Thick drops climbing uphill on an oscillating substrate
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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2018
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| Online Access: | https://eprints.nottingham.ac.uk/48865/ |
| _version_ | 1848797866593615872 |
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| author | Bradshaw, Joel Billingham, John |
| author_facet | Bradshaw, Joel Billingham, John |
| author_sort | Bradshaw, Joel |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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 π (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. |
| first_indexed | 2025-11-14T20:10:41Z |
| format | Article |
| id | nottingham-48865 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:10:41Z |
| publishDate | 2018 |
| publisher | Cambridge University Press |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-488652018-08-07T04:30:17Z https://eprints.nottingham.ac.uk/48865/ Thick drops climbing uphill on an oscillating substrate Bradshaw, Joel Billingham, John 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 π (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. Cambridge University Press 2018-04-10 Article PeerReviewed application/pdf en https://eprints.nottingham.ac.uk/48865/1/Paper_final.pdf Bradshaw, Joel and Billingham, John (2018) Thick drops climbing uphill on an oscillating substrate. Journal of Fluid Mechanics, 840 . pp. 131-153. ISSN 1469-7645 https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/thick-drops-climbing-uphill-on-an-oscillating-substrate/2A8582E4FBAB0E152D46D4B89FE5905B doi:10.1017/jfm.2018.71 doi:10.1017/jfm.2018.71 |
| spellingShingle | Bradshaw, Joel Billingham, John Thick drops climbing uphill on an oscillating substrate |
| title | Thick drops climbing uphill on an oscillating substrate |
| title_full | Thick drops climbing uphill on an oscillating substrate |
| title_fullStr | Thick drops climbing uphill on an oscillating substrate |
| title_full_unstemmed | Thick drops climbing uphill on an oscillating substrate |
| title_short | Thick drops climbing uphill on an oscillating substrate |
| title_sort | thick drops climbing uphill on an oscillating substrate |
| url | https://eprints.nottingham.ac.uk/48865/ https://eprints.nottingham.ac.uk/48865/ https://eprints.nottingham.ac.uk/48865/ |