreads on a solid substrate, it is unclear what the correct boundary conditions are to impose at the moving contact line. The classical no-slip condition is generally acknowledged to lead to a non-integrable singularity at the moving contact line, which a slip condition, associated with a small slip parameter, ${\it\lambda}$, serves to alleviate. In this paper, we discuss what occurs as the slip parameter, ${\it\lambda}$, tends to zero. In particular, we explain how the zero-slip limit should be discussed in consideration of two distinguished limits: one where time is held constant, $t=O(1)$, and one where time tends to infinity at the rate $t=O(|\!\log {\it\lambda}|)$. The crucial result is that in the case where time is held constant, the ${\it\lambda}\rightarrow 0$ limit converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. However, if ${\it\lambda}\rightarrow 0$ and $t\rightarrow \infty$, then contact line slippage is a leading-order singular effect.

JFM classification

Interfacial Flows (free surface): Contact lines Low-Reynolds-number flows: Lubrication theory Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 772 , 10 June 2015 , pp. 107 - 126
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Copyright
© 2015 Cambridge University Press 

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