REVIEW ARTICLE

Towards a theoretical picture of dense granular flows down inclines Unlike most fluids, granular materials include coexisting solid, liquid or gaseous regions, which produce a rich variety of complex flows. Dense flows down inclines preserve this complexity but remain simple enough for detailed analysis. In this review we survey recent advances in this rapidly evolving area of granular flow, with the aim of providing an organized, synthetic review of phenomena and a characterization of the state of understanding. The perspective that we adopt is influenced by the hope of obtaining a theory for dense, inclined flows that is based on assumptions that can be tested in physical experiments and numerical simulations, and that uses input parameters that can be independently measured. We focus on dense granular flows over three kinds of inclined surfaces: flat-frictional, bumpy-frictional and erodible. The wealth of information generated by experiments and numerical simulations for these flows has led to meaningful tests of relatively simple existing theories.

R. DELANNAY1*, M. LOUGE2, P. RICHARD1, N. TABERLET1 AND A. VALANCE1 1

Groupe Matière Condensée et Matériaux, UMR CNRS 6626, Université Rennes I, Campus de Beaulieu, F-35042 Rennes, France 2 Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York 14853, USA *e-mail: [email protected]

Dense flows of grains driven by gravity down inclines occur in nature and in industrial processes. Natural examples include rock avalanches and dune slides. Applications are found in the chemical, mining and pharmaceutical industries. Dense flows of macroscopic grains show features that are commonly observed in cooled or in transformed amorphous solids1, such as jamming and glass transition, yield, crystallization and first-order phase transition from a disordered to an ordered state. For example, the inhomogeneous flow of bulk metallic glasses is localized in a few narrow shear bands2. Individual grains are to granular mechanics what atoms are to material science. However, the complexity and distribution of the shape and size of granular materials make it daunting to predict their flow behaviour from material properties. Thus, tractable analyses of granular matter require grain properties to be represented by parameters, such as restitution or friction coefficients, that characterize overall grain behaviour, but not the detailed solid mechanics within a single grain. In general, fundamental understanding of granular mechanics must be rooted in simple model systems, such as the dense flows of monodisperse spheres considered in this review. Although significant progress has been made in describing these flows, they still challenge our understanding and remain the subject of active research. Kinetic theories for hard spheres that interact through uncorrelated, instantaneous, binary collisions exploit the analogies nature materials | VOL 6 | FEBRUARY 2007 | www.nature.com/naturematerials

between the colliding macroscopic particles and the molecules of a dense gas to derive constitutive laws for flows of inelastic grains. They have had some success in describing relatively dilute flows or moderately dense flows in the absence of gravity3–5. However, the assumptions on which they are based can fail in the presence of strong volumetric forces. Repeated collisions or persistent contacts between multiple particles can then occur throughout much of the flow. Empirical constitutive equations are only beginning to be obtained in such situations, and none have yet been predicted from first principles. Recent physical experiments and numerical simulations have yielded a wealth of information on dense granular flows driven by gravity down inclined channels. Different boundary conditions at the base and at the lateral walls are found to lead to remarkably distinct flow behaviour. We restrict our attention to steady, fully developed (SFD) flows of identical spherical grains of diameter d and material density ρs driven by the gravitational acceleration g on flat6, bumpy7 or erodible8 planar bases inclined at an angle α to the horizontal. By fully developed we mean that flow quantities, such as the flow height H perpendicular to the base or the mass flow rate Q, do not change in the flow direction. The flows are either unbounded in the direction normal to the plane of shearing9 or are channelled by flat, vertical sidewalls8. Our objective in the first section is to identify indisputable observations from experiments and numerical simulations that a theory should reproduce. This summary of salient flow features complements the review of experiments that appears in the paper by the GDR MiDi group10, to which some of us contributed, and constitutes a more focused companion to the general review of dense flows written by Pouliquen and Chevoir11. The reviews of rapid granular flows by Campbell12 and Goldhirsch13 have little to say about dense, inclined flows. Dense flows down flat, frictional inclines are supported on a shear layer at their base in which collisional transfer of momentum 99

REVIEW ARTICLE Box 1 Numerical methods Two types of simulation, both referred to as discrete element methods, are used for dense flows. Soft particle, or MD, simulations involve rigid, overlapping particles interacting at compliant elastic, frictional contacts with additional dissipative elements that provide a normal coefficient of restitution of less than unity14,25. The equations of motion are solved for the individual particles with a time step that is small enough to resolve the deformation of the contact elements during the course of the particle interactions. Contact dynamic simulations treat the particles as rigid, non-overlapping and frictional and, at each time step, obtain by iteration a solution to the underdetermined algebraic system for the forces of interaction26. Despite the fact that, to speed the simulation, the soft particles interact through normal springs that are more compliant than real contacts and that the solution to the iteration at each time step in the contact dynamics simulation is not unique, the methods seem to agree and, as far as can be determined, also with the physical experiments. For example, the simulations of Walton24 over a flat, frictional base are consistent with profiles observed by Louge and Keast6. Those of Silbert et al.9 capture the essential behaviour recorded by Pouliquen7 for flows on a bumpy base. The numerical results of Bi et al.27 corroborate data of Taberlet et al.8 on an erodible heap bounded by sidewalls. Finally, Hanes and Walton23 report good qualitative agreement between physical experiments and numerical simulations for channelled flows down a bumpy incline.

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chain. Most theories rely on incorporating some observed features of the inclined flow for closure. The last section looks towards future work. Discrete numerical simulations of the individual particles of the flow have been developed to the point at which they can test the assumptions on which theories are based, and provide guidance on the additional modelling that is required. In addition, experimental techniques are now available that can measure quantities of interest within the flow. With the assistance of such simulations and experiments, truly predictive theories for inclined flows should be developed soon. PHENOMENA

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Figure 1 A typical facility for producing dense granular flows down inclines. Once the inclination angle α is set to within 0.1°, the overall mass flow rate is typically found by weighing the output collected over a known time, and the height is recorded with a laser7. Other techniques include capacitance probes that measure the local volume fraction and the integrated volume fraction through the height6, fibre-optic sensors at the sidewalls20 or in the interior23 that provide profiles of mean velocity u along the flow and velocity fluctuations, and high-speed video imaging that gives similar information at the wall in three-dimensional flows23 or in the single layer of a two-dimensional flow of spheres or disks between lateral boundaries3,21,22.

supports the less agitated material above. Dense flows over bumpy, frictional inclines are fluidized throughout their height with a core of relatively uniform density that can occupy much of their interior. Sidewall friction, however, can influence this simple picture and can support flows over rough bases at increasingly high angles of inclination. Sidewall friction can also stabilize flows over the erodible surface of a heap at angles of inclination much greater than the angle of repose. The second section focuses on theory. Theories for dense inclined flows usually incorporate both rate-dependent collisional contributions and rate-independent frictional contributions to the shear and normal stresses. The former are associated with brief interaction between pairs of particles; the latter are associated with repeated or enduring contacts between multiple particles, often in a 100

In this section we summarize salient results from laboratory experiments and numerical simulations. Dense dry flows of spherical grains down inclines depend crucially on the nature of the base, the presence of sidewalls, and the height of the flow. To present a complete description of the corresponding effects, we focus on three generic flows, namely those on flat, bumpy and erodible bases. Each kind has a different behaviour, as well as a different range of angles of inclination at which their velocity profiles and heights are SFD on time and length scales much longer than the inverse shear rate and the flow height. Recent experimental and numerical data have highlighted these differences, which have not always been fully understood. For dense inertial flows of granular materials, a challenge in experiments is to exclude interactions that do not involve mechanical properties of the grains or the boundaries. Thus, adhesive, cohesive, electrostatic, van der Waals, capillary or magnetic forces are avoided, usually by controlling humidity, avoiding triboelectric or magnetic materials and selecting nearly spherical grains with diameters on the order of a millimetre. Brittle materials such as glass beads are forced to flow until their surface adopts steady frictional properties6. With these precautions, the impulsive contacts of a wide variety of nearly spherical solids are captured by the simple model of Walton14, which involves a coefficient of sliding friction μ and coefficients of normal and tangential restitution e and β. These parameters, which are averages over the microscopic properties of the grain material, can be measured independently15–17 or predicted in terms of material properties18,19. For relatively dense flows that involve impulsive contacts with little dissipation, these parameters provide a successful link between numerical simulations, microgravity experiments, and kinetic theory5. However, the presence of enduring contacts in dense, gravity-driven flows makes it more difficult to describe the mechanical properties of the grains so simply. In that case, normal restitution is less significant, and simulations using different contact models can yield different numerical results, although the nature materials | VOL 6 | FEBRUARY 2007 | www.nature.com/naturematerials

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Figure 2 Snapshots of periodic MD simulations of 10,000 grains in a domain bounded by frictionless sidewalls separated by W = 20d along z, streamwise length L = 25d, inclination α = 20°, a friction coefficient μ = 0.8 between grains, and other grain interaction parameters found in Bi et al.27. Colours represent the mean velocity made dimensionless with gravity and grain diameter. Shown are flows on a flat, frictional base (a) and a bumpy base made of cylinders parallel to z (b).

corresponding flow behaviours are qualitatively similar9 (see also Box 1). As will be described in our discussion of theories, present research involves attempts to bridge the gap between macroscopic flow behaviour and microscopic material properties when longlasting contacts are present. Figure 1 shows a typical setup that produces a dense, inclined flow. So far, the measurements that have been made in such devices (see, for example, refs 6, 7, 20–23) are limited. Numerical simulations produce more detailed information on velocities, volume fraction and stresses9,24. They reveal the role of contact parameters such as friction and normal and tangential restitution, and of the nature of the base and the lateral walls. Such numerical simulations of granular flows focus on individual particles of a flow as they travel between and interact in collisions25,26 (see Box 1). We now describe the essential features of the three flows of interest. To illustrate and contrast their behaviours, we accompany our discussion with figures taken from numerical simulations of the three types of flow27.

long-period upward-moving waves observed by Louge and Keast6 or in the sidewall-stabilized heaps described in the section below. Walton24 used molecular dynamics (MD) simulations that were periodic in the flow and lateral directions to observe SFD flows on a flat, frictional base at a value of tanα that was lower than the coefficient of sliding friction μ that he adopted for collisions with the base and among grains. He confirmed that Q was proportional to H3/2 and found that the shear layer grows thicker with increasing normal coefficient of restitution e, but that, surprisingly, Q also decreases with e as a greater proportion of the flow experiences collisional dissipation. Because in Walton’s simulations all forces between the grains and the base are exerted at contacts, and the value of tanα is less than the coefficient of sliding friction, not all contacts are sliding at the base; instead, some have tangential contact forces less than the normal forces, consistent with Coulomb’s law24. However, because physical flows are observed to take place at angles of inclination with tangents greater than the coefficient of sliding friction6,15, the frictional forces in the experiments may involve static as well as sliding friction, or something more complicated29.

FLAT, FRICTIONAL BASE BUMPY, RIGID BASE

On a flat, frictional base, flows of nearly spherical grains with H > 5d (ref. 24) feature a thin, agitated basal shear layer that supports a more passive, denser overburden at relatively low inclinations6 (Fig. 2a). In a shear layer of thickness less than 3d, grains roll around the axis perpendicular to the plane of shear but frustrate each other’s spin through collisions. There, the mean velocity does not vanish at the base (Fig. 3a), the solid volume fraction ν is smaller than in the overburden (Fig. 3b), and the strength of the velocity fluctuations grows with the thickness of the flow and the angle of inclination6. SFD flows occur in the range of tanα between the measured static and sliding friction coefficients between a grain and the base. In shallow SFD flows with a height less than ten particle diameters, Louge and Keast6 measured a mass flow rate Q that was proportional to the product of H3/2 and an increasing function of α. Their results agreed at least qualitatively with those of Johnson et al.28 for flows of similar heights in a similar experimental facility. In such shallow flows, the lateral boundaries did not seem to be significant6. However, these sidewalls affect the momentum balance for H > 10d, for example within the nature materials | VOL 6 | FEBRUARY 2007 | www.nature.com/naturematerials

Dense SFD flows over rigid bases covered with bumps on the scale of order d require steeper inclinations than their counterparts on a flat, frictional plane. As Fig. 4a shows, they exist in a range of thicknesses and inclination angles forming two regions, in which SFD flows have a different character7,30–32. Numerical simulations of dense, deep flows performed in cells that are periodic in the flow and lateral directions show that in the absence of lateral walls the flows are mobilized through their height9 and often have a negligible mean velocity at the base (Figs 2b and 3a). In region I of Fig. 4a, this complete mobilization results in Q proportional to H5/2. There, flows consist of three zones: a thin top layer where ν decreases abruptly towards the free surface; a core, in which ν is remarkably constant but decreases with increasing α; and a basal layer with thickness less than 10d above the bottom boundary (Figs 2b and 3b). In the basal layer, the profiles of mean velocity and volume fraction depend on the nature of the base33,34. If the flow is sufficiently thick to establish a core, profiles in the latter are independent of the bottom boundary34, with the energy of the velocity fluctuations proportional to the square of the shear 101

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rate. This is the scaling expected from a collisional theory. As Fig. 3 shows, the core above a bumpy base has the same shear rate and volume fraction as the overburden above a flat, frictional base. The MD simulations of Silbert et al.9 further reveal that profiles are independent of e, whereas μ affects the profile of the mean velocity u only when μ is less than one, but it does not affect ν. This suggests that the value of ν in the core is independent of the dynamics of the flow and that the dissipation of fluctuation energy is dominated by frictional losses. Flows over bumpy boundaries are often influenced by the presence of sidewalls. The extensive measurements of Ancey35 provide a clear indication of this. Numerical simulations by Hanes and Walton23, performed in conjunction with physical experiments, showed that sidewalls can dissipate fluctuation energy and produce profiles of fluctuation energy in the interior with a completely different character from that at the sidewalls. The lateral boundaries in the experiments of Hungr and Morgenstern36 on deep, very rapid, dense flows may be responsible for their observation that such flows were not sheared throughout their height. This localization of shearing at the base may also be due to their large H/d, which is much greater than in Pouliquen’s experiments7. In general, more experiments are needed at large H/d. ERODIBLE BASE

Grains flowing at the surface of a quasi-static granular bed are frequently observed in industrial processes and in nature. 102

Figure 4 Diagrams of dimensionless flow height against tanα, highlighting similarities between simulations and experiments. a, Symbols represent simulations of Silbert et al.30 at transitions between flow regimes. Lines are visual fits. Circles correspond to Hstop. The flow is unsteady to the right of the line joining the square symbols. The triangles and dashed line subdivide SFD flows in two regions, I and II, that show different behaviours. In region I the flow is thick enough to establish a core of invariant solid volume fraction. In region II thinner flows with H near Hstop or, equivalently, with α reduced to less than the starting angle, show more complex profiles that depend on the nature of the bottom boundary. As Fig. 3 shows, the boundary also induces layering of the particles in its neighbourhood. This is particularly pronounced in dense flows that involve a single layer of spheres or disks between vertical walls3,21,22. b, Similar flow diagram from Pouliquen’s experiments7. Circles represent Hstop and squares all other recorded heights. The solid and dashed curves are predictions for Hstop and for the minimum height to establish a core, respectively31. The right-hand vertical dashed line marks the maximum angle of inclination predicted for steady flows31.

Interestingly, the gas, liquid and solid behaviours of granular materials are present at the same time in such flows. They are strongly influenced by sidewalls27,37. Most studies concerning these flows have been conducted in two different configurations. The first is the rotating drum10,38,39, a cylinder, partly filled with granular material, rotating about its axis at a controlled speed. Because the rotating drum does not permit fully developed flows, except perhaps at its midpoint, we focus instead on the second configuration, of which nature materials | VOL 6 | FEBRUARY 2007 | www.nature.com/naturematerials

REVIEW ARTICLE Taberlet et al.8 offer the simplest example. These authors describe “side-wall stabilized heaps” (SSHs), where sidewalls of effective friction μw decrease the shear stress in the flow (Fig. 5), leading to a ratio of shear to normal stress at the base of the mobilized layer of approximate thickness H that is less than the tangent of the angle of repose αr of the pile (Fig. 6b, inset): (S/N)base = tanα − μw(H/W) ≤ tanαr

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Sidewalls have a strong influence on both flow inclination and other characteristics. The typical velocity profile consists of three distinct parts (Figs 5 and 6). Within the quasi-static pile, grains experience a slow intermittent motion with a convex velocity profile, in agreement with earlier experiments on flows over an erodible base40. Above this pile, the velocity profile becomes more linear. Near the free surface, the profile levels off in a gaseous layer dominated by collisional interactions. Finally, the packing fraction increases progressively with distance from the free surface (Fig. 6). This is fundamentally different from the behaviour of the volume fraction in deep, dense flows over a bumpy base in the absence of frictional sidewalls (Fig. 2b). The relation between mass flow rate and height in the SSH also differs from that on a bumpy, rigid base, for which the flow rate is proportional to H5/2. In the SSH, the flow rate is proportional to H − Hc, where Hc is the critical height at which the flow jams. Ancey35 had already noted the transition between the two kinds of flow in his experiments. He also observed volume fraction profiles similar to those in Fig. 6. Recent numerical simulations27 have shown that a flow rate linear in H and with a variable volume fraction in the core can be obtained on a bumpy base with frictional sidewalls. Conversely, when frictionless sidewalls are introduced, the essential features of the SSH are replaced by those on a bumpy base27. The distinction between the two kinds of flow is therefore due to sidewall friction rather than to the nature of the base (erodible or bumpy). Moreover, it is the friction at the walls, not the confinement and ordering that these walls impose, that rules the flow. Recent experiments37 have shown that steady flows on piles at relatively small inclination are entirely controlled by sidewall effects, even for channels 600 particle diameters wide. The question remains whether it is possible to create rectilinear SFD flows on an erodible pile in the absence of confining sidewalls. Daerr and Douady32 reported that, without lateral confinement, flows on an unstable heap spread laterally from the point at which they are mobilized. In contrast, Félix and Thomas41 observed how rectilinear SFD flows of polydisperse grains form natural levees that confine the slide in the lateral direction. Thus, rectilinear flows on an erodible base may require a channel to remain SFD. THEORY

In this section we view the development of theories for dense granular flows down inclines from the perspective of the kinetic theory of dense gases. We adopt this perspective for several reasons. First, although the simplest form of the kinetic theory, which assumes that particle interactions are instantaneous, binary and uncorrelated, fails in general to describe dense flows with repeated or enduring contacts between particles, there are indications that such a theory can predict some aspects of these flows. For example, in inclined flows that involve a dense, passive mass of grains riding on a basal shear layer24,36, the collisional flow in that layer is amenable to such a description42. In addition, simulations of dense flows34 that almost certainly involve multiple contacts show features of the constitutive relations for shear and normal stresses from the simplest kinetic theory, provided that a suitable dependence of the pair distribution on ν can be found (Fig. 7). nature materials | VOL 6 | FEBRUARY 2007 | www.nature.com/naturematerials

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Figure 5 Snapshot of MD simulations with frictional sidewalls (μ = 0.8), 60,000 grains and α = 40° (refs 8, 27). Grains are placed in a horizontal cell that is periodic in the flow direction. Tilting the cell triggers flows that evolve towards a steady state over a wide range of inclination angles. The lower part of the granular assembly remains quasi-static, while a SFD flow parallel to the bottom develops on top. Such periodic simulations replicate the formation of SSHs in experiments, where grains are poured between confining walls at a given flow rate8,35,39. An SSH occurs when the flow rate exceeds a critical value that depends on channel width and material properties (mainly friction coefficients). Grains poured into the channel become progressively trapped, contributing to a slow increase in the free surface, until a rectilinear SFD flow occurs on top of the pile at an inclination that exceeds the angle of repose of the material.

Second, the simplest form of the kinetic theory can be systematically extended to include repeated, correlated collisions with the BBGKY equations43, and enduring interactions between multiple particles can in principle be described by incorporating long-range forces into the expressions for the transport coefficients44. That is, a theoretical framework already exists within the kinetic theory for describing dense, granular flows down inclines. Third, our view is that an understanding of the physics of the macroscopic phenomena requires an understanding of the physics at the particle scale and the formulation of the linkage between the micro and macro scales. An ab initio theory of this nature requires a minimum of assumptions. Consequently, when its predictions are tested, it is clear why the theory fails, although it may not be obvious how to change the assumptions to improve its performance. Thus, in this section we first consider ab initio microscopic constitutive theories based on the kinetic theory of colliding grains45, as well as variants in which pair distribution is adjusted to improve flow predictions46. We then describe mixed theories that assume coexisting collisional and persistent contacts31,47–51. We neither describe nor evaluate theories for dense inclined flows based on activation energy52,53. However, we mention constitutive relations with frictional and collisional contributions based on dimensional analysis and input from experiments10,54. Kinetic theory for a dense gas of hard, smooth, elastic spheres has been extended to apply to granular materials by including small amounts of dissipation55–57. In this case, the constitutive relations 103

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for stress τ are expressed in terms o f the mean velocity u as in a newtonian fluid: τ = [−p + (λ − 2η/3)∇∙u]1 + η[(∇u) + (∇u)T]

(2)

In addition to equations of conservation of mass and linear momentum, a balance equation determines the evolution and spatial variation of the energy E per unit mass of the macroscopic velocity fluctuations as it is conducted from point to point in the flow, produced by the mean shear rate and dissipated in collisions: ρsν(∂E/∂t + u.∇E) = −∇∙q + τ : ∇u − γ

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The pressure p = ρsν(1 + 4νg12)T, the viscosities λ and η and the conductivity κ are given in terms of ν, the ‘granular temperature’ T = 2E/3 and the pair distribution at contact g12(ν). For nearly elastic spheres, the rate of collisional dissipation of fluctuation energy is γ = (24/π)(ρsν2g12/d)(1 − e)T 3/2

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Small amounts of friction in collisions can be incorporated into the theory, through an effective coefficient of restitution, without changing its structure58; however, significant collisional friction requires the consideration of the balance equations for angular momentum and the energy of the fluctuations in particle spin59. Increased dissipation due to friction or small restitution requires an even more complicated description that involves balance equations for higher moments of the velocity fluctuations, higher spatial gradients and higher powers of lower gradients60–62. In general, a granular flow will slip relative to a boundary; so, in addition to being dissipated in collisions at the boundaries, fluctuation energy can be generated by slip63. That is, boundaries can either provide or remove fluctuation energy from the flow. 104

The simplest kinetic theories have been applied to SFD flows down bumpy inclines45,51,64. Solutions in the absence of sidewalls that are relatively dense throughout their height are found to exist for bumpy bases that provide fluctuation energy to the flow. In such solutions, the energy of the velocity fluctuations typically decreases with distance from the base, and the solid volume fraction has a maximum in the interior of the flow. Flows with these features have been observed in experiments23,65 and simulations23 with sidewalls. However, in such flows there is significant slip at the base, a variation in the volume fraction through the height, and sensitivity to the effective coefficient of restitution. None of these are features of the slow dense flows over bumpy inclines observed in numerical simulations of inclined flows without sidewalls9,30,66. For a dissipative bumpy base, the kinetic theory predicts that, in the absence of sidewalls, the fluctuation energy increases and the volume fraction decreases with distance from the base. These flows are not dense through their entire height, and, because of the high volume fraction and low fluctuation energy near the base, it is unlikely that the collisions between particles are instantaneous and binary and that the positions and velocities of the colliding particles are uncorrelated. That is, the predictions of the simplest theory are incompatible with the assumptions on which it is based. There have been several attempts to extend the simple kinetic theory to include higher volume fractions, greater dissipation in collisions, and multiple or enduring contacts. Bocquet et al.46 adopt the dense limit of the simple kinetic theory but, by analogy with the behaviour of supercooled liquids in the neighbourhood of the glass transition, they introduce an anomalous density dependence of the viscosity close to random close-packed density. Then the energy balance, subjected to zero energy flux at top and bottom, has solutions only for a flow thickness above a lower limit that varies with the inclination in a way similar to that observed in the experiments and simulations. When flow is possible, a given flow thickness can occur over a range of inclination angles, in contrast to the dense limit of the simple kinetic theory, in which the thickness is a unique function of inclination42. With a no-slip condition imposed at the base, the predicted profiles of velocity, energy and volume fraction have the qualitative features of those nature materials | VOL 6 | FEBRUARY 2007 | www.nature.com/naturematerials

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Figure 7 Predictions of the kinetic theory for stresses on planes parallel to the flow compared with numerical simulations of a two-dimensional inclined flow of identical spheres34. Stresses are made dimensionless with the material surface density ρs defined here as the mass of a grain divided by its cross-section, the two-dimensional ‘granular temperature’ T = E, the grain diameter d and the mean strain rate du /dy; E is the fluctuation kinetic energy divided by the grain mass and ν is the surface fraction. Lines are predictions of Jenkins and Richman86 for two different expressions of the pair distribution function at contact. The Verlet–Levesque distribution87 (dotted line) clearly fails at high surface fraction; Mitarai and Nakanishi34 replace it with a free-volume theory that Luding88 matched empirically to the Verlet–Levesque expression87 at low ν. Agreement is good in the ‘core’ of the flow where ν is invariant along y (open symbols), but the theory can fail near the base (solid symbols), particularly at low angles of inclination where enduring contacts are more prevalent. Angles of inclination are 20° (circles), 21° (triangles), 22° (squares) and 23° (diamonds). a, Two-dimensional normal stress N. b, Shear stress S having here the dimensions of a force per unit length.

measured in the simulations, including a maximum velocity that varies with H3/2(sinα)1/2 and, for specific conditions, a relatively uniform volume fraction through the height of the flow. However, the volume fractions predicted by the theory are much lower than those recorded in experiments or simulations. Friction in enduring sliding contacts has been incorporated into kinetic theories by adding a heuristic frictional contribution to the collisional stress of the simple kinetic theory47. Johnson et al.28 and Louge and Keast6 use such a decomposition to interpret the results of their experiments on flow over a flat, frictional base. Johnson et al.28 relate the enduring part of the shear stress to the enduring part of the normal stress through a constant coefficient of internal friction. They close the theory with a functional relationship between the enduring part of the normal stress and the volume fraction. When the resulting equations and boundary conditions are solved for SFD dense flows, they obtain rough agreement between the predicted and measured mass flow rate, mass per unit basal area, and inclination angle and velocity profiles that involve a region of localized shearing near the base. Louge and Keast6 focus on the region of intense shearing near the flat, frictional base, and phrase and solve a boundary-value problem for the flow velocity and energy. They also decompose the stress into parts associated with collisional interactions and enduring frictional contacts, and take the ratio of the enduring shear stress to enduring normal stress to be a coefficient of internal friction. In contrast with Johnson et al.28, they use the balance of force normal to the flow to determine the normal component of the contact force, assume that the total shear stress works to produce fluctuation energy, and close the theory by determining the volume fraction in the shear layer by using the balance of angular momentum appropriate to an nature materials | VOL 6 | FEBRUARY 2007 | www.nature.com/naturematerials

SFD flow. The theory captures the dependence of effective friction on Froude number (Fig. 8). Louge31 proposes a similar theory in an analysis of relatively deep, dense flows down a bumpy incline. In considering the extended region of uniform volume fraction in the interior of the flow, he uses an algebraic balance between the production and dissipation of energy, and determines the fraction of the shear stress due to enduring contacts as a function of the inclination angle and the coefficient of internal friction. Then he calculates profiles of the average and fluctuation velocities from the constitutive relation for the collisional shear stress and the vertical momentum balance, respectively. When the values of the internal friction and coefficient of collisional restitution are appropriately specified, the predicted relation between mass flow rate and angle of inclination agree with experimental data7, and the predicted profiles of average and fluctuation velocity are very near those reported by Silbert and colleagues9. Ancey and Evesque49 operate in a similar context but relate the collisional and frictional contributions to the shear stress through an assumption about the uniformity of the energy dissipation through the height of the flow. The difficulty with theories that decompose the stress into parts associated with collisions and enduring contacts is that information in addition to the coefficient of internal friction has to be specified so as to close the theory. Aranson and Tsimring and Volfson et al.50,67,68 address this closure directly by introducing an evolution equation for the fraction of enduring contacts. By analogy with the transition between solid and liquid phases, they take the fixed points of their evolution equation to correspond to all enduring contacts, all collisional contacts, and a fraction that is a function of inclination angle. This function varies between 0 and 1 105

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Figure 8 Ratio of shear and normal stress (effective friction) on a flat, frictional wall versus Froude number Fr based on mean velocity and weight6. For SFD flows, the effective friction is tanα. Dashed lines represent theoretical predictions at constant solid volume fractions (shown) within the thin agitated basal shear layer. If this volume fraction remained constant with α, then effective friction would decrease with Fr between the friction μE for long-lasting contacts and its counterpart μI for short impacts on the base, thus producing unstable flows. A balance of angular momentum (solid curve) for grains in the basal layer resolves this paradox by setting the basal volume fraction, thus predicting an effective friction consistent with data (circles). For identical spheres, this theory forbids flows in the shaded region, where basal volume fraction exceeds random jammed packing.

non-local dissipation of fluctuation energy in chains and clusters of particles. The idea that a correlation length is important to the understanding and description of dense granular flows is also emphasized by other researchers. Inspired by da Cruz et al.73, the GDR MiDi group10 focuses on data from experiments and discrete simulation for rather rigid contacts and derives, from the data, dimensionless relations between the shear stress, normal stress, average flow velocity and mass density that apply to dense flows in several different geometries. These may be phrased as empirical constitutive relations for the shear stress and normal stress in steady, homogeneous shearing. Using these relations, the authors show to what extent their data on dense inclined flows can be interpreted in terms of a mixing length. Jop et al.74 generalized this approach to tensorial constitutive relations, although experimental justification is at present limited to unidirectional flows of monodisperse spheres that do not slip at boundaries. Motivated by numerical simulations of the run-out of deep, dense avalanches75, Campbell76 follows Babic et al.77 and characterizes the influence of the deformation of the contacts on the rheology of dense shearing flows. In complementary work, Zhang and Rauenzahn78 measure the temporal relaxation of contact duration and provide an evolution equation for the stress based on this. Finally, Ertas and Halsey79 interpret the rheology of dense, inclined flows as resulting from the formation of large-scale, coherent eddies of grains whose scale is determined by the second gradient of the velocity. In short, theories for dense inclined flow all recognize the need to account for correlations between the velocities and/or locations of the particles associated with the enduring frictional contacts, but they differ in the ways in which they incorporate them into the theory. Our view is that the existing constitutive relations of the kinetic theory need be modified only by the incorporation of the correlations associated with the enduring and/or repeated contacts. However, the appropriate form of this modification has yet to be determined. CONCLUSIONS AND OUTLOOK

as the inclination angle varies between a value below which a flow with purely collisional contacts is linearly unstable, and a value above which a static pile with purely enduring contacts is linearly unstable. However, although the theory has the capacity to predict the limiting thickness below which inclined flow is impossible, and other features of shallow and intermittent avalanches, it has yet to be applied to deep, dense SFD flows down inclines. The existence of spatial correlations between particles in dense flows has been explicitly incorporated into a flow theory by Mills and his co-authors48,69–71. They decompose the shear stress into a rate-independent frictional part associated with transient force chains, a local viscous part associated with mobile grains, and a non-local viscous part that accounts for the interactions between the mobile particles and the particles in the chains. The normal stress is also assumed to possess a static and a non-local part. The non-local parts are proportional to the integral of the viscous shear stress, normalized by a correlation length that is related to the spacing of the force chains along the flow. The static shear stress is related to the static normal stress by a coefficient of effective friction. The theory has the capacity to predict the variation in flow rate with flow thickness and angle of inclination, profiles of linear velocity and constant volume fraction, and the variation of the volume fraction with the angle of inclination seen in experiments on relatively shallow, dense, inclined flows, both over a rigid, bumpy base and over the surface of a heap. However, it makes no attempt to predict the fluctuation energy profile. In contrast, Rajchenbach72 emphasizes the energetics, and explains the linear velocity profiles observed in the dense, shallow flows as being due to the complete 106

Numerical simulations have now been tested against physical experiments on dense granular flows down inclines, and, despite the relative simplicity of the particle interaction models, they have predicted the global features of such flows. Figures 4 and 6 are examples. With the confidence that the simulations are faithful to the physics, the detailed information provided by them can serve as the basis for ab initio models based on the particle interactions and appropriate statistical characterizations of velocity and spatial distributions. In general, simulations and experiments have highlighted the importance of sidewalls. They influence relatively thick flows and are responsible for maintaining flows on steep erodible heaps. One important challenge is to understand their influence on the profiles of solid volume fraction. For example, no theory yet explains why this quantity varies with height in an SSH but remains essentially constant in the core of a mobilized flow in the absence of sidewalls. Basal boundary morphology influences the flow in the neighbourhood of the base in ways that remain poorly understood. At the base, several assumptions of the simplest kinetic theory no longer hold. Force can be transmitted between grains through contacts that are not impulsive, layering is associated with anisotropy in the spatial distribution of pairs of particles, and the balance of angular momentum may have a role. Although the influence of boundaries can diminish with distance as, for example, in the core of flows on a bumpy base, theories and simulations should pay greater attention to the features of the flow near the base. nature materials | VOL 6 | FEBRUARY 2007 | www.nature.com/naturematerials

REVIEW ARTICLE The role of friction is pervasive but unresolved. For example, it is unknown why the effective friction exerted by the walls on the SSH is so high, and how its magnitude is related to the friction between individual grains and the sidewalls. It is unclear whether sliding friction on a flat base is sufficient to describe the interaction with the base. Here, numerical simulations should be used to produce flows on flat frictional boundaries with sidewalls. From these it could be determined why flows exist at angles of inclinations with tangents greater than the coefficient of sliding friction at the base. In the interior of dense flows, theories should explain why the friction between individual grains is much more important than normal restitution, and should derive new constitutive relations for the stresses and for the transport and dissipation of fluctuation energy that account for the multiple, correlated, long-lasting contacts that particles often experience. Other specific questions remain. Can SFD flows exist over an erodible base without lateral boundaries? In narrow channels of width on the order of a few grain diameters, does arching lead to deviations from the behaviour of a wider SSH? What causes flows on a bumpy base to develop a thinner basal shear layer at very high speeds? We expect that further progress will be facilitated by the introduction of new experimental techniques and the additional exploitation of some existing techniques. These all have the common feature of permitting measurements in the interior of the flow. They include positron emission particle tracking80, magnetic resonance imaging81, diffusive wave scattering82, fluid index matching83, confocal imaging84 and X-ray tomography85. We have seen that several theories have the capability of reproducing the observed features of relatively shallow, dense inclined flows. Now is the time to distinguish between them. So far, numerical simulations have demonstrated the phenomena over various ranges of parameter space. They should now be used to test the essential features of the proposed theories, to eliminate as many as possible from consideration, and to focus on those that seem to be securely founded on the appropriate physics. Finally, although we have focused this review on relatively shallow, slow and dense granular flows over bumpy, flat or erodible bases, other regimes with greater heights and speed deserve future attention, despite the experimental and numerical challenges they elicit. We expect that rich physics have yet to be discovered on the stability, transitions and multiplicity among granular flow regimes with greater scale and speed. doi:10.1038/nmat1813 References 1. Maloney, C. & Lemaître, A. Universal breakdown of elasticity at the onset of material failure. Phys. Rev. Lett. 93, 195501 (2004). 2. Johnson, W. L. Bulk glass-forming metallic alloys: science and technology. Mater. Res. Soc. Bull. 24, 42–56 (1999). 3. Azanza, E., Chevoir, F. & Moucheront, P. Experimental study of collisional granular flows down an inclined plane. J. Fluid Mech. 400, 199–227 (1999). 4. Forterre, Y. & Pouliquen, O. Longitudinal vortices in granular flow. Phys. Rev. Lett. 86, 5886–5889 (2001). 5. Xu, H., Louge, M., & Reeves, A. Solutions of the kinetic theory for bounded collisional granular flows. Continuum Mech. Thermodyn. 15, 321–349 (2003). 6. Louge, M. Y. & Keast, S. C. On dense granular flows down flat frictional inclines. Phys. Fluids 13, 1213–1233 (2001). 7. Pouliquen, O. Scaling laws in granular flows down a bumpy inclined plane. Phys. Fluids 11, 542–548 (1999). 8. Taberlet, N. et al. Super stable granular heap in thin channel. Phys. Rev. Lett. 91, 264301 (2003). 9. Silbert, L. E. et al. Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E 64, 51302 (2001). 10. GDR MiDi. On dense granular flows. Eur. Phys. J. E 14, 341–365 (2004). 11. Pouliquen, O. & Chevoir, F. Dense flows of dry granular materials. C. R. Acad. Sci. Paris, Phys. 3, 163–175 (2002). 12. Campbell, C. S. Rapid granular flows. Annu. Rev. Fluid Mech. 22, 57–92 (1990). 13. Goldhirsch, I. Rapid granular flows. Annu. Rev. Fluid Mech. 35, 267–292 (2003). 14. Walton, O. R. in Particulate Two-Phase Flows (ed. Roco, M.) Ch. 25, 884–911 (Butterworth-Heinemann, Boston, 1993).

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15. Foerster, S. F., Louge, M. Y., Chang, H. & Allia, K. Measurements of the collision properties of small spheres. Phys. Fluids 6, 1108–1115 (1994). 16. Kharaz, A. H., Gorham, D. A. & Salman, A. D. An experimental study of the elastic rebound of spheres. Powder Technol. 120, 281–291 (2001). 17. Louge, M. Y. & Adams, M. E. Anomalous behavior of normal kinematic restitution in the oblique impacts of a hard sphere on an elastoplastic plate. Phys. Rev. E 65, 021303 (2002). 18. Thornton, C. Coefficient of restitution for collinear collisions of elastic perfectly plastic spheres. J. Appl. Mech. 64, 383–386 (1997). 19. Zhang, X. & Vu-Quoc, L. A method to extract the mechanical properties of particles in collision based on a new elasto-plastic normal force–displacement model. Mech. Mater. 34, 779–794 (2002). 20. Ahn, H., Brennen, C. E. & Sabersky, R. H. Measurements of velocity, velocity fluctuations, density, and stresses in chute flows of granular materials. J. Appl. Mech. 58, 792–803 (1991). 21. Drake, T. G. Granular flow: physical experiments and their implications for microstructural theories. J. Fluid Mech. 225, 121–151 (1991). 22. Berton, G., Delannay, R., Richard, P., Taberlet, N. & Valance, A. Two-dimensional inclined chute flows: transverse motion and segregation. Phys. Rev. E 68, 051303 (2003). 23. Hanes, D. M. & Walton, O. R. Simulations and physical measurements of glass spheres flowing down a bumpy incline. Powder Tech. 109, 133–144 (2000). 24. Walton, O. R. Numerical simulation of inclined chute flows of monodisperse, inelastic, frictional spheres. Mech. Mater. 16, 239–247 (1993). 25. Luding, S. in The Physics of Granular Media (eds Hinrichsen, H. & Wolf, D.) 299–324 (Wiley-VCH, Weinheim, 2004). 26. Moreau, J. J. Some numerical methods in multibody dynamics: applications to granular materials. Eur. J. Mech. A 13, 93–114 (1994). 27. Bi, W., Delannay, R., Richard, P., Taberlet, N. & Valance, A. 2D and 3D confined granular chute flows: experimental and numerical results. J. Phys. Condens. Matter 17, 1–24 (2005). 28. Johnson, P. C., Nott, P. & Jackson, R. Frictional–collisional equations of motion for particulate flows and their application to chutes. J. Fluid Mech. 210, 501–535 (1990). 29. Baumberger, T. & Caroli, C. Multicontact solid friction: a macroscopic probe of pinning and dissipation on the mesoscopic scale. Mater. Res. Soc. Bull. 23, 41–46 (1998). 30. Silbert, S. L., Landry, J. W. & Grest, G. S. Granular flow down a rough inclined plane; transition between thin and thick piles. Phys. Fluids 15, 1–10 (2003). 31. Louge, M. Y. Model for dense granular flows down bumpy inclines. Phys. Rev. E 67, 061303 (2003). 32. Daerr, A. & Douady, S. Two types of avalanche behaviour in granular media. Nature 399, 241–243 (1999). 33. Silbert, L. E., Grest, G. S. & Plimpton, S. J. Boundary effects and self-organization in dense granular flows. Phys. Fluids 14, 1–10 (2002). 34. Mitarai, N. & Nakanishi, H. Bagnold scaling, density plateau, and kinetic theory analysis of dense granular flow. Phys. Rev. Lett. 94, 128001 (2005). 35. Ancey, C. Dry granular flows down an inclined channel: experimental investigation on the frictional–collisional regime. Phys. Rev. E 65, 011304 (2001). 36. Hungr, O. & Morgenstern, N. R. Experiments on the flow behaviour of granular materials at high velocity in an open channel. Géotechnique 34, 405–413 (1984). 37. Jop, P., Forterre, Y. & Pouliquen, O. Crucial role of side walls for granular surface flows: consequences for rheology. J. Fluid Mech. 541, 167–192 (2005). 38. Rajchenbach, J. Granular flows. Adv. Phys. 49, 229–256 (2000). 39. Khakhar, D. V., Orpe, A. V., Andersen, P. & Ottino, J. M. Surface flows of granular materials: model and experiments in heap formation. J. Fluid Mech. 441, 255–264 (2001). 40. Komatsu, T. S., Inagaki, S., Nakagawa, N. & Nasuno, S. Creep motion in a granular pile exhibiting steady surface flow. Phys. Rev. Lett. 86, 1757–1760 (2001). 41. Félix, G. & Thomas, N. Relation between dry granular flow regimes and morphology of deposits: formation of levees in pyroclastic deposits. Earth Planet. Sci. Lett. 221, 197–213 (2004). 42. Jenkins, J. T. in Dynamics: Models and Kinetic Methods for Non-Equilibrium Many Bodied Systems (ed. Karkheck, J.) 313–323 (Kluwer, Dordrecht, 2000). 43. Chapman, S. & Cowling, T. G. The Mathematical Theory of Nonuniform Gases. (Cambridge Univ. Press, Cambridge, 1970). 44. Kirkwood, J. G. Selected Topics in Statistical Mechanics (Gordon & Breach, New York, 1967). 45. Jenkins, J. T. Rapid granular flow down inclines. Appl. Mech. Rev. 47, S240–S244 (1994). 46. Bocquet, L., Errami, L. J. & Lubensky, T. C. A hydrodynamic model of a jammed-to-flowing transition in gravity driven granular materials. Phys. Rev. Lett. 89, 184301 (2002). 47. Savage, S. B. in Mechanics of Granular Materials: New Models and Constitutive Relations (eds Jenkins, J. T. & Satake, M.) 261–282 (Elsevier, Amsterdam, 1983). 48. Mills, P., Tixier, M. & Loggia, D. Model for stationary dense granular flow along an inclined wall. Europhys. Lett. 45, 733–738 (1999). 49. Ancey, C. & Evesque, P. Frictional–collisional regime for granular suspension flow down an inclined channel. Phys. Rev. E 62, 8349–8360 (2000). 50. Aranson, I. S. & Tsimring, L. S. Continuum description of partially fluidized granular flows. Phys. Rev. E 65, 061303 (2002). 51. Anderson, K. G. & Jackson, R. A. Comparison of the solutions of some proposed equations of motion of granular materials for fully developed flow down inclined planes. J. Fluid Mech. 241, 145–168 (1992). 52. Pouliquen, O., Forterre, Y. & Ledizes, S. Dense granular flow down incline as a self-activated process. Adv. Complex Syst. 4, 441–450 (2001). 53. Lemaitre, A. Origin of a repose angle: kinetics of rearrangement for granular materials. Phys. Rev. Lett. 89, 064303 (2002). 54. Pouliquen, O. On the shape of granular fronts down rough inclined planes. Phys. Fluids 11, 1956–1958 (1999). 55. Jenkins, J. T. & Savage, S. B. A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130, 187–202 (1983). 56. Lun, C. K. K., Savage, S. B., Jeffrey, D. J. & Chepurniy, N. Kinetic theories for granular flow: inelastic particles in Couette flow and slightly inelastic particles in a general flow field. J. Fluid Mech. 140, 223–256 (1984).

107

REVIEW ARTICLE 57. Jenkins, J. T. & Richman, M. W. Grad’s 13-moment system for a dense gas of inelastic spheres. Arch. Rat. Mech. Anal. 87, 355–377 (1985). 58. Jenkins, J. T. & Zhang, C. Kinetic theory for nearly elastic, slightly frictional spheres. Phys. Fluids 14, 1228–1235 (2002). 59. Lun, C. K. K. Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres. J. Fluid Mech. 223, 539–559 (1991). 60. Chou, C.-S. & Richman, M. W. Constitutive theory for homogeneous granular shear flows of highly elastic spheres. Physica A 259, 430–448 (1998). 61. Sela, N. & Goldhirsch, I. Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. J. Fluid Mech. 361, 41–74 (1998). 62. Montanero, J. M., Garzo, V., Santos, A. & Brey, J. J. Kinetic theory of simple granular shear flows of smooth hard spheres. J. Fluid Mech. 389, 391–411 (1999). 63. Richman, M. W. Boundary conditions based on a modified Maxwellian velocity distribution for flows of identical, smooth, nearly elastic spheres. Acta Mech. 75, 227–240 (1988). 64. Richman, M. W. & Marciniec, R. P. Gravity-driven granular flows of smooth, inelastic spheres down bumpy inclines. J. Appl. Mech. 57, 1036–1043 (1990). 65. Nishimura, K., Kosugi, K. & Nakagawa, M. Experiments on ice-sphere flows along an inclined chute. Mech. Mater. 16, 205–209 (1993). 66. Silbert, L. E., Ertas, D., Grest, G. S., Halsey, T. C. & Levine, D. Analogies between granular jamming and the liquid-glass transition. Phys. Rev. E 65, 051307 (2002). 67. Aranson, I. S. & Tsimring, L. S. Continuum description of avalanches in granular media. Phys. Rev. E 64, 020301 (2001). 68. Wolfson, D., Tsimring, L. S. & Aranson, I. S. Partially fluidized shear granular flows: continuum theory and MD simulations. Phys. Rev. Lett. 90, 254301 (2003). 69. Mills, P., Tixier, M. & Loggia, D. Influence of roughness and dilatancy for dense granular flow along an inclined wall. Eur. Phys. J. E 1, 5–8 (2000). 70. Chevoir, F., Prochnow, M., Jenkins, J. T. & Mills, P. in Powders and Grains 01 (ed. Kishino, Y.) 373–376 (Balkema, Lisse, 2001). 71. Bonamy, D. & Mills, P. Diphasic non-local model for granular surface flows. Europhys. Lett. 63, 42–48 (2003). 72. Rajchenbach, J. Dense, rapid flows of inelastic grains under gravity. Phys. Rev. Lett. 90, 144302 (2003). 73. da Cruz, F., Emam, S., Prochnow, M., Roux, J.-N. & Chevoir, F. Rheophysics of dense granular materials: discrete simulation of plane shear flows. Phys. Rev. E 72, 021309 (2005). 74. Jop, P., Forterre, Y. & Pouliquen, O. A constitutive law for dense granular flows. Nature 441, 727–730 (2006). 75. Campbell, C. S, Cleary, P. & Hopkins, M. A. Long run-out landslides: a study by computer simulation. J. Geophys. Res. 100, 8267–8283 (1995). 76. Campbell, C. S. Granular shear flow at the elastic limit. J. Fluid Mech. 465, 261–291 (2002).

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77. Babic, M., Shen, H. H. & Shen, H. T. The stress tensor in granular shear flows of uniform, deformable disks at high solids concentrations. J. Fluid Mech. 219, 81–118 (1990). 78. Zhang, D. Z. & Rauenzahn, R. M. Stress relaxation in dense and slow flows. J. Rheol. 44, 1019–1041 (2000). 79. Ertas, D. & Halsey, T. C. Granular gravitational collapse and chute flow. Europhys. Lett. 60, 931–937 (2002). 80. Wildman, R. D., Huntley, J. M., Hansen, J.-P. & Parker, D. J. Granular temperature profiles in three-dimensional vibrofluidized granular beds. Phys. Rev. E 63, 061311 (2001). 81. Fukushima, E. Granular flow studies by NMR: a chronology. Adv. Complex Syst. 4, 1–5 (2001). 82. Dixon, P. K. & Durian, D. J. Speckle visibility spectroscopy and variable granular fluidization. Phys. Rev. Lett. 90 184302 (2003). 83. Pouliquen, O., Belzons, M. & Nicolas, M. Fluctuating particle motion during shear induced granular compaction. Phys. Rev. Lett. 91, 014301 (2003). 84. Toiya, M., Stambaugh, J. & Losert, W. Transient and oscillatory granular shear flow. Phys. Rev. Lett. 93, 088001 (2004). 85. Richard, P., Philippe, P., Barbe, F., Bourlès, T. X. & Bideau, D. Analysis by x-ray microtomography of a granular packing undergoing compaction. Phys. Rev. E 68, 020301 (2003). 86. Jenkins, J. T. & Richman, M. Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks. Phys. Fluids 28, 3485–3494 (1985). 87. Verlet, L. & Levesque, D. Integral equations for classical fluids. III. The hard discs system. Mol. Phys. 46, 969–980 (1982). 88. Luding, S. Global equation of state of two-dimensional hard sphere system. Phys. Rev. E 63, 042201 (2001).

Acknowledgements We thank Daniel Bideau, Gérard Le Caër, Luc Oger, Nathalie Thomas, and our colleagues in the Groupement de Recherche Milieux Divises (GDR MiDi) for valuable discussions. We thank James T. Jenkins for contributing several paragraphs on merits of the kinetic theory, and Namiko Mitarai for providing data shown in Fig. 7. The preparation of this review was assisted by financial support from the GDR MiDi and US–France Cooperative Research grant INT-0233212. Our research in dense, inclined flows is sponsored by the French Ministry of Education and Research (ACI PCN (INSU): Écoulements gravitaires: modélisation des processus), the CNRS (PNRN: Programme National des Risques Naturels, écoulements gravitaires), and NASA grants NCC3-468, NAG3-2705, NCC3-797 and NAG3-2353. Correspondence and requests for materials should be addressed to R.D.

Competing financial interests The authors declare that they have no competing financial interests.

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