In this paper, we overview examples of chaos in granular flows. We begin by reviewing several remarkable behaviors that have intrigued researchers over the past few decades, and we then focus on three areas in which chaos plays an intrinsic role in granular behavior. First, we discuss pattern formation in vibrated beds, which we show is a direct result of chaotic scattering combined with dynamical dissipation. Next, we consider stick-slip motion, which involves chaotic scattering on the micro-scale, and which results in complex and as yet unexplained peculiarities on the macro-scale. Finally, we examine granular mixing, which we show combines micro-scale chaotic scattering and macro-scale stick-slip motion into behaviors that are well described by dynamical systems tools, such as iterative mappings.
References
1.
E.
Altshuler
, O.
Ramos
, E.
Martínez
, A. J.
Batista-Leyva
, A.
Rivera
, and K. E.
Bassler
, “Sandpile formation by revolving rivers
,” Phys. Rev. Lett.
91
, 014501
(2003
).2.
P. B.
Umbanhowar
, F.
Melo
, and H. L.
Swinney
, “Localized excitations in a vertically vibrated granular layer
,” Nature
382
, 793
–796
(1996
).3.
T.
Shinbrot
and F. J.
Muzzio
, “Mixing and segregation in tumbling blenders
,” in Encyclopedia of Pharmaceutical Science and Technology
, edited by J.
Swarbrick
(CRC Press
, NY
, 2013
).4.
J.
Eggers
, “Sand as Maxwell's demon
,” Phys. Rev. Lett.
83
, 5322
–5325
(1999
).5.
E.
Ott
and T.
Tel
, “Chaotic scattering: An introduction
,” Chaos
3
, 417
–26
(1993
).6.
T.
Shinbrot
, “Competition between randomizing impacts and inelastic collisions in granular pattern formation
,” Nature
389
, 574
–576
(1997
).7.
8.
C.
Bizon
, M. D.
Shattuck
, J. B.
Swift
, W. D.
McCormick
, and H. L.
Swinney
, “Patterns in 3D vertically oscillated granular layers: Simulation and experiment
,” Phys. Rev. Lett.
80
, 57
–60
(1998
).9.
T.
Shinbrot
, “Synchronization of coupled maps and stable windows
,” Phys. Rev. E
50
, 3230
–3233
(1994
).10.
T.
Shinbrot
and F. J.
Muzzio
, “Noise to order
,” Nature
410
, 251
–258
(2001
).11.
A. J.
Liu
and S. R.
Nagel
, “Nonlinear dynamics: Jamming is not just cool anymore
,” Nature
396
, 21
–22
(1998
).12.
T. S.
Majmudar
, M.
Sperl
, S.
Luding
, and R. P.
Behringer
, “Jamming transition in granular systems
,” Phys. Rev. Lett.
98
, 058001
(2007
).13.
T. S.
Komatsu
, S.
Inagaki
, N.
Nakagawa
, and S.
Nasuno
, “Creep motion in a granular pile exhibiting steady surface flow
,” Phys. Rev. Lett.
86
, 1757
–1760
(2001
).14.
B.
Percier
, S.
Manneville
, J. N.
McElwaine
, S. W.
Morris
, and N.
Taberlet
, “Lift and drag forces on an inclined plow moving over a granular surface
,” Phys. Rev. E
84
, 051302
(2011
).15.
J. W.
Carson
, “Silo failures: Case histories and lessons learned
,” in Handbook of Conveying and Handling of Particulate Solids
, edited by A.
Levy
and H.
Kalman
(Elsevier Science
, 2001
), pp. 153
–166
.16.
B.
Miller
, C.
O'Hern
, and R. P.
Behringer
, “Stress fluctuations for continuously sheared granular materials
,” Phys. Rev. Lett.
77
, 3110
–3113
(1996
).17.
Traffic and Granular Flow'99—Social, Traffic, and Granular Dynamics
, edited by D.
Helbing
, H. J.
Herrmann
, M.
Schreckenberg
, and D. E.
Wolf
.18.
D.
Helbing
, I.
Farkas
, and T.
Vicsek
, “Simulating dynamical features of escape panic
,” Nature
407
, 487
–490
(2000
).19.
A.
John
, A.
Schadschneider
, D.
Chowdhury
, and K.
Nishinari
, “Trafficlike collective movement of ants on trails: Absence of jammed phase
,” Phys. Rev. Lett.
102
, 108001
(2009
).20.
E.
Altshuler
, O.
Ramos
, Y.
Nuñez
, J.
Fernandez
, A. J.
Batista-Leyva
, and C.
Noda
, “Symmetry breaking in escaping ants
,” Am. Nat.
166
, 643
–649
(2005
).21.
R.
Burridge
and L.
Knopoff
, “Model and theoretical seismicity
,” Bull. Seismol. Soc. Am.
57
, 341
–371
(1967
).22.
Z.
Olami
, J. H. S.
Feder
, and K.
Christensen
, “Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes
,” Phys. Rev. Lett.
68
, 1244
–1247
(1992
).23.
J. S.
Langer
, J. M.
Carlson
, C. R.
Myers
, and B. E.
Shaw
, “Slip complexity in dynamic models of earthquake faults
,” Proc. Natl. Acad. Sci. U.S.A.
93
, 3825
–3829
(1996
).24.
H. J.
Melosh
, “The mechanics of large rock avalanches
,” Rev. Eng. Geol.
7
, 41
–50
(1987
).25.
J. J.
McGuire
and T. H.
Jordan
, “Further evidence for the compound nature of slow earthquakes: The Prince Edward Island earthquake of April 28, 1997
,” J. Geophys. Res.
105
, 7819
–7828
, doi: (2000
).26.
C.
Kisslinger
, “Processes during the Matsushiro swarm as revealed by leveling, gravity, and spring-flow observations
,” Geology
3
, 57
–62
(1975
).27.
T.
Shinbrot
, N. H.
Kim
, and N. N.
Thyagu
, “Electrostatic precursors to granular slip events
,” Proc. Natl. Acad. Sci. U.S.A.
109
, 10806
–10810
(2012
).28.
S.
Nasuno
, A.
Kudrolli
, and J. P.
Gollub
, “Friction in granular layers: Hysteresis and precursors
,” Phys. Rev. Lett.
79
, 949
–952
(1997
).29.
S.
Nasuno
, A.
Kudrolli
, A.
Bak
, and J. P.
Gollub
, “Time-resolved studies of stick-slip friction in sheared granular layers
,” Phys. Rev. E
58
, 2161
–2171
(1998
).30.
O.
Ramos
, E.
Altshuler
, and K. J.
Måløy
, “Avalanche prediction in a self-organized pile of beads
,” Phys. Rev. Lett.
102
, 078701
(2009
).31.
E. J.
Banigan
, M. K.
Illich
, D. J.
Stace-Naughton
, and D. A.
Egolf
, “The chaotic dynamics of jamming
,” Nat. Phys.
9
, 288
–292
(2013
).32.
D. J.
Pine
, J. P.
Gollub
, J. F.
Brady
, and A. M.
Leshansky
, “Chaos and threshold for irreversibility in sheared suspensions
,” Nature
438
, 997
–1000
(2005
).33.
L.
Corté
, P. M.
Chaikin
, J. P.
Gollub
, and D. J.
Pine
, “Random organization in periodically driven systems
,” Nat. Phys.
4
, 420
–424
(2008
).34.
N. J.
Wagner
and J. F.
Brady
, “Shear thickening in colloidal dispersions
,” Phys. Today
27
–32
(2009
).35.
J. D.
Paulsen
, N. C.
Keim
, and S. R.
Nagel
, “Multiple transient memories in experiments on sheared non-Brownian suspensions
,” Phys. Rev. Lett.
113
, 068301
(2014
).36.
C.
Josserand
, A. V.
Tkachenko
, D. M.
Mueth
, and H. M.
Jaeger
, “Memory effects in granular materials
,” Phys. Rev. Lett.
85
, 3632
–3635
(2000
).37.
J.
Kaiser
, “Untersuchungen über das Auftreten von Geräuschen beim Zugversuch
,” Ph. D. dissertation (Technische Hochschule München
, 1950
).38.
H.
Miyamoto
, H.
Yano
, D. J.
Scheeres
, S.
Abe
, O.
Barnouin-Jha
, A. F.
Cheng
, H.
Demura
, R. W.
Gaskell
, N.
Hirata
, M.
Ishiguro
, T.
Michikami
, A. M.
Nakamura
, R.
Nakamura
, J.
Saito
, and S.
Sasaki
, “Regolith migration and sorting on asteroid Itokawa
,” Science
316
, 1011
–1014
(2007
).39.
J. J.
McCarthy
, E. J.
Wolf
, T.
Shinbrot
, G.
Metcalfe
, and J. M.
Ottino
, “Mixing of granular materials in slowly rotated containers
,” Am. Inst. Chem. Eng. J.
42
, 3351
–3353
(1996
).40.
G.
Metcalfe
, T.
Shinbrot
, J. J.
McCarthy
, and J. M.
Ottino
, “Avalanche mixing of granular solids
,” Nature
374
, 39
–41
(1995
).41.
D. V.
Khakhar
, J. J.
McCarthy
, T.
Shinbrot
, and J. M.
Ottino
, “Transverse flow and mixing of granular materials in a rotating cylinder
,” Phys. Fluids
9
, 31
–43
(1997
).42.
T.
Shinbrot
, A. W.
Alexander
, M.
Moakher
, and F. J.
Muzzio
, “Chaotic granular mixing
,” Chaos
9
, 611
–620
(1999
).43.
M.
Rudolph
, T.
Shinbrot
, and R. M.
Lueptow
, “A model of mixing and transport in wavy Taylor-Couette flow
,” Physica D
121
, 163
–174
(1998
).44.
T.
Shinbrot
, L.
Bresler
, and J. M.
Ottino
, “Control of transport in a chaotic lattice
,” Physica D
93
, 191
–209
(1996
).45.
T.
Shinbrot
and F. J.
Muzzio
, “Nonequilibrium patterns in granular mixing and segregation
,” Phys. Today
25
–30
(2000
).46.
P. E.
Shaw
, “The electrical charges from like solids
,” Nature
118
, 659
–660
(1926
).47.
T.
Pähtz
, H. J.
Herrmann
, and T.
Shinbrot
, “Why do particle clouds generate electric charges?
,” Nat. Phys.
6
, 364
–368
(2010
).48.
H. T.
Baytekin
, A. Z.
Patashinski
, M.
Branicki
, B.
Baytekin
, S.
Soh
, and B. A.
Grzybowski
, “The mosaic of surface charge in contact electrification
,” Science
333
, 308
–312
(2011
).49.
T.
Siu
, J.
Cotton
, G.
Mattson
, and T.
Shinbrot
, “Self-sustaining charging of identical colliding particles
,” Phys. Rev. E
89
, 052208
(2014
).50.
51.
52.
See, e.g.,
N. N.
Thyagu
, A.
Vasilenko
, A.
Voyiadjis
, B. J.
Glasser
, and T.
Shinbrot
, “Stuck in traffic: Patterns of powder adhesion
,” Eur. Phys. J. E
35
, 105
(2012
).53.
T.
Siu
, W.
Pittman
, J.
Cotton
, and T.
Shinbrot
, “Nonlinear granular electrostatics
,” Granular Matter
(in press).54.
E.
Aharonov
and D.
Sparks
, “Stick-slip motion in simulated granular layers
,” J. Geophys. Res.
109
, B09306
(2004
).55.
S. C.
Yang
and S. S.
Hsiau
, “The simulation and experimental study of granular materials discharged from a silo with the placement of inserts
,” Powder Technol.
120
, 244
–255
(2001
).56.
D.
Schulze
, “Feeders and flow-promoting devices
,” in Silos: Fundamentals of Theory, Behaviour and Design
, edited by C. J.
Brown
and J.
Nielsen
(CRC Press
, 1998
).57.
I.
Zuriguel
, A.
Janda
, A.
Garcimartın
, C.
Lozano
, R.
Arevalo
, and D.
Maza
, “Silo clogging reduction by the presence of an obstacle
,” Phys. Rev. Lett.
107
, 278001
(2011
).58.
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