Walkers are dual objects comprising a bouncing droplet dynamically coupled to an underlying Faraday wave at the surface of a vibrated bath. In this paper, we study the wave-mediated interaction of two walkers launched at one another, both experimentally and theoretically. Different outcomes are observed in which either the walkers scatter or they bind to each other in orbits or promenade-like motions. The outcome is highly sensitive to initial conditions, which is a signature of chaos, though the time during which perturbations are amplified is finite. The vertical bouncing dynamics, periodic for a single walker, is also strongly perturbed during the interaction, owing to the superposition of the wave contributions of each droplet. Thanks to a model based on inelastic balls coupled to the Faraday waves, we show that this perturbed vertical dynamics is the source of horizontal chaos in such a system.
References
1.
A.
Andersen
, J.
Madsen
, C.
Reichelt
, S. R.
Ahl
, B.
Lautrup
, C.
Ellegaard
, M. T.
Levinsen
, and T.
Bohr
, “Double-slit experiment with single wave-driven particles and its relation to quantum mechanics
,” Phys. Rev. E
92
, 013006
(2015
). 2.
J.
Arbelaiz
, A. U.
Oza
, and J. W.
Bush
, “Promenading pairs of walking droplets: Dynamics and stability
,” Phys. Rev. Fluids
3
, 013604
(2018
). 3.
J.
Bechhoefer
, V.
Ego
, S.
Manneville
, and B.
Johnson
, “An experimental study of the onset of parametrically pumped surface waves in viscous fluids
,” J. Fluid Mech.
288
, 325
–350
(1995
). 4.
C.
Borghesi
, J.
Moukhtar
, M.
Labousse
, A.
Eddi
, E.
Fort
, and Y.
Couder
, “Interaction of two walkers: Wave-mediated energy and force
,” Phys. Rev. E
90
, 063017
(2014
). 5.
J. W. M.
Bush
, “Pilot-wave hydrodynamics
,” Annu. Rev. Fluid Mech.
47
, 269
–292
(2015
). 6.
J. W. M.
Bush
, A. U.
Oza
, and J.
Moláček
, “The wave-induced added mass of walking droplets
,” J. Fluid Mech.
755
, R7
(2014
). 7.
Y.
Couder
and E.
Fort
, “Single-particle diffraction and interference at a macroscopic scale
,” Phys. Rev. Lett.
97
, 154101
(2006
). 8.
Y.
Couder
, S.
Protiere
, E.
Fort
, and A.
Boudaoud
, “Dynamical phenomena: Walking and orbiting droplets
,” Nature
437
, 208
–208
(2005
). 9.
A. P.
Damiano
, P. -T.
Brun
, D. M.
Harris
, C. A.
Galeano-Rios
, and J. W.
Bush
, “Surface topography measurements of the bouncing droplet experiment
,” Exp. Fluids
57
, 163
(2016
). 10.
M.
Di Bernardo
, C. J.
Budd
, A. R.
Champneys
, P.
Kowalczyk
, A. B.
Nordmark
, G. O.
Tost
, and P. T.
Piiroinen
, “Bifurcations in nonsmooth dynamical systems
,” SIAM Rev.
50
, 629
–701
(2008
). 11.
M.
Durey
and P. A.
Milewski
, “Faraday wave–droplet dynamics: Discrete-time analysis
,” J. Fluid Mech.
821
, 296
–329
(2017
). 12.
A.
Eddi
, A.
Decelle
, E.
Fort
, and Y.
Couder
, “Archimedean lattices in the bound states of wave interacting particles
,” Europhys. Lett.
87
, 56002
(2009
). 13.
A.
Eddi
, E.
Fort
, F.
Moisy
, and Y.
Couder
, “Unpredictable tunneling of a classical wave-particle association
,” Phys. Rev. Lett.
102
, 240401
(2009
). 14.
A.
Eddi
, E.
Sultan
, J.
Moukhtar
, E.
Fort
, M.
Rossi
, and Y.
Couder
, “Information stored in faraday waves: The origin of a path memory
,” J. Fluid Mech.
674
, 433
–463
(2011
). 15.
A.
Eddi
, D.
Terwagne
, E.
Fort
, and Y.
Couder
, “Wave propelled ratchets and drifting rafts
,” Europhys. Lett.
82
, 44001
(2008
). 16.
B.
Filoux
, M.
Hubert
, P.
Schlagheck
, and N.
Vandewalle
, “Walking droplets in linear channels
,” Phys. Rev. Fluids
2
, 013601
(2017
). 17.
E.
Fort
, A.
Eddi
, A.
Boudaoud
, J.
Moukhtar
, and Y.
Couder
, “Path-memory induced quantization of classical orbits
,” Proc. Natl. Acad. Sci.
107
, 17515
–17520
(2010
). 18.
C. A.
Galeano-Rios
, P. A.
Milewski
, and J. -M.
Vanden-Broeck
, “Non-wetting impact of a sphere onto a bath and its application to bouncing droplets
,” J. Fluid Mech.
826
, 97
–127
(2017
). 19.
T.
Gilet
, “Dynamics and statistics of wave-particle interactions in a confined geometry
,” Phys. Rev. E
90
, 052917
(2014
). 20.
T.
Gilet
, “Quantumlike statistics of deterministic wave-particle interactions in a circular cavity
,” Phys. Rev. E
93
, 042202
(2016
). 21.
T.
Gilet
and J. W.
Bush
, “Chaotic bouncing of a droplet on a soap film
,” Phys. Rev. Lett.
102
, 014501
(2009
). 22.
T.
Gilet
, N.
Vandewalle
, and S.
Dorbolo
, “Completely inelastic ball
,” Phys. Rev. E
79
, 055201
(2009
). 23.
D. M.
Harris
and J. W. M.
Bush
, “Droplets walking in a rotating frame: From quantized orbits to multimodal statistics
,” J. Fluid Mech.
739
, 444
–464
(2014
). 24.
D. M.
Harris
, T.
Liu
, and J. W.
Bush
, “A low-cost, precise piezoelectric droplet-on-demand generator
,” Exp. Fluids
56
, 83
(2015
). 25.
D. M.
Harris
, J.
Moukhtar
, E.
Fort
, Y.
Couder
, and J. W.
Bush
, “Wavelike statistics from pilot-wave dynamics in a circular corral
,” Phys. Rev. E
88
, 011001
(2013
). 26.
L.
Tadrist
, J.-B.
Shim
, and T. G. P.
Schlagheck
, “Faraday instability and subthreshold faraday waves: Surface waves emitted by walkers
,” J. Fluid Mech.
848
, 906
–945
(2018
). 27.
M.
Labousse
, A. U.
Oza
, S.
Perrard
, and J. W.
Bush
, “Pilot-wave dynamics in a harmonic potential: Quantization and stability of circular orbits
,” Phys. Rev. E
93
, 033122
(2016
). 28.
M.
Labousse
, S.
Perrard
, Y.
Couder
, and E.
Fort
, “Build-up of macroscopic eigenstates in a memory-based constrained system
,” New J. Phys.
16
, 113027
(2014
). 29.
J.
Luck
and A.
Mehta
, “Bouncing ball with a finite restitution: Chattering, locking, and chaos
,” Phys. Rev. E
48
, 3988
(1993
). 30.
P. A.
Milewski
, C. A.
Galeano-Rios
, A.
Nachbin
, and J. W. M.
Bush
, “Faraday pilot-wave dynamics: Modelling and computation
,” J. Fluid Mech.
778
, 361
–388
(2015
). 31.
J.
Moláček
and J. W. M.
Bush
, “Drops bouncing on a vibrating bath
,” J. Fluid Mech.
727
, 582
–611
(2013
). 32.
J.
Moláček
and J. W. M.
Bush
, “Drops walking on a vibrating bath: Towards a hydrodynamic pilot-wave theory
,” J. Fluid Mech.
727
, 612
–647
(2013
). 33.
A.
Nachbin
, P. A.
Milewski
, and J. W. M.
Bush
, “Tunneling with a hydrodynamic pilot-wave model
,” Phys. Rev. Fluids
2
, 034801
(2017
). 34.
A. U.
Oza
, D. M.
Harris
, R. R.
Rosales
, and J. W.
Bush
, “Pilot-wave dynamics in a rotating frame: On the emergence of orbital quantization
,” J. Fluid Mech.
744
, 404
–429
(2014
). 35.
A. U.
Oza
, R. R.
Rosales
, and J. W.
Bush
, “A trajectory equation for walking droplets: Hydrodynamic pilot-wave theory
,” J. Fluid Mech.
737
, 552
–570
(2013
). 36.
A. U.
Oza
, E.
Siéfert
, D. M.
Harris
, J.
Moláček
, and J. W.
Bush
, “Orbiting pairs of walking droplets: Dynamics and stability
,” Phys. Rev. Fluids
2
, 053601
(2017
). 37.
S.
Perrard
, M.
Labousse
, M.
Miskin
, E.
Fort
, and Y.
Couder
, “Self-organization into quantized eigenstates of a classical wave-driven particle
,” Nat. Commun.
5
, 3219
(2014
). 38.
S.
Protiere
, S.
Bohn
, and Y.
Couder
, “Exotic orbits of two interacting wave sources
,” Phys. Rev. E
78
, 036204
(2008
). 39.
S.
Protière
, A.
Boudaoud
, and Y.
Couder
, “Particle–wave association on a fluid interface
,” J. Fluid Mech.
554
, 85
–108
(2006
). 40.
G.
Pucci
, D. M.
Harris
, L. M.
Faria
, and J. W.
Bush
, “Walking droplets interacting with single and double slits
,” J. Fluid Mech.
835
, 1136
–1156
(2018
). 41.
G.
Pucci
, P. J.
Sáenz
, L. M.
Faria
, and J. M. W.
Bush
, “Non-specular reflection of walking droplets
,” J. Fluid Mech.
804
, R3
(2016
). 42.
P. J.
Sáenz
, T.
Cristea-Platon
, and J. W.
Bush
, “Statistical projection effects in a hydrodynamic pilot-wave system
,” Nat. Phys.
14
, 315
(2018
). 43.
N.
Sampara
and T.
Gilet
, “Two-frequency forcing of droplet rebounds on a liquid bath
,” Phys. Rev. E
94
, 053112
(2016
). 44.
D.
Shirokoff
, “Bouncing droplets on a billiard table
,” Chaos
23
, 013115
(2013
). 45.
L. D.
Tambasco
, D. M.
Harris
, A. U.
Oza
, R. R.
Rosales
, and J. W.
Bush
, “The onset of chaos in orbital pilot-wave dynamics
,” Chaos
26
, 103107
(2016
). 46.
D.
Terwagne
, F.
Ludewig
, N.
Vandewalle
, and S.
Dorbolo
, “The role of the droplet deformations in the bouncing droplet dynamics
,” Phys. Fluids
25
, 122101
(2013
). 47.
N.
Vandewalle
, D.
Terwagne
, K.
Mulleners
, T.
Gilet
, and S.
Dorbolo
, “Dancing droplets onto liquid surfaces
,” Phys. Fluids
18
, 091106
(2006
). © 2018 Author(s).
2018
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