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Free rings of bouncing droplets: stability and dynamics

Published online by Cambridge University Press:  05 October 2020

Miles M. P. Couchman Open the ORCID record for Miles M. P. Couchman [Opens in a new window]  and
John W. M. Bush Open the ORCID record for John W. M. Bush [Opens in a new window]
Miles M. P. Couchman
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA
John W. M. Bush*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA02139, USA
*
†Email address for correspondence: [email protected]
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Abstract

We present the results of a combined experimental and theoretical investigation of the stability of rings of millimetric droplets bouncing on the surface of a vibrating liquid bath. As the bath's vibrational acceleration is increased progressively, droplet rings are found to destabilize into a rich variety of dynamical states including steady rotational motion, periodic radial or azimuthal oscillations and azimuthal travelling waves. The instability observed is dependent on the ring's initial radius and drop number, and whether the drops are bouncing in- or out-of-phase relative to their neighbours. As the vibrational acceleration is further increased, more exotic dynamics emerges, including quasi-periodic motion and rearrangement into regular polygonal structures. Linear stability analysis and simulation of the rings based on the theoretical model of Couchman et al. (J. Fluid Mech., vol. 871, 2019, pp. 212–243) largely reproduce the observed behaviour. We demonstrate that the wave amplitude beneath each drop has a significant influence on the stability of the multi-droplet structures: the system seeks to minimize the mean wave amplitude beneath the drops at impact. Our work provides insight into the complex interactions and collective motions that arise in bouncing-droplet aggregates.

JFM classification

Drops and Bubbles: Drops Waves/Free-surface Flows: Faraday waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 903 , 25 November 2020 , A49
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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
Aref, H., Newton, P. K., Stremler, M. A., Tokieda, T. & Vainchtein, D. L. 2003 Vortex crystals. Adv. Appl. Mech. 39, 1–79.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 (1163), 505–515.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.CrossRefGoogle ScholarPubMed
de Broglie, L. 1956 Une tentative d'interprétation causale et nonlinéaire de la mechanique ondulatoire: la théorie de la double solution. Gautier-Villars.Google Scholar
Bush, J. W. M. 2015 Pilot-wave hydrodynamics. Annu. Rev. Fluid Mech. 47, 269–292.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
Campbell, L. J. & Ziff, R. M. 1979 Vortex patterns and energies in a rotating superfluid. Phys. Rev. B 20 (5), 1886–1902.CrossRefGoogle Scholar
Celli, M., Lacomba, E. A. & Pérez-Chavela, E. 2011 On polygonal relative equilibria in the N-vortex problem. J. Math. Phys. 52 (10), 103101.CrossRefGoogle Scholar
Colin, S., Durt, T. & Willox, R. 2017 de Broglie's double solution program: 90 years later. Annales de la Fondation Louis de Broglie 42, 19–71.Google Scholar
Couchman, M. M. P., Turton, S. E. & Bush, J. W. M. 2019 Bouncing phase variations in pilot-wave hydrodynamics and the stability of droplet pairs. J. Fluid Mech. 871, 212–243.CrossRefGoogle Scholar
Couder, Y., Fort, E., Gautier, C.-H. & Boudaoud, A. 2005 a From bouncing to floating: noncoalescence of drops on a fluid bath. Phys. Rev. Lett. 94 (17), 177801.CrossRefGoogle ScholarPubMed
Couder, Y., Protiere, S., Fort, E. & Boudaoud, A. 2005 b Dynamical phenomena: walking and orbiting droplets. Nature 437 (7056), 208.CrossRefGoogle ScholarPubMed
Crowdy, D. 1999 A class of exact multipolar vortices. Phys. Fluids 11 (9), 2556–2564.CrossRefGoogle Scholar
Crowdy, D. 2003 Polygonal N-vortex arrays: a stuart model. Phys. Fluids 15 (12), 3710–3717.CrossRefGoogle Scholar
Crowdy, D. & Cloke, M. 2002 Stability analysis of a class of two-dimensional multipolar vortex equilibria. Phys. Fluids 14 (6), 1862–1876.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
Durey, M. & Milewski, P. A. 2017 Faraday wave–droplet dynamics: discrete-time analysis. J. Fluid Mech. 821, 296–329.CrossRefGoogle Scholar
Durkin, D. & Fajans, J. 2000 Experiments on two-dimensional vortex patterns. Phys. Fluids 12 (2), 289–293.CrossRefGoogle Scholar
Ebeling, W., Erdmann, U., Dunkel, J. & Jenssen, M. 2000 Nonlinear dynamics and fluctuations of dissipative Toda chains. J. Stat. Phys. 101, 443–457.CrossRefGoogle Scholar
Eddi, A., Boudaoud, A. & Couder, Y. 2011 a 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. 2011 b Information stored in Faraday waves: the origin of a 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 (4), 44001.CrossRefGoogle Scholar
Filoux, B., Hubert, M. & Vandewalle, N. 2015 Strings of droplets propelled by coherent waves. Phys. Rev. E 92 (4), 041004.CrossRefGoogle 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, 97–127.CrossRefGoogle Scholar
Galeano-Rios, C. A., Milewski, P. A. & Vanden-Broeck, J.-M. 2019 Quasi-normal free-surface impacts, capillary rebounds and application to Faraday walkers. J. Fluid Mech. 873, 856–888.CrossRefGoogle Scholar
Grzybowski, B. A., Stone, H. A. & Whitesides, G. M. 2000 Dynamic self-assembly of magnetized, millimetre-sized objects rotating at a liquid–air interface. Nature 405, 1033–1036.CrossRefGoogle Scholar
Harris, D. M. & Bush, J. W. M. 2015 Generating uniaxial vibration with an electrodynamic shaker and external air bearing. J. Sound Vib. 334, 255–269.CrossRefGoogle 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
Harris, D. M., Quintela, J., Prost, V., Brun, P.-T. & Bush, J. W. M. 2017 Visualization of hydrodynamic pilot-wave phenomena. J. Vis. (Visualization) 20, 13–15.CrossRefGoogle Scholar
Havelock, T. H. 1931 LII. The stability of motion of rectilinear vortices in ring formation. Lond. Edin. Dublin Phil. Mag. J. Sci. 11 (70), 617–633.CrossRefGoogle Scholar
Kossin, J. P. & Schubert, W. H. 2004 Mesovortices in hurricane Isabel. Bull. Am. Meteorol. Soc. 85 (2), 151–153.CrossRefGoogle Scholar
Krishnamurthy, V. S., Wheeler, M. H., Crowdy, D. G. & Constantin, A. 2019 Steady point vortex pair in a field of stuart-type vorticity. J. Fluid Mech. 874, R1.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Miles, J. & Henderson, D. 1990 Parametrically forced surface waves. Annu. Rev. Fluid Mech. 22 (1), 143–165.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, 361–388.CrossRefGoogle Scholar
Moláček, J. & Bush, J. W. M. 2013 a Drops bouncing on a vibrating bath. J. Fluid Mech. 727, 582–611.CrossRefGoogle Scholar
Moláček, J. & Bush, J. W. M. 2013 b Drops walking on a vibrating bath: towards a hydrodynamic pilot-wave theory. J. Fluid Mech. 727, 612–647.CrossRefGoogle Scholar
Morikawa, G. K. & Swenson, E. V. 1971 Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 14 (6), 1058–1073.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, 552–570.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
Protière, S., Bohn, S. & Couder, Y. 2008 Exotic orbits of two interacting wave sources. Phys. Rev. E 78 (3), 036204.CrossRefGoogle ScholarPubMed
Protière, S., Boudaoud, A. & Couder, Y. 2006 Particle–wave association on a fluid interface. J. Fluid Mech. 554, 85–108.CrossRefGoogle Scholar
Protière, S., Couder, Y., Fort, E. & Boudaoud, A. 2005 The self-organization of capillary wave sources. J. Phys.: Condens. Matter 17 (45), S3529–S3535.Google Scholar
Rousseau, E., Ponarin, D., Hristakos, L., Avenel, O., Varoquaux, E. & Mukharsky, Y. 2009 Addition spectra of Wigner islands of electrons on superfluid helium. Phys. Rev. B 79 (4), 045406.CrossRefGoogle Scholar
Saarikoski, H., Reimann, S. M., Harju, A. & Manninen, M. 2010 Vortices in quantum droplets: analogies between boson and fermion systems. Rev. Mod. Phys. 82 (3), 2785–2834.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
Schecter, D. A., Dubin, D. H. E., Fine, K. S. & Driscoll, C. F. 1999 Vortex crystals from 2D Euler flow: experiment and simulation. Phys. Fluids 11 (4), 905–914.CrossRefGoogle Scholar
Sungar, N., Sharpe, J. P., Pilgram, J. J., Bernard, J. & Tambasco, L. D. 2018 Faraday-Talbot effect: alternating phase and circular arrays. Chaos 28 (9), 096101.CrossRefGoogle ScholarPubMed
Tadrist, L., Shim, J.-B., Gilet, T. & Schlagheck, P. 2018 Faraday instability and subthreshold Faraday waves: surface waves emitted by walkers. J. Fluid Mech. 848, 906–945.CrossRefGoogle Scholar
Thomson, J. J. 1883 A Treatise on the Motion of Vortex Rings: An Essay to which the Adams Prize was Adjudged in 1882, in the University of Cambridge. Macmillan.Google Scholar
Thomson, S. J., Couchman, M. M. P. & Bush, J. W. M. 2020 a Collective vibrations of confined levitating droplets. Phys. Rev. Fluids 5, 083601.CrossRefGoogle Scholar
Thomson, S. J., Durey, M. & Rosales, R. R. 2020 b Collective vibrations of a hydrodynamic active lattice. Proc. R. Soc. A 476, 20200155.CrossRefGoogle ScholarPubMed
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.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 (8), 082002.CrossRefGoogle Scholar
Yarmchuk, E. J., Gordon, M. J. V. & Packard, R. E. 1979 Observation of stationary vortex arrays in rotating superfluid helium. Phys. Rev. Lett. 43 (3), 214–217.CrossRefGoogle Scholar

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