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Validation and modification of asymptotic analysis of slow and rapid droplet spreading by numerical simulation

Published online by Cambridge University Press:  09 January 2013

Yi Sui  and
Peter D. M. Spelt
Yi Sui*
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK School of Engineering and Materials Science, Queen Mary University of London, London E1 4NS, UK
Peter D. M. Spelt
Affiliation:
Laboratoire de la Mécanique des Fluides et d’Acoustique, CNRS, Ecully, France Département Mécanique, Université Claude Bernard Lyon 1, Villeurbanne, France
*
†Email address for correspondence: y.sui@qmul.ac.uk
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Abstract

Using a slip-length-based level-set approach with adaptive mesh refinement, we have simulated axisymmetric droplet spreading for a dimensionless slip length down to $O(1{0}^{\ensuremath{-} 4} )$. The main purpose is to validate, and where necessary improve, the asymptotic analysis of Cox (J. Fluid Mech., vol. 357, 1998, pp. 249–278) for rapid droplet spreading/dewetting, in terms of the detailed interface shape in various regions close to the moving contact line and the relation between the apparent angle and the capillary number based on the instantaneous contact-line speed, $\mathit{Ca}$. Before presenting results for inertial spreading, simulation results are compared in detail with the theory of Hocking & Rivers (J. Fluid Mech., vol. 121, 1982, pp. 425–442) for slow spreading, showing that these agree very well (and in detail) for such small slip-length values, although limitations in the theoretically predicted interface shape are identified; a simple extension of the theory to viscous exterior fluids is also proposed and shown to yield similar excellent agreement. For rapid droplet spreading, it is found that, in principle, the theory of Cox can predict accurately the interface shapes in the intermediate viscous sublayer, although the inviscid sublayer can only be well presented when capillary-type waves are outside the contact-line region. However, $O(1)$ parameters taken to be unity by Cox must be specified and terms be corrected to ${\mathit{Ca}}^{+ 1} $ in order to achieve good agreement between the theory and the simulation, both of which are undertaken here. We also find that the apparent angle from numerical simulation, obtained by extrapolating the interface shape from the macro region to the contact line, agrees reasonably well with the modified theory of Cox. A simplified version of the inertial theory is proposed in the limit of negligible viscosity of the external fluid. Building on these results, weinvestigate the flow structure near the contact line, the shear stress and pressure along the wall, and the use of the analysis for droplet impact and rapid dewetting. Finally, we compare the modified theory of Cox with a recent experiment for rapid droplet spreading, the results of which suggest a spreading-velocity-dependent dynamic contact angle in the experiments. The paper is closed with a discussion of the outlook regarding the potential of using the present results in large-scale simulations wherein the contact-line region is not resolved down to the slip length, especially for inertial spreading.

JFM classification

Interfacial Flows (free surface): Contact lines Drops and Bubbles: Drops and Bubbles Interfacial Flows (free surface): Interfacial Flows (free surface)

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Type
Papers
Information
Journal of Fluid Mechanics , Volume 715 , 25 January 2013 , pp. 283 - 313
Copyright
©2013 Cambridge University Press

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