Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-26T08:38:06.987Z Has data issue: false hasContentIssue false
  • English
  • Français

The wave-induced added mass of walking droplets

Published online by Cambridge University Press:  22 August 2014

John W. M. Bush ,
Anand U. Oza  and
Jan Moláček
John W. M. Bush*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Anand U. Oza
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Jan Moláček
Affiliation:
Max Planck Institute for Dynamics and Self-Organization, 37077 Göttingen, Germany
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

It has recently been demonstrated that droplets walking on a vibrating fluid bath exhibit several features previously thought to be peculiar to the microscopic realm. The walker, consisting of a droplet plus its guiding wavefield, is a spatially extended object. We here examine the dependence of the walker mass and momentum on its velocity. Doing so indicates that, when the walker’s time scale of acceleration is long relative to the wave decay time, its dynamics may be described in terms of the mechanics of a particle with a speed-dependent mass and a nonlinear drag force that drives it towards a fixed speed. Drawing an analogy with relativistic mechanics, we define a hydrodynamic boost factor for the walkers. This perspective provides a new rationale for the anomalous orbital radii reported in recent studies.

JFM classification

Drops and Bubbles: Drops Waves/Free-surface Flows: Faraday waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 755 , 25 September 2014 , R7
Copyright
© 2014 Cambridge University Press 

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

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
Bohm, D. J. & Hiley, B. J. 1982 The de Broglie pilot wave theory and the further development of new insights arising out of it. Found. Phys. 12 (10), 1001–1016.CrossRefGoogle Scholar
Boyer, T. H. 2011 Any classical description of nature requires classical electromagnetic zero-point radiation. Am. J. Phys. 79, 1163–1167.Google Scholar
de Broglie, L. 1926 Ondes et Mouvements. Gauthier-Villars.Google Scholar
de Broglie, L. 1956 Une Tentative d’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 (4), 1–23.Google Scholar
Bush, J. W. M. 2015 Pilot-wave hydrodynamics. Annu. Rev. Fluid Mech. 47 (in press).CrossRefGoogle Scholar
Chebotarev, L. 2000 Introduction: the de Broglie–Bohm–Vigier approach in quantum mechanics. In Jean-Pierre Vigier and the Stochastic Interpretation of Quantum Mechanics (ed. Jeffers, S., Lehnert, B., Abramson, N. & Chebotarev, L.), Berkeley Press.Google Scholar
Couder, Y. & Fort, E. 2006 Single-particle diffraction and interference at a macroscopic scale. Phys. Rev. Lett. 97, 154101.Google Scholar
Couder, Y., Protière, S., Fort, E. & Boudaoud, A. 2005 Walking and orbiting droplets. Nature 437, 208.Google Scholar
Eddi, A., Fort, E., Moisy, F. & Couder, Y. 2009 Unpredictable tunneling of a classical wave–particle association. Phys. Rev. Lett. 102, 240401.CrossRefGoogle ScholarPubMed
Eddi, A., Moukhtar, J., Perrard, S., Fort, E. & Couder, Y. 2012 Level splitting at macroscopic scale. Phys. Rev. Lett. 108, 264503.CrossRefGoogle ScholarPubMed
Eddi, A., Terwagne, D., Fort, E. & Couder, Y. 2008 Wave propelled ratchets and drifting rafts. Europhys. Lett. 82, 44001.Google Scholar
Faraday, M. 1831 On a peculiar class of acoustical figures, and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 299–340.Google Scholar
Fort, E., Eddi, A., Boudaoud, A., Moukhtar, J. & Couder, Y. 2010 Path-memory induced quantization of classical orbits. Proc. Natl Acad. Sci. 107 (41), 17515–17520.Google Scholar
Haisch, B. & Rueda, A. 2000 On the relation between a zero-point-field-induced inertial effect and the Einstein–de Broglie formula. Phys. Rev. A 268, 421–427.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.Google Scholar
Labousse, M. & Perrard, S. 2014 Non Hamiltonian features of a classical pilot-wave dynamics. Phys. Rev. E (in press).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. 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
de la Peña, L. & Cetto, A. M. 1996 The Quantum Dice: An Introduction to Stochastic Electrodynamics. Kluwer Academic.CrossRefGoogle Scholar
Perrard, S., Labousse, M., Fort, E. & Couder, Y. 2014a Chaos driven by interfering memory. Phys. Rev. Lett. (in press).Google Scholar
Perrard, S., Labousse, M., Miskin, M., Fort, E. & Couder, Y. 2014b 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.CrossRefGoogle Scholar
Protière, S., Couder, Y., Fort, E. & Boudaoud, A. 2005 The self-organization of capillary wave sources. J. Phys.: Condens. Matter 17 (45), S3529–S3535.Google Scholar
Rueda, A. & Haisch, B. 2005 Gravity and the quantum vacuum inertia hypothesis. Ann. Phys. (Leipzig) 14 (8), 479–498.Google Scholar
Wind-Willassen, Ø., Moláček, J., Harris, D. M. & Bush, J. W. M. 2013 Exotic states of bouncing and walking droplets. Phys. Fluids 25, 082002.Google Scholar