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A model for Faraday pilot waves over variable topography

Published online by Cambridge University Press:  06 December 2016

Luiz M. Faria
Luiz M. Faria*
Affiliation:
Department of Mathematics, Massachusets Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: lfaria@mit.edu
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Abstract

Couder et al. (Nature, vol. 437 (7056), 2005, p. 208) discovered that droplets walking on a vibrating bath possess certain features previously thought to be exclusive to quantum systems. These millimetric droplets synchronize with their Faraday wavefield, creating a macroscopic pilot-wave system. In this paper we exploit the fact that the waves generated are nearly monochromatic and propose a hydrodynamic model capable of quantitatively capturing the interaction between bouncing drops and a variable topography. We show that our reduced model is able to reproduce some important experiments involving the drop–topography interaction, such as non-specular reflection and single-slit diffraction.

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 811 , 25 January 2017 , pp. 51 - 66
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Copyright
© 2016 Cambridge University Press 

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References

Andersen, A., Madsen, J., Reichelt, C., Ahl, S. R., Lautrup, B., Ellegaard, C., Levinsen, M. T. & Bohr, T. 2015 Double-slit experiment with single wave-driven particles and its relation to quantum mechanics. Phys. Rev. E 92 (1), 013006.Google Scholar
Bach, R., Pope, D., Liou, Sy.-H. & Batelaan, H. 2013 Controlled double-slit electron diffraction. New J. Phys. 15 (3), 033018.Google Scholar
Batelaan, H., Jones, E., Huang, W. C.-W. & Bach, R. 2016 Momentum exchange in the electron double-slit experiment. J. Phys. Conf. Series 701, 012007.CrossRefGoogle 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
Blanchette, F. 2016 Modeling the vertical motion of drops bouncing on a bounded fluid reservoir. Phys. Fluids 28 (3), 032104.Google Scholar
de Broglie, L. 1930 An Introduction to the Study of Wave Mechanics. Methuen & Co. Ltd..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 (15), 154101.CrossRefGoogle Scholar
Couder, Y., Protiere, S., Fort, E. & Boudaoud, A. 2005 Dynamical phenomena: walking and orbiting droplets. Nature 437 (7056), 208–208.CrossRefGoogle ScholarPubMed
Crommie, M. F., Lutz, C. P. & Eigler, D. M. 1993 Confinement of electrons to quantum corrals on a metal surface. Science 262 (5131), 218–220.CrossRefGoogle ScholarPubMed
Damiano, A. P., Brun, P.-T., Harris, D. M., Galeano-Rios, C. A. & Bush, J. W. M. 2016 Surface topography measurements of the bouncing droplet experiment. Exp. Fluids 57 (10), 163.Google Scholar
Dubertrand, R., Hubert, M., Schlagheck, P., Vandewalle, N., Bastin, T. & Martin, J.2016 Scattering theory of walking droplets in the presence of obstacles. ArXiv preprint arXiv:1605.02370.Google Scholar
Eddi, A., Fort, E., Moisy, F. & Couder, Y. 2009 Unpredictable tunneling of a classical wave–particle association. Phys. Rev. Lett. 102 (24), 240401.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
Fort, E., Eddi, A., Boudaoud, A., Moukhtar, J. & Couder, Y. 2010 Path-memory induced quantization of classical orbits. Proc. Natl Acad. Sci. USA 107 (41), 17515–17520.Google Scholar
Gilet, T. 2014 Dynamics and statistics of wave–particle interactions in a confined geometry. Phys. Rev. E 90 (5), 052917.Google Scholar
Harris, D. M.2015 The pilot-wave dynamics of walking droplets in confinement. PhD thesis, Massachusetts Institute of Technology.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 (1), 011001.Google Scholar
Kumar, K. 1996 Linear theory of faraday instability in viscous liquids. Proc. R. Soc. Lond. A 452, 1113–1126.Google Scholar
Labousse, M.2014 Étude d’une dynamique à mémoire de chemin: une expérimentation théorique. PhD thesis, Université Pierre et Marie Curie UPMC Paris VI.Google Scholar
Leveque, R. J. 2007 Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, vol. 98. SIAM.Google Scholar
Milewski, P. A., Galeano-Rios, C. A., Nachbin, A. & Bush, J. W. M. 2015 Faraday pilot-wave dynamics: modelling and computation. J. Fluid Mech. 778, 361–388.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
Nachbin, A., Milewski, P. A. & Bush, J. W. M. 2016 Tunneling with a hydrodynamic pilot-wave model. Phys. Rev. Fluids; (submitted for publication).Google Scholar
Oza, A. U., Harris, D. M., Rosales, R. R. & Bush, J. W. M. 2014 Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization. J. Fluid Mech. 744, 404–429.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. Nat. Commun. 5, 3219.Google Scholar
Protière, S., Boudaoud, A. & Couder, Y. 2006 Particle–wave association on a fluid interface. J. Fluid Mech. 554, 85–108.Google Scholar
Pucci, G., Harris, D. M., Faria, L. M. & Bush, J. W. M. 2016a Walking droplets passing through single- and double-slits. J. Fluid Mech. (to be submitted).Google Scholar
Pucci, G., Sáenz, P. J., Faria, L. M. & Bush, J. W. M. 2016b Non-specular reflection of walking droplets. J. Fluid Mech. 804, R3.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.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar