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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 p ∗ ∗ ∗ 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- p 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 y^ directed along x^ is positive, where x^ if the unit vector A feature of the core of dense, steady, fully-developed along the flow and y^ 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 writtenp in dimensionless form in terms of ve- locity 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, inp 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 in- variant 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 P¨oschel, 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 P¨oschel, 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 p 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 Predictionsp 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 y 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.

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