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Faraday pilot-wave dynamics: modelling and computation

  • Paul A. Milewski (a1), Carlos A. Galeano-Rios (a2), André Nachbin (a2) and John W. M. Bush (a3)

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.

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Ambrose, D. M., Bona, J. L. & Nicholls, D. P. 2012 Well-posedness for water waves with viscosity. J. Discrete Continuous Dyn. Syst. B 17, 11131137.
Bacchiagaluppi, G. & Valentini, A. 2009 Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference. Cambridge University Press.
Bach, R., Pope, D., Liou, S. & Batelaan, H. 2013 Controlled double-slit electron diffraction. New J. Phys. 15, 033018.
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, 505515.
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.
de Broglie, L. 1926 Ondes et mouvements. Gauthier-Villars.
de Broglie, L. 1930 An Introduction to the Study of Wave Mechanics. Methuen.
de Broglie, L. 1956 Une interprétation causale et non linéaire de la Mécanique ondulatoire: la théorie de la double solution. Gauthier-Villars.
de Broglie, L. 1987 Interpretation of quantum mechanics by the double solution theory. Ann. Fond. Louis de Broglie 12, 123.
Bush, J. W. M. 2010 Quantum mechanics writ large. Proc. Natl Acad. Sci. USA 107, 1745517456.
Bush, J. W. M. 2015 Pilot-wave hydrodynamics. Annu. Rev. Fluid Mech. 47, 269292.
Couder, Y. & Fort, E. 2006 Single particle diffraction and interference at a macroscopic scale. Phys. Rev. Lett. 97, 154101.
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.
Couder, Y., Protière, S., Fort, E. & Boudaoud, A. 2005b Walking and orbiting droplets. Nature 437, 208.
Crommie, M. F., Lutz, C. P. & Eigler, D. M. 1993a Imaging standing waves in a two-dimensional electron gas. Nature 363, 524527.
Crommie, M. F., Lutz, C. P. & Eigler, D. M. 1993b Confinement of electrons to quantum corrals on a metal surface. Science 262, 218220.
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, 12971302.
Eddi, A., Decelle, A., Fort, E. & Couder, Y. 2009a Archimedean lattices in the bound states of wave interacting particles. Europhys. Lett. 87, 56002.
Eddi, A., Fort, E., Moisy, F. & Couder, Y. 2009b Unpredictable tunneling of a classical wave-particle association. Phys. Rev. Lett. 102, 240401.
Eddi, A., Moukhtar, J., Perrard, J., Fort, E. & Couder, Y. 2012 Level splitting at a macroscopic scale. Phys. Rev. Lett. 108, 264503.
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, 433463.
Eddi, A., Terwagne, D., Fort, E. & Couder, Y. 2008 Wave propelled ratchets and drifting rafts. Europhys. Lett. 82, 44001.
Fort, E., Eddi, A., Boudaoud, A., Moukhtar, J. & Couder, Y. 2010 Path memory induced quantization of classical orbits. Proc. Natl Acad. Sci. USA 107, 1751517520.
Gilet, T. & Bush, J. W. M. 2009a The fluid trampoline: droplets bouncing on a soap film. J. Fluid Mech. 625, 167203.
Gilet, T. & Bush, J. W. M. 2009b Chaotic bouncing of a droplet on a soap film. Phys. Rev. Lett. 102, 014501.
Harris, D. M. & Bush, J. W. M. 2014 Droplets walking in a rotating frame: from quantized orbits to multimodal statistics. J. Fluid Mech. 739, 444464.
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.
Holmes, M. H. 1995 Introduction to Perturbation Methods, Texts in Applied Mathematics, vol. 20. Springer.
Kumar, K. 1996 Linear theory of Faraday instability in viscous liquids. Proc. R. Soc. Lond. A 452, 11131126.
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 4968.
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.
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.
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), 11021114.
Moláček, J. & Bush, J. W. M. 2012 A quasi-static model of drop impact. Phys. Fluids 24, 127103.
Moláček, J. & Bush, J. W. M. 2013a Drops bouncing on a vibrating bath. J. Fluid Mech. 727, 582611.
Moláček, J. & Bush, J. W. M. 2013b Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory. J. Fluid Mech. 727, 612647.
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, 404429.
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.
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, 552570.
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.
Protière, S., Bohn, S. & Couder, Y. 2008 Exotic orbits of two interacting wave sources. Phys. Rev. E 78, 36204.
Protière, S., Boudaoud, A. & Couder, Y. 2006 Particle-wave association on a fluid interface. J. Fluid Mech. 554, 85108.
Protière, S., Couder, Y., Fort, E. & Boudaoud, A. 2005 The self-organization of capillary wave sources. J. Phys.: Condens. Matter 17, 35293535.
Walker, J. 1978 Drops of liquid can be made to float on the liquid. What enables them to do so? Sci. Am. 238, 151158.
Wang, Z. & Milewski, P. A. 2012 Dynamics of gravity–capillary solitary waves in deep water. J. Fluid Mech. 708, 480501.
Wind-Willassen, O., Moláček, J., Harris, D. M. & Bush, J. W. M. 2013 Exotic states of bouncing and walking droplets. Phys. Fluids 25, 082002.
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Journal of Fluid Mechanics
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