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Faraday instability in floating liquid lenses: the spontaneous mutual adaptation due to radiation pressure

Published online by Cambridge University Press:  14 May 2013

G. Pucci ,
M. Ben Amar  and
Y. Couder
G. Pucci*
Affiliation:
Matière et Systèmes Complexes, Université Paris Diderot–Paris 7, CNRS–UMR 7057, Bâtiment Condorcet, 75013 Paris, France
M. Ben Amar
Affiliation:
Laboratoire de Physique Statistique, Ecole Normale Supérieure, UPMC Univ. Paris 6, Université Paris Diderot–Paris 7, CNRS–UMR 8550, 24 rue Lhomond, 75005 Paris, France
Y. Couder
Affiliation:
Matière et Systèmes Complexes, Université Paris Diderot–Paris 7, CNRS–UMR 7057, Bâtiment Condorcet, 75013 Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

Fluid dynamics instabilities are usually investigated in two types of situations, either confined in cells with fixed boundaries, or free to grow in open space. In this article we study the Faraday instability triggered in a floating liquid lens. This is an intermediate situation in which a hydrodynamical instability develops in a domain with flexible boundaries. The instability is observed to be initially disordered with fluctuations of both the wave field and the lens boundaries. However, a slow dynamics takes place, leading to a mutual adaptation so that a steady regime is reached with a stable wave field in a stable lens contour. The most recurrent equilibrium lens shape is elongated with the Faraday wave vector along the main axis. In this self-organized situation an equilibrium is reached between the radiation pressure exerted by Faraday waves on the borders and their capillary response. The elongated shape is obtained theoretically as the exact solution of a Riccati equation with a unique control parameter and compared with the experiment.

JFM classification

Drops and Bubbles: Drops Waves/Free-surface Flows: Faraday waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 725 , 25 June 2013 , pp. 402 - 427
Copyright
©2013 Cambridge University Press 

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References

Agam, O. & Altshuler, B. L. 2001 Scars in parametrically excited surface waves. Physica A 302, 310–317.Google Scholar
Ben Amar, M. 1992 Plumes in Hele-Shaw cells. Phys. Fluids A 4, 2641–2650.CrossRefGoogle Scholar
Bénard, H. 1900 Les tourbillons cellulaires dans une nappe liquide transportent de la chaleur par convection en régime permanent. Ann. Chim. Phys. 7 (23), 62–144.Google Scholar
Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225, 505–515.Google Scholar
Boudaoud, A., Couder, Y. & Ben Amar, M. 1999a Self-adaptation in vibrating soap films. Phys. Rev. Lett. 82, 3847–3850.Google Scholar
Boudaoud, A., Couder, Y. & Ben Amar, M. 1999b A self-adaptive oscillator. Eur. Phys. J. B 9, 159–165.Google Scholar
Burgess, D. & Foster, M. R. 1990 Analysis of the boundary conditions for a Hele-Shaw bubble. Phys. Fluids A 2, 1105–1117.Google Scholar
Ciliberto, S. & Gollub, J. P. 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381–398.Google Scholar
Couder, Y. 2000 Viscous fingering as an archetype for growth patterns. In Perspective in Fluid Dynamics (ed. Batchelor, G. K., Moffat, H. K. & Worster, M. G.), pp. 53–104. Cambridge University Press.Google Scholar
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383–409.Google Scholar
Faraday, M. 1831 On the forms and states assumed by fluids in contact with vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 319–340.Google Scholar
de Gennes, P. G., Brochard-Wyart, F. & Quéré, D. 2002 Capillarity and Wetting Henomena: Drops, Bubbles, Pearls and Waves. Springer.Google Scholar
Jeng, U.-S., Esibov, L., Crow, L. & Steyrel, A. 1998 Viscosity effect on capillary waves at liquid interfaces. J. Phys.: Condens. Matter 10, 4955–4962.Google Scholar
Kudrolli, A., Abraham, M. C. & Gollub, J. P. 2001 Scarred patterns in surface waves. Phys. Rev. E 63, 026208.CrossRefGoogle ScholarPubMed
Kudrolli, A. & Gollub, J. P. 1996 Patterns and spatiotemporal chaos in parametrically forced surface waves: a systematic survey at large aspect ratio. Physica D 97, 133–154.CrossRefGoogle Scholar
Kumar, K. 1996 Linear theory of Faraday instability in viscous liquids. Proc. R. Soc. A 452, 1113–1126.Google Scholar
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.Google Scholar
Kundu, P. K. & Cohen, I. M. 2004 Fluid Mechanics. Elsevier Academic.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Butterworth-Heinemann/Elsevier.Google Scholar
Langmuir, I. 1933 Oil lenses on water and the nature of monomolecular expanded films. J. Chem. Phys. 1, 756–776.Google Scholar
Longuet-Higgins, M. S. & Stewart, R. W. 1964 Radiation stress in water waves; a physical discussion, with applications. Deep-Sea Res. 11, 529–562.Google Scholar
Marquardt, F., Harris, J. G. E. & Girvin, S. M. 2006 Dynamical multistability induced by radiation pressure in high-finesse micromechanical optical cavities. Phys. Rev. Lett. 96, 103901.Google Scholar
Meystre, P., Wright, E. M., McCullen, J. D. & Vignes, E. 1985 Theory of radiation-pressure-driven interferometers. J. Opt. Soc. Am. B 2, 1830–1840.CrossRefGoogle Scholar
Muller, H. W., Wittmer, H., Wagner, C., Albers, J. & Knorr, K. 1997 Analytic stability theory for Faraday waves and the observation of the harmonic surface response. Phys. Rev. Lett. 78, 2357–2360.CrossRefGoogle Scholar
Noblin, X., Buguin, A. & Brochard-Wyart, F. 2002 Fast dynamics of floating triple lines. Langmuir 18, 9350–9356.Google Scholar
Park, C. W. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291–308.Google Scholar
Pucci, G., Fort, E., Ben Amar, M. & Couder, Y. 2011 Mutual adaptation of a Faraday instability pattern with its flexible boundaries in floating fluid drops. Phys. Rev. Lett. 106, 024503.CrossRefGoogle ScholarPubMed
Ramshankar, R., Berlin, D. & Gollub, J. P. 1990 Transport by capillary waves. Part I. Particle trajectories. Phys. Fluids 2, 1955–1965.Google Scholar
Rayleigh, Lord 1916 On the dynamics of revolving fluids. Proc. R. Soc. Lond. A 92, 148–154.Google Scholar
Reinelt, D. A. 1987 Interface conditions for two-phase displacement in Hele-Shaw cells. J. Fluid Mech. 183, 219–234.Google Scholar
Residori, S., Guarino, A. & Bortolozzo, U. 2007 Two-mode competition in Faraday instability. Eur. Phys. Lett. 77, 44003.Google Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of fluid into a porous medium or Hele-Shaw cell containing a more viscous fluid. Proc. R. Soc. Lond. A 245, 312–329.Google Scholar
Tan, M. K., Friend, J. R., Matar, O. K. & Yeo, L. Y. 2010 Capillary wave motion excited by high frequency surface acoustic waves. Phys. Fluids 22, 112112.Google Scholar
Turner, J. S. 1962 The starting plume in neutral surroundings. J. Fluid Mech. 13, 356–368.Google Scholar
Vega, J. M., Rudiger, S. & Viñals, J. 2004 Phenomenological model of weakly damped Faraday waves and the associated mean flow. Phys. Rev. E 70, 046306.Google Scholar
Zhang, W. & Viñals, J. 1997 Pattern formation in weakly damped parametric surface waves. J. Fluid Mech. 336, 301–330.Google Scholar