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Quasi-normal free-surface impacts, capillary rebounds and application to Faraday walkers

Published online by Cambridge University Press:  01 July 2019

C. A. Galeano-Rios Open the ORCID record for C. A. Galeano-Rios [Opens in a new window] ,
P. A. Milewski Open the ORCID record for P. A. Milewski [Opens in a new window]  and
J.-M. Vanden-Broeck
C. A. Galeano-Rios*
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
P. A. Milewski
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
J.-M. Vanden-Broeck
Affiliation:
Department of Mathematics, University College London, London WC1E 6BT, UK
*
†Email address for correspondence: C.A.Galeano.Rios@bath.ac.uk
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Abstract

We present a model for capillary-scale objects that bounce on a fluid bath as they also translate horizontally. The rebounding objects are hydrophobic spheres that impact the interface of a bath of incompressible fluid whose motion is described by linearised quasi-potential flow. Under a quasi-normal impact assumption, we demonstrate that the problem can be decomposed into an axisymmetric impact onto a quiescent bath surface, and the unforced evolution of the surface waves. We obtain a walking model that is free of impact parametrisation and we apply this formulation to model droplets walking on a vibrating bath. We show that this model accurately reproduces experimental reports of bouncing modes, impact phases and time-dependent wave field topography for bouncing and walking droplets. Moreover, we revisit the modelling of horizontal drag during droplet impacts to incorporate the effects of the changes in the pressed area during droplet–surface contacts. Finally, we show that this model captures the recently discovered phenomenon of superwalkers.

JFM classification

Waves/Free-surface Flows: Capillary waves Drops and Bubbles: Drops
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 873 , 25 August 2019 , pp. 856 - 888
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Copyright
© 2019 Cambridge University Press 

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References

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, 505–515.Google Scholar
Beyer, J. & Friedrich, R. 1995 Faraday instability: linear analysis for viscous fluids. Phys. Rev. E 51 (2), 1162.Google Scholar
Bush, J. W. M. 2015 Pilot-wave hydrodynamics. Annu. Rev. Fluid Mech. 47, 269–292.Google 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.Google Scholar
Bush, J. W. M. & Hu, D. L. 2006 Walking on water: biolocomotion at the interface. Annu. Rev. Fluid Mech. 38, 339–369.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.Google 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, 177801.Google Scholar
Couder, Y., Protière, S., Fort, E. & Boudaoud, A. 2005b Walking and orbiting droplets. Nature 437, 208.Google Scholar
Damiano, A. P.2015 Surface topography measurements of the bouncing droplet experiment. Master’s thesis, École Polytechnique Fédérale de Lausanne.Google 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.Google Scholar
Dorbolo, S., Terwagne, D., Vandewalle, N. & Gilet, T. 2008 Resonant and rolling droplet. New J. Phys. 10 (11), 113021.Google Scholar
Durey, M.2018 Faraday wave–droplet dynamics: a hydrodynamic quantum analogue. PhD thesis, University of Bath.Google Scholar
Durey, M. & Milewski, P. A. 2017 Faraday wave–droplet dynamics: discrete-time analysis. J. Fluid Mech. 821, 296–329.Google Scholar
Durey, M., Milewski, P. A. & Bush, J. W. M. 2018 Dynamics, emergent statistics, and the mean-pilot-wave potential of walking droplets. Chaos 28 (9), 096108.Google Scholar
Eddi, A., Sultan, R., Moukhtar, J., Fort, E., Rossi, M. & Couder, Y. 2011 Information stored in Faraday waves: the origin of a path memory. J. Fluid Mech. 674, 433–463.Google Scholar
Galeano-Rios, C. A.2016 Hydrodynamic pilot-waves: analytical modeling approaches to the interaction of drops and surface waves. PhD thesis, Instituto Nacional de Matemática Pura e Aplicada.Google Scholar
Galeano-Rios, C. A., Couchman, M. M. P., Caldairou, P. & Bush, J. W. M. 2018 Ratcheting droplet pairs. Chaos 28 (9), 096112.Google Scholar
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.Google Scholar
Gilet, T. & Bush, J. W. M. 2009 The fluid trampoline: droplets bouncing on a soap film. J. Fluid Mech. 625, 167–203.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
Ho, D. T., Bliven, L. F., Wanninkhof, R. I. K. & Schlosser, P. 1997 The effect of rain on air–water gas exchange. Tellus B 49 (2), 149–158.Google Scholar
Howison, S. D., Ockendon, J. R. & Wilson, S. K. 1991 Incompressible water-entry problems at small deadrise angles. J. Fluid Mech. 222, 215–230.Google Scholar
Korobkin, A. A. & Pukhnachov, V. V. 1988 Initial stage of water impact. Annu. Rev. Fluid Mech. 20 (1), 159–185.Google Scholar
Lee, D.-G. & Kim, H.-Y. 2008 Impact of a superhydrophobic sphere onto water. Langmuir 24, 142–145.Google Scholar
Milewski, P., Galeano-Rios, C. A., Nachbin, A. & Bush, J. W. M. 2015 Faraday pilot-wave dynamics: modelling and computation. J. Fluid Mech. 778, 361–388.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
Moore, M. R., Howison, S. D., Ockendon, J. R. & Oliver, J. M. 2012 Three-dimensional oblique water-entry problems at small deadrise angles. J. Fluid Mech. 711, 259–280.Google Scholar
Nachbin, A. 2018 Walking droplets correlated at a distance. Chaos 28 (9), 096110.Google Scholar
Nachbin, A., Milewski, P. A. & Bush, J. W. M. 2017 Tunneling with a hydrodynamic pilot-wave model. Phys. Rev. Fluids 2 (3), 034801.Google 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.Google Scholar
Perrard, S., Labousse, M., Miskin, M., Fort, E. & Couder, Y. 2014 Self-organization into quantized eigenstates of a classical wave-driven particle. Nat. Commun. 5, 3219.Google Scholar
Prosperetti, A. 1976 Viscous effects on small-amplitude surface waves. Phys. Fluids 19 (2), 195–203.Google Scholar
Protière, S., Boudaoud, A. & Couder, Y. 2006 Particle–wave association on a fluid interface. J. Fluid Mech. 554, 85–108.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.Google Scholar
Purvis, R. & Smith, F. T. 2005 Droplet impact on water layers: post-impact analysis and computations. Phil. Trans. R. Soc. Lond. A 363 (1830), 1209–1221.Google Scholar
Sampara, N. & Gilet, T. 2016 Two-frequency forcing of droplet rebounds on a liquid bath. Phys. Rev. E 94 (5), 053112.Google Scholar
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.Google Scholar
Terwagne, D., Ludewig, F., Vandewalle, N. & Dorbolo, S. 2013 The role of droplet deformations in the bouncing droplet dynamics. Phys. Fluids 25, 122101.Google Scholar
Terwagne, D., Vandewalle, N. & Dorbolo, S. 2007 Lifetime of a bouncing droplet. Phys. Rev. E 76, 056311.Google Scholar
Trefethen, L. M. & Panton, R. L. 1990 Some unanswered questions in fluid mechanics. Appl. Mech. Rev. 43 (8), 153–170.Google Scholar
Valani, R. N., Slim, A. C. & Simula, T.2018 Superwalkers. arXiv:1807.06879v1.Google Scholar
Valani, R. N., Slim, A. C. & Simula, T.2019 Superwalking droplets. Phys. Rev. Lett. (submitted) arXiv:1807.06879v2.Google Scholar
Wagner, H. 1932 Über Stoß-und Gleitvorgänge an der Oberfläche von Flüssigkeiten. Z. Angew. Math. Mech. 12 (4), 193–215.Google Scholar
Wind-Willassen, O., Moláek, J., Harris, D. M. & Bush, J. W. M. 2013 Exotic states of bouncing and walking droplets. Phys. Fluids 25, 0820023.Google Scholar
Zhang, W. & Vinals, J. 1997 Pattern formation in weakly damped parametric surface waves driven by two frequency components. J. Fluid Mech. 341, 225–244.Google Scholar
Zhbankova, S. L. & Kolpakov, A. V. 1990 Collision of water drops with a plane water surface. Fluid Dyn. 25 (3), 470–473.Google Scholar

Galeano-Rios et al. supplementary movie 1

Bouncer: Drop and waves (Fig. 4a)

Download Galeano-Rios et al. supplementary movie 1(Video)
Video 1.4 MB

Galeano-Rios et al. supplementary movie 2

Bouncer: Wave field (Fig. 4a)

Download Galeano-Rios et al. supplementary movie 2(Video)
Video 1.5 MB

Galeano-Rios et al. supplementary movie 3

Walker: Drop and waves (Fig. 4b)

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Video 44.2 MB

Galeano-Rios et al. supplementary movie 4

Walker: Wave field (Fig. 4b)

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Video 11.7 MB

Galeano-Rios et al. supplementary movie 5

Superbouncer: Drop and waves (Fig. 12a)

Download Galeano-Rios et al. supplementary movie 5(Video)
Video 1.5 MB

Galeano-Rios et al. supplementary movie 6

Superbouncer: Wave field (Fig. 12a)

Download Galeano-Rios et al. supplementary movie 6(Video)
Video 1.8 MB

Galeano-Rios et al. supplementary movie 7

Superwalker: Drop and waves (Fig. 12b)

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Video 19.9 MB

Galeano-Rios et al. supplementary movie 8

Superwalker: Wave field (Fig. 12b)

Download Galeano-Rios et al. supplementary movie 8(Video)
Video 5.4 MB