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On the contact region of a diffusion-limited evaporating drop: a local analysis

Published online by Cambridge University Press:  18 December 2013

S. J. S. Morris
S. J. S. Morris*
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
*
†Email address for correspondence: [email protected]
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Abstract

Motivated by experiments showing that a sessile drop of volatile perfectly wetting liquid initially advances over the substrate, but then reverses, we formulate the problem describing the contact region at reversal. Assuming a separation of scales, so that the radial extent of this region is small compared with the instantaneous radius $a$ of the apparent contact line, we show that the time scale characterizing the contact region is small compared with that on which the bulk drop is evolving. As a result, the contact region is governed by a boundary-value problem, rather than an initial-value problem: the contact region has no memory, and all its properties are determined by conditions at the instant of reversal. We conclude that the apparent contact angle $\theta $ is a function of the instantaneous drop radius $a$, as found in the experiments. We then non-dimensionalize the boundary-value problem, and find that its solution depends on one parameter $\mathscr{L}$, a dimensionless surface tension. According to this formulation, the apparent contact angle is well-defined: at the outer edge of the contact region, the film slope approaches a limit that is independent of the curvature of bulk drop. In this, it differs from the dynamic contact angle observed during spreading of non-volatile drops. Next, we analyse the boundary-value problem assuming $\mathscr{L}$ to be small. Though, for arbitrary $\mathscr{L}$, determining $\theta $ requires solving the steady diffusion equation for the vapour, there is, for small $\mathscr{L}$, a further separation of scales within the contact region. As a result, $\theta $ is now determined by solving an ordinary differential equation. We predict that $\theta $ varies as ${a}^{- 1/ 6} $, as found experimentally for small drops ($a\lt 1~\mathrm{mm} $). For these drops, predicted and measured angles agree to within 10–30 %. Because the discrepancy increases with $a$, but $\mathscr{L}$ is a decreasing function of $a$, we infer that some process occurring outside the contact region is required to explain the observed behaviour of larger drops having $a\gt 1~\mathrm{mm} $.

JFM classification

Interfacial Flows (free surface): Capillary flows Interfacial Flows (free surface): Contact lines Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 739 , 25 January 2014 , pp. 308 - 337
Copyright
©2013 Cambridge University Press 

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