An Underlying Asymmetry within Particle-size Segregation

2.8 MB·English

by  Kasper van der Vaart

An Underlying Asymmetry within Particle-size Segregation
K. van der Vaart,1,∗ P. Gajjar,2 G. Epely-Chauvin,1 N. Andreini,1 J.M.N.T. Gray,2 and C. Ancey1
1Environmental Hydraulics Laboratory, E´cole Polytechnique F´ed´erale de Lausanne, E´cublens, Lausanne, Switzerland
2School of Mathematics and Manchester Centre for Nonlinear Dynamics, University of Manchester, Manchester, UK
(Dated: May 11, 2015)
Weexperimentallystudyparticlescaledynamicsduringsegregationofabidispersemixtureunder
oscillatory shear. Large and small particles show an underlying asymmetry that is dependent on
the local particle concentration, with small particles segregating faster in regions of many large
5 particles and large particles segregating slower in regions of many small particles. Wequantify the
1 asymmetryonbulkandparticlescales, andcaptureittheoretically. Thisgivesnewphysicalinsight
0 intosegregation and reveals a similarity with sedimentation, traffic flow and particle diffusion.
2
PACSnumbers: 45.70.Mg,05.45.-a,47.57.Gc
y
a
M
The natural tendency of granular media to self- mixture of borosilicate glass spheres (ρ = 2.23 g/cm3)
p
organizewhenagitatedor shearedproduces a richdiver- with diameters d = 8 mm and d = 4 mm. The larger
8 l s
sity of complex and beautiful patterns [1–3]. Although particles are placed at the bottom, the surface flattened

] itiscounter-intuitivethatthe componentsofaheteroge- and the smaller particles placed on top. The sidewalls
t
f neousmixturewillreadilyseparate,thispropertyhasse- oscillate whilst remaining parallel, applying a periodic
o
rious technical implications as the cause of product non- shear γ(t) = γ sin(ωt) [32] as shown in Figure 1. The
s 0
. uniformity in many industrial processes [4–6] and also corresponding shear rate γ˙(t) = γ0ωcos(ωt), frequency
at playsapivotalroleintheenhancedrun-outoflargescale ω = 2π/T rads−1, period T = 13 s and strain ampli-
m geophysicalgranularflows,suchasdebris-flows,pyroclas- tude γ = tanθ . The sidewalls displace to a maxi-
0 max
- ticflowsandsnowavalanches[7–10]. Afirmknowledgeof mum angle θmax = ±30°, giving a maximum shear rate
d
the segregationprocess is thus of universal importance. of γ˙ = γ ω and a maximum grain displacement ampli-
n 0 0
o Although there has been considerable recent progress tude A = hγ0. The angle is decreased to θmax = ±10°
c in developing continuum based segregation models for fortheparticletrajectorydatainordertoslowdownthe
[ sheared granular flows [11–16], the individual particle segregation and increase the temporal resolution. Non-

2 dynamics are still poorly understood. Discrete Particle dimensional time tˆ= t/T corresponds to the number of
v Method (DPM) simulations [17–20] produce a wealth of elapsed cycles. We follow a sample using RIMS, with
9 micro-scaleinformation,butaremodelsinthemselves. It the index-matchedliquida mixture ofbenzylalcoholand
7 is vital to directly measure particle segregation dynam- ethanol (viscosity µ = 3 mPas) containing a fluorescent
8
ics in real experiments, but such an analysis is difficult dye(rhodamine). Thelowviscosityoftheinterstitialliq-
6
0 with conventional techniques such as binning and side- uid means that fluid drag forces are small compared to
. wall observation [21–24]. Non-intrusive imaging tech- both gravitational forces and the applied shear (Stokes
1
niques, such as X-ray tomography [25] and refractive number>1 [33]). The mixture is lit with a 532 nm laser
0
5 index-matched scanning (RIMS) [26, 27] allow examina- sheetperpendiculartotheoscillatingwalls,givingastack
1 tion of the interior of a granular medium, with RIMS of vertical cross sections. A scan is performed after each
: recently developing into a powerful tool for examining fulloscillationwith the shearboxin the uprightposition.
v
i monodisperse and bidisperse flows [29–31]. In particu- Theimageswereprocessedusingconvolution[34]togive
X
lar, the work of Harrington et al. [31] on the emergence
r of granular segregation demonstrates how particle scale
a
analysis can give new physical insights.
Inthis Letter, we analyzeparticle scaledynamics dur-
ing segregation of a bidisperse mixture under oscillatory
shear. We find that the behavior of small and large par-
ticles exhibits anasymmetry relatedto the localparticle
concentration, with small grains moving faster through
regions of many large particles and large particles ris-
ing slower through regions of many small particles. This
asymmetryisquantifiedonbothparticleandbulklength
scales, and it is shown how to incorporate the behavior FIG. 1: (color online). Left: The experimental setup. A raw
within the theoretical framework. data image is shown and a cross-section of a reconstructed
Methods.—A shearbox 51 mm deep and 37 mm wide sample with 3mm and 6 mm beads. Right: cross-sections at
is filled to a height h = 87± 3 mm with a bidisperse different times during an experiment.
2
three dimensional particle positions, which are coarse- than the subsequent transition from the mixed state to
grained in order to determine a continuous volume frac- the final segregated state. This points to two separate
tion[35]. Somesidewalleffectsexist,withsmallparticles processes that are likely to be related to the distinct be-
preferentially located near the stationary vertical walls, havior of small and large particles.
butthis does notaffectthe overallsegregation. The hor- Particle dynamics.—We are thus motivated to study
izontal particle motion is diffusive, hence the concentra- a single small particle segregating in a mixture of large
tionisspatiallyaveragedtogiveauniformconcentration particlesand a single largeparticle segregatingina mix-
in the x-y plane. We observeno convectionrolls [36], al- ture of small particles, which we refer to as Φ = 0+%
though for |φ | > 45° geometrical squeezing was seen and Φ = 100−% mixtures respectively. The trajectories
max
to cause convection. of the two particles, shown in Fig. 3(a), are quite differ-
Results.—ThetypicalbehaviorisshowninFig.1: The ent: (i) the large particle segregates roughly three times
initialstatewithlargeparticlesonthebottomevolvesto slower than the small particle; and (ii) the large particle
a final state with large particles on top, because small rises smoothly at an almost constant speed, whereas the
particlessinkandlargeparticlesrise. Interestingly,some small particle shows a stepwise motion with steps of the
large particles remain below when all the others have order of dl. This suggests that the small particle falls
reached the top. These particles are not stuck but rise through gaps in the large particle matrix under gravity,
at a slower rate than the ones that have reached the top typically traversing just a single layer.
before them. Although this has been inferred before, it In order to more precisely understand the nature of
has not yet been explained [24]. these trajectories, we study the displacement after τˆ cy-
We define a segregationtime tˆ as the time needed for cles: ∆r(τˆ) = r(tˆ+τˆ) −r(tˆ). The root mean square
s
the vertical centers of mass of the two species to reach a displacement (RMSD) σ(τˆ) = ph∆r2(τˆ)i is plotted
steadystate,asshowninFig.2(a). Werecordtˆ formix- in Fig. 3(b). The dynamics are diffusive (logarithmic
s
tures with varying global volume fraction of small parti- slope 1/4) for both particles at short timescales and su-
clesΦ(%)=V /(V +V ),whilekeepingthetotalmixture perdiffusive(logarithmicslope1/2)atlongertimescales.
s l s
volumeconstant. Figure2(b)showsthattˆ scaleslinearly Thecrossoverlengthscalebetweenthe diffusive andseg-
s
with Φ, i.e. with more small particles in the mixture the regation(super-diffusive)regimesforthesmallparticleis
segregation is slower [20]. Similar trends were observed approximately dl, which corresponds to the typical seg-
over the entire range of angles that can be accessed in regation step size of the small particle. The crossover
our setup. This behavior is usually given a two-part ex-
planation: At low Φ, small particles move slower when
there are more small particles [37]. At high Φ, it takes a 8 y4 (a) 101 20x 10−3 (b)
longer time for large particles to travel to the top when 6 dl3 wν10
thelayerofsmallparticlesabovethemisthicker[20,24].
Iisnigbnoothrede.xpSloanhaotwiondsotthheesbeeehxapvliaonraotifotnhsecoomthbeirnespaetciaens dzl4 21 x/2dl 3 στ(ˆ)/dl100 00 Φ(5%0) 100
intermediate Φ? A clue is given by [24] which reported 2 10−1 lsamraglel;;ΦΦ==5500%%
thatforaΦ=50%mixturethetransitionfromthe state large;Φ=100−%
small;Φ=0+%
with small particles on top to a mixed state was faster 00 200 400 600 800 1000 100 101 102
0.4 tˆ τˆ
(c) (d) (e)
0.3
10 250
(a) (b) eaxspyemrimmeetnrtic /dl0.2
8 200 symmetric ς
0.1
z
6 150
dl tˆs 0
4 100 0 200 4000 200 4000 200 400
tˆ tˆ tˆ
2 small 50
large FIG. 3: (color online) Individual particle dynamics for small
00 50 100 150 00 50 100 particles (blue,gray) andlarge particles (black)with θmax =
tˆ Φ(%) 10°. (a) Vertical trajectories of a small particle segregating
±in a Φ = 0+% mix; and a large particle segregating in a
−
FIG. 2: (a) Time evolution of the vertical center of mass po- Φ=100 % mix. Inset: Particle movement in the horizontal
sition (n1 Pni=1zi) for large and small particles in Φ = 25% plane. (b) RMSD σ(τˆ) for different mixtures (see legend),
(black), 50% (dark gray) and 75% (light gray) mixtures. with the solid line a fit of σs = √D0τˆ+ws2τˆ2 at Φ = 50%
θmax = 30°. (b)SegregationtimetˆsasafunctionofΦ;solid (shifted for clarity). The dotted lines show the slopes 1/4
line is a±fit for the symmetric model with Sr = 0.016, while and 1/2. Inset: wν(Φ) for large (ν = l) and small particles
thedashedlineisafitforasymmetricmodelwithSr =0.030 (ν = s). (c)-(e) ς for single cycles in Φ = 0+%, 50% and
−
and κ=0.89. 100 % mixturesrespectively.
3
length scale for the large particle is lower, roughly 0.2d andlargegrainshaveadifferentsignificanceontheover-
l
(0.4d ), and is likely to be related to the scale of the re- all segregation dynamics at different Φ. At high Φ, the
s
arrangementsofthesurroundingsmallparticles. Tocon- significantdynamicsareofthe‘slow’largeparticle,which
firm this, we measure the RMSD per cycle ς =ph∆r2i, are governed by the scale of rearrangements of the sur-
asshowninFigs.3(c)and3(e). Thetypicalvalueofς for rounding small particles. At low Φ, it is the ‘fast’ small
thesinglelargeparticleliesjustbelowς forthesurround- particlewhichissignificantasitcanmakebigsegregation
ing small particles [Fig. 3(e)]. Although the displace- stepsbetweenlargeparticlelayers. AtanintermediateΦ
ments ς for the single small particle experiences large both processes combine; small particles slow down, be-
variations,as a resultoffalling throughlayers,the mean causelayeringdisappears,while largeparticlesspeedup,
valueisofthesameorderasthatofthesurroundinglarge because the scale of rearrangements increases.
particles [Fig. 3(c)]. Displacementstatistics.—Tostudythisbehavioratthe
The plot of σ(τˆ) for a Φ = 50% mixture in Fig. 3(b) particle scale for each species (ν = l,s), we measure
shows that the curves lie between those for Φ = 0+% theconditionalprobabilitiesP(∆zν|φ)oftheverticaldis-
and Φ = 100−%, but with a comparable amount of placement∆zν giventhatthelocalsmallparticlevolume
segregation. Fitting each of the curves with σ (τˆ) = fraction is φ. Note that shear-gradients [38] do not play
ν
pD0τˆ+wν2τˆ2 with diffusion coefficient D0 allows us to a role, because of the linear shear profile that is applied.
examine the segregation velocities w for large (ν = l) Here, φ = 0 corresponds to regions of only large parti-
ν
and small (ν = s) particles at different Φ. The inset of cles and φ = 1 to only small particles. The results in
Fig. 3(b) shows that w (Φ) decreases with increasing Φ, Figs. 4(a) and 4(b) demonstrate that large particles are
s
whereas w (Φ) increases to a maximum at Φ=50% and less likely to segregate at high φ compared to small par-
l
then decreases, although not to zero, at Φ=100−%. To ticlessegregatingatlowφ. Inthe following,wewillrefer
understand the peak in wl(Φ), we plot ς for Φ = 50% to this as“asymmetry”. Similarto the data for wl(Φ) in
in Fig. 3(d). The values of ς for both small and large the inset of Fig. 3(b), we observe that the large particles
particles increase with respect to the Φ = 100−% mix, have their greatest displacement at an intermediate φ.
however, the large particle movement is still less com- The effect ofasymmetry at a mesoscalecanbe seenin
pared to the small particles. the temporal development of φ(z,tˆ) for a Φ=50% mix-
At this point we can hypothesize an explanation for tureinFig.5(a). Twoimportantfeaturesexist: (i)small
the trend in Fig. 2(b): the individual dynamics of small particlesreachthebottomoftheflowfastercomparedto
largeparticlesreachingthetop;(ii)largeparticlesappear
to rise predominantlytogether(indicated by the bandof
P(∆zl|φ) P(∆zs|φ) low φ). The first feature is easily explained by asymme-
0.15 0.15
(a) (b) try: small particles beginning the experiment near the
interface between the two species quickly travel to the
0.1 0.1
bottom through the large particle matrix, in accordance
0.05 0.05 with P(∆z |φ). The second feature is possibly linked to
s
∆z/dll 0 ∆z/dsl 0 tahteanlaringeteprmaretdicilaetsehcaovnicnegntaramtiaoxni.mum segregation speed
−0.05 −0.05 Theoretical comparison.—Currentapproachestomod-
elingsizesegregationuseanadvection-diffusionequation
−0.1 −0.1
for φ [40]:
∂φ ∂ ∂ ∂φ
0 0.5 1 0 0.5 1 +div(φu)− (cid:0)qF(φ)(cid:1)= (cid:0)D (cid:1), (1)
0.03 0.03 ∂t ∂z ∂z ∂z
(c) (d)
/A0.02 /A0.02 where u is the bulk velocity field, q = q(ωγ0,g) is the
zi|l zi|s mean segregationspeed, g is gravity, D is the diffusivity
∆ 0.01 ∆ 0.01
|h |h and F(φ) is the flux function, which determines the de-
0 0 pendence of the segregationflux on φ. The simplest flux
0 0.5 1 0 0.5 1
φ φ function has a quadratic form
F(φ)=φ(1−φ). (2)
FIG. 4: (color online). (a)-(b) Conditional probabilities
P(∆zlφ) and P(∆zsφ). White curves are ∆zl and ∆zs . This is employed in a number of models [12–14] and is
| | h i h i
(c)-(d) ∆zl /Aand ∆zs /Aasafunctionofφ,witherror- symmetric about φ=0.5, dictating that small and large
|h i| |h i|
bars indicating the standard error of the mean. Dashed and
particles behave identically, but in opposite directions.
solid lines are plots of Eq. (5) for quadratic and cubic flux
Recently, asymmetric flux functions were introduced by
functions F(φ) with Sr =0.008 and Sr =0.015 respectively.
[41] with the simplest being a cubic form
The values of Sr were scaled to account for the lower shear
rate γ0ω at θmax = 10°. F(φ)=A φ(1−φ)(1−κφ), (3)
± κ
4
z 1(a) |h∆zsi| at low φ to tracking errors, when small particles
h0.5 move more than their radius and their displacement is
not recorded, thereby lowering the measured value.
0
1 Discussion.—We analyze particle motion in a segre-
(b)
z0.5 0 0.5 1.0 gatingbidispersemixtureunderoscillatoryshearanddis-
cover an underlying asymmetry in the behavior of large
0 and small particles. The small particle motion is step-
1
(c) like, falling down through the large particle matrix typ-
z0.5 ically one layer at a time. On the other hand, the large
particlemotionissmootherbutslower,andlinkedtothe
0
0 20 40 60 80 100 120 scale of rearrangements of the surrounding small parti-
tˆ
cles. The asymmetric motion of the large and smallpar-
FIG. 5: (color online). (a) Temporal development of φ(z,tˆ) ticles combine to give a characteristic dependence of the
versusnormalizedflowheightz/hforaΦ=50%mixturewith particle segregation speeds on the local volume fraction.
θmax = 30°. (b)-(c) Theoretical predictions from Eq. (4). Large particles segregate slower in the presence of many
±
(b) Prediction using the symmetric flux function (2), with small particles, while small particles segregate faster in
Sr = 0.016 and Sr/Dr = 20.9 [39]. (c) Prediction using the the presence of many large particles. We also observe
asymmetricfluxfunction (3),with Sr =0.030, Sr/Dr =29.6 that large particles move quickest when close to other
and κ=0.89.
large particles at intermediate concentrations, a process
reminiscent of a collective motion [42]. The underlying
asymmetry also manifests at meso and bulk scales. In
where asymmetry parameter 0 6 κ < 1, and normaliza- the developmentof φ(z,tˆ), smallparticles reachthe bot-
tion constant A gives the same amplitude as the sym- tom of the flow faster than large particles reach the top.
κ
metric flux function. The applied shear gives a velocity The segregation time increases linearly when a mixture
profileu=(u(z,t),0,0). Incombinationwiththelateral contains a larger fraction of small particles. Although
spatial uniformity of φ, this means that the transport there is no direct evidence that the observedasymmetry
term in Eq. (1) is zero. Equation (1) reduces to: persistsforcontinuousshear,StaronandPhillips[20]re-
port that the segregation time under steady shear also
∂φ ∂ ∂ ∂φ increases linearly with the total concentration of small
− (cid:0)SrF(φ)(cid:1)= (cid:0)Dr (cid:1), (4)
∂t ∂zˆ ∂zˆ ∂zˆ particles. These insights give a new understanding of
segregationinshearedsystems,withthedynamicbehav-
where zˆ = z/A, and S = qT/A, D = DT/A2 are
r r ior of two species being inherently different.
non-dimensional segregation and diffusive-remixing co-
Models for segregation have typically considered the
efficients respectively. The symmetric and asymmetric
motion of the large and small grains to be identical.
modelswereleastsquaresfittedtothedatainFig.2(b)to
However, an experimentally determined cubic flux [41]
obtainS =0.016forthesymmetricmodelandκ=0.89,
r bringsasymmetricbehaviorforthetwospeciesandgives
S =0.030fortheasymmetricmodel. IntegratingEq.(4)
r good agreement on both particle and bulk scales. This
givestheφevolutioninFigs.5(b)and5(c). Qualitatively,
draws parallels with the use of asymmetric flux func-
Fig.5(c)reproducestheexperimentalresultonsomecrit-
tions to model asymmetry in sedimentation [43], traffic
ical points: (i) the difference in time between the arrival
flows [44, 45] and diffusion across membranes [46]. For
ofsmallparticlesatthebottomandlargeparticlesatthe
example, in the sedimentation of suspensions, particles
topofthe flow;(ii)thecollectiverisingoflargeparticles;
settle faster when travelingtogether,but the settling ve-
and(iii)alowerφinthebottomhalfoftheflownearthe
locitygoestozeromorerapidlythanalineardecreaseat
end of the experiment, indicating that some large parti-
high concentrations [43]. Similarly, the velocity of cars
cles are still inside the small particle matrix, segregating
in traffic approaches zero non-linearly at high car den-
veryslowly. Thesefeaturesarenotfoundinthesymmet-
sities [45]. The commonality between these processes is
ric resultin Fig.5(b). The theoreticaldisplacementsper
their discrete nature, but interestingly, size segregation
cycle are given by
is the only process that consists of two discrete species.
F(φ) F(φ) Further work is needed to analyze particle scale mo-
(cid:12)∆zˆl(cid:12)=Sr , (cid:12)∆zˆs(cid:12)=Sr , (5) tion for continuously sheared flows, e.g. down chutes
(cid:12) (cid:12) 1−φ (cid:12) (cid:12) φ
and within rotating drums, to determine whether asym-
and are shown alongside the experimental data in metry persists and what formthe segregationflux takes.
Figs. 4(c) and 4(d). The trend is clearly better pre- Thedistinctsegregationdynamicsofthetwospeciesalso
dictedbytheasymmetricflux,whichisabletoreproduce leadstoquestionsastoapossiblerelationwithotherdy-
both the peak in |h∆z i| around φ=0.5 and the nonlin- namic processes such as dynamic heterogeneity [48].
l
ear decrease of |h∆z i|. We attribute the discrepancy of Acknowledgements.—The authors would like to ac-
s
5
knowledge enlightening discussions with Marco Ra- G. Moser, M. Monnereau, J. M. N.T. Gray, and C. An-
maioli, Karen Daniels, Anthony Thornton and Matthias cey, Phys.Fluids A 23, 013301 (2011).
Schr¨oter, as well as technical and analytical assis- [24] L.A.GolickandK.E.Daniels,Phys.Rev.E80,042301
(2009).
tance from Justine Caillet. Financial support for
[25] S.McDonald,D.Harris,andP.Withers,Int.J.Material
this work came from the Swiss SNF grant 200021-
Res. 103, 162 (2012).
149441 and NERC grants NER/A/S/2003/00439 and
[26] J. A. Dijksman, F. Rietz, K. A. L˝orincz, M. van Hecke,
NE/E003206/1,as well as EPSRC grants EP/I019189/1
and W. Losert, Rev.Sci. Instrum. 83, 011301 (2012).
and EP/K00428X/1. [27] S. Wiederseiner, N. Andreini, G. Epely-Chauvin, and
C. Ancey,Exp.Fluids 50, 1183 (2011), ISSN0723-4864.
[28] M. Iskander, Modelling with Transparent Soils, Springer
Series in Geomechanics and Geoengineering (Springer,
2010).
∗ Electronic address: kasper.vandervaart@epfl.ch [29] M. Herrera, S. McCarthy, S. Slotterback, E. Cephas,
[1] I.S.AransonandL.S.Tsimring,GranularPatterns (Ox- W. Losert, and M. Girvan, Phys. Rev. E 83, 061303
ford UniversityPress, 2008). (2011).
[2] J.M.OttinoandD.V.Khakhar,Annu.Rev.FluidMech. [30] S.Slotterback,M.Mailman,K.Ronaszegi,M.vanHecke,
32, 55 (2000). M. Girvan, and W. Losert, Phys. Rev. E 85, 021309
[3] D.Pihler-Puzovi´candT.Mullin,Proc.Roy.Soc.A469, (2012).
20130203 (2013). [31] M. Harrington, J. H. Weijs, and W. Losert, Phys. Rev.
[4] S.C. Williams, Powder Technol. 2, 13 (1968). Lett. 111, 078001 (2013).
[5] J. Johanson, Chem. Eng. May,183 (1978). [32] A. M. Scott and J. Bridgwater, Ind. Eng. Chem. Fund.
[6] T.ShinbrotandF.J.Muzzio,Phys.Today53,25(2000). 14, 22 (1975).
[7] R.M. Iverson, Rev.Geophys 35, 245 (11007). [33] C. Cassar, M. Nicolas and O. Pouliquen, Phys. Fluids
[8] C.G.Johnson,B.P.Kokelaar,R.M.Iverson,M.Logan, 17, 103301 (2005).
R. G. LaHusen, and J. M. N. T. Gray, J. Geophys. Res. [34] M.D.Shattuck,R.A.Ingale,andP.M.Reis,AIPConf.
117, 013301 (2012). Proc. 1145, 43 (2009).
[9] D. M. Palladino and G. A. Valentine, J. Volcanol. [35] T.Weinhart,R.Hartkamp,A.R.Thornton,andS.Lud-
Geotherm. Res. 69, 343 (11005), ISSN0377-0273. ing, Phys.Fluids 25, 070605 (2013).
[10] J. McElwaine and K. Nishimura, in IUTAM Sympo- [36] J. R.Royerand P. M. Chaikin, P. Natl. Acad. Sci. USA
sium on Segregation in Granular Flows, edited by A. D. 112, 49 (2015).
Rosato and D. L. Blackmore (Springer Netherlands, [37] A. Jha, J. Gill, and V. Puri, Particul. Sci. Technol. 26,
2000), vol. 81 of Solid Mechanics and Its Applications, 482 (2008).
pp.81–88. [38] Y. Fan and K.M. Hill, NewJ. Phys. 13, 095009 (2011).
[11] J. Bridgwater, W. Foo, and D. Stephens, Powder Tech- [39] The ratio Sr/Dr is found by least squares matching the
nol. 41, 147 (1985). steady end state.
[12] J.M.N.T.GrayandA.R.Thornton,P.R.Soc.A461, [40] J.Gray,P.Gajjar, andB.P.Kokelaar,ComptesRendus
1447 (2005). Physique16, 73 (2015).
[13] L. B. H. May, L. A. Golick, K. C. Phillips, M. Shearer, [41] P.Gajjar andJ.M.N.T.Gray,J.FluidMech.757,297
and K. E. Daniels, Phys.Rev.E 81, 051301 (2010). (2014).
[14] Y. Fan, C. P. Schlick, P. B. Umbanhowar, J. M. Ottino, [42] T. Vicsek and A.Zafeiris, Phys. Rep.517, 71 (2012).
and R.M. Lueptow, J. Fluid Mech. 741, 252 (2014). [43] G. Batchelor, J. Fluid Mech. 52, 245 (1972).
[15] B. Marks, P. Rognon, and I. Einav, J. Fluid Mech. 690, [44] M.J.LighthillandG.B.Whitham,Proc.Roy.Soc.Lond.
4100 (2012). A229, 317 (1955).
[16] K.M.HillandS.D.Tan,J.FluidMech.756,54(2014). [45] 75 Years of the Fundamental Diagram for Traffic Flow
[17] J. Bridgwater, Particuology 8, 563 (2010). Theory: Greenshields Symposium, no. E-C149 in Trans-
[18] A. Thornton, T. Weinhart, S. Luding, and O. Bokhove, portation Research Circular (Transportation Research
Int.J. Mod. Phys.C 23, 1240014 (2012). Board of the National Academies, Washington, DC,
[19] Y. Fan, P. B. Umbanhowar, J. M. Ottino, and R. M. USA,2011).
Lueptow, Proc. Roy.Soc. A 469, 20130235 (2013). [46] N. Packard and R. Shaw, Working Paper, Santa Fe
[20] L.StaronandJ.Phillips,Phys.Fluids26,033302(2014). Institution, Santa Fe Institute, 13100 Hyde Park
[21] S.B. SavageandC. K.K.Lun,J. Fluid Mech.189, 311 Road, Santa Fe, New Mexico 87501, USA (2004), URL
(1988). http://www.santafe.edu/media/workingpapers/04-12-040.pdf.
[22] J.VallanceandS.Savage,inIUTAM symposium on seg- [47] T. S. Majmudar and R. P. Behringer, Nature 435, 1079
regation in granular materials, edited by A. D. Rosato (2005).
andD.L.Blackmore(Dordrecht: Kluwer,2000),pp.31– [48] R. Candelier, O. Dauchot, and G. Biroli, Phys. Rev.
51. Lett. 102, 088001 (2009).
[23] S. Wiederseiner, N. Andreini, G. Epely-Chauvin,