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Thick drops climbing uphill on an oscillating substrate

Published online by Cambridge University Press:  07 February 2018

J. T. Bradshaw  and
J. Billingham Open the ORCID record for J. Billingham [Opens in a new window]
J. T. Bradshaw
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
J. Billingham*
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
*
†Email address for correspondence: [email protected]
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Abstract

Experiments have shown that a liquid droplet on an inclined plane can be made to move uphill by sufficiently strong, vertical oscillations (Brunet et al., Phys. Rev. Lett., vol. 99, 2007, 144501). In this paper, we study a two-dimensional, inviscid, irrotational model of this flow, with the velocity of the contact lines a function of contact angle. We use asymptotic analysis to show that, for forcing of sufficiently small amplitude, the motion of the droplet can be separated into an odd and an even mode, and that the weakly nonlinear interaction between these modes determines whether the droplet climbs up or slides down the plane, consistent with earlier work in the limit of small contact angles (Benilov and Billingham, J. Fluid Mech. vol. 674, 2011, pp. 93–119). In this weakly nonlinear limit, we find that, as the static contact angle approaches $\unicode[STIX]{x03C0}$ (the non-wetting limit), the rise velocity of the droplet (specifically the velocity of the droplet averaged over one period of the motion) becomes a highly oscillatory function of static contact angle due to a high frequency mode that is excited by the forcing. We also solve the full nonlinear moving boundary problem numerically using a boundary integral method. We use this to study the effect of contact angle hysteresis, which we find can increase the rise velocity of the droplet, provided that it is not so large as to completely fix the contact lines. We also study a time-dependent modification of the contact line law in an attempt to reproduce the unsteady contact line dynamics observed in experiments, where the apparent contact angle is not a single-valued function of contact line velocity. After adding lag into the contact line model, we find that the rise velocity of the droplet is significantly affected, and that larger rise velocities are possible.

JFM classification

Interfacial Flows (free surface): Capillary flows Interfacial Flows (free surface): Contact lines Drops and Bubbles: Drops
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 840 , 10 April 2018 , pp. 131 - 153
Copyright
© 2018 Cambridge University Press 

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