Kinetic theory for sheared granular flows

Rapid granular flows are far-from-equilibrium-driven dissipative systems where the interaction between the particles dissipates energy, and so a continuous supply of energy is required to agitate the particles and facilitate the rearrangement required for the flow. This is in contrast to flows of molecular fluids, which are usually close to equilibrium, where the molecules are agitated by thermal fluctuations. Sheared granular flows form a class of flows where the energy required for agitating the particles in the flowing state is provided by the mean shear. These flows have been studied using the methods of kinetic theory of gases, where the particles are treated in a manner similar to molecules in a molecular gas, and the interactions between particles are treated as instantaneous energy-dissipating binary collisions. The validity of the assumptions underlying kinetic theory, and their applicability to the idealistic case of dilute sheared granular flows are first discussed. The successes and challenges for applying kinetic theory for realistic dense sheared granular flows are then summarised.     

Free cooling phase-diagram of hard-spheres with short- and long-range interactions

We study the stability, the clustering and the phase-diagram of free cooling granular gases. The systems consist of mono-disperse particles with additional non-contact (long-range) interactions, and are simulated here by the event-driven molecular dynamics algorithm with discrete (short-range shoulders or wells) potentials (in both 2D and 3D). Astonishingly good agreement is found with a mean field theory, where only the energy dissipation term is modified to account for both repulsive or attractive non-contact interactions. Attractive potentials enhance cooling and structure formation (clustering), whereas repulsive potentials reduce it, as intuition suggests. The system evolution is controlled by a single parameter: the non-contact potential strength scaled by the fluctuation kinetic energy (granular temperature). When this is small, as expected, the classical homogeneous cooling state is found. However, if the effective dissipation is strong enough, structure formation proceeds, before (in the repulsive case) non-contact forces get strong enough to undo the clustering (due to the ongoing dissipation of granular temperature). For both repulsive and attractive potentials, in the homogeneous regime, the cooling shows a universal behaviour when the (inverse) control parameter is used as evolution variable instead of time. The transition to a non-homogeneous regime, as predicted by stability analysis, is affected by both dissipation and potential strength. This can be cast into a phase diagram where the system changes with time, which leaves open many challenges for future research.     

Energy fluctuations in vibrated and driven granular gases

We investigate the behavior of energy fluctuations in several models of granular gases maintained in a non-equilibrium steady state. In the case of a gas heated from a boundary, the inhomogeneities of the system play a predominant role. Interpreting the total kinetic energy as a sum of independent but not identically distributed random variables, it is possible to compute the probability density function (pdf) of the total energy. Neglecting correlations and using the analytical expression for the inhomogeneous temperature profile obtained from the granular hydrodynamic equations, we recover results that have previously been observed numerically and that had been attributed to the presence of correlations. In order to separate the effects of spatial inhomogeneities from those ascribable to velocity correlations, we have also considered two models of homogeneously thermostated gases: in this framework it is possible to reveal the presence of non-trivial effects due to velocity correlations between particles. Such correlations stem from the inelasticity of collisions. Moreover, the observation that the pdf of the total energy tends to a Gaussian in the large system limit suggests that they are also due to the finite size of the system.     

Cooling kinetics of a granular gas of viscoelastic particles

The evolution of a granular gas of viscoelastic particles in the homogeneous cooling state is studied. The velocity distribution function of granular particles and the time dependence of the mean kinetic energy of particles (granular temperature) are found. The noticeable deviation of the distribution function from the Maxwell distribution and its non-monotonous evolution are established. The perturbation theory with respect to the small dispersion parameter is elaborated and the analytical expressions for the asymptotic time dependence of the velocity distribution function and the granular gas temperature are derived.     

A Fractal Model for the Effective Thermal Conductivity of Granular Flow with Non—uniform Particles

The equipartition of energy applied in binary mixture of granular flow is extended to granular flow with non-uniform particles.Based on the fractal characteristic of granular flow with non-uniform particles as well as energy equipartition,a fractal velocity distribution function and a fractal model of effective thermal conductivity are derived.Thermal conduction resulted from motions of particles in the granular flow,as well as the effect of fractal dimension on effective thermal conductivity,is discussed.     

Mathematical simulation of the space-time evolution of fission-fragment plasma tracks

A theoretical investigation of the space-time evolution of tracks created by heavy charged particles in noble gases is given, taking into account electron and ion diffusion, electron energy relaxation due to collisions, molecular ion formation and electron-ion recombination due to ternary collisions. Results of a mathematical simulation in helium are presented.     

Introduction to the focus issue on granular materials

In a review paper [H. M. Jaeger, S. R. Nagel, and R. P. Behringer, "Granular solids, liquids and gases," Rev. Mod. Phys. 68, 1259-1273 (1996)] a few years ago, we wrote about granular material as a distinctive form of matter that exhibits behavior rather different from that of ordinary solids, liquids, or gases. We traced this distinction to three characteristic properties. First, the individual particles making up a granular material are typically large so that thermal energy is irrelevant compared to gravitational energy. Consequently, concepts from equilibrium statistical mechanics are often not applicable. Second, the interactions between particles are frictional and can be mobilized to different degrees depending on the preparation history, giving rise to memory effects, i.e., a static pile will remember how it was formed. Third, when particles collide they do so inelastically so that a "gas" of particles will slow down and come to rest in clumps. In the intervening years, the research on granular matter has progressed rapidly and this may be a good time to ask what we have learned since that article was written. In this spirit, the present special issue of the journal Chaos assembles a spectrum of papers discussing recent developments in the field. (c) 1999 American Institute of Physics.     

准静态颗粒介质的弹性势能弛豫分析

颗粒体系是典型的多体相互作用体系, 具有多重的能量亚稳态. 对于准静态颗粒体系, 引入构型颗粒温度T_c描述弹性势能涨落. 本文认为平衡的体系具有一定的构型颗粒温度T_a, 其量值反映了其结构特征. 当外界扰动激发的构型颗粒温度超出T_a时, 产生不可逆过程. 通过对应力松弛过程的分析, 发现(T_c-T_a)激发了弹性弛豫, 且(T_c-T_a)越大则松弛过程中应力变化越大, 最终构型颗粒温度T_c→T_a时,宏观应力松弛结束,体系达到新的能量亚稳态.     

Granular Leidenfrost effect in vibrated beds with bumpy surfaces

The effects of subjecting a bed of granular materials to horizontal vibrations by a bumpy oscillating surface have been investigated computationally in this study. The behaviour of the granular bed is determined by the vibration conditions applied which include the vibrating frequency and amplitude as well as the bumpiness of the oscillating surface. Under sufficiently vigorous vibration conditions, the granular Leidenfrost effect whereby the entire granular bed is levitated above the vibrating base by a layer of highly energetic particles may be observed. Granular temperature profiles of systems that exhibit the granular Leidenfrost effect indicate an unequal distribution of energy between particles near the vibrating base and those in the bulk. A bumpy oscillating surface was also observed to be more effective at introducing perturbations and transferring energy into a granular bed. The granular Leidenfrost effect can be induced by the application of larger grain sizes of particles constituting the bumpy vibrating base under vibration conditions that are normally insufficient for the onset of the effect. Lastly, a phase diagram which can be utilized for predicting the behaviours of granular beds that are subjected to oscillations by various types of bumpy surfaces has been constructed based on the simulation results obtained.     

Evolution of Energy in Submerged Granular Column Collapse

The evolution of energy in subaerial and subaqueous granular column collapses is studied.Employing the refractive index matching method and planar laser-induced fluorescence technique,we obtain granular and liquid images simultaneously in a single experiment of subaqueous flow.Particle image velocimetry and particle tracking velocimetry are used to process the data for the fluid and granular phase.We find stepwise decreases in the total kinetic energy of the granular material.The stage of rapidly falling energy corresponds to large transverse changes in the direction of the massive granular particles.Moreover,in this stage,a major fraction of the granular kinetic energy transferred from the granular potential energy is lost or transferred.Interestingly,compared with dry granular flow,the existence of an ambient liquid seems to reduce the total dissipated energy,which may be the reason why previous studies observed similar granular runout distances in subaqueous and dry granular collapses.     

Brownian motion in granular gases of viscoelastic particles

A theory is developed of Brownian motion in granular gases (systems of many macroscopic particles undergoing inelastic collisions), where the energy loss in inelastic collisions is determined by a restitution coefficient ɛ. Whereas previous studies used a simplified model with ɛ = const, the present analysis takes into account the dependence of the restitution coefficient on relative impact velocity. The granular temperature and the Brownian diffusion coefficient are calculated for a granular gas in the homogeneous cooling state and a gas driven by a thermostat force, and their variation with grain mass and size and the restitution coefficient is analyzed. Both equipartition principle and fluctuation-dissipation relations are found to break down. One manifestation of this behavior is a new phenomenon of “relative heating” of Brownian particles at the expense of cooling of the ambient granular gas.     

Spontaneous ratchet effect in a granular gas

The spontaneous clustering of a vibrofluidized granular gas is employed to generate directed transport in two different compartmentalized systems: a granular fountain in which the transport takes the form of convection rolls, and a granular ratchet with a spontaneous particle current perpendicular to the direction of energy input. In both instances, transport is not due to any system-intrinsic anisotropy, but arises as a spontaneous collective symmetry breaking effect of many interacting granular particles. The experimental and numerical results are quantitatively accounted for within a flux model.     

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.     

Cellular model for the compaction of a vertically tapped granular column

A cellular model for the compaction of granular material is described. It takes into account horizontal redistribution as well as vertical transfer of particles. Parameters are the width of the horizontal redistribution and the settling probability. Numerical simulations of the behaviour of a granular column in a container are shown as an example, and the evolution of some characteristic features over time has been followed for some typical configurations. Experimental results for the time evolution of the density can be reproduced for a settling probability proportional to the unoccupied spaces for particles in the lower cells. Received 3 March 2000 and Received in final form 21 July 2000     

三维颗粒气体相分离现象

颗粒体系是一类复杂的耗散体系.在颗粒气体中,耗散性质会使其内部形成局部的凝聚,类似于真实气体中亚稳分解形成的液滴,因此被认为是颗粒气液两相分离的过程. 零重力环境下二维颗粒气体相分离现象已有成熟的流体静力学理论解释,将该理论模型推广到三维情形,发现相分离现象依然存在且具有同样的不稳定性根源,通过理论计算给出了三维相分离发生的具体条件. 同时,用分子动力学方法模拟检验了理论结果,并给出了三维颗粒气体相分离的新形貌. 关键词： 颗粒气体 耗散 相分离 分子动力学模拟     

Steady State Properties of One-Dimensional Non-uniform Granular Gases Subjected to Gaussian White Noise Driving

We present a model of non-uniform granular gases in one-dimensional case, whose granularity distribution has the fractal characteristic. We have studied the nonequilibrium properties of the system by means of Monte Carlo method. When the typical relaxation time T of the Brownian process is greater than the mean collision time To, the energy evolution of the system exponentially decays, with a tendency to achieve a stable asymptotic value, and the system finally reaches a nonequilibrium steady state in which the velocity distribution strongly deviates from the Gaussian one. Three other aspects have also been studied for the steady state： the visualized change of the particle density, the entropy of the system and the correlations in the velocity of particles. And the results of simulations indicate that the system has strong spatial clustering; Furthermore, the influence of the inelasticity and inhomogeneity on dynamic behaviors have also been extensively investigated, especially the dependence of the entropy and the correlations in the velocity of particles on the restitute coefficient e and the fractal dimension D.     

Particle dynamics simulations of triboelectric charging in granular insulator systems

Particle dynamics simulations are carried out to study triboelectric charging in granular systems composed of a single insulating material. The simulations implement a model in which electrons trapped in localized high energy states can be transferred during collisions to low energy states in the other particle. It is shown that this effect alone can generate electrostatic charging in the system, and cause net electron transfer from larger particles to smaller particles. The magnitude of charging is small for systems of a single particle size but becomes much greater for a system with polydispersal particle sizes, due to the net electron transfer from larger to smaller particles. The negative charge of smaller particles, and positive charge of larger particles has been observed in field studies and laboratory experiments of granular systems.     

垂直振动激发下颗粒物质的能量耗散

本文对垂直振动激发下的铜和玻璃珠两种密集颗粒样品,通过实时测量样品盒的加速度和振台对它的作用力,研究了样品的平均耗散功率.实验发现该耗散功率在给定振动强度下随振动频率的变化曲线是一个峰值在几十赫兹的宽峰结构,在给定频率下随强度的变化具有幂率规律,其幂值小于简谐受迫振动模型的幂值2.这些结果将有助于研究测定颗粒流体和气体的迁移系数,以及它们在不同运动模式下的能耗差异.     

颗粒介质弹性的弛豫

颗粒介质是复杂的多体相互作用体系, 其弹性源自内部的力链结构, 弹性能量处在亚稳态, 具有复杂的弛豫行为. 在常规作用下, 颗粒介质往往呈现明显的弹性弛豫. 应力松弛是应变恒定时应力的衰减现象, 弹性弛豫是应力松弛的主要原因. 在前期工作基础上, 从弹性势能面和双颗粒温度热力学角度分析了弹性弛豫的机理, 量化了弹性应力演化不可逆过程; 基于双颗粒温度热力学计算得到了弹性能、颗粒温度和应力的演化, 其中应力松弛的计算结果与实验结果基本一致, 讨论了颗粒温度初值和输运系数的影响. 指出, 开展力链结构及其动力学研究是揭示宏观弹性弛豫机理的关键.     

Entropy, entropy flux and entropy rate of granular materials

The aim of this work is to analyze the entropy, entropy flux and entropy rate of granular materials within the frameworks of the Boltzmann equation and continuum thermodynamics. It is shown that the entropy inequality for a granular gas that follows from the Boltzmann equation differs from the one of a simple fluid due to the presence of a term which can be identified as the entropy density rate. From the knowledge of a non-equilibrium distribution function-valid for processes closed to equilibrium-it is obtained that the entropy density rate is proportional to the internal energy density rate divided by the temperature, while the entropy flux is equal to the heat flux vector divided by the temperature. A thermodynamic theory of a granular material is also developed whose objective is the determination of the basic fields of mass density, momentum density and internal energy density. The constitutive laws are restricted by the principle of material frame indifference and by the entropy principle. Through the exploitation of the entropy principle with Lagrange multipliers, it is shown that the results obtained from the kinetic theory for granular gases concerning the entropy density rate and entropy flux are valid in general for processes close to equilibrium of granular materials, where linearized constitutive equations hold.