Inelastic Gases Research Articles

Extended kinetic theory applied to pressure-controlled shear flows of frictionless spheres between rigid, bumpy planes.

We numerically investigate, through discrete element simulations, the steady flow of identical, frictionless spheres sheared between two parallel, bumpy planes in the absence of gravity and under a fixed normal load. We measure the spatial distributions of solid volume fraction, mean velocity, intensity of agitation and stresses, and confirm previous results on the validity of the equation of state and the viscosity predicted by the kinetic theory of inelastic granular gases. We also directly measure the spatial distributions of the diffusivity and the rate of collisional dissipation of the fluctuation kinetic energy, and successfully test the associated constitutive relations of the extended kinetic theory, i.e., a kinetic theory which includes the role of velocity correlations. We then phrase and numerically integrate a system of differential equations governing the flow, with suitably modified boundary conditions. We show a remarkable qualitative and quantitative agreement with the results of the discrete simulations. In particular, we study the effect of (i) the coefficient of collisional restitution, (ii) the imposed load and (iii) the bumpiness of the planes on the profiles of the hydrodynamic fields, the ratio of shear stress-to-pressure and the gap between the bumpy planes. Finally, we predict the critical value of the imposed load above which crystallization occurs, based on the value of the solid volume fraction near the boundaries obtained from the numerical solution of the kinetic theory. This notably reproduces what we observe in the discrete simulations.

Mathematical Investigation of 1D Discontinuity Waves in Dilute Granular Gases

The propagation of acceleration waves in dilute granular gases was investigated. Acceleration waves propagating in elastic gases, mixtures, and other materials are widely studied in the literature, but not in granular gases. A thirteen-moment theory for granular gas was considered in the framework of Grad’s theory. The spatially homogeneous solutions were determined, and the hyperbolicity of the model is discussed. The propagation of acceleration waves in a non-constant state was investigated; the amplitude of the fastest mode was derived, and the critical time was evaluated. The acceleration wave propagation velocity in inelastic gases was shown to be lower than in elastic gases.

Higher-order moment theories for dilute granular gases of smooth hard spheres

Grad’s method of moments is employed to develop higher-order Grad moment equations – up to the first 26 moments – for dilute granular gases within the framework of the (inelastic) Boltzmann equation. The homogeneous cooling state of a freely cooling granular gas is investigated with the Grad 26-moment equations in a semi-linearized setting and it is shown that the granular temperature in the homogeneous cooling state still decays according to Haff’s law while the other higher-order moments decay on a faster time scale. The nonlinear terms of the fully contracted fourth moment are also considered and, by exploiting the stability analysis of fixed points, it is shown that these nonlinear terms have a negligible effect on Haff’s law. Furthermore, an even larger Grad moment system, which includes the fully contracted sixth moment, is also scrutinized and the stability analysis of fixed points is again exploited to conclude that even the inclusion of the scalar sixth-order moment into the Grad moment system has a negligible effect on Haff’s law. The constitutive relations for the stress and heat flux (i.e. the Navier–Stokes and Fourier relations) are derived through the Grad 26-moment equations and compared with those obtained via the Chapman–Enskog expansion and via computer simulations. The linear stability of the homogeneous cooling state is analysed through the Grad 26-moment system and various subsystems by decomposing them into longitudinal and transverse systems. It is found that one eigenmode in both longitudinal and transverse systems in the case of inelastic gases is unstable. By comparing the eigenmodes from various theories, it is established that the 13-moment eigenmode theory predicts that the unstable heat mode of the longitudinal system remains unstable for all wavenumbers below a certain coefficient of restitution, while any other higher-order moment theory shows that this mode becomes stable above some critical wavenumber for all values of the coefficient of restitution. In particular, the Grad 26-moment theory leads to a smooth profile for the critical wavenumber, in contrast to the other considered theories. Furthermore, the critical system size obtained through the Grad 26-moment theory is in excellent agreement with that obtained through existing theories.

Anomalous transport of impurities in inelastic Maxwell gases.

A mixture of dissipative hard grains generically exhibits a breakdown of kinetic energy equipartition. The undriven and thus freely cooling binary problem, in the tracer limit where the density of one species becomes minute, may exhibit an extreme form of this breakdown, with the minority species carrying a finite fraction of the total kinetic energy of the system. We investigate the fingerprint of this non-equilibrium phase transition, akin to an ordering process, on transport properties. The analysis, performed by solving the Boltzmann kinetic equation from a combination of analytical and Monte Carlo techniques, hints at the possible failure of hydrodynamics in the ordered region. As a relevant byproduct of the study, the behaviour of the second- and fourth-degree velocity moments is also worked out.

High-energy tail of the velocity distribution of driven inelastic Maxwell gases

A model of homogeneously driven dissipative system, consisting of a collection of N particles that are characterized by only their velocities, is considered. Adopting a discrete time dynamics, at each time step, a pair of velocities is randomly selected. They undergo inelastic collision with probability p. With probability , energy of the system is changed by changing the velocities of both the particles independently according to , where η is a Gaussian noise drawn independently for each particle as well as at each time steps. For the case , although the energy of the system seems to saturate (indicating a steady state) after time steps of O(N), it grows linearly with time after time steps of , indicating the absence of a eventual steady state. For , the system reaches a steady state, where the average energy per particle and the correlation of velocities are obtained exactly. In the thermodynamic limit of large N, an exact equation is obtained for the moment generating function. In the limit of nearly elastic collisions and weak energy injection, the velocity distribution is shown to be a Gaussian. Otherwise, for , the high-energy tail of the velocity distribution is Gaussian, with a different variance, while for the velocity distribution has an exponential tail.

A numerical study of the Navier–Stokes transport coefficients for two-dimensional granular hydrodynamics

A numerical study that aims to analyze the thermal mechanisms of unsteady, supersonic granular flow by means of hydrodynamic simulations of the Navier–Stokes granular equation is reported in this paper. 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 the solutions are compared with event-driven simulations of the same system under the same conditions, by analyzing the density, temperature and velocity field. Important differences are found between the two approaches, leading to interesting implications. In particular, the heat transfer mechanism coupled to the density gradient, which is a distinctive feature of inelastic granular gases, is responsible for a major discrepancy in the temperature field and hence in the diffusion mechanisms.

Formation of singularities in solutions to ideal hydrodynamics of freely cooling inelastic gases

We consider solutions to the hyperbolic system of equations of ideal granular hydrodynamics with conserved mass, total energy and finite momentum of inertia and prove that these solutions generically lose the initial smoothness within a finite time in any space dimension n. Furthermore, in the one-dimensional case we introduce a solution depending only on the spatial coordinate outside of a ball containing the origin and prove that this solution under rather general assumptions on initial data cannot be global in time too. Then we construct an exact axially symmetric solution with separable time and space variables having a strong singularity in the density component beginning from the initial moment of time, whereas other components of solution are initially continuous.

Velocity Distributions in Inelastic Granular Gases with Continuous Size Distributions

We study by numerical simulation the property of velocity distributions of granular gases with a power-law size distribution, driven by uniform heating and boundary heating. It is found that the form of velocity distribution is primarily controlled by the restitution coefficient η and q, the ratio between the average number of heatings and the average number of collisions in the system. Furthermore, we show that uniform and boundary heating can be understood as different limits of q, with q ≫ 1 and q ≤ 1, respectively.

Hydrodynamics of Inelastic Maxwell Models

An overview of recent results pertaining to the hydrodynamic description (both Newtonian and non-Newtonian) of granular gases described by the Boltzmann equation for inelastic Maxwell models is presented. The use of this mathematical model allows us to get exact results for different problems. First, the Navier--Stokes constitutive equations with explicit expressions for the corresponding transport coefficients are derived by applying the Chapman--Enskog method to inelastic gases. Second, the non-Newtonian rheological properties in the uniform shear flow (USF) are obtained in the steady state as well as in the transient unsteady regime. Next, an exact solution for a special class of Couette flows characterized by a uniform heat flux is worked out. This solution shares the same rheological properties as the USF and, additionally, two generalized transport coefficients associated with the heat flux vector can be identified. Finally, the problem of small spatial perturbations of the USF is analyzed with a Chapman--Enskog-like method and generalized (tensorial) transport coefficients are obtained.

Fluctuating hydrodynamics for dilute granular gases

Starting from the kinetic equations for the fluctuations and correlations of a dilute gas of inelastic hard spheres or disks, a Boltzmann-Langevin equation for the one-particle distribution function of the homogeneous cooling state is constructed. This equation is the linear Boltzmann equation with a fluctuating white-noise term. Balance equations for the fluctuating hydrodynamic fields are derived. New fluctuating forces appear as compared with the elastic limit. The particular case of the transverse velocity field is investigated in detail. Its fluctuations can be described by means of a Langevin equation, but exhibiting two main differences with the Landau-Lifshitz theory: the noise is not white, and its second moment is not determined by the shear viscosity. This shows that the fluctuation-dissipation relations for molecular fluids do not straightforwardly carry over to inelastic gases. The theoretical predictions are shown to be in good agreement with molecular-dynamics simulation results.

Formation of density singularities in ideal hydrodynamics of freely cooling inelastic gases: A family of exact solutions

We employ granular hydrodynamics to investigate a paradigmatic problem of clustering of particles in a freely cooling dilute granular gas. We consider large-scale hydrodynamic motions where the viscosity and heat conduction can be neglected, and one arrives at the equations of ideal gas dynamics with an additional term describing bulk energy losses due to inelastic collisions. We employ Lagrangian coordinates and derive a broad family of exact nonstationary analytical solutions that depend only on one spatial coordinate. These solutions exhibit a new type of singularity, where the gas density blows up in a finite time when starting from smooth initial conditions. The density blowups signal formation of close-packed clusters of particles. As the density blow-up time tc is approached, the maximum density exhibits a power law ∼(tc−t)−2. The velocity gradient blows up as ∼−(tc−t)−1 while the velocity itself remains continuous and develops a cusp (rather than a shock discontinuity) at the singularity. The gas temperature vanishes at the singularity, and the singularity follows the isobaric scenario: the gas pressure remains finite and approximately uniform in space and constant in time close to the singularity. An additional exact solution shows that the density blowup, of the same type, may coexist with an “ordinary” shock, at which the hydrodynamic fields are discontinuous but finite. We confirm stability of the exact solutions with respect to small one-dimensional perturbations by solving the ideal hydrodynamic equations numerically. Furthermore, numerical solutions show that the local features of the density blowup hold universally, independently of details of the initial and boundary conditions.

Formation and evolution of density singularities in hydrodynamics of inelastic gases

We use hydrodynamics to investigate clustering in a gas of inelastically colliding spheres. The hydrodynamic equations exhibit a finite-time density blowup, where the gas pressure remains finite. The density blowup signals the formation of close-packed clusters. The blowup dynamics is universal and describable by exact analytic solutions continuable beyond the blowup time. These solutions show that dilute hydrodynamic equations yield a powerful effective description of a granular gas flow with close-packed clusters, described as finite-mass pointlike singularities of the density. This description is similar in spirit to the description of shocks in ordinary ideal gas dynamics.

Capturing blast waves in granular flow

In this paper we continue the analysis of compressible Euler equations for inelastic granular gases described by a granular equation of state due to Goldshtein and Shapiro [Goldshtein A, Shapiro M. Mechanics of collisional motion of granular materials. Part 1: General hydrodynamic equations. J Fluid Mech 1995;282:75–114], and an energy loss term accounting for inelastic collisions. We study the hydrodynamics of blast waves in granular gases by means of a fifth-order accurate scheme that resolves the evolution under different restitution coefficients. We have observed and analyzed the formation of a cluster region near the contact wave using the one-dimensional and two-dimensional versions of our code.

Energy and number of collision fluctuations in inelastic gases

The two-dimensional Inelastic Maxwell Model (IMM) is studied by numerical simulations. It is shown how the inelasticity of collisions together with the fluctuations of the number of collisions undergone by a particle lead to energy fluctuations. These fluctuations are associated to a shrinking of the available phase space. We find the asymptotic scaling of these energy fluctuations and show how they affect the tail of the velocity distribution during long time intervals. We stress that these fluctuations relax like power laws on much slower time scales than the usual exponential relaxations taking place in kinetic theory.

The characteristic function method applied to molecular dynamics of inelastic granular gases

In this work we study the dissipative dynamics of a system of smooth hard spheres with a constant restitution coefficient and small number of particles in the homogeneous cooling regime. We focus on the velocity distribution from simulations in the Molecular Dynamics approach. The main goal of this paper is to present a methodology based on the application of the characteristic function technique and the so-called W function, introduced by Lévy to measure the distance of distributions from the Gaussian. We use this methodology to (i) characterize asymptotic stationary states independently of initial conditions; (ii) study the multiple dependence of these stationary states on the number of particles, density regimes and boundary conditions for a fixed restitution coefficient; (iii) discuss the existence of a limit state in the thermodynamic limit; (iv) address the open problem of the convergence of a Sonine expansion in the highly dissipative regime and (v) measure the overpopulated high energy tails. Moreover, we investigate in what sense the theoretical results related to the Enskog–Boltzmann equation can be reproduced.

On high-energy tails in inelastic gases

We study the formation of high-energy tails in a one-dimensional kinetic model for granular gases, the so-called Inelastic Maxwell Model. We introduce a time-discretized version of the stochastic process, and show that continuous time implies larger fluctuations of the particles energies. This is due to a statistical relation between the number of inelastic collisions undergone by a particle and its average energy. This feature is responsible for the high-energy tails in the model, as shown by computer simulations and by analytical calculations on a linear Lorentz model.

Velocity distributions in dilute granular systems

We investigate the idea that velocity distributions in granular gases are determined mainly by eta, the coefficient of restitution and q, which measures the relative importance of heating (or energy input) to collisions. To this end, we study by numerical simulation the properties of inelastic gases as functions of eta, concentration phi, and particle number N with various heating mechanisms. For a wide range of parameters, we find Gaussian velocity distributions for uniform heating and non-Gaussian velocity distributions for boundary heating. Comparison between these results and velocity distributions obtained by other heating mechanisms and for a simple model of a granular gas without spatial degrees of freedom, shows that uniform and boundary heating can be understood as different limits of q, with q>>1 and q < or approximately 1 respectively. We review the literature for evidence of the role of q in the recent experiments.

Power-law velocity distributions in granular gases

The kinetic theory of granular gases is studied for spatially homogeneous systems. At large velocities, the equation governing the velocity distribution becomes linear, and it admits stationary solutions with a power-law tail, f (v) approximately v(-sigma) . This behavior holds in arbitrary dimension for arbitrary collision rates including both hard spheres and Maxwell molecules. Numerical simulations show that driven steady states with the same power-law tail can be realized by injecting energy into the system at very high energies. In one dimension, we also obtain self-similar time-dependent solutions where the velocities collapse to zero. At small velocities there is a steady state and a power-law tail but at large velocities, the behavior is time dependent with a stretched exponential decay.

Granular matter: A wonderful world of clusters in far-from-equilibrium systems

We review several theoretical and experimental methods of modeling and investigating granular matter far from equilibrium. The theoretical methods include an extension of the classical Boltzmann equation to inelastic gases, scalar internal degrees of freedom, and Hamiltonian-like grain–grain interactions; the experimental technique is concerned with thermal properties of electrically conducting clusters. We discuss the results, focusing on phenomena nonexistent in physics of gases, fluids or solids, e.g. anomalous temperature gradients or electric resistance. One of the models is used to study the interplay between classical and self-organized criticality.

Stationary States and Energy Cascades in Inelastic Gases

We find a general class of nontrivial stationary states in inelastic gases where, due to dissipation, energy is transferred from large velocity scales to small velocity scales. These steady states exist for arbitrary collision rules and arbitrary dimension. Their signature is a stationary velocity distribution f(v) with an algebraic high-energy tail, f(v) approximately v(-sigma). The exponent sigma is obtained analytically and it varies continuously with the spatial dimension, the homogeneity index characterizing the collision rate, and the restitution coefficient. We observe these stationary states in numerical simulations in which energy is injected into the system by infrequently boosting particles to high velocities. We propose that these states may be realized experimentally in driven granular systems.