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International audience; We investigate the steady motion of solid particles through successive jumps over a horizontal, rigid, bumpy bed driven by the shearing of a viscous fluid in the absence of turbulence, lubrication forces and collisions above the bed. We employ a discrete element method for the particles coupled to a mean field continuum model for the fluid to run quasi-two-dimensional simulations that we compare with the predictions of a simple model which assumes that all the particles follow identical periodic trajectories determined by the intensity of the shearing and compatible with previously suggested laws relating the particle velocities before and after the impact with the bed. We solve the periodic model both numerically and analytically, and identify the solutions that are linearly stable to small perturbations. We show that the stable solutions of the periodic model are in qualitative and quantitative agreement with the discrete simulations, as long as the number of moving particles in the system is not too large. The discrete simulations further reveal that there are two distinct families of particle trajectories, and that the simple periodic model is actually a good representation of the more energetic particles, that spend most of their time in the upper flow layers where they can gain momentum from the flow.

We perform discrete numerical simulations at constant volume of dense, steady, homogeneous flows of true cylinders interacting via Hertzian contacts, with and without friction, in the absence of preferential alignment. We determine the critical values of solid volume fraction and average number of contacts per particle above which rate-independent components of the stresses develop, along with a sharp increase in the fluctuations of angular velocity. We show that kinetic theory, extended to account for velocity correlation at solid volume fractions larger than 0.49, can quantitatively predict the measured fluctuations of translational velocity, at least for sufficiently rigid cylinders, for any values of the cylinder aspect ratio and friction here investigated. The measured pressure above and below the critical solid volume fraction is in agreement with a recent theory originally intended for spheres that conjugates extended kinetic theory, the finite duration of collisions between soft particles and the development of an elastic network of long-lasting contacts responsible for the rate-independency of the flows in the supercritical regime. Finally, we find that, for sufficiently rigid cylinders, the ratio of shear stress to pressure in the subcritical regime is a linear function of the ratio of the shear rate to a suitable measure of the fluctuations of translational velocity, in qualitative accordance with kinetic theory, with an intercept that increases with friction. A decrease in the particle stiffness gives rise to nonlinear effects that greatly diminishes the stress ratio.

We perform discrete element simulations of freely cooling, dense granular materials, previously sheared at a constant rate. Particles are identical, frictional spheres interacting via linear springs and dashpots and the solid volume fraction is constant and equal to 60% during both shearing and cooling. We measure the average and the distributions of contacts per particle and the anisotropy of the contact network. We observe that the granular material, at the beginning of cooling, can be shear-jammed, fragile or unjammed. The initial state determines the subsequent evolution of the dense assembly into either an anisotropic solid, an isotropic or an anisotropic fluid, respectively. While anisotropic solids and isotropic fluids rapidly reach an apparent final steady configuration, the microstructure continues to evolve for anisotropic fluids. We explain this with the presence of vortices in the flow field that counteract the randomizing and structure-annihilating effect of collisions. We notice, in accordance with previous findings, that the initial fraction of mechanically stable particles permits to distinguish between shear-jammed, fragile or unjammed states and, therefore, determine beforehand the fate of the freely evolving granular materials. We also find that the fraction of mechanically stable particles is in a one-to-one relation with the average number of contacts per particle. The latter is, therefore, a variable that must be incorporated in continuum models of granular materials, even in the case of unjammed states, where it was widely accepted that the solid volume fraction was sufficient to describe the geometry of the system.

We first phrase a boundary-value problem for a dense, steady, fully-developed, gravitational flow of identical inelastic spheres over in inclined bumpy base in the absence of sidewalls. We then obtain approximate analytical solutions for the profiles of the solid volume fraction, the strength of the velocity fluctuations, and the mean velocity of the flow. We compare these with those obtained in numerical solutions of the exact equations.

We predict the threshold for continuing saltation of spheres in a turbulent fluid that explicitly accounts for the influence of fluid drag, lubrication forces, bed roughness, and interparticle cohesion. This reduces the need for the fitting parameters employed in existing formulations. The theory is based on a highly idealized model of steady saltation as a collection of particles that follow the same average, periodic trajectory—a succession of identical jumps, collisions with the bed, and rebounds from it. The saltation threshold is first derived in the limit of large particle inertia and, then, extended to infer results when the viscous forces of the fluid and interparticle cohesion are not negligible. The theory is successfully compared with existing discrete element simulations of spheres interacting with a Reynolds-averaged turbulent fluid and with wind tunnel experiments in a range of particle-to-fluid density ratios and particle Reynolds numbers for saltation in terrestrial and extraterrestrial conditions.

We experimentally investigate the gravitational-driven motion of a heavy object inside a vertical 2D assembly of identical, plastic cylinders arranged in a regular, triangular lattice. The bottom of the assembly is in contact with a rough plate whose horizontal, sinusoidal motion induces the formation of shear bands in the granular solid, aligned with the edges of the lattice. The intruder sinks when the width of the shear band is as large as its size and halts once the regular configuration of the grains is recovered. The resulting vertical motion of the intruder is random and intermittent, as in disordered granular or colloidal systems near jamming, with alternate flows and blockades. We show, in analogy with earthquakes, that the relation between the size and the duration of the flowing events follows a power-law with an exponent larger than one, and that the statistics of their size is compatible with the Gutenberg–Richter law. We also show that the probability density function of times between flowing events is similar to the Omori law governing the distribution of aftershock sequences following large earthquakes. Finally, the analysis of the velocity fluctuations of the intruder points to a transition from a strong to a weak contact network in the ordered granular assembly, similar to the transition from jammed to fragile states in disordered systems.

We employ kinetic theory, extended to incorporate the influence of velocity correlations, friction and particle stiffness, and a model for rate-independent, elastic components of the stresses at volume fractions larger than a critical value, in an attempt to reproduce the results of discrete-element numerical simulations of steady, fully developed, dissipative, collisional shearing flows over and within inclined, erodible, fragile beds. The flows take place between vertical, frictional sidewalls at different separations with sufficient total particle flux so that differently inclined, erodible beds result. Numerical solutions of the spanwise-averaged differential equations of the theory and the associated boundary conditions are seen to be capable of reproducing profiles of stresses, solid volume fraction, average velocity and the strength of the particle velocity fluctuations, both in the rapid collisional flow above the bed and in the slower creeping flow within the bed. The indication is that extended kinetic theory has the unique ability to faithfully describe steady, inhomogeneous, granular shearing flows, ranging from dilute to extremely dense, using balances of momentum and energy and employing boundary conditions that are associated with the balances, with a small number of physically determined, microscopic parameters.

We make use of recent extensions of kinetic theory for dense, dissipative shearing flows to phrase and solve boundary-value problems for steady flows of a dense aggregate of identical frictional spheres sheared in a gravitational field between horizontal, rigid, bumpy boundaries by the upper boundary or in the absence of gravity between two coaxial, bumpy cylinders by the inner cylinder. In both scenarios, the resulting flow consists of a region of rapid, collisional flow and a denser region of slower flow in which more enduring particle contacts play a role. In the denser region, or bed, we assume that the collisional production of energy is negligible and the anisotropy of the contact forces influences the shear stress and the pressure in the same way. We show profiles of average velocity and provide relationships between the thickness of the fast flow, the gravitational acceleration (if present), the velocity of the moving boundary, the shear stress and the confining pressure.

As random close packing is approached in a granular medium of frictionless spheres, the pressure, shear stress, and second normal stress become singular with exponents 5/2, 5/2, and 7/4, as predicted by a kinetic theory and confirmed by prior discrete element modeling simulations.

Boundary conditions for the slip velocity of inelastic, frictionless spheres interacting with bumpy walls are derived via discrete element method simulations of Couette granular flows. The bumpiness is created by gluing spheres identical to those flowing in a regular hexagonal array to a flat plane. Depending on the particle inelasticity and bumpiness, the characteristics of the flow range from simple shearing to plug flow. At low bumpiness—small distance between the wall-particles—the ratio of particle shear stress to pressure is a non-linear function of the slip velocity and presents a maximum. At high bumpiness, the bumpy plane behaves as a flat, frictional surface and the stress ratio saturates to a constant value for large slip velocity.

Discrete element numerical simulations of unsteady, homogeneous flows have been performed by shearing a fixed volume of identical, soft, frictional spheres. A constant, global, shear rate was instantly applied to particles that are initially at rest, non interacting, and randomly distributed. The granular material exhibits either large or small fluctuations in the evolving pressure, depending whether the average number of contacts per particle (coordination number) is less or larger than a critical value. When the coordination number is less than the critical value, the amplitude of the pressure fluctuations is dependent on the shear rate, whereas, it is rate-independent in the opposite case, signatures, according to the case, of fluid-like and solid-like behaviour. The same critical coordination number has been previously found to represent the minimum value at which rate-independent components of stresses develop in steady, simple shearing and the jamming transition in isotropic random packings. The observed complex behaviour of the measured pressure in the fluid–solid transition can be predicted with a constitutive model involving the coordination number, the particle stiffness and the intensity of particle agitation.

We propose corrections to the kinetic theory for fluidity in granular flows near random close packing, in order to take account of anisotropy and increases in granular temperature.

We use previous results from discrete element simulations of simple shear flows of rigid, identical spheres in the collisional regime to show that the volume fractiondependence of the stresses is singular at the shear rigidity. Here, we identify the shear rigidity, which is a decreasing function of the interparticle friction, as the maximum volume fraction beyond which a random collisional assembly of grains cannot be sheared without developing force chains that span the entire domain. In the framework of extended kinetic theory, i.e., kinetic theory that accounts for the decreasing in the collisional dissipation due to the breaking of molecular chaos at volume fractions larger than 0.49, we also show that the volume fraction-dependence of the correlation length (measure of the velocity correlation) is singular at random close packing, independent of the interparticle friction. The difference in the singularities ensures that the ratio of the shear stress to the pressure at shear rigidity is different from zero even in the case of frictionless spheres: we identify that with the yield stress ratio of granular materials, and we show that the theoretical predictions, once the different singularities are inserted into the functions of extended kinetic theory, are in excellent agreement with the results of numerical simulations.

We compare the predictions of extended kinetic theory (EKT), where the roles of surface friction and correlation in fluctuation velocities are taken into account, with discrete element simulations of steady, fully-developed, inclined flows of identical spheres over bumpy bases, in the presence and absence of flat, frictional sidewalls. We show that the constitutive relation for the pressure of EKT must be modified in the proximity of the boundary, because of the influence of excluded volume and shielding associated with collisions of particles with the boundary itself. We also note that currently available boundary conditions for flows over bumpy planes in kinetic theory underestimate the energy dissipation. These two observations explain the lack of agreement of EKT with the simulations, in terms of the maximum angles of inclination for which steady, fully-developed flows are possible. That is, for some high angles of inclination, EKT does not have solutions, while steady flows are predicted in DEM. However, whenever a solution to the system of differential equations of EKT does exist, the predicted distributions of velocity, solid volume fraction and granular temperature satisfactorily match the numerical measurements. The incompressible, algebraic approximation of EKT, which ignores the conduction of energy in the energy balance, admits solutions for a wider range of angles of inclination, as in the simulations, but fails to reproduce the quantitative and qualitative behaviour of solid volume fraction and granular temperature in the two conductive layers at the top and bottom of the flow. When frictional sidewalls are added to the domain, we show that the spanwise ratio of shear stress to pressure is linearly distributed in the dense core region of the flow, confirming that the sidewalls exert, on average, a Coulomb-like resistance to the flow with an effective friction coefficient which is less than half the actual particle-wall friction.

We apply the extended kinetic theory (EKT) to the dense, simple shear flow of inelastic hard spheres. EKT is a phenomenological extension of kinetic theory which aims at incorporating in the simplest possible way the role of pre-collisional velocity correlations which are likely to occur at a concentration larger than the freezing point. The main effect of that correlation is the decrease in the rate at which fluctuating energy is dissipated in inelastic collisions. We use previously published results of numerical simulations performed using an event-driven algorithm to obtain analytical expressions for the radial distribution function at contact (which diverges at a concentration lower than the value at random close packing for sheared inelastic spheres) and the correlation length (i.e., the decreasing factor of the dissipation rate) at different values of the coefficient of restitution. With those, we show that when the diffusion of fluctuating energy of the particles is negligible, EKT implies that three branches of the analytical relation between the ratio of the shear stress to the pressure and the concentration (granular rheology) exist. Hence, for a certain value of the stress ratio, up to three corresponding values of the concentration are possible, with direct implications on the existence of multiple solutions to steady granular flows.

The prediction of the transport of sediments in streams is of crucial importance for many geophysical and industrial applications. Most of the available formulas for sediment transport are empirical and apply to situations near initiation, where a few erratic particles are seen jumping and rolling over an immobile bed. However, they are commonly adopted for predicting massive transport of sediments, although more rigorous approaches exist. The latter make use of constitutive relations from kinetic theories of granular gases, but require the numerical integrations of complicated, nonlinear differential equations, hence discouraging their usage for practical purposes. A new, explicit formula for predicting intense sediment transport is proposed here, based on kinetic theories of granular gases and incorporating in a simple yet rigorous way the possibility of turbulent suspension of the particles. It is shown that this formula, unlike others, can quantitatively reproduce physical experiments on steady, uniform flows of natural and artificial particles and water over horizontal, movable beds taken from the literature. These findings suggest that granular physics is now mature enough to provide practical tools in fields that were so far mainly empirically oriented. DOI: 10.1061/(ASCE)HY.1943-7900.0000686. © 2013 American Society of Civil Engineers. CE Database subject headings: Sediment transport; Sheet flow; Kinetics; Turbulence; Predictions. Author keywords: Sediment transport; Sheet flow; Kinetic theories. Introduction and Theory Most research on sediment transport has emphasized the forces that the liquid component exerts on the particles, rather than on the particle-particle interactions. This is partially due to the fact that the laboratory experiments were mostly conducted, for prac- tical reasons, at small values of water discharge, close to the incep- tion of particle motion, where interparticle forces are negligible. At higher values of water discharge, massive transport of sediments takes place instead, and those forces cannot be ignored. Sophisti

We make use of discrete-element-method numerical simulations of inelastic frictionless cylinders in simple shearing at different length-to-diameter ratios and solid volume fractions to analyze the rate of collisional dissipation of the fluctuation kinetic energy. We show that the nonspherical geometry of the particles is responsible, by inducing rotation, for increasing the dissipation rate of the fluctuation kinetic energy with respect to that for frictionless spheres. We also suggest that the partial alignment of the cylinders induced by shearing concurs with the particle inelasticity in generating correlation in the velocity fluctuations and thus affecting the collisional dissipation rate as the solid volume fraction increases. Finally, we propose simple phenomenological modifications to the expression of the collisional dissipation rate of kinetic theory of granular gases to take into account our findings.

This paper focuses on the mechanical behaviour of granular systems under shearing, unsteady conditions. The results of numerical simulations of time evolving, homogeneous, shear flows of an assembly of frictional spheres, under constant volume conditions are illustrated. Simulations have been performed considering three volume fractions corresponding to fluid, solid and near-to-critical conditions at steady state. The three systems follow very different evolutionary paths, in terms of pressure, coordination number and stress ratio. Fluid-like and solid-like systems exhibit large and small fluctuations, respectively, in those quantities. A critical value of the coordination number seems to govern the transition from fluid to solid.

SUMMARY This paper aims at extending the well-known critical state concept, associated with quasi-static conditions, by accounting for the role played by the strain rate when focusing on the steady, simple shear flow of a dry assembly of identical, inelastic, soft spheres. An additional state variable for the system, the granular temperature, is accounted for. The granular temperature is related to the particle velocity fluctuations and measures the agitation of the system. This state variable, as is in the context of kinetic theories of granular gases, is assumed to govern the response of the material at large strain rates and low concentrations. The stresses of the system are associated with enduring, frictional contacts among particles involved in force chains and nearly instantaneous collisions. When the first mechanism prevails, the material behaves like a solid, and constitutive models of soil mechanics hold, whereas when inelastic collisions dominate, the material flows like a granular gas, and kinetic theories apply. Considering a pressure-imposed flow, at large values of the normal stress and small values of the shear rate, the theory predicts a nonmonotonic shear rate dependence of the stress ratio at the steady state, which is likely to govern the evolution of landslides. Copyright © 2013 John Wiley & Sons, Ltd.

This work deals with the evaluation of the most suitable expression for the motion resistance of a debris flow. In particular, it focuses on inertial debris flows, i.e., granular-fluid mixtures in which the particle inertia dominates both the fluid viscous force and turbulence; it provides, through an order-of-magnitude analysis, the criterion to be satisfied for a debris flow to be considered inertial and shows that most of real-scale debris flows match this description. The analytical relation between flow depth, depth-averaged velocity, and tangent of the angle of inclination of the free surface is then used in steady, uniform flow conditions to approximate the flow resistance in depth-averaged mathematical models of debris flows. That resistance formula is tested against experimental results on the longitudinal profile of steady, fully saturated waves of water and gravel over both rigid and erodible beds, and against field measurements of real events. The notable agreement, especially in compa...

We derive expressions for the rates of erosion and deposition at the interface between a dense, dry, inclined granular flow and an erodible bed. In obtaining these, we assume that the interface between the flowing grains and the bed moves with the speed of a pressure wave in the flow, for deposition, or with the speed of a disturbance through the contacting particles in the bed, for erosion. We employ the expressions for the rates of erosion and deposition to show that after an abrupt change in the angle of inclination of the bed the characteristic time for the motion of the interface is much shorter than the characteristic time of the flow. This eliminates the need for introducing models of erosion and deposition rate in the mass balance; and the instantaneous value of the particle flux is the same function of the instantaneous value of the flow depth as in a steady, uniform flow.

We apply the continuum equations of a kinetic theory to predict the features of uniform, steady, inclined flows of identical, frictional, inelastic spheres over a rigid, bumpy base between vertical, frictional side walls. Numerical solutions of these equations over a range of mass flow rates exhibit features seen in physical experiments and numerical solutions in the absence of side walls. For the densest flows, we employ a phenomenological extension of kinetic theory that involves a length scale associated with particle correlations. When a dense flow is thick enough, an algebraic balance between the production and dissipation of fluctuation energy reproduces the relation between mass flow rate and mass hold-up obtained when solving the boundary-value problem of the extended theory.

Using the results of recent numerical simulations, we extend an existing kinetic theory for dense flows of identical, nearly elastic, frictionless spheres to identical, very dissipative, frictional spheres. The existing theory incorporates an additional length scale in the expression for the collisional rate of dissipation; this length scale is identified with the size of a cluster of correlated particles. Parameters of the theory for very dissipative, frictional spheres are set using the results of physical experiments on inclined flows of spheres over a rigid, bumpy base in the absence of sidewalls. The resulting theory is then tested against the results of physical experiments on flows of the same material over the surface of an erodible heap when frictional sidewalls are present.

A simple two-phase model for steady fully developed flows of particles and water over erodible inclined beds is developed for situations in which the water and particles have the same depth. The rheology of the particles is based on recent numerical simulations and physical experiments, the rheology of the fluid is based on an eddy viscosity, and the interaction between the particles and the fluid is through drag and buoyancy. Numerical solutions of the resulting differential equations and boundary conditions provide velocity profiles of the fluid and particles, the concentration profile of the particles, and the depth of the flow at a given angle of inclination of the bed. Simple approximations permit analytical expressions for the flow velocities and the depth of flow to be obtained that agree with the numerical solutions and those measured in experiments.

We perform discrete element simulations of homogeneous shearing of frictionless cylinders and show that the particles are characterized by orientational order and form a granular liquid crystal. For elongated and flat cylinders, the alignment is in the plane of shearing, while cylinders having an aspect ratio equal to 1 and 0.8 show no orientational order. We show that the particle pressure is insensitive to the cylinder aspect ratio and well predicted by the kinetic theory of granular gases, with a singularity in the radial distribution function at contact different from that for frictionless spheres. The numerical results quantitatively agree with physical experiments on different geometries. The particle shear stress is affected by orientational anisotropy. We postulate that, for frictionless cylinders, the viscosity is roughly due to the motion of the orientationally disordered fraction of the particles, and show that it is proportional, through the order parameter, to the expression of kinetic theory. Finally, we suggest that the orientational order is the result of the competing effects of the shear rate, which induces alignment, and the granular temperature, which ramdomizes.

International audience; We determine approximate, analytical solutions for average, periodic trajectories of particles that are accelerated by the turbulent shearing of a fluid between collisions with a hydrodynamically rough bed. We indicate how the viscosity of the fluid may influence the collisions with the bed. The approximate solutions compare well with periodic solutions for average periodic trajectories over rigid-bumpy and erodible beds that are generated numerically. The analytic solutions permit the determination of the relations between the particle flux and the strength of the shearing flow over a range of particle and fluid properties that vary between those for sand in air and sand in water.

A recent simple analytical approach to the problem of steady, uniform transport of sediment by a turbulent shearing fluid dominated by interparticle collisions is extended to the case in which the mean turbulent lift may partially or totally support the weight of the sediment. We treat the granular–fluid mixture as a continuum and make use of constitutive relations of kinetic theory of granular gases to model the particle phase and a simple mixing-length approach for the fluid. We focus on pressure-driven flows over horizontal, erodible beds and divide the flow itself into layers, each dominated by different physical mechanisms. This permits a crude analytical integration of the governing equations and to obtain analytical expressions for the distribution of particle concentration and velocity. The predictions of the theory are compared with existing laboratory measurements on the flow of glass spheres and sand particles in water. We also show how to build a regime map to distinguish between collisional, turbulent-collisional, and fully turbulent suspensions.

We reanalyze previous experimental measurements of solid volume fraction, mean velocity, and velocity fluctuations in collisional suspensions of plastic cylinders and water flowing over inclined, erodible beds. We show that the particle pressure scales with the granular temperature, as predicted by kinetic theory of granular gases. The assumption that the particle shear stress is also well predicted by kinetic theory permits us to determine the fluid shear stress and the effective fluid viscosity from the experiments. The fluid viscosity can be decomposed into turbulent and granularlike components: the turbulent viscosity can be modeled using a mixing length, which is a decreasing function of the local volume fraction and does not depend upon the distance from the bed; the granularlike viscosity, associated with the transfer of momentum due to the conjugate motion of the fluid mass added to the particles, can be modeled by replacing the particle density with the density of the added fluid mass in the viscosity of kinetic theory.

This work deals with the simple shear flow of neutrally buoyant, rigid, frictionless spheres immersed in a viscous fluid that exchange momentum through inelastic collisions. It is shown how kinetic theories are able to provide a full analytical description of the flow, once the influence of the viscous fluid is taken into account in a simple way through the dependence of the collisional coefficient of restitution on the Stokes number. This allows the capture of the characteristics of the experiments performed by Bagnold 60 years ago and the interpretation of the macroviscous and inertial regimes described by the same author as the limits for the coefficient of restitution equal to zero and to the value valid in absence of the viscous fluid, respectively. DOI: 10.1061/(ASCE)HY.1943-7900.0000701. © 2013 American Society of Civil Engineers. CE Database subject headings: Shear flow; Kinetics; Rheology; Coefficients. Author keywords: Simple shear flow; Kinetic theory; Granular-fluid mixture; Rheology.

We extend models for granular flows based on the kinetic theory beyond the critical volume fraction at which a rate-independent contribution to the stresses develops. This involves the incorporation of a measure of the duration of the particle interaction before and after this volume fraction. At volume fractions less than the critical, the stress components contain contributions from momentum exchanged in collisions that are influenced by the particle elasticity. At volume fractions greater than the critical, the stress components contain both static contributions from particle elasticity and dynamic contributions from the momentum transfer associated with the release of elastic energy by the breaking of force chains. A simple expression for the duration of a collision before and after the critical volume fraction permits a smooth transition between the two regimes and predictions for the components of the stress in steady, homogeneous shearing that are in good agreement with the results of numerical simulations. Application of the theory to steady, inhomogeneous flows reproduces the features of such flows seen in numerical simulations and physical experiments.

International audience; We use existing 3D Discrete Element simulations of simple shear flows of spheres to evaluate the radial distribution function at contact that enables kinetic theory to correctly predict the pressure and the shear stress, for different values of the collisional coefficient of restitution. Then, we perform 3D Discrete Element simulations of plane flows of frictionless, inelastic spheres, sheared between walls made bumpy by gluing particles in a regular array, at fixed average volume fraction and distance between the walls. The results of the numerical simulations are used to derive boundary conditions appropriated in the cases of large and small bumpiness. Those boundary conditions are, then, employed to numerically integrate the differential equations of Extended Kinetic Theory, where the breaking of the molecular chaos assumption at volume fraction larger than 0.49 is taken into account in the expression of the dissipation rate. We show that the Extended Kinetic Theory is in very good agreement with the numerical simulations, even for coefficients of restitution as low as 0.50. When the bumpiness is increased, we observe that some of the flowing particles are stuck in the gaps between the wall spheres. As a consequence, the walls are more dissipative than expected, and the flows resemble simple shear flows, i.e., flows of rather constant volume fraction and granular temperature.

We outline the development of theory to describe, dense, collisional shearing flows of identical compliant spheres. We begin with two simple theories: one for rigid, nearly elastic spheres that interact through instantaneous, binary collisions; the other for compliant spheres that interact through multiple, enduring contacts. We then join the two extremes by adding compliance to the collisions and collisions to the spheres in enduring contact. Finally, we compare the predictions of the resulting theory with the results of discrete numerical simulations of steady, homogeneous shearing of compliant frictional spheres.

We make use of recent extensions of kinetic theory of granular gases to include the role of particle stiffness in collisions to deal with pressure-imposed shearing flows between bumpy planes in relative motion, in which the solid volume fraction and the intensity of the velocity fluctuations are not uniformly distributed in the domain. As in previous numerical simulations on the flow of disks in an annular shear cell, we obtain an exponential velocity profile in the region where the volume fraction exceeds the critical value at which a rate-independent contribution to the stresses arises. We also show that the thickness of the inertial region, where the solid volume fraction is less than the critical value, and the shear stress at the moving boundary are determined functions of the relative velocity of the boundaries.

We analyze steady, uniform flow down inclines in which collisions between particles dominate the transfer of momentum and energy in a region of intensely sheared particles at the base of the flow. At the top of the region is an erodible interface that separates two regimes involving colliding particles. Those below the interface participate in the shearing, those above the interface are too concentrated to be sheared. At the bottom of the region, the particles of the flow collide with the fixed particles of a bumpy boundary. Gravity acts throughout the region of shear. The shear stress transmitted through the shear layer to the material above it is determined in terms of the velocity of this material and the total height of the flow.

This work deals with the propagation of granular-liquid waves over rigid beds, originated by the sudden removal of a sluice gate in a rectangular, inclined flume. In particular, we experimentally investigate the role of the initial volume ratio of granular material---monodispersed plastic cylinders---to water, the flume width, and the bed roughness on the time evolution of the granular front. Due to the presence of the interstitial liquid, we observed previously unreported types of collapse: (i) discontinuous flows, where the granular material stops after an initial spreading, and then flows again when the liquid, initially slower than the particles, reaches the front and remobilizes it; (ii) flows evolving into uniformly progressive waves at an angle of inclination of the flume well below the angle of repose of the dry granular material. We also noticed an unusual influence of the lateral confinement on the wave propagation. Indeed, the constant front velocity in the uniformly progressive state decreases when the channel width increases. We claim that the latter observation and the presence of discontinuous flows, strongly support the idea that only two-phase, stratified mathematical models can predict the behavior of unsteady, granular-liquid mixtures at high concentration, such as debris flows.

This work focuses on the mechanical response of dry granular materials under steady, simple shear conditions. In particular, the goal is to obtain a complete rheology able to describe the material behavior within the entire range of concentrations for which the flow can be considered dense. The total stress is assumed to be the linear sum of a frictional and a kinetic component. The frictional and the kinetic contributions are modeled in the context of the critical state theory and the kinetic theory of dense granular gases, respectively; in the latter, the correlated motion among the particles, which is likely to occur at high concentration, is also included. In accordance with recent findings on disordered granular packings, the frictional component of stresses is assumed to vanish when the concentration is below the random loose packing. According to this approach, four nondimensional quantities govern steady, simple shear flows: the concentration, the shear to normal stress ratio, the ratio of the time scales associated with the motion perpendicular and parallel to the flow, and the ratio between the particle stiffness and the normal stress. The present theory allows us to reproduce, in a notable way, both numerical simulations on simple shear flows of disks and physical experiments on incline flows of glass spheres taken from the literature.

We simplify a recent theory for steady, gravity‐driven flow of a highly concentrated granular‐fluid mixture, with the assumption that the flow is shallow, and we solve, in an approximate way, for the variation of height in a steady, non‐uniform, inclined flow over a rigid bed in absence of sidewalls. We then analyze the influence of the approximations on the shape of the wave.

We simplify a two-phase theory proposed by Berzi and Jenkins for the uniform motion of a granular-fluid mixture to obtain explicit, analytical relations between the tangent of the angle of inclination of the free surface, the average particle (fluid) velocity and the particle (fluid) depth. Those expressions, valid, in principle, only in uniform flow conditions, can then be employed to express the motion resistance for the particles and the fluid in mathematical models of non-uniform flow, as customary in Hydraulics. The advantages of those formulas with regard to previous, widely employed expressions are also discussed.

We extend a recent theory for steady uniform gravity-driven flow of a highly concentrated granular-fluid mixture over an erodible bed between frictional sidewalls. We first include angles of inclination greater than the angle of repose of the particles; then, we introduce a boundary condition for flow over a rigid bumpy bed. We compare the predictions of the resulting theory with the volume flow rates, depths and angles of inclination measured in the experiments on dry and variously saturated flows over rigid and erodible boundaries. Finally, we employ the resulting theory, with the assumption that the flow is shallow, to solve, in an approximate way, for the variation of height and average velocities along a steady non-uniform inclined flow of a granular-fluid mixture that moves over a rigid bumpy bed. The solutions exhibit features of the flow seen in the experiments – for example, a dry bulbous snout in advance of the fluid, whose length increases with increasing number of the particles and that disappears with increasing velocity – for which satisfactory explanations were lacking.

We extend a simple two-phase model for a steady fully developed flow of particles and water over an erodible inclined bed to situations in which the water and particles do not have the same depth. The rheology of the particles is based on recent numerical simulations and physical experiments, the rheology of the fluid is based on an eddy viscosity, and the interaction between the particles and the fluid is through drag and buoyancy. Simple approximations permit analytical expressions for the flow velocities and the depth of flow to be obtained that satisfactorily reproduce those measured in experiments.

This paper focuses on the evaluation of the regime transition in the surface flow of a neutrally buoyant suspension. The revisiting of the 1954 Bagnold paper made by Hunt et al. (2002) showed that the choice of the non-dimensional parameter afterwards known as Bagnold number for evaluating the transition between macroviscous and grain-inertial regime is at least doubtful. Here, a different approach, based on the universal stability parameter of Hanks (1963a, b), is proposed. The application of this transition criterion to the case of the incompressible uniform surface flow of a liquid-solid mixture shows that the Reynolds number is the parameter that really controls the transition between the flow regimes. The critical value of the Reynolds number is a function of the solid volume fraction of the mixture. This value is derived assuming that the critical value of the Hanks stability parameter does not change when referring to the ensemble averaged momentum equations of the continuous phase rather than to the classical Navier-Stokes equations. The present analysis shows that highly concentrated neutrally buoyant suspensions can be characterized by the macroviscous regime even if the Reynolds number is not small, up to an order of 10. Thus, one should pay attention in applying at real scale laws obtained through experiments performed at laboratory scale.

We provide numerical solutions to the momentum and energy balance of a kinetic theory for the steady, collisional shearing of identical, inelastic, frictional spheres between two different types of boundaries—rigid-bumpy and erodible, in the absence of gravity. A rigid-bumpy boundary is a source of fluctuation energy for the flow, an erodible boundary is a sink. As a consequence, the characteristics of shearing between two rigid-bumpy boundaries, two erodible boundaries, and a rigid-bumpy and an erodible boundary are all different. Here, we display these differences and relate them to measurements of inhomogeneous shearing and the development of shear bands in laboratory experiments.

Correction for ‘Steady shearing flows of deformable, inelastic spheres’ by Diego Berzi et al., Soft Matter, 2015, 11, 4799–4808.

We apply the constitutive relations of kinetic theory of granular gases to the transport of cohesionless sediments driven by a gravitational liquid turbulent stream in steady uniform conditions. The sediment-laden flow forms self-equilibrated mechanisms of resistance at the bed surface, below which the sediments are at rest. This geo-physical process takes place quite often in streams at moderate slope and may be interpreted through tools common to fluid mechanics and particle physics. Taking into account the viscous dissipation of the fluctuation energy of the particles, and using approximate methods of integration of the governing differential equations, permit to obtain a set of simple formulas for predicting how depths and flow rates adjust to the angle of inclination of the bed, without requiring additional tuning parameters besides the particle and fluid properties. The agreement with laboratory experiments performed with either plastic cylinders or gravel in water is remarkable. We also provide quantitative criteria to determine the range of validity of the theory, i.e., the values of the Shields number and the angle of inclination of the bed for which the particle stresses can be mostly ascribed to collisional exchange of momentum.

We study flows of inelastic spheres on the surface of an erodible bed between frictional sidewalls and distinguish two regions in such flows: a dilute, diffuse region, neighboring the free surface, for which we solve a boundary-value problem based on the kinetic theory, and a dense algebraic layer, in which there is an approximate algebraic balance between production and dissipation of fluctuation energy. We take into account correlated motions between the particles at high volume fractions and employ the trapezoidal rule to solve, in an approximate way, for the flow quantities in the diffuse layer. Using boundary conditions of no-slip and yield at the bed and vanishing of the stresses and the energy flux at the free surface, we obtain analytical predictions of flow depth and mass flow rate that compare favorably with the results of experiments performed on glass spheres flowing on the surface of a heap and in half-filled rotating drums.