Hostname: page-component-7b9c58cd5d-6tpvb Total loading time: 0 Render date: 2025-03-14T20:33:17.123Z Has data issue: false hasContentIssue false
  • English
  • Français

Droplets walking in a rotating frame: from quantized orbits to multimodal statistics

Published online by Cambridge University Press:  23 December 2013

Daniel M. Harris  and
John W. M. Bush
Daniel M. Harris
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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present the results of an experimental investigation of a droplet walking on the surface of a vibrating rotating fluid bath. Particular attention is given to demonstrating that the stable quantized orbits reported by Fort et al. (Proc. Natl Acad. Sci., vol. 107, 2010, pp. 17515–17520) arise only for a finite range of vibrational forcing, above which complex trajectories with multimodal statistics arise. We first present a detailed characterization of the emergence of orbital quantization, and then examine the system behaviour at higher driving amplitudes. As the vibrational forcing is increased progressively, stable circular orbits are succeeded by wobbling orbits with, in turn, stationary and drifting orbital centres. Subsequently, there is a transition to wobble-and-leap dynamics, in which wobbling of increasing amplitude about a stationary centre is punctuated by the orbital centre leaping approximately half a Faraday wavelength. Finally, in the limit of high vibrational forcing, irregular trajectories emerge, characterized by a multimodal probability distribution that reflects the persistent dynamic influence of the unstable orbital states.

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 739 , 25 January 2014 , pp. 444 - 464
Copyright
©2013 Cambridge University Press 

References

Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. A 225, 505–515.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, 1–23.Google Scholar
de Bruyn, J. R., Lewis, B. C., Shattuck, M. D. & Swinney, H. L. 2001 Spiral patterns in oscillated granular layers. Phys. Rev. E 63, 041305.Google Scholar
Bush, J. W. M. 2010 Quantum mechanics writ large. Proc. Natl Acad. Sci. 107, 17455–17456.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., Fort, E., Gautier, C. H. & Boudaoud, A. 2005 From bouncing to floating: non-coalescence of drops on a fluid bath. Phys. Rev. Lett. 94, 177801.Google Scholar
Crommie, M. F., Lutz, C. P. & Eigler, D. M. 1993 Confinement of electrons to quantum corrals on a metal surface. Science 262, 218–220.CrossRefGoogle ScholarPubMed
Eddi, A., Fort, E., Moisy, F. & Couder, Y. 2009 Unpredictable tunneling of a classical wave–particle association. Phys. Rev. Lett. 102, 240401.Google Scholar
Eddi, A., Moukhtar, J., Perrard, S., Fort, E. & Couder, Y. 2012 Level splitting at 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 path memory. J. Fluid Mech. 674, 433–463.CrossRefGoogle Scholar
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 the forms and states of fluids on vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 319–340.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.CrossRefGoogle Scholar
Goldman, D. I. 2002 Pattern formation and fluidization of vibrated granular layers, and grain dynamics and jamming in a water fluidized bed. PhD thesis, University of Texas at Austin, Austin, TX.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
Kumar, K. 1996 Linear theory of Faraday instability in viscous fluids. Proc. R. Soc. 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.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., Harris, D. M., Rosales, R. R. & Bush, J. W. M. 2013a Pilot-wave dynamics in a rotating frame: on the emergence of orbital quantization. (under review).Google Scholar
Oza, A., Rosales, R. R. & Bush, J. W. M. 2013b A trajectory equation for walking droplets. J. Fluid Mech. 737, 552–570.Google Scholar
Oza, A., Wind-Willassen, Ø., Harris, D. M., Rosales, R. R. & Bush, J. W. M. 2013c Exotic orbits in hydrodynamic pilot-wave theory (in preparation).Google Scholar
Perrard, S., Labousse, M., Miskin, M., Fort, E. & Couder, Y. 2013 Memory driven wave–particle self-organization (under review).Google Scholar
Protière, S., Boudaoud, A. & Couder, Y. 2006 Particle wave association on a fluid interface. J. Fluid Mech. 554, 85–108.Google Scholar
Reis, P. M., Ingale, R. A. & Shattuck, M. D. 2007 Forcing independent velocity distributions in an experimental granular fluid. Phys. Rev. E 75, 051311.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.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