Skip to main content Accessibility help
×
Home
Hostname: page-component-684bc48f8b-kl86h Total loading time: 0.403 Render date: 2021-04-11T13:47:06.122Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

Bouncing phase variations in pilot-wave hydrodynamics and the stability of droplet pairs

Published online by Cambridge University Press:  17 May 2019

Miles M. P. Couchman
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Sam E. Turton
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 integrated experimental and theoretical investigation of the vertical motion of millimetric droplets bouncing on a vibrating fluid bath. We characterize experimentally the dependence of the phase of impact and contact force between a drop and the bath on the drop’s size and the bath’s vibrational acceleration. This characterization guides the development of a new theoretical model for the coupling between a drop’s vertical and horizontal motion. Our model allows us to relax the assumption of constant impact phase made in models based on the time-averaged trajectory equation of Moláček and Bush (J. Fluid Mech., vol. 727, 2013b, pp. 612–647) and obtain a robust horizontal trajectory equation for a bouncing drop that accounts for modulations in the drop’s vertical dynamics as may arise when it interacts with boundaries or other drops. We demonstrate that such modulations have a critical influence on the stability and dynamics of interacting droplet pairs. As the bath’s vibrational acceleration is increased progressively, initially stationary pairs destabilize into a variety of dynamical states including rectilinear oscillations, circular orbits and side-by-side promenading motion. The theoretical predictions of our variable-impact-phase model rationalize our observations and underscore the critical importance of accounting for variability in the vertical motion when modelling droplet–droplet interactions.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

Access options

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

References

Arbelaiz, J., Oza, A. U. & Bush, J. W. M. 2018 Promenading pairs of walking droplets: dynamics and stability. Phys. Rev. Fluids 3 (1), 013604.CrossRefGoogle Scholar
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.Google Scholar
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 (6), 063017.Google ScholarPubMed
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
Bush, J. W. M. 2015a The new wave of pilot-wave theory. Phys. Today 68 (8), 4753.CrossRefGoogle Scholar
Bush, J. W. M. 2015b Pilot-wave hydrodynamics. Annu. Rev. Fluid Mech. 47, 269292.CrossRefGoogle Scholar
Bush, J. W. M., Couder, Y., Gilet, T., Milewski, P. A. & Nachbin, A. 2018 Introduction to focus issue on hydrodynamic quantum analogs. Chaos 28 (9), 096001.CrossRefGoogle ScholarPubMed
Bush, J. W. M., Oza, A. U. & Moláček, J. 2014 The wave-induced added mass of walking droplets. J. Fluid Mech. 755, R7.CrossRefGoogle Scholar
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 (17), 177801.CrossRefGoogle Scholar
Couder, Y., Protiere, S., Fort, E. & Boudaoud, A. 2005b Dynamical phenomena: walking and orbiting droplets. Nature 437 (7056), 208.CrossRefGoogle Scholar
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 (10), 163.CrossRefGoogle Scholar
Dubertrand, R., Hubert, M., Schlagheck, P., Vandewalle, N., Bastin, T. & Martin, J. 2016 Scattering theory of walking droplets in the presence of obstacles. New J. Phys. 18 (11), 113037.Google Scholar
Durey, M. & Milewski, P. A. 2017 Faraday wave–droplet dynamics: discrete-time analysis. J. Fluid Mech. 821, 296329.CrossRefGoogle Scholar
Eddi, A., Boudaoud, A. & Couder, Y. 2011a Oscillating instability in bouncing droplet crystals. Europhys. Lett. 94 (2), 20004.CrossRefGoogle Scholar
Eddi, A., Decelle, A., Fort, E. & Couder, Y. 2009 Archimedean lattices in the bound states of wave interacting particles. Europhys. Lett, 87 (5), 56002.CrossRefGoogle Scholar
Eddi, A., Sultan, E., Moukhtar, J., Fort, E., Rossi, M. & Couder, Y. 2011b Information stored in Faraday waves: the origin of a 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 (4), 44001.CrossRefGoogle Scholar
Faria, L. M. 2017 A model for Faraday pilot waves over variable topography. J. Fluid Mech. 811, 5166.CrossRefGoogle Scholar
Filoux, B., Hubert, M. & Vandewalle, N. 2015 Strings of droplets propelled by coherent waves. Phys. Rev. E 92 (4), 041004.Google ScholarPubMed
Galeano-Rios, C. A., Couchman, M. M. P., Caldairou, P. & Bush, J. W. M. 2018 Ratcheting droplet pairs. Chaos 28 (9), 096112.CrossRefGoogle ScholarPubMed
Galeano-Rios, C. A., Milewski, P. A. & Vanden-Broeck, J.-M. 2017 Non-wetting impact of a sphere onto a bath and its application to bouncing droplets. J. Fluid Mech. 826, 97127.CrossRefGoogle Scholar
Gilet, T., Vandewalle, N. & Dorbolo, S. 2009 Completely inelastic ball. Phys. Rev. E 79 (5), 055201.Google ScholarPubMed
Hammond, R. T. 2010 Relativistic particle motion and radiation reaction in electrodynamics. Electron. J. Theoret. Phys. 7 (23), 221258.Google Scholar
Harris, D. M. & Bush, J. W. M. 2015 Generating uniaxial vibration with an electrodynamic shaker and external air bearing. J. Sound Vib. 334, 255269.Google Scholar
Harris, D. M., Liu, T. & Bush, J. W. M. 2015 A low-cost, precise piezoelectric droplet-on-demand generator. Exp. Fluids 56 (4), 83.CrossRefGoogle Scholar
Labousse, M., Oza, A. U., Perrard, S. & Bush, J. W. M. 2016 Pilot-wave dynamics in a harmonic potential: quantization and stability of circular orbits. Phys. Rev. E 93 (3), 033122.Google Scholar
Lieber, S. I., Hendershott, M. C., Pattanaporkratana, A. & Maclennan, J. E. 2007 Self-organization of bouncing oil drops: two-dimensional lattices and spinning clusters. Phys. Rev. E 75 (5), 056308.Google ScholarPubMed
Miles, J. & Henderson, D. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22 (1), 143165.CrossRefGoogle Scholar
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.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
Nachbin, A. 2018 Walking droplets correlated at a distance. Chaos 28 (9), 096110.CrossRefGoogle ScholarPubMed
Nachbin, A., Milewski, P. A. & Bush, J. W. M. 2017 Tunneling with a hydrodynamic pilot-wave model. Phys. Rev. Fluids 2 (3), 034801.CrossRefGoogle Scholar
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.CrossRefGoogle 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, 552570.CrossRefGoogle Scholar
Oza, A. U., Siéfert, E., Harris, D. M., Moláček, J. & Bush, J. W. M. 2017 Orbiting pairs of walking droplets: dynamics and stability. Phys. Rev. Fluids 2, 053601.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. Phys. Fluids 26 (8), 082101.CrossRefGoogle Scholar
Perrard, S. & Labousse, M. 2018 Transition to chaos in wave memory dynamics in a harmonic well: deterministic and noise-driven behavior. Chaos 28 (9), 096109.CrossRefGoogle Scholar
Protière, S., Bohn, S. & Couder, Y. 2008 Exotic orbits of two interacting wave sources. Phys. Rev. E 78 (3), 036204.Google ScholarPubMed
Protière, S., Boudaoud, A. & Couder, Y. 2006 Particle–wave association on a fluid interface. J. Fluid Mech. 554, 85108.CrossRefGoogle Scholar
Protière, S., Couder, Y., Fort, E. & Boudaoud, A. 2005 The self-organization of capillary wave sources. J. Phys.: Condens. Matter 17 (45), S3529S3535.Google Scholar
Pucci, G., Sáenz, P. J., Faria, L. M. & Bush, J. W. M. 2016 Non-specular reflection of walking droplets. J. Fluid Mech. 804, R3.CrossRefGoogle Scholar
Sáenz, P. J., Pucci, G., Goujon, A., Cristea-Platon, T., Dunkel, J. & Bush, J. W. M. 2018 Spin lattices of walking droplets. Phys. Rev. Fluids 3 (10), 100508.CrossRefGoogle Scholar
Tadrist, L., Sampara, N., Schlagheck, P. & Gilet, T. 2018a Interaction of two walkers: perturbed vertical dynamics as a source of chaos. Chaos 28 (9), 096113.CrossRefGoogle Scholar
Tadrist, L., Shim, J.-B., Gilet, T. & Schlagheck, P. 2018b Faraday instability and subthreshold Faraday waves: surface waves emitted by walkers. J. Fluid Mech. 848, 906945.CrossRefGoogle Scholar
Terwagne, D., Ludewig, F., Vandewalle, N. & Dorbolo, S. 2013 The role of the droplet deformations in the bouncing droplet dynamics. Phys. Fluids 25 (12), 122101.CrossRefGoogle Scholar
Turton, S. E., Couchman, M. M. P. & Bush, J. W. M. 2018 A review of the theoretical modeling of walking droplets: towards a generalized pilot-wave framework. Chaos 28 (9), 096111.CrossRefGoogle Scholar
Walker, J. 1978 The amateur scientist. Drops of liquid can be made to float on liquid. What enables them to do so? Sci. Am. 238 (6), 151.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 (8), 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: 27
Total number of PDF views: 347 *
View data table for this chart

* Views captured on Cambridge Core between 17th May 2019 - 11th 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.

Bouncing phase variations in pilot-wave hydrodynamics and the stability of droplet pairs
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.

Bouncing phase variations in pilot-wave hydrodynamics and the stability of droplet pairs
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.

Bouncing phase variations in pilot-wave hydrodynamics and the stability of droplet pairs
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *