Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-26T03:52:40.969Z Has data issue: false hasContentIssue false
  • English
  • Français

Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization

Published online by Cambridge University Press:  13 March 2014

Anand U. Oza ,
Daniel M. Harris ,
Rodolfo R. Rosales  and
John W. M. Bush
Anand U. Oza
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Daniel M. Harris
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Rodolfo R. Rosales
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
John W. M. Bush*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
†Email address for correspondence: bush@math.mit.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We present the results of a theoretical investigation of droplets walking on a rotating vibrating fluid bath. The droplet’s trajectory is described in terms of an integro-differential equation that incorporates the influence of its propulsive wave force. Predictions for the dependence of the orbital radius on the bath’s rotation rate compare favourably with experimental data and capture the progression from continuous to quantized orbits as the vibrational acceleration is increased. The orbital quantization is rationalized by assessing the stability of the orbital solutions, and may be understood as resulting directly from the dynamic constraint imposed on the drop by its monochromatic guiding wave. The stability analysis also predicts the existence of wobbling orbital states reported in recent experiments, and the absence of stable orbits in the limit of large vibrational forcing.

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 744 , 10 April 2014 , pp. 404 - 429
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

Bacciagaluppi, G. & Valentini, A. 2009 Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference. Cambridge University Press.CrossRefGoogle Scholar
Bateman, H. 1944 Partial Differential Equations of Mathematical Physics. Dover.Google Scholar
Bell, J. S. 1988 Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press.Google Scholar
de Broglie, L. 1926 Ondes et Mouvements. 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
Burinskii, A. 2008 The Dirac–Kerr–Newman electron. Grav. Cosmol. 40 (2), 109–122.CrossRefGoogle Scholar
Bush, J. W. M. 2010 Quantum mechanics writ large. Proc. Natl Acad. Sci. USA 107 (41), 17455–17456.CrossRefGoogle Scholar
Cohen-Tannoudji, C., Diu, B. & Laloë, F. 1977 Quantum Mechanics. John Wiley & Sons.Google Scholar
Couder, Y. & Fort, E. 2006 Single-particle diffraction and interference at a macroscopic scale. Phys. Rev. Lett. 97, 154101.CrossRefGoogle Scholar
Couder, Y., Protière, S., Fort, E. & Boudaoud, A. 2005 Walking and orbiting droplets. Nature 437, 208.CrossRefGoogle ScholarPubMed
Crommie, M., Lutz, C. & Eigler, D. 1993 Confinement of electrons to quantum corrals on a metal surface. Science 262, 5131.CrossRefGoogle ScholarPubMed
Eddi, A., Fort, E., Moisy, F. & Couder, Y. 2009 Unpredictable tunnelling 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., Sultan, E., Moukhtar, J., Fort, E., Rossi, M. & Couder, Y. 2011 Information stored in Faraday waves: the origin of path memory. J. Fluid Mech. 675, 433–463.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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
Kumar, K. 1996 Linear theory of Faraday instability in viscous fluids. Proc. R. Soc. Lond. A 452, 1113–1126.Google Scholar
Moláček, J. & Bush, J. W. M. 2013a Drops bouncing on a vibrating bath. J. Fluid Mech. 727, 582–611.CrossRefGoogle 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.CrossRefGoogle Scholar
Oza, A. U., Bush, J. W. M. & Rosales, R. R. 2014a Orbital stability in hydrodynamic pilot-wave theory (in preparation).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.CrossRefGoogle Scholar
Oza, A. U., Wind-Willassen, Ø., Harris, D. M., Rosales, R. R. & Bush, J. W. M. 2014b Pilot-wave hydrodynamics in a rotating frame: exotic orbits. Physics of Fluids (submitted).CrossRefGoogle 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., Boudaoud, A. & Couder, Y. 2006 Particle-wave association on a fluid interface. J. Fluid Mech. 554, 85–108.CrossRefGoogle Scholar
Walker, J. 1978 Drops of liquid can be made to float on the liquid. What enables them to do so?. Sci. Am. 238 (6), 151–158.CrossRefGoogle Scholar
Watson, G. N. 1966 A Treatise on the Theory of Bessel Functions. 2nd edn. Cambridge University Press.Google Scholar
Weinstein, A. & Pounder, J. R. 1945 An electromagnetic analogy in mechanics. Am. Math. Mont. 52 (8), 432–438.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.CrossRefGoogle Scholar