“Phonon” Conductivity Along a Column of Spheres in Contact Relation to Volume Fraction Invariance in the Core of Granular ﬂows Down Inclines

Granular Matter manuscript No. (will be inserted by the editor)

“Phonon” conductivity along a column of spheres in contact Relation to volume fraction invariance in the core of granular ﬂows down inclines

Michel Y. Louge

Received: June 14, 2011 / Accepted: date

Abstract We calculate energy conduction and dissipa- as tion along a column of spheres linked with linear springs √ ∗ ∗ ∗ du and dashpots to illustrate how grains in simultaneous τ = f1 T , (1) xyI dy∗ contact may produce a constant “phonon” conductiv- ∗ 2 ity of granular fluctuation kinetic energy. In the core of ∗ ∗ du τ = −f4T + f6 , (2) dense unconfined granular flows down bumpy inclines, yyI dy∗ we show that phonon conductivity dominates its coun- √ dT ∗ dν terpart calculated from gas kinetic theory. However, the q∗ = −f T ∗ − f T ∗3/2 , (3) I 2 dy∗ 5 dy∗ volume dissipation rate of phonon fluctuation energy is ∗ ∗3/2 of the same order as the kinetic theory prediction. γI = f3T , (4)

Keywords Granular conduction · Dense flows where f1–f6 are functions of ν, the pair distribution function at contact g12(ν), and properties characteriz- PACS 44.10.+i · 46.40.-f · 45.70.Mg · 62.30.+d ing binary impacts [2–4]. (In Eqs. (1) and (2), we define τij as the component of the cartesian stress tensor τ on a surface of normal j directed along i. We also adopt the convention that compressive normal stresses are nega- 1 Introduction tive, and that the shear stress on a surface of normal yˆ directed along xˆ is positive, where xˆ if the unit vector A feature of the core of dense, steady, fully-developed along the flow and yˆ is the downward normal to its free free-surface flows of spheres of diameter d and mate- surface). rial density ρs driven by gravity g down a bumpy base If stresses only include “collisional” contributions inclined at an angle α is that their solid volume frac- from Eqs. (1) and (2) (denoted by the subscript “I”), tion ν appears independent of distance from the free then momentum balances for the granular flows men- surface [1]. If so, the following derivation shows that tioned earlier are strict invariance of ν with depth contradicts any generic dτ ∗ /dy∗ = −ν sin α (5) granular kinetic theory that produces “collisional” con- xyI ∗ stitutive laws for shear stress τxy = τ (ρsgd), normal I xyI along the x-direction of flow, and stress τ = τ ∗ (ρ gd), flux of fluctuation energy q = yyI yyI s I ∗ 3/2 ∗ 3/2 ∗ ∗ q ρs(gd) and dissipation rate γI = γ ρs(gd) /d dτ /dy = −ν cos α. (6) I I yyI that are written√ in dimensionless form in terms of velocity u = u∗ gd, and granular temperature T = T ∗gd in the downward depth y = y∗d from the free surface. Equations (5) and (6) integrate to τ ∗ /τ ∗ = tan α at xyI yyI M. Louge any depth. Substituting this result in Eq. (1) recasts Sibley School of Mechanical & Aerospace Engineering the equation of state (2) in terms of T , ν and tan α, Cornell University, Ithaca, NY 14850, USA E-mail: [email protected] τ ∗ = −f T ∗, (7) yyI 7 2 Michel Y. Louge where the function agitated beds, Hostler and Brennen [13] reach a similar q conclusion that pressure waves and stress are propa- 2 2 f7 = 2f4/(1 + 1 + 4f4f6 tan α/f1 ) (8) gated along ephemeral chains of particles. Jia, et al [14] and Jia [15] refine the observations of exists as long as the argument of the square root is Liu and Nagel [11,12] by distinguishing two modes in positive. signals transmitted through a laterally confined bed at If ν is indeed independent of y∗ in the core, then the random jammed state subjected to uniaxial com- substitution of Eqs. (1), (3)-(4) and (7) into the fluctu- pression. The first, labeled “E”, carries a narrow coher- ation energy balance ent band of relatively low frequencies with a correla- ∗ ∗ ∗ ∗ tion length on the order of a few d. “E” is reproducible −dqI /dy + τxyI du /dy − γI = 0 (9) and reversible, although it is attenuated through the yields the paradox assembly. The second, labeled “S”, is acoustic speckle of higher frequencies. Tournat, et al [16] attribute the 2 ∗2 2 2 f1f2/(2f7 ) + y (tan α − f1f3/f7 ) = 0, (10) non-linearity of the latter to clapping contacts in the weak particle network, which can open and close on which clearly contradicts independence of ν with y∗. length scales < d. Finally, Behringer and Kondic [17] Because the first term of Eq. (10) arises from dq /dy in I show that elastic energy is the dominant mode of fluc- Eq. (9), ν is strictly independent of y∗ in the core iff the tuation energy storage in dense sheared systems. These energy flux is also invariant. Because, using Eqs. (6) and observations suggest that fluctuation energy, which acts (7), the derivative dT ∗/dy∗ = ν cos α/f is constant in 7 as coherent “E” vibrations of low frequency, is trans- the core iff ν is constant, the adoption of a Fourier flux mitted along strong load-bearing chains of length > d q = −κ dT/dy requires that conductivity be invariant I I aligned, on average, with the principal axis of stress. as well, in√ contradiction with kinetic theories predicting ∗ ∗ κI = f2 T . (If ν is depth-invariant, the form of f5 matters not to the argument). 2 Kinematic restitution This paper suggests a possible origin for such invariant conductivity. To do so, we first analyze a mech- To relate this notion to invariant phonon conductivity, anism of kinematic restitution involving several grains. we model grains in simultaneous contact along the prin- We hypothesize that, in dense flows, fluctuation energy cipal axis as a one-dimensional chain of spheres of iden- is mainly transferred across “chains” of particles in si- tical mass m linked by linear springs of constant β and multaneous contact, rather than by ephemeral binary linear dashpots of constant δ (Fig. 1). Although real collisions. This approach underlines the role of contact grains interact through non-linear Hertzian contacts, model parameters in dense flows, which Pöschel, et al this calculation is instructive, albeit cruder than the [5] and Campbell [6,7] have noted. We then exploit this works of Hinch and Saint-Jean on Newton’s cradle [18], simple analysis to derive conduction and dissipation of of Coste, et al [19] on solitary waves in a Hertzian col- acoustic energy in such chains. umn, of Falcon, et al [20] on a column of spheres im- There is a rich literature on sound and vibration pacting a wall, or of Pöschel, et al [21] on the onset of in granular materials. In a granular assembly of disks, detachment in a column of visco-hertzian spheres and Radjai, et al [8] distinguish load-bearing “strong” and associated experiment [22]. Nonetheless, it is relevant dissipative “weak” particle networks. Contacts within to linear contact models like “L2” and “L3” in Silbert, the former are non-sliding and transmit the largest forces, et al [1]. whereas nearly all sliding dissipation takes place within In this problem, independent parameters form the the latter. Majmudar and Behringer [9] confirm that √ group Peb ≡ βm/δ. Numerical simulations often use purely sheared granular assemblies of disks transmit such spring and dashpot to produce a constant coeffi- force preferentially along the principal axis of stress. In cient of normal restitution e in binary collisions, simply sheared Brownian suspensions, the contact pair q correlation is similarly anisotropic, whereby two par- 2 e = exp[−π/ 2Peb − 1]. (11) ticles have greater contact probability along the com- pressional axis than the extensional one [10]. In exper- In the model of Fig. 1, the center displacement of iments on sound transmission through loosely packed sphere i away from its equilibrium position si is denoted p static beds of glass beads, Liu and Nagel [11] note sen- by ξi. Because in most cases `/d 2βd/mg, the sitivity of acoustic propagation to small changes in the distance between two successive sphere centers under packing. Liu [12] interpret this as evidence that a small static equilibrium remains ∆s ≈ d despite gravity, as number of force chains carry most of the vibration. In ξ(s = 0; g 6= 0) d. The force exerted by sphere (i−1) “Phonon” conductivity along a column of spheres in contact 3

1 0.5 incident e » 0.26 sphere e » 0.72 0.5 t† = p Pe 0

β δ † t ¶

/ 0 p Pe l † ¶x -0.5

- 2p Pe s -0.5 i+1 2p Pe F i+1 Δs Pe = 8Ö5/p Pe = 80Ö5/p si -1 -1 Fi Δs 1 10 100 1000 1 10 100 1000 † † si-1 t t Fig. 2 Predictions√ of the continuum model for values of Pe shown and Peb = 4 5. The left graph represents a collision of only two spheres, `/d = 1; the right is for a column of 11 ξ † s spheres, `/d = 10. We ignore t > 2πPe (right of dashed line), base after which the entire column comes apart.

Fig. 1 Model for a one-dimensional column of identical spheres linked by identical springs and dashpots. The bottom 1 1 sphere is ﬁxed to the base and the top impacts the column. 80Ö5 e Pe0 = 4Ö5 The small left sketch illustrates the local displacement ξ away e e 20Ö5 e Pe0 = 20Ö5 e Pe0 = 2Ö5 from the equilibrium position s of the second sphere burrow- Rioual 0.5 0.5 ing into the ﬁrst. The shaded incident sphere at s = ` hits e Pe0 = Ö5 4Ö5 e Pe0 = 80Ö5 the column at t = 0. Ö5 e Pe0 = 21

e_eff Rioual 0 0 20 40 0 0 100 200 Peb /d on i is Fi = −β(ξi − ξi−1) − δd(ξi − ξi−1)/dt. Thus, the l dynamics of sphere i is governed by Fig. 3 Predictions of the continuum model. Left: column of ` = d (solid line) and exact solution of Eq. (11) for a binary 2 collision (dashed line). Right: normal restitution coefficient d ξ d 100000 m i = β(ξ − 2ξ + ξ ) + δ ξ − 2ξ + ξ for the top sphere in Fig. 1 versus relative column length `/d dt2 i+1 i i−1 dt i+1 i i−1 for values of Peb shown. Symbols10000 represent80Ö5 Rioual’s simula- (12) tions [25] with binary restitution e = 0.9 or, according to 1000 20Ö5 Eq. (11), Peb ≈ 21; the heavy line is Eq. (16) for this Peb. t† 4Ö5 Defining a mass per unit length ρ ≡ m/∆s, a one- 100 Ö5 dimensional stiffness σ ≡ β∆s, and a one-dimensional 10 viscosity µ ≡ δ∆s, and making ∆s → 0, we obtain initial velocity (∂ξ/∂t) = −u in the region s ∈ t=0 1 0 1 10 100 ∂2ξ ∂2ξ ∂3ξ [` − d/2, `]. Using separation of variables,h/d the solution = c2 + D , (13) is ξ† = P∞ ξ†, where ∂t2 ∂s2 ∂t∂s2 j=0 j j 2 and the expression † 2 (−1) Pej Peb ξj = − sin × π 1 q 2 Pe (j + 2 ) Pe − 1 j F = −σ∂ξ/∂s − µ∂2ξ/∂s∂t (14) j h 1 i h t† i h t† q i sin πj + s† exp − sin Pe2 − 1 . (15) for the force exerted on a control segment. This clas- 2 Pe2 Pe2 j sical substitution of a continuum Eq. (13) for its dis- j j † 2 crete counterpart (12) (see for example Leibig [23] for In this expression, ξ ≡ 2ξc /u0D, Pe ≡ Peb(2/π)(`/d), † † 2 a linear undamped 2D unsheared system, or Somfai, et Pej ≡ Pe/(j + 1/2), s ≡ s/`, and t ≡ 2tc /D. Note 2 al [24] for a linear approximation of a 1D chain with that, for Pej < 1, sin[...] → sinh[...] and (Pej − 1) → 2 Hertzian contacts) ignores small wavelengths on the (1 − Pej ). Thus, if it exists, the frequency of the j-th 2 2 1/2 2 scale of d, but focuses instead on propagation along mode is ωj = (2c /D)(Pe − 1) /Pe . Figure 2 shows p p j j greater distances. In Eq. (13), c ≡ σ/ρ = 6β/πρsd the predicted dimensionless velocity at the top of the and D ≡ µ/ρ = 6δ/πρsd have units of velocity and column, s = `. The coefficient of restitution is the ratio diffusion coefficient, respectively, so Peb = cd/D has of the first velocity peak to u0 (Fig. 2), the structure of a Pécletnumber. On the free surface 1 ∂ξ ∂ξ† at s = `, the boundary condition F = 0, ∀t requires e = − = − † . (16) ∂ξ/∂s = 0. At the base, the column is anchored, so u0 ∂t s=` ∂t s†=1 ξ = 0 for s = 0. Surprisingly, applying the continuum model to a col- We first exploit Eq. (13) to predict the response umn of only two spheres, ` = d (Fig. 3, left), agrees of the column initially at rest, ξ = 0, ∀s, t < 0, to well with the exact solution of Eq. (11). As the col- the impact of a single sphere, which we model as an umn height or, equivalently, Pe increases, e decreases 4 Michel Y. Louge

below. Then, the wave travels through the column, re- recoil rest flects at the base, and returns energy to the top sphere. 0 The latter then detaches, catches up with the incident sphere, and delivers it a final jolt. Because the ini- dx†/dt† t† = p Pe tial rebound does not dissipate energy throughout the -0.5 column, more energy is eventually recovered than pre- compression 2p Pe dicted by Eq. (15), for which contacts are compressive s = everywhere (ξ < 0, ∀s) until the first peak in the time- -1 l † † history of −(∂ξ /∂t )s†=1 (Fig. 2). Despite this subtle process, Eq. (16) captures the increasing s/l reduction of normal restitution with increasing column 0 length (Fig. 3, right). It predicts that spheres in si- † x multaneous contact do not restitute energy locally, but rather conduct it to their neighbors [26]. In this, it -5 differs from dissipation in a column where individual spheres undergo pairwise interactions at constant restitution [27]. If a granular chain subject to an impact at -10 one end remains connected on a time scale greater than 0 200 400 t† the binary collision time, we expect Eq. (13) to capture the propagation of energy on wavelengths > d. Fig. 4 Time histories of dimensionless velocity (top) and displacement (bottom) for a column of nine spheres initially at rest and impacted by a like sphere (`/d = 10) with √ 3 Phonon conductivity Peb = 4 5. The top graph displays three stages in the velocity of the impacting sphere: an initial deceleration dur- ing which the column is progressively compressed, followed We use the model in section 2 to analyze the transfer by a long rest on top of the column, and a recoil culminat- of fluctuation energy through a continuously connected ing in a restitution of e ≈ 0.26. The bottom graph shows displacement curves at increasing values along the column: chain of spheres. First, we seek complex traveling waves s/` = 0.01, 0.02, 0.05, 0.1, 0.25, 0.5, 0.75 and 1. Note that, be- solutions to Eq. (13) † cause ξ < 0, the column is everywhere in compression until ˜ ˜ recoil at t† ≈ 2πPe. ξ = ξ0 exp[ı(ωt − ks)], (17) where ı2 = −1 and tildes denote complex numbers. The resulting dispersion relation is rapidly (Fig. 3, right), and the time to the first peak, 2 2 2˜2 where kinematic restitution takes place, gets longer. For ω − ıDk ω − c k = 0. (18) Pe > 10, which represents, for√ example, a column of Its solutions indicate that the waves satisfy three spheres or more at Peb = 4 5, the restitution cy- ˜ cle includes a short compression period, most of which ξ = ξ0 exp[−s/λ] exp[ı(ωt − Ks)], (19) is independent of Pe, a remarkably long rest and, fi- with dimensionless dissipation length nally, a short recoil just ahead of the incipient exten- √ √ sion of the entire column, ξ > 0, ∀s (Fig. 2, right, and c 2 1 + λ = √ p√ , (20) Fig. 4). Because the model predicts fragmentation of D 1 + − 1 the column at first recoil, t† ≈ 2πPe (i.e., t ' 2`/c = p dimensionless wavenumber ` 2πρsd/(3β)), Eq. (13) is not meaningful thereafter. √ p√ Conversely, despite its use of linear springs that allow D 1 + + 1 K = √ √ , (21) tensile forces at contact, the calculation remains faith- c 2 1 + ful to the physics of granular contacts prohibiting such 1/2 2 tensile forces, since it predicts no sphere separation any- dimensionless frequency ≡ ωD/c , and long wave- where in the column until first recoil (Fig. 4). length group velocity Rioual [25] carried out numerical simulations of the lim(dω/dK) = c. (22) column in Fig. 1. As Fig. 3 (right, symbols) shows, he →0 found that the incident sphere recoils with greater en- In this model, we view the core of a dense granular ergy than Eq. (16) predicts (heavy line). The reason flow on a bumpy incline as parallel rectilinear conduc- is that the incident sphere rebounds slightly just af- tive columns of length ` aligned with the major prin- ter impact from energy reflected off the first contact cipal axis of stress. On any plane perpendicular to the “Phonon” conductivity along a column of spheres in contact 5

2 2 4 2 latter, there are 6ν/πd columns per unit area. The in- 0 = (4Pe − 1)/4Pe ≈ 1/Pe 1, and corresponds to stantaneous energy conducted per unit time through a Jia’s “E” wave [14]. For this most energetic mode, we column is the sum of two terms. The first is the prod- find the dimensionless conductivity uct of the fluctuation kinetic energy per unit length κ 3 ` (ρ/2)(∂ξ/∂t˜ )2 and the instantaneous advection veloc- κ∗ ≡ √E ' νβ∗1/2 , (28) E ρ d gd π d ity ∂ξ/∂t˜ . The second is the working of the force F˜ s in Eq. (14) through this velocity. Thus, upon differen- upon neglecting in Eq. (26) and, from Eq. (27), the tiating Eq. (19) with respect to s and t, the complex dimensionless dissipation rate energy flux through planes perpendicular to the major dγ 3π2 νδ∗T ∗ principal axis of stress is γ∗ ≡ E = , (29) E 3/2 2 ρs(gd) 8 (`/d) ˜ 3 ˜ 6ν hρ∂ξ ˜ ∂ξ i √ q˜E = + F = ∗ ∗ p ∗ ∗ πd2 2 ∂t ∂t where β ≡ βd/mg, δ ≡ (δ/m) d/g and Peb = β /δ . 6ν h ρ 1 σ i Staron [29] found that the length of coherent struc- ω2ξ˜2 − ı ωξ˜+ + ıK ı − µ . (23) πd2 2 λ ω tures in dense granular flows is inversely related to a Because the advection term in square brackets is first Savage number I capturing the relative importance of order in ξ0, we neglect it compared with the working grain inertia and confining stress [30], as Pouliquen [31] of the force. We define the complex phonon temper- had observed and Halsey and Erta¸s[32] conjectured. ature T˜ as (2/3) the local fluctuation kinetic energy Her data is fitted by `/d ' (a/I)1/a with a ' 4. per mass traveling in phase with the wave in Eq. (19), Recognizing that dense flows experience frictional (3/2)T˜ ≡ −(1/2)ı(∂ξ/∂t˜ )2. Then, the complex phonon stresses in addition to those in Eqs. (1) and (2), Louge [33] conductivityκ ˜E is calculated the fraction η of the total shear stress that involves enduring contacts in the core of steady, fully- q˜E 3 1 developed, dense flows down bumpy inclines, κ˜E = − ≈ ρsνD + ı , (24) (∂T˜ /∂s) 2 1/2 r 2 n 1 µE o µE µE µE 2 1/2 η = µ − − 1 − + , (30) in which we used ρ = ρs(π/6)d , σ/ω = ρD/ and E tan α 2F 2 F tan α 4F 2 µ = ρD. For the traveling fluctuation energy wave Ξ˜ = where F ≡ (f f )1/2/f and µ is internal friction for Ξ0 exp[ı(ωt−kex)], we identify the one-dimensional com- 1 2 4 E 2 2 grains in enduring contact (typically µ ' 0.5). He plex energy balance ρsν(∂Ξ/∂t˜ ) =κ ˜E(∂ T˜ /∂s ) with E an unsteady heat equation that includes conduction closed by relating η to ν using η ' (ν−ν0)/(ν1−ν0) with and dissipation, ν0 ' 0.545 and ν1 ' 0.595, calculated a temperature increasing with distance from the free surface ∂Ξ ∂2T Ξ ∗ ∗ ρsν = κE 2 − ρsν , (25) T = y ν cos α(1 − η tan α/µ )/f , (31) ∂t ∂s τe E 4 so the (real) conductivity is which leads to

2 1/2 p 3 c I = (νf4) tan α(1 − η)/f1 1 − η tan α/µE. (32) κ = ρ ν (1 + ). (26) E 2 s ω For nearly elastic frictionless spheres, Jenkins and Rich- 2 2 √ Note that κE has similar frequency dependence as phonon man found f1 = 8ν g12{1+π[1+5/(8νg12)] /12}/(5 π), √ 2 2 conduction in a solid at low temperature [28]. Because, f2 = (4/ π)ν g12{1 + (9π/32)[1 + 5/(12νg12)] }, f3 = √ 2 2 in a dense flow, energy is mainly conducted along chains (12/ π)ν g12(1 − e ), and f4 = ν(1 + 4νg12) [2]. Com- aligned with the major principal axis of stress [9] with bining Eq. (28)–(32), phonon conductivity dominates unit orientation ê, we expect an anisotropic phonon its collisional counterpart; for example, with µE = 0.5 conductivity tensor ∼ κ ê ⊗ ê. E and parameters of system “L3” from Silbert,√ et al [1] 2 ? 5 ? The relaxation time τe = D/c from Eq. (25) then (e = 0.88, β = 2 10 , δ = 25, Peb = 8 5), κE/κI ' corresponds to a volumetric energy dissipation rate 200/y∗1/2 1. However, phonon and collisional dissipation are of the same order for typical depths, γ /γ ' 3 c2 E I γ = ρ νT , (27) 12/y∗1/2. Because I, η, ν and `/d are invariant in the E 2 s D core of dense inclined flows, so is κE. Because the to- in which we identify fluctuation energy with (3/2)T . tal conductivity κ = κI + κE ' κE, our “toy model” Following an impact on the column, the response mode predicts that it is invariant, thus revolving the paradox with greatest energy has the lowest frequency ω0. It is in Eq. (10). Finally, the comparable magnitudes of γI obtained by taking i = 0 in Eq. (15) i.e., (i = 0) ≡ and γE might hint why Jenkins [34] could predict core 6 Michel Y. Louge invariance of ν by postulating a different dissipation 12. Liu, C., Spatial patterns of sound propagation in sand, length scale than what kinetic theory prescribes. Phys. Rev. B 50, 782–794 (1994). In summary, we noted that granular kinetic theo- 13. Hostler, S. R., C. E. Brennen, Pressure wave propagation in a granular bed, Phys. Rev. E 72, 031303 1–13 (2005). ries predicting a temperature-dependent Fourier con- 14. Jia, X., C. Caroli, B. Velicky, Ultrasound propagation ductivity are incompatible with the strict invariance of in externally stressed granular media, Phys. Rev. Lett. 82, volume fraction that is observed in the core of dense, 1863–1866 (1999). steady, fully-developed granular flows down inclines. 15. Jia, X., Codalike multiple scattering of elastic waves in dense granular media, Phys. Rev. Lett. 93, 154303 1–4 Toward resolving this paradox, we derived an expres- (2004). sion for kinematic restitution in the impact of a sphere 16. Tournat, V., Zaitsev, V., Gusev, V., Nazarov, V., Béquin, on a column of like spheres compressed through linear P., Castagnède,B., Probing weak forces in granular media springs and dashpots (Eqs. 15-16), and we exploited through nonlinear dynamic dilatancy: Clapping contacts and polarization anisotropy, Phys. Rev. Lett. 92, 085502 the resulting balance equation for energy propagated 1–4 (2004). in the column to derive a phonon-like conductivity κE 17. Kondic, L., Behringer, R. P., Elastic energy, fluctuations that is independent of temperature (Eq. 28) and a dis- and temperature for granular materials, Europhys. Lett. 67, sipation rate (Eq. 29). If dense granular flows feature 205–211 (2004). 18. Hinch, E. J., S. Saint-Jean, The fragmentation of a line of such columns aligned along the direction eˆ of the major balls by an impact, Proc. R. Soc. Lond. A 455, 3201–3220 principal stress axis on time scales greater than the bi- (1999). nary collision time, then we expect phonon conductivity 19. Coste, C., E. Falcon, S. Fauve, Solitary waves in a chain ∼ κ eˆ ⊗ eˆ to dominate its kinetic theory counterpart. of beads under Hertz contact, Phys. Rev. E 56, 6104–6117 E (1997). 20. Falcon, E., C. Laroche, S. Fauve, C. Coste, Collision of a Acknowledgements The author misses Isaac Goldhirsch’s 1-D column of beads with a wall, Eur. Phys. J. B 5, 111–131 camaraderie and scholarship. He is grateful to R. Behringer, (1998). E. Clément, D. Dean, R. Delannay, E. Garcia, J. Jenkins, V. 21. Pöschel, T, T. Schwager, C. Salueña,Onset of fluidization Kumaran, S. Luding, J. McElwaine, T. Pöschel, P. Richard, in vertically shaken granular material, Phys. Rev. E 62, C. Thornton, and A. Valance for stimulating discussions. This 1361–1367 (2000). research was supported in part by the National Science Foun- 22. Renard, S., T. Schwager, T. Pöschel, C. Salueña,Verti- dation under Grant No. PHY99-0794. cally shaken column of spheres. Onset of fluidization, Eur. Phys. J. E 4, 233–239 (2001). 23. Leibig, M., Model for the propagation of sound in granular materials, Phys. Rev. E 49, 1647–1656 (1994). References 24. Somfai, E., J.-N. Roux, J. H. Snoeijer, M. van Hecke, W. van Saarloos, Elastic wave propagation in confined granular 1. Silbert, L. E., D. Erta¸s, G. S. Grest, T. C. Halsey, D. systems, Phys. Rev. E 72, 021301 1–18 (2005). Levine, S. J. Plimpton, Granular flow down an inclined 25. Rioual, F., Etude´ de quelques aspects du transport éolien: plane: Bagnold scaling and rheology, Phys. Rev. E 64, processus de saltation et formation des rides, Ph.D. thesis, 051302 1–14 (2001). Universitéde Rennes 1 (2002). 2. Jenkins, J. T., M. W. Richman, Grad’s 13-moment system 26. Rajchenbach, J., Dense, rapid flows of inelastic grains for a dense gas of inelastic spheres, Arch. Rat. Mech. Anal. under gravity, Phys. Rev. Lett. 90, 144302 1–4 (2003). 87 355–377 (1985). 27. Pöschel, T., N. V. Brilliantov, Extremal collision se- 3. Kumaran, V., Dense granular flow down an inclined plane: quences of particles on a line: Optimal transmission of ki- from kinetic theory to granular dynamics, J. Fluid Mech. netic energy, Phys. Rev. E 63, 021505 1–9 (2001). 599 121–168 (2008). 28. Gurevich, V. L., Transport in phonon systems, North- 4. V. Garzó,V., J. W. Dufty, Dense fluid transport for in- Holland, Amsterdam, p. 115 (1986). elastic hard spheres, Phys. Rev. E 59, 5895–5911 (1999). 29. Staron, L., Correlated motion in the bulk of dense gran- 5. Pöschel, T., Salueña,C., T. Schwager, Scaling properties ular flows, Phys. Rev. E 77, 051304 1–6 (2008). of granular materials, Phys. Rev. E 64, 011308 (2001). 30. da Cruz, F., S. Emam, M. Prochnow, J.-N. Roux, 6. Campbell, C., Granular shear flows at the elastic limit, J. F. Chevoir, Rheophysics of dense granular materials: Dis- Fluid Mech. 465, 261–291 (2002). crete simulation of plane shear flows, Phys. Rev. E 72, 7. Campbell, C., A problem related to the stability of force 021309 1–17 (2005). chains, Granular Matter 5, 129–134 (2003). 31. Pouliquen, O., Velocity correlations in dense granular 8. Radjai, F., D. E. Wolf, M. Jean, J.-J. Moreau, Bimodal flows, Phys. Rev. Lett. 93, 248001 1–4 (2004). character of stress transmission in granular packings, Phys. 32. Erta¸s, D., T. C. Halsey, Coherent structures in dense Rev. Lett. 80, 61–64 (1998). granular flows, arXiv:cond-mat/0506170v1 (2005). 9. Majmudar, T. S., R. P. Behringer, Contact force measure- 33. Louge, M. Y., Model for dense granular flows down ments and stress-induced anisotropy in granular materials, bumpy inclines, Phys. Rev. E, 67 061303 1–11 (2003). Nature 435, 1079–1082, doi:10.1038/nature03805 (2005). 34. Jenkins, J. T., Dense inclined flows of inelastic spheres, Granular Matter 10 47–52 (2007). 10. Morris, J. F., B. Katyal, Microstructure from simulated Brownian suspension flows at large shear rate, Phys. Fluids 14, 1920–1937 (2002). 11. Liu, C., S. R. Nagel, Sound in a granular material: Disor- der and nonlinearity, Phys. Rev. B 48, 15646–15650 (1993).