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Faraday pilot-wave dynamics: modelling and computation

Published online by Cambridge University Press:  31 July 2015

Paul A. Milewski ,
Carlos A. Galeano-Rios ,
André Nachbin  and
John W. M. Bush
Paul A. Milewski*
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
Carlos A. Galeano-Rios
Affiliation:
IMPA/National Institute of Pure and Applied Mathematics, Est. D. Castorina, 110, Rio de Janeiro, RJ 22460-320, Brazil
André Nachbin
Affiliation:
IMPA/National Institute of Pure and Applied Mathematics, Est. D. Castorina, 110, Rio de Janeiro, RJ 22460-320, Brazil
John W. M. Bush
Affiliation:
Department of Mathematics, MIT, Cambridge, MA, USA
*
†Email address for correspondence: p.a.milewski@bath.ac.uk
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Abstract

A millimetric droplet bouncing on the surface of a vibrating fluid bath can self-propel by virtue of a resonant interaction with its own wave field. This system represents the first known example of a pilot-wave system of the form envisaged by Louis de Broglie in his double-solution pilot-wave theory. We here develop a fluid model of pilot-wave hydrodynamics by coupling recent models of the droplet’s bouncing dynamics with a more realistic model of weakly viscous quasi-potential wave generation and evolution. The resulting model is the first to capture a number of features reported in experiment, including the rapid transient wave generated during impact, the Doppler effect and walker–walker interactions.

JFM classification

Waves/Free-surface Flows: Capillary waves Drops and Bubbles: Drops Waves/Free-surface Flows: Faraday waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 778 , 10 September 2015 , pp. 361 - 388
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Copyright
© 2015 Cambridge University Press 

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References

Ambrose, D. M., Bona, J. L. & Nicholls, D. P. 2012 Well-posedness for water waves with viscosity. J. Discrete Continuous Dyn. Syst. B 17, 1113–1137.CrossRefGoogle Scholar
Bacchiagaluppi, G. & Valentini, A. 2009 Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference. Cambridge University Press.Google Scholar
Bach, R., Pope, D., Liou, S. & Batelaan, H. 2013 Controlled double-slit electron diffraction. New J. Phys. 15, 033018.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
Borghesi, C., Moukhtar, J., Labousse, M., Eddi, A., Fort, E. & Couder, Y. 2014 The interaction of two walkers: wave-mediated energy and force. Phys. Rev. E 90, 063017.Google Scholar
de Broglie, L. 1926 Ondes et mouvements. Gauthier-Villars.Google Scholar
de Broglie, L. 1930 An Introduction to the Study of Wave Mechanics. Methuen.Google Scholar
de Broglie, L. 1956 Une interprétation causale et non linéaire de la Mécanique ondulatoire: la théorie de la double solution. Gauthier-Villars.Google Scholar
de Broglie, L. 1987 Interpretation of quantum mechanics by the double solution theory. Ann. Fond. Louis de Broglie 12, 1–23.Google Scholar
Bush, J. W. M. 2010 Quantum mechanics writ large. Proc. Natl Acad. Sci. USA 107, 17455–17456.Google Scholar
Bush, J. W. M. 2015 Pilot-wave hydrodynamics. Annu. Rev. Fluid Mech. 47, 269–292.CrossRefGoogle Scholar
Couder, Y. & Fort, E. 2006 Single particle diffraction and interference at a macroscopic scale. Phys. Rev. Lett. 97, 154101.CrossRefGoogle Scholar
Couder, Y., Fort, E., Gautier, C.-H. & Boudaoud, A. 2005a From bouncing to floating: noncoalescence of drops on a fluid bath. Phys. Rev. Lett. 94, 177801.Google Scholar
Couder, Y., Protière, S., Fort, E. & Boudaoud, A. 2005b Walking and orbiting droplets. Nature 437, 208.Google Scholar
Crommie, M. F., Lutz, C. P. & Eigler, D. M. 1993a Imaging standing waves in a two-dimensional electron gas. Nature 363, 524–527.CrossRefGoogle Scholar
Crommie, M. F., Lutz, C. P. & Eigler, D. M. 1993b Confinement of electrons to quantum corrals on a metal surface. Science 262, 218–220.Google Scholar
Dias, F., Dyachenko, A. I. & Zakharov, V. E. 2008 Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions. Phys. Lett. A 372, 1297–1302.Google Scholar
Eddi, A., Decelle, A., Fort, E. & Couder, Y. 2009a Archimedean lattices in the bound states of wave interacting particles. Europhys. Lett. 87, 56002.Google Scholar
Eddi, A., Fort, E., Moisy, F. & Couder, Y. 2009b Unpredictable tunneling of a classical wave-particle association. Phys. Rev. Lett. 102, 240401.Google Scholar
Eddi, A., Moukhtar, J., Perrard, J., Fort, E. & Couder, Y. 2012 Level splitting at a macroscopic scale. Phys. Rev. Lett. 108, 264503.Google Scholar
Eddi, A., Sultan, E., Moukhtar, J., Fort, E., Rossi, M. & Couder, Y. 2011 Information stored in Faraday waves: the origin of a path memory. J. Fluid Mech. 674, 433–463.Google Scholar
Eddi, A., Terwagne, D., Fort, E. & Couder, Y. 2008 Wave propelled ratchets and drifting rafts. Europhys. Lett. 82, 44001.Google Scholar
Fort, E., Eddi, A., Boudaoud, A., Moukhtar, J. & Couder, Y. 2010 Path memory induced quantization of classical orbits. Proc. Natl Acad. Sci. USA 107, 17515–17520.CrossRefGoogle Scholar
Gilet, T. & Bush, J. W. M. 2009a The fluid trampoline: droplets bouncing on a soap film. J. Fluid Mech. 625, 167–203.Google Scholar
Gilet, T. & Bush, J. W. M. 2009b Chaotic bouncing of a droplet on a soap film. Phys. Rev. Lett. 102, 014501.Google Scholar
Harris, D. M. & Bush, J. W. M. 2014 Droplets walking in a rotating frame: from quantized orbits to multimodal statistics. J. Fluid Mech. 739, 444–464.Google Scholar
Harris, D. M., Moukhtar, J., Fort, E., Couder, Y. & Bush, J. W. M. 2013 Wavelike statistics from pilot-wave dynamics in a circular corral. Phys. Rev. E 88, 011001,1–5.Google Scholar
Holmes, M. H. 1995 Introduction to Perturbation Methods, Texts in Applied Mathematics, vol. 20. Springer.Google Scholar
Kumar, K. 1996 Linear theory of Faraday instability in viscous liquids. Proc. R. Soc. Lond. 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
Labousse, M., Perrard, S., Couder, Y. & Fort, E. 2014 Build-up of macroscopic eigenstates in a memory-based constrained system. New. J. Phys. 16, 113027.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Milewski, P. A. & Tabak, E. G. 1999 A pseudospectral procedure for the solution of nonlinear wave equations with examples from free-surface flows. SIAM J. Sci. Comput. 21 (3), 1102–1114.CrossRefGoogle Scholar
Moláček, J. & Bush, J. W. M. 2012 A quasi-static model of drop impact. Phys. Fluids 24, 127103.Google Scholar
Moláček, J. & Bush, J. W. M. 2013a Drops bouncing on a vibrating bath. J. Fluid Mech. 727, 582–611.Google Scholar
Moláček, J. & Bush, J. W. M. 2013b Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory. J. Fluid Mech. 727, 612–647.Google Scholar
Oza, A. U., Harris, D. M., Rosales, R. R. & Bush, J. W. M. 2014a Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization. J. Fluid Mech. 744, 404–429.Google Scholar
Oza, A. U., Harris, D. M., Rosales, R. R. & Bush, J. W. M. 2014b Pilot-wave dynamics in a rotating frame: exotic orbits. Phys. Fluids 26, 082101.Google Scholar
Oza, A. U., Rosales, R. R. & Bush, J. W. M. 2013 A trajectory equation for walking droplets: hydrodynamic pilot-wave theory. J. Fluid Mech. 737, 552–570.Google Scholar
Perrard, S., Labousse, M., Miskin, M., Fort, E. & Couder, Y. 2014 Self-organization into quantized eigenstates of a classical wave-driven particle. Nature Commun. 5, 3219.CrossRefGoogle ScholarPubMed
Protière, S., Bohn, S. & Couder, Y. 2008 Exotic orbits of two interacting wave sources. Phys. Rev. E 78, 36204.Google ScholarPubMed
Protière, S., Boudaoud, A. & Couder, Y. 2006 Particle-wave association on a fluid interface. J. Fluid Mech. 554, 85–108.Google Scholar
Protière, S., Couder, Y., Fort, E. & Boudaoud, A. 2005 The self-organization of capillary wave sources. J. Phys.: Condens. Matter 17, 3529–3535.Google Scholar
Walker, J. 1978 Drops of liquid can be made to float on the liquid. What enables them to do so? Sci. Am. 238, 151–158.CrossRefGoogle Scholar
Wang, Z. & Milewski, P. A. 2012 Dynamics of gravity–capillary solitary waves in deep water. J. Fluid Mech. 708, 480–501.CrossRefGoogle Scholar
Wind-Willassen, O., Moláček, J., Harris, D. M. & Bush, J. W. M. 2013 Exotic states of bouncing and walking droplets. Phys. Fluids 25, 082002.CrossRefGoogle Scholar