Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-17T08:29:34.827Z Has data issue: false hasContentIssue false
  • English
  • Français

Drops climbing uphill on an oscillating substrate

Published online by Cambridge University Press:  07 March 2011

E. S. BENILOV  and
J. BILLINGHAM
E. S. BENILOV*
Affiliation:
Department of Mathematics, University of Limerick, Ireland
J. BILLINGHAM
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
*
†Email address for correspondence: eugene.benilov@ul.ie
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent experiments by Brunet, Eggers & Deegan (Phys. Rev. Lett., vol. 99, 2007, p. 144501 and Eur. Phys. J., vol. 166, 2009, p. 11) have demonstrated that drops of liquid placed on an inclined plane oscillating vertically are able to climb uphill. In the present paper, we show that a two-dimensional shallow-water model incorporating surface tension and inertia can reproduce qualitatively the main features of these experiments. We find that the motion of the drop is controlled by the interaction of a ‘swaying’ (odd) mode driven by the in-plane acceleration and a ‘spreading’ (even) mode driven by the cross-plane acceleration. Both modes need to be present to make the drop climb uphill, and the effect is strongest when they are in phase with each other.

JFM classification

Interfacial Flows (free surface): Capillary flows Drops and Bubbles: Drops Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 674 , 10 May 2011 , pp. 93 - 119
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Benilov, E. S. 2010 Drops climbing uphill on a slowly oscillating substrate. Phys. Rev. E 82, 026320.CrossRefGoogle ScholarPubMed
Brunet, P., Eggers, J. & Deegan, R. D. 2007 Vibration-induced climbing of drops. Phys. Rev. Lett. 99, 144501.CrossRefGoogle ScholarPubMed
Brunet, P., Eggers, J. & Deegan, R. D. 2009 Motion of a drop driven by substrate vibrations. Eur. Phys. J. Special Topics 166, 11–14.CrossRefGoogle Scholar
Ceniceros, H. D. & Hou, T. Y. 1998 Convergence of a non-stiff boundary integral method for interfacial flows with surface tension. Math. Comput. 67, 137–182.CrossRefGoogle Scholar
Daniel, S., Chaudhury, M. K. & de Gennes, P.-G. 2005 Vibration-actuated drop motion on surfaces for batch microfluidic processes. Langmuir 21, 4240–4248.CrossRefGoogle ScholarPubMed
Davis, S. H. 2002 Interfacial fluid dynamics. In Perspectives in Fluid Dynamics: A Collective Introduction to Current Research (ed. Batchelor, G. K., Moffatt, H. K. & Worster, M. G.), pp. 1–51. Cambridge University Press.Google Scholar
Dold, J. W. 1992 An efficient surface-integral algorithm applied to unsteady gravity waves. J. Comput. Phys. 103, 90–115.CrossRefGoogle Scholar
Hocking, L. M. 1987 The damping of capillary gravity-waves at a rigid boundary. J. Fluid Mech. 179, 253–266.CrossRefGoogle Scholar
Hocking, L. M. & Davis, S. H. 2002 Inertial effects in time-dependent motion of thin films and drops. J. Fluid Mech. 467, 1–17.CrossRefGoogle Scholar
King, A. C. 1991 Moving contact lines in slender fluid wedges. Q. J. Mech. Appl. Maths 44, 173–192.CrossRefGoogle Scholar
Noblin, X., Kofman, R. & Celestini, F. 2009 Ratchetlike motion of a shaken drop. Phys. Rev. Lett. 102, 194504.CrossRefGoogle ScholarPubMed