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Thick drops climbing uphill on an oscillating substrate

Bradshaw, Joel; Billingham, John

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Authors

Joel Bradshaw

JOHN BILLINGHAM john.billingham@nottingham.ac.uk
Professor of Theoretical Mechanics



Abstract

Experiments have shown that a liquid droplet on an inclined plane can be made to move uphill by suffciently strong, vertical oscillations (Brunet, Eggers, and Deegan, Phys. Rev. Lett. 99, 2007). 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. 674, 2011). In this weakly nonlinear limit,we find that as the static contact angle approaches ? (the non-wetting limit), the rise nvelocity 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.

Citation

Bradshaw, J., & Billingham, J. (2018). Thick drops climbing uphill on an oscillating substrate. Journal of Fluid Mechanics, 840, https://doi.org/10.1017/jfm.2018.71

Journal Article Type Article
Acceptance Date Dec 24, 2017
Online Publication Date Feb 7, 2018
Publication Date Apr 10, 2018
Deposit Date Jan 3, 2018
Publicly Available Date Mar 29, 2024
Journal Journal of Fluid Mechanics
Print ISSN 0022-1120
Electronic ISSN 1469-7645
Publisher Cambridge University Press
Peer Reviewed Peer Reviewed
Volume 840
DOI https://doi.org/10.1017/jfm.2018.71
Public URL https://nottingham-repository.worktribe.com/output/924497
Publisher URL https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/thick-drops-climbing-uphill-on-an-oscillating-substrate/2A8582E4FBAB0E152D46D4B89FE5905B

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