Granular elasticity, an elasticity theory useful for calculating static stress distribution in granular media, is generalized to the dynamic case by including the plastic contribution of the strain. A complete hydrodynamic theory is derived based on the hypothesis that granular medium turns transiently elastic when deformed. This theory includes both the true and the granular temperatures, and employs a free energy expression that encapsulates a full jamming phase diagram, in the space spanned by pressure, shear stress, density and granular temperature. For the special case of stationary granular temperatures, the derived hydrodynamic theory reduces to {\em hypoplasticity}, a state-of-the-art engineering model.
Although fully elastic when static, granular media become transiently elastic when being slowly sheared -- during which both the elastic energy and stress relax. Starting from this observation, we cogently derive the framework for granular hydrodynamics, a set of differential equations consistent with general principles of physics, especially reversible and irreversible thermodynamics. In addition, an expression for the granular elastic energy is reviewed and further discussed.
Granular solid hydrodynamics, constructed to describe quasi-elastic and plastic motion of granular solid, is shown also capable of accounting for the rheology of granular dense flow. This makes it a unified, though still qualitative, hydrodynamic description, enabling one to tackle fluidization and jamming, the hysteretic transition between elasto-plastic motion and uniform dense flow.
"Granular elasticity," useful for calculating static stress distributions in granular media, is generalized by including the effects of slowly moving, deformed grains. The result is a hydrodynamic theory for granular solids that agrees well with models from soil mechanics.
A unified continuum-mechanical theory has been until now lacking for granular media, some believe it could not exist. Derived employing the hydrodynamic approach, GSH is such a theory, though as yet a qualitative one. The behavior being accounted for includes static stress distribution, elastic wave, elasto-plastic motion, the critical state and rapid dense flow. The equations and application to a few typical experiments are presented here.
Compaction is considered and embedded into broader granular behavior. Reversible compaction is related to the pressure exerted by agitated grains, a quantity relevant to dense flow. Irreversible compaction is derived from the loss of elastic deformation, the physics behind elasto-plastic flows.
Vibrating walls, used to maintain the temperature in a granular gas, modify the system strongly. Most conspicuously, the usual one-peak velocity distribution splits into two, asymmetrically positioned. A surgical repair of the usual hydrodynamic description is presented that provides an account for, and an understanding of, the situation.
We present a numerical simulation of a granular material using hydrodynamic equations. We show that, in the absence of external forces, such a system phase-separates into high density and low density regions. We show that this separation is dependent on the inelasticity of collisions, and comment on the mechanism for this clustering behavior. Our results are compatible with the granular clustering seen in experiments and molecular dynamic simulations of inelastic hard disks.
We show a case of steady flow in a granular gas that, for small shear rates, is accurately described by Navier-Stokes hydrodynamics, even for high inelasticity. The (low density) granular gas is composed of identical inelastic spheres and is fluidized by a temperature field produced by a stochastic volume force plus thermal walls. We prove that the theoretical Navier-Stokes transport coefficients exhibit good agreement with numerical solutions of the corresponding the kinetic equation (direct simulation Monte Carlo method) in the steady state.
We examine the hydrodynamics of a granular gas using numerical simulation. We demonstrate the appearance of shearing and clustering instabilities predicted by linear stability analysis, and show that their appearance is directly related to the inelasticity of collisions in the material. We discuss the rate at which these instabilities arise and the manner in which clusters grow and merge.
Using a hydrodyamic model of granular flows, we present very long time simulations of a granular fluid in two dimensions without gravity and with periodic boundary conditions in a square domain. Depending upon the values of the viscosity, thermal conductivity and dissipation, we find for intermediate times a metastable clustering state. For longer times the system is attracted to either a shear band or a vortex state. Our results are in general agreement with molecular dynamics simulations.
We investigate hydrodynamic noise in a dilute granular gas during the homogeneous cooling state, by means of a proper application of the Direct Simulation Monte Carlo (DSMC) algorithm. The DSMC includes a source of randomization which is not present in Molecular Dynamics (MD) for inelastic hard disks. Notwithstanding this difference, a fair quantitative agreement is found, including a violation of the fluctuation-dissipation relation for the noise amplitude of the same order observed in MD. This study suggests that deterministic collision dynamics is not an essential ingredient to reproduce, u…
Monodisperse granular flows often develop regions with hexagonal close packing of particles. We investigate this effect in a system of inelastic hard spheres driven from below by a "thermal" plate. Molecular dynamics simulations show, in a wide range of parameters, a close-packed cluster supported by a low-density region. Surprisingly, the steady-state density profile, including the close-packed cluster part, is well described by a variant of Navier-Stokes granular hydrodynamics (NSGH). We suggest a simple explanation for the success of NSGH beyond the freezing point.
In this supplementary paper we present some details on the solid-liquid interface detection, the deduction of the non-equilibrium free energy, the analysis of the granular temperature and energy per mode, a validation of the small slope approximation, a description of the Langevin dynamics, and the error analysis.
We present simulations of granular flows in a modified Couette cell, using a continuum model recently proposed for dense granular flows. Based on a friction coefficient, which depends on an inertial number, the model captures the positions of the wide shear bands. We show that a smooth transition in velocity-profile shape occurs when increasing the height of the granular material, leading to a differential rotation of the central part close to the surface. The numerical predictions are in qualitative agreement with previous experimental results. The model provides predictions for the increase …
{\sc gsh} is a continuum mechanical theory constructed to qualitatively account for a broad range of granular phenomena. To probe and demonstrate its width, simple solutions of {\sc gsh} are related to granular phenomena and constitutive models, including (i) for vanishing shear rates: static stress distribution and propagation of elastic waves; (ii) at slow rates: critical state, shear band, the models of hypoplasticity and barodesy; (iii) at higher rates: the MIDI-model, rapid dense flow in the Bagnold regime. A unified, densely correlated understanding of granular physics emerges as a resul…
Propotional paths as summed up by the Goldscheider Rule (GR) -- stating that given a constant strain rate, the evolution of the stress maintains the ratios of its components -- is a characteristics of elasto-plastic motion in granular media. Barodesy, a constitutive relation proposed recently by Kolymbas, is a model that, with GR as input, successfully accounts for data from soil mechanical experiments. Granular solid hydrodynamics (GSH), a theory derived from general principles of physics and two assumptions about the basic behavior of granular media, is constructed to qualitatively account f…
Comment on the paper J. Solsvik and E. Manger, "Kinetic theory models for granular mixtures with unequal granular temperature: Hydrodynamic velocity," Phys. Fluids \textbf{33}, 043321 (2021).
A simple exactly solvable kinetic model for the non-linear inelastic hard sphere Boltzmann equation is used to explore the relevance of hydrodynamics for a granular gas. The equation predicts a non-trivial homogeneous cooling state (HCS), including algebraic decay at large velocities. The linearized kinetic equation for small perturbations about the HCS is solved exactly. It is shown that the hydrodynamic excitations exist in this linearized dynamics, and their detailed form is shown to agree with results from the Chapman-Enskog method up through Navier-Stokes order. The existence of the hydro…
A simulation model is presented, comprising elastic spheres with a short range attraction. Besides conservative forces, radial- and shear friction, and radial noise are added. The model can be used to simulate colloids, granular solids and powders, and the parameters may be related to experimental systems via the range of attraction and the adhesion energy. The model shares the simplicity and speed of Dissipative Particle Dynamics (DPD), yet the predictions are rather non-trivial. We demonstrate that the model predicts the correct scaling relations for fracture of granular solids, and we prese…
We analyze the transport properties of a low density ensemble of identical macroscopic particles immersed in an active fluid. The particles are modeled as inelastic hard spheres (granular gas). The non-homogeneous active fluid is modeled by means of a non-uniform stochastic thermostat. The theoretical results are validated with a numerical solution of the corresponding the kinetic equation (direct simulation Monte Carlo method). We show a steady flow in the system that is accurately described by Navier-Stokes (NS) hydrodynamics, even for high inelasticity. Surprisingly, we find that the deviat…
Hydrodynamic equations for granular gases driven by the Fokker-Planck operator are derived. Transport coefficients appeared in Navier-Stokes order change from the values of a free cooling state to those of a steady state.
We consider dense rapid shear flow of inelastically colliding hard disks. Navier-Stokes granular hydrodynamics is applied accounting for the recent finding \cite{Luding,Khain} that shear viscosity diverges at a lower density than the rest of constitutive relations. New interpolation formulas for constitutive relations between dilute and dense cases are proposed and justified in molecular dynamics (MD) simulations. A linear stability analysis of the uniform shear flow is performed and the full phase diagram is presented. It is shown that when the inelasticity of particle collision becomes large…
The {\em hydrodynamic} approach to a continuum mechanical description of granular behavior is reviewed and elucidated. By considering energy and momentum conservation simultaneously, the general formalism of {\em hydrodynamics} provides a systematic method to {derive} the structure of constitutive relations, including all gradient terms needed for nonuniform systems. An important input to arrive at different relations (say, for Newtonian fluid, solid and granular medium) is the energy, especially the number and types of its variables. Starting from a careful examination of the physics underlyi…
We study the Navier-Stokes steady states for a low density monodisperse hard sphere granular gas (i.e, a hard sphere ideal monatomic gas with inelastic interparticle collisions). We present a classification of the uniform steady states that can arise from shear and temperature (or energy input) applied at the boundaries (parallel walls). We consider both symmetric and asymmetric boundary conditions and find steady states not previously reported, including sheared states with linear temperature profiles. We provide explicit expressions for the hydrodynamic profiles for all these steady states. …
We investigate the inelastic hard disk gas sheared by two parallel bumpy walls (Couette-flow). In our molecular dynamic simulations we found a sensitivity to the asymmetries of the initial condition of the particle places and velocities and an asymmetric stationary state, where the deviation from (anti)symmetric hydrodynamic fields is stronger as the normal restitution coefficient decreases. For the better understanding of this sensitivity we carried out a linear stability analysis of the former kinetic theoretical solution [Jenkins and Richman: J. Fluid. Mech. {\bf 171} (1986)] and found it t…
Weakly nonlinear analysis of a two dimensional sheared granular flow is carried out under the Lees-Edwards boundary condition. We derive the time dependent Ginzburg-Landau (TDGL) equation of a disturbance amplitude starting from a set of granular hydrodynamic equations and discuss the bifurcation of the steady amplitude in the hydrodynamic limit.
In a granular solid, mechanical equilibrium requires a delicate balance of forces at the disordered grain scale. To understand how macroscopic rigidity can emerge in this amorphous solid, it is crucial that we understand how Newton's laws pass from the disordered grain scale to the laboratory scale. In this work, we introduce an exact discrete calculus, in which Newton's laws appear as differential relations at the scale of a single grain. Using this calculus, we introduce gauge variables which describe identically force- and torque-balanced configurations. In a first, intrinsic formulation, w…
The anisotropic, stress-dependent velocity of elastic waves in glass beads -- as observed by Y. Khidas and X. Jia, see [Phys. Rev. E, 81:021303, Feb. 2010] -- is shown to be well accounted for by ``granular solid hydrodynamics,'' a broad-range macroscopic theory of granular behavior. As the theory makes no reference to fabric anisotropy, the influence of which on sound is in doubt.
A recently introduced model describing -on a 1d lattice- the velocity field of a granular fluid is discussed in detail. The dynamics of the velocity field occurs through next-neighbours inelastic collisions which conserve momentum but dissipate energy. The dynamics can be described by a stochastic equation in full phase space, or through the corresponding Master Equation for the time evolution of the probability distribution. In the hydrodynamic limit, equations for the average velocity and temperature fields with fluctuating currents are derived, which are analogous to those of granular fluid…
We present in this Letter experimental results on the bidimensional flow field around a cylinder penetrating into dense granular matter together with drag force measurements. A hydrodynamic model based on extended kinetic theory for dense granular flow reproduces well the flow localization close to the cylinder and the corresponding scalings of the drag force, which is found to not depend on velocity, but linearly on the pressure and on the cylinder diameter and weakly on the grain size. Such a regime is found to be valid at a low enough "granular" Reynolds number.
The approach to the critical state -- the transition from partially elastic to perfectly plastic behavior -- is considered the most characteristic of granular phenomena in soil mechanics. By identifying the critical state as the steady-state solution of the elastic strain, and presenting the main results as transparent, analytic expressions, the physics of this important phenomenon is clarified.
We study the transition of a granular packing from liquid to solid bonding in the course of drying. The particles are initially wetted by a liquid brine and the cohesion of the packing is ensured by capillary forces, but the crystallization of the solute transforms the liquid bonds into partially cemented bonds. This transition is evidenced experimentally by measuring the compressive strength of the samples at regular intervals of times. Our experimental data reveal three regimes: 1) Up to a critical degree of saturation, no solid bonds are formed and the cohesion remains practically constant;…
The rheology of a granular shear flow is studied in a quasi-2d rotating cylinder. Measurements are carried out near the midpoint along the length of the surface flowing layer where the flow is steady and non-accelerating. Streakline photography and image analysis are used to obtain particle velocities and positions. Different particle sizes and rotational speeds are considered. We find a sharp transition in the apparent viscosity ($η$) variation with rms velocity ($u$). In the fluid-like region above the depth corresponding to the transition point (higher rms velocities) there is a rapid incre…
We present a hydrodynamic theoretical model for "Brazil nut" size segregation in granular materials. We give analytical solutions for the rise velocity of a large intruder particle immersed in a medium of monodisperse fluidized small particles. We propose a new mechanism for this particle size-segregation due to buoyant forces caused by density variations which come from differences in the local "granular temperature". The mobility of the particles is modified by the energy dissipation due to inelastic collisions and this leads to a different behavior from what one would expect for an elastic …
These are notes prepared for presentation at the workshop "Challenges in Granular Matter" at the Abdus Salam Institute for Theoretical Physics, Trieste, August 2001. Revisions and figures will be added at a later date. Many features of real granular fluids under rapid flow are replicated by a system of smooth hard spheres with inelastic collisions. For such a system, it is tempting to apply standard methods of kinetic theory and hydrodynamics to calculate properties of interest. The domain of validity for such methods is a priori uncertain due to the inelasticity, but recent systemmatic studie…
The penetration by a gravity driven impact of a solid sphere into a granular medium is studied by two-dimensional simulations. The scaling laws observed experimentally for both the final penetration depth and the stopping time with the relevant physical parameters are here recovered numerically without the consideration of any solid friction. Collisional processes are thus found as essential in explaining the physics of the qualitatively observed phenomena whereas frictional processes can only be considered as secondary effects in the granular penetration by impact.
We investigate hydrodynamic fluctuations in a 2D granular fluid excited by a vibrating base and in the presence of gravity, focusing on the transverse velocity modes. Since the system is inhomogeneous, we measure fluctuations in horizontal layers whose width is smaller than the characteristic hydrodynamic lengths: they can be considered as almost-homogeneous subsystems. The large time decay of autocorrelations of modes is exponential and compatible with vorticity diffusion due to shear viscosity, as in equilibrium fluids. The velocity structure factor, which strongly deviates from the equilibr…
The evolution to the steady state of a granular gas subject to simple shear flow is analyzed by means of computer simulations. It is found that, regardless of its initial preparation, the system reaches (after a transient period lasting a few collisions per particle) a non-Newtonian (unsteady) hydrodynamic regime, even at strong dissipation and for states where the time scale associated with inelastic cooling is shorter than the one associated with the irreversible fluxes. Comparison with a simplified rheological model shows a good agreement.
The basis for a hydrodynamic description of granular gases is discussed for a low density gas of smooth, inelastic hard spheres. The more fundamental mesoscopic description is taken to be the nonlinear Boltzmann kinetic equation. Two arguments are presented in favor of a hydrodynamics for granular gases. The first one is the concept of a "normal" solution and its explicit approximate construction via the Chapman-Enskog method. The second is the demonstration of hydrodynamic modes in the spectrum of the generator for the dynamics of small spatial perturbations of the homogeneous reference state…
In this review, a theoretical description is provided for the solid (granular) phase of the gas-solid flows that are the focus of this book. Emphasis is placed on the fundamental concepts involved in deriving a macroscopic hydrodynamic description for the granular material in terms of the hydrodynamic fields (species densities, flow velocity, and the granular temperature) from a prescribed "microscopic" interaction among the grains. To this end, the role of the interstitial gas phase, body forces such as gravity, and other coupling to the environment are suppressed and retained only via a poss…
A numerical study is presented to analyze the thermal mechanisms of unsteady, supersonic granular flow, by means of hydrodynamic simulations of the Navier-Stokes granular equations. For this purpose a paradigmatic problem in granular dynamics such as the Faraday instability is selected. Two different approaches for the Navier-Stokes transport coefficients for granular materials are considered, namely the traditional Jenkins-Richman theory for moderately dense quasi-elastic grains, and the improved Garzó-Dufty-Lutsko theory for arbitrary inelasticity, which we also present here. Both solutions …
We investigate a lattice model representing a granular gas in a thin channel. We deduce the hydrodynamic description for the model from the microscopic dynamics in the large system limit, including the lowest finite-size corrections. The main prediction from hydrodynamics, when finite-size corrections are neglected, is the existence of a steady "uniform longitudinal flow" (ULF), with the granular temperature and the velocity gradient both uniform and directly related. Extensive numerical simulations of the system show that such a state can be observed in the bulk of a finite-size system by att…
Despite extensive studies on either smooth granular-fluid flow or the solid-like deformation at the slow limit, the change between these two extremes remains largely unexplored. By systematically investigating the fluctuations of tightly packed grains under steady shearing, we identify a transition zone with prominent stick-slip avalanches. We establish a state diagram, and propose a new dimensionless shear rate based on the speed dependence of inter-particle friction and particle size. With fluid-immersed particles confined in a fixed volume and forced to "flow" at viscous numbers $J$ decades…
A hydrodynamic theory is formulated for buoyancy-driven ("thermal") granular convection, recently predicted in molecular dynamic simulations and observed in experiment. The limit of a dilute flow is considered. The problem is fully described by three scaled parameters. The convection occurs via a supercritical bifurcation, the inelasticity of the collisions being the control parameter. The theory is expected to be valid for small Knudsen numbers and nearly elastic grain collisions.
It has been recently shown (Fouxon et al. 2007) that, in the framework of ideal granular hydrodynamics (IGHD), an initially smooth hydrodynamic flow of a granular gas can produce an infinite gas density in a finite time. Exact solutions that exhibit this property have been derived. Close to the singularity, the granular gas pressure is finite and almost constant. This work reports molecular dynamics (MD) simulations of a freely cooling gas of nearly elastically colliding hard disks, aimed at identifying the "attempted" density blowup regime. The initial conditions of the simulated flow mimic t…
Confined granular fluids, placed in a shallow box that is vibrated vertically, can achieve homogeneous stationary states thanks to energy injection mechanisms that take place throughout the system. These states can be stable even at high densities and inelasticities allowing for a detailed analysis of the hydrodynamic modes that govern the dynamics of granular fluids. Analyzing the decay of the time correlation functions it is shown that there is a crossover between a quasielastic regime in which energy evolves as a slow mode, to a inelastic regime, with energy slaved to the other conserved fi…
We study the structural properties of two-dimensional granular packings prepared by random deposition from a source line. We consider a class of random ballistic deposition models based on single-particle relaxation rules controlled by a critical angle, and we show that these local rules can be formulated as rolling friction in the framework of dynamic methods for the simulation of granular materials. We find that a packing prepared by random deposition models is generically unstable, and undergoes dynamic rearrangements. As a result, the dynamic method leads systematically to a higher solid f…
Segregation induced by a thermal gradient of an impurity in a driven low-density granular gas is studied. The system is enclosed between two parallel walls from which we input thermal energy to the gas. We study here steady states occurring when the inelastic cooling is exactly balanced by some external energy input (stochastic force or viscous heating), resulting in a uniform heat flux. A segregation criterion based on Navier-Stokes granular hydrodynamics is written in terms of the tracer diffusion transport coefficients, whose dependence on the parameters of the system (masses, sizes and coe…
Thermal wall is a convenient idealization of a rapidly vibrating plate used for vibrofluidization of granular materials. The objective of this work is to incorporate the Knudsen temperature jump at thermal wall in the Navier-Stokes hydrodynamic modeling of dilute granular gases of monodisperse particles that collide nearly elastically. The Knudsen temperature jump manifests itself as an additional term, proportional to the temperature gradient, in the boundary condition for the temperature. Up to a numerical pre-factor of order unity, this term is known from kinetic theory of elastic gases. We…
We describe a special class of steady Couette flows in dilute granular gases admitting a non-Newtonian hydrodynamic description for strong dissipation. The class occurs when viscous heating exactly balances inelastic cooling, resulting in a uniform heat flux. It includes the Fourier flow of ordinary gases and the simple or uniform shear flow (USF) of granular gases as special cases. The rheological functions have the same values as in the USF and generalized thermal conductivity coefficients can be identified. These points are confirmed by molecular dynamics simulations, Monte Carlo simulation…
