Skip to main content Accessibility help
×
Home
Hostname: page-component-78dcdb465f-f64jw Total loading time: 0.366 Render date: 2021-04-15T15:42:10.496Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

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

Published online by Cambridge University Press:  23 December 2013

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
Corresponding
E-mail address:

Abstract

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.

Type
Papers
Copyright
©2013 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below.

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, 505515.CrossRefGoogle 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, 123.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.CrossRefGoogle ScholarPubMed
Bush, J. W. M. 2010 Quantum mechanics writ large. Proc. Natl Acad. Sci. 107, 1745517456.CrossRefGoogle Scholar
Couder, Y. & Fort, E. 2006 Single particle diffraction and interference at a macroscopic scale. Phys. Rev. Lett. 97, 154101.CrossRefGoogle 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.CrossRefGoogle Scholar
Crommie, M. F., Lutz, C. P. & Eigler, D. M. 1993 Confinement of electrons to quantum corrals on a metal surface. Science 262, 218220.CrossRefGoogle 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., 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, 433463.CrossRefGoogle Scholar
Eddi, A., Terwagne, D., Fort, E. & Couder, Y. 2008 Wave propelled ratchets and drifting rafts. Europhys. Lett. 82, 44001.CrossRefGoogle Scholar
Faraday, M. 1831 On the forms and states of fluids on vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. 121, 319340.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), 1751517520.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.CrossRefGoogle Scholar
Kumar, K. 1996 Linear theory of Faraday instability in viscous fluids. Proc. R. Soc. A 452, 11131126.CrossRefGoogle Scholar
Moláček, J. & Bush, J. W. M. 2013a Drops bouncing on a vibrating bath. J. Fluid Mech. 727, 582611.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, 612647.CrossRefGoogle 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, 552570.CrossRefGoogle 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, 85108.CrossRefGoogle 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, 151158.CrossRefGoogle 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

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 245 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 15th April 2021. This data will be updated every 24 hours.

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.

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

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

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

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *