Skip to main content
×
×
Home

Faraday wave–droplet dynamics: discrete-time analysis

  • Matthew Durey (a1) and Paul A. Milewski (a1)
Abstract

A droplet may ‘walk’ across the surface of a vertically vibrating bath of the same fluid, due to the propulsive interaction with its wave field. This hydrodynamic pilot-wave system exhibits many dynamics previously believed to exist only in the quantum realm. Starting from first principles, we derive a discrete-time fluid model, whereby the bath–droplet interactions are modelled as instantaneous. By analysing the stability of the fixed points of the system, we explain the dynamics of a walking droplet and capture the quantisations for multiple-droplet interactions. Circular orbits in a harmonic potential are studied, and a double quantisation of chaotic trajectories is obtained through systematic statistical analysis.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Faraday wave–droplet dynamics: discrete-time analysis
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Faraday wave–droplet dynamics: discrete-time analysis
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Faraday wave–droplet dynamics: discrete-time analysis
      Available formats
      ×
Copyright
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Corresponding author
Email address for correspondence: m.durey@bath.ac.uk
References
Hide All
Abramowitz, M. & Stegun, I. 1964 Handbook of Mathematical Functions. Dover.
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 Interaction of two walkers: wave-mediated energy and force. Phys. Rev. E 90, 063017.
de Broglie, L. 1926 Ondes et Mouvements. Gauthier-Villars.
Bush, J. W. M. 2015 Pilot-wave hydrodynamics. Annu. Rev. Fluid Mech. 47, 269292.
Bush, J. W. M., Oza, A. U. & Moláček, J. 2014 The wave-induced added mass of walking droplets. J. Fluid Mech. 755, R7.
Catllá, A. J., Schaeffer, D. G., Witelski, T. P., Monson, E. E. & Lin, A. L. 2008 On spiking models for synaptic activity and impulsive differential equations. SIAM Rev. 50 (3), 553569.
Couder, Y. & Fort, E. 2006 Single-particle diffraction and interference at a macroscopic scale. Phys. Rev. Lett. 97, 1541017.
Couder, Y. & Fort, E. 2011 Probabilities and trajectories in a classical wave–particle duality. J. Phys.: Conf. Ser. 361, 012001.
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.
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 (163), 17.
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., Fort, E., Moisy, F. & Couder, Y. 2009 Unpredictable tunneling of a classical wave–particle association. Phys. Rev. Lett. 102, 240401.
Eddi, A., Moukhtar, J., Perrard, S., Fort, E. & Couder, Y. 2012 Level splitting at 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.
Filoux, B., Hubert, M. & Vandewalle, N. 2015 Strings of droplets propelled by coherent waves. Phys. Rev. E 92, 041004(R).
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), 1751517520.
Gilet, T. & Bush, J. W. M. 2009 Chaotic bouncing of a droplet on a soap film. Phys. Rev. Lett. 102, 014501.
Gilet, T., Terwagne, D., Vandewalle, N. & Dorbolo, S. 2008 Dynamics of a bouncing droplet onto a vertically vibrated interface. Phys. Rev. Lett. 100, 167802.
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.
Hartigan, J. A. 1975 Clustering Algorithms. Wiley.
Hartigan, J. A. & Wong, M. A. 1979 A K-means clustering algorithm. J. R. Stat. Soc. Ser. C Appl. Stat. 28 (1), 100108.
Hubert, M., Robert, D., Caps, H., Dorbolo, S. & Vandewalle, N. 2015 Resonant and antiresonant bouncing droplets. Phys. Rev. E 91, 023017.
Kodinariya, T. M. & Makwana, P. R. 2013 Review on determining number of cluster in k-means clustering. Intl J. Adv. Res. Comput. Sci. Management Studies 1 (6), 9095.
Kumar, K. 1996 Linear theory of Faraday instability in viscous fluids. 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., Oza, A., Perrard, S. & Bush, J. W. M. 2016 Pilot-wave dynamics in a harmonic potential: quantization and stability of circular orbits. Phys. Rev. E 93, 033122.
Labousse, M. & Perrard, S. 2014 Non-Hamiltonian features of a classical pilot-wave system. Phys. Rev. E 90, 022913.
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.
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, 361388.
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., Rosales, R. R. & Bush, J. W. M. 2013 A trajectory equation for walking droplets: hydrodynamic pilot-wave theory. J. Fluid Mech. 737, 552570.
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. Phys. Fluids 26, 082101.
Perrard, S., Labousse, M., Fort, E. & Couder, Y. 2014a Chaos driven by interfering memory. Phys. Rev. Lett. 113, 104101.
Perrard, S., Labousse, M., Miskin, M., Fort, E. & Couder, Y. 2014b Self-organization into quantized eigenstates of a classical wave-driven particle. Nature Comm. 5, 3219.
Protière, S., Bohn, S. & Couder, Y. 2008 Exotic orbits of two interacting wave sources. Phys. Rev. E 78, 036204.
Protière, S., Boudaoud, A. & Couder, Y. 2006 Particle–wave association on a fluid interface. J. Fluid Mech. 554, 85108.
Walker, J. 1978 Drops of liquid can be made to float on the liquid. What enables them to do so? Sci. Am. 238, 151158.
Wind-Willassen, Ø., Moláček, J., Harris, D. M. & Bush, J. W. M. 2013 Exotic states of bouncing and walking droplets. Phys. Fluids 25, 082002.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed