Transport properties of dense dissipative hard-sphere fluids for arbitrary energy loss models

Kinetic Theory of Polydisperse Granular Mixtures: Influence of the Partial Temperatures on Transport Properties-A Review

It is well-recognized that granular media under rapid flow conditions can be modeled as a gas of hard spheres with inelastic collisions. At moderate densities, a fundamental basis for the determination of the granular hydrodynamics is provided by the Enskog kinetic equation conveniently adapted to account for inelastic collisions. A surprising result (compared to its molecular gas counterpart) for granular mixtures is the failure of the energy equipartition, even in homogeneous states. This means that the partial temperatures Ti (measuring the mean kinetic energy of each species) are different to the (total) granular temperature T. The goal of this paper is to provide an overview on the effect of different partial temperatures on the transport properties of the mixture. Our analysis addresses first the impact of energy nonequipartition on transport which is only due to the inelastic character of collisions. This effect (which is absent for elastic collisions) is shown to be significant in important problems in granular mixtures such as thermal diffusion segregation. Then, an independent source of energy nonequipartition due to the existence of a divergence of the flow velocity is studied. This effect (which was already analyzed in several pioneering works on dense hard-sphere molecular mixtures) affects to the bulk viscosity coefficient. Analytical (approximate) results are compared against Monte Carlo and molecular dynamics simulations, showing the reliability of kinetic theory for describing granular flows.

Enskog kinetic theory of rheology for a moderately dense inertial suspension

The Enskog kinetic theory for moderately dense inertial suspensions under simple shear flow is considered as a model to analyze the rheological properties of the system. The influence of the background fluid on suspended particles is modeled via a viscous drag force plus a Langevin-like term defined in terms of the background temperature. In a previous paper [Hayakawa et al., Phys. Rev. E 96, 042903 (2017)10.1103/PhysRevE.96.042903], Grad's moment method with the aid of a linear shear-rate expansion was employed to obtain a theory which gave good agreement with the results of event-driven Langevin simulations of hard spheres for low densities and/or small shear rates. Nevertheless, the previous approach had a limitation of not being applicable to the high-shear-rate and high-density regime. Thus, in the present paper, we extend the previous work and develop Grad's theory including higher-order terms in the shear rate. This improves significantly the theoretical predictions, a quantitative agreement between theory and simulation being found in the high-density region (volume fractions smaller than or equal to 0.4).

Enskog kinetic theory for multicomponent granular suspensions

The Navier-Stokes transport coefficients of multicomponent granular suspensions at moderate densities are obtained in the context of the (inelastic) Enskog kinetic theory. The suspension is modeled as an ensemble of solid particles where the influence of the interstitial gas on grains is via a viscous drag force plus a stochastic Langevin-like term defined in terms of a background temperature. In the absence of spatial gradients, it is shown first that the system reaches a homogeneous steady state where the energy lost by inelastic collisions and viscous friction is compensated for by the energy injected by the stochastic force. Once the homogeneous steady state is characterized, a normal solution to the set of Enskog equations is obtained by means of the Chapman-Enskog expansion around the local version of the homogeneous state. To first order in spatial gradients, the Chapman-Enskog solution allows us to identify the Navier-Stokes transport coefficients associated with the mass, momentum, and heat fluxes. In addition, the first-order contributions to the partial temperatures and the cooling rate are also calculated. Explicit forms for the diffusion coefficients, the shear and bulk viscosities, and the first-order contributions to the partial temperatures and the cooling rate are obtained in steady-state conditions by retaining the leading terms in a Sonine polynomial expansion. The results show that the dependence of the transport coefficients on inelasticity is clearly different from that found in its granular counterpart (no gas phase). The present work extends previous theoretical results for dilute multicomponent granular suspensions [Khalil and Garzó, Phys. Rev. E 88, 052201 (2013)10.1103/PhysRevE.88.052201] to higher densities.

Shear-banding instability in arbitrarily inelastic granular shear flows

One prototypical instability in granular flows is the shear-banding instability, in which a uniform granular shear flow breaks into alternating bands of dense and dilute clusters of particles having low and high shear (shear stress or shear rate), respectively. In this work, the shear-banding instability in an arbitrarily inelastic granular shear flow is analyzed through the linear stability analysis of granular hydrodynamic equations closed with Navier-Stokes-level constitutive relations. It is shown that the choice of appropriate constitutive relations plays an important role in predicting the shear-banding instability. A parametric study is carried out to study the effect of the restitution coefficient, channel width, and mean density. Two global criteria relating the control parameters are found for the onset of the shear-banding instability.

Thermal conductivity at the high-density limit and the levitating granular cluster

The granular Leidenfrost state consists of a dense granular cluster levitating above a hot granular gas. The density of particles inside the cluster can be very high and even close to the density of crystalline packing. To describe this state theoretically, one needs to know the density dependence of constitutive relations (pressure, heat losses, thermal conductivity) at these very high densities. However, the accurate expression for the coefficient of thermal conductivity is lacking. In this work, the constitutive relations were measured at high densities in molecular dynamics simulations in three different settings: a uniform freely cooling dense granulate (to measure heat losses), a uniform ensemble of elastically colliding particles (to measure pressure), and a dense granular medium between two thermal walls under gravity (to measure thermal conductivity). Next, the hydrodynamic equations with the resulting expressions were solved to describe the levitating cluster state in various parameter regimes. Separate molecular dynamics simulations were performed to test the theoretical predictions and measure the density and temperature profiles of the granular Leidenfrost state, and a good agreement with theoretical results was observed.

Impact of roughness on the instability of a free-cooling granular gas

A linear stability analysis of the hydrodynamic equations with respect to the homogeneous cooling state is carried out to identify the conditions for stability of a granular gas of rough hard spheres. The description is based on the results for the transport coefficients derived from the Boltzmann equation for inelastic rough hard spheres [Phys. Rev. E 90, 022205 (2014)PLEEE81539-375510.1103/PhysRevE.90.022205], which take into account the complete nonlinear dependence of the transport coefficients and the cooling rate on the coefficients of normal and tangential restitution. As expected, linear stability analysis shows that a doubly degenerate transversal (shear) mode and a longitudinal ("heat") mode are unstable with respect to long enough wavelength excitations. The instability is driven by the shear mode above a certain inelasticity threshold; at larger inelasticity, however, the instability is driven by the heat mode for an inelasticity-dependent range of medium roughness. Comparison with the case of a granular gas of inelastic smooth spheres confirms previous simulation results about the dual role played by surface friction: while small and large levels of roughness make the system less unstable than the frictionless system, the opposite happens at medium roughness. On the other hand, such an intermediate window of roughness values shrinks as inelasticity increases and eventually disappears at a certain value, beyond which the rough-sphere gas is always less unstable than the smooth-sphere gas. A comparison with some preliminary simulation results shows a very good agreement for conditions of practical interest.

Rheology of dilute cohesive granular gases

Rheology of a dilute cohesive granular gas is theoretically and numerically studied. The flow curve between the shear viscosity and the shear rate is derived from the inelastic Boltzmann equation for particles having square-well potentials in a simple shear flow. It is found that (i) the stable uniformly sheared state only exists above a critical shear rate and (ii) the viscosity in the uniformly sheared flow is almost identical to that for uniformly sheared flow of hard core granular particles. Below the critical shear rate, clusters grow with time, in which the viscosity can be approximated by that for the hard-core fluids if we replace the diameter of the particle by the mean diameter of clusters.

Kinetic theory of shear thickening for a moderately dense gas-solid suspension: From discontinuous thickening to continuous thickening

The Enskog kinetic theory for moderately dense gas-solid suspensions under simple shear flow is considered as a model to analyze the rheological properties of the system. The influence of the environmental fluid on solid particles is modeled via a viscous drag force plus a stochastic Langevin-like term. The Enskog equation is solved by means of two independent but complementary routes: (i) Grad's moment method and (ii) event-driven Langevin simulation of hard spheres. Both approaches clearly show that the flow curve (stress-strain rate relation) depends significantly on the volume fraction of the solid particles. In particular, as the density increases, there is a transition from the discontinuous shear thickening (observed in dilute gases) to the continuous shear thickening for denser systems. The comparison between theory and simulations indicates that while the theoretical predictions for the kinetic temperature agree well with simulations for densities φ≲0.5, the agreement for the other rheological quantities (the viscosity, the stress ratio, and the normal stress differences) is limited to more moderate densities (φ≲0.3) if the inelasticity during collisions between particles is not large.

Phase separation in driven granular gases: exploring the elusive character of nonequilibrium steady states

The emergence of patterns and phase separation in many-body systems far from thermal equilibrium is discussed using the example of driven granular gases. It is shown that phase separation follows a similar mechanism as in the systems of active Brownian particles. Depending on the quantities chosen for observation, it may or may not be easy to find functionals analogous to the free energy in equilibrium statistical physics. We argue that although such functionals can always be derived from the dynamics, it is of only limited value for predicting relevant aspects of the nonequilibrium steady state of the system. Consequently, although there is indeed a 'principle' governing the selection of collective nonequilibrium steady states (and the corresponding large deviation functional can be identified), it is not generally useful for predicting the behaviour of the system.

Granular rotor as a probe for a nonequilibrium bath

This study numerically and analytically investigates the dynamics of a rotor under viscous or dry friction as a nonequilibrium probe of a granular gas. In order to demonstrate the role of the rotor as a probe for a nonequilibrium bath, the molecular dynamics (MD) simulation of the rotor is performed under viscous or dry friction surrounded by a steady granular gas under gravity. A one-to-one map between the velocity distribution function (VDF) of the granular gas and the angular distribution function for the rotor is theoretically derived. The MD simulation demonstrates that the one-to-one map accurately infers the local VDF of the granular gas from the angular VDF of the rotor, and vice versa.

Enstrophy cascades in two-dimensional dense granular flows

Employing two-dimensional molecular dynamics simulations of dense granular materials under simple shear deformations, we investigate vortex structures of particle rearrangements. Introducing vorticity fields as a measure of local spinning motions of the particles, we observe their heterogeneous distributions, where statistics of vorticity fields exhibit the highly non-Gaussian behavior and typical domain sizes of vorticity fields significantly increase if the system is yielding under quasistatic deformations. In such dense granular flows, a power-law decay of vorticity spectra can be observed at mesoscopic scale, implying anomalous local structures of kinetic energy dissipation. We explain the power-law decay, or enstrophy cascades in dense granular materials, by a dimensional analysis, where the dependence of vorticity spectra not only on the wave number, but also on the shear rate, is well explained. From our dimensional analyses, the scaling of granular temperature and rotational kinetic energy is also predicted.

Transport coefficients of solid particles immersed in a viscous gas

Transport properties of a suspension of solid particles in a viscous gas are studied. The dissipation in such systems arises from two sources: inelasticity in particle collisions and viscous dissipation due to the effect of the gas phase on the particles. Here we consider a simplified case in which the mean relative velocity between the gas and solid phases is taken to be zero, such that "thermal drag" is the only remaining gas-solid interaction. Unlike the previous, more general, treatment of the drag force [Garzó et al., J. Fluid Mech. 712, 129 (2012)]JFLSA70022-112010.1017/jfm.2012.404, here we take into account contributions to the (scaled) transport coefficients η^{*} (shear viscosity), κ^{*} (thermal conductivity), and μ^{*} (Dufour-like coefficient) coming from the temperature dependence of the (dimensionless) friction coefficient γ^{*} characterizing the amplitude of the drag force. At moderate densities, the thermal drag model (which is based on the Enskog kinetic equation) is solved by means of the Chapman-Enskog method and the Navier-Stokes transport coefficients are determined in terms of the coefficient of restitution, the solid volume fraction, and the friction coefficient. The results indicate that the effect of the gas phase on η^{*} and μ^{*} is non-negligible (especially in the case of relatively dilute systems) while the form of κ^{*} is the same as the one obtained in the dry granular limit. Finally, as an application of these results, a linear stability analysis of the hydrodynamic equations is carried out to analyze the conditions for stability of the homogeneous cooling state. A comparison with direct numerical simulations shows a good agreement for conditions of practical interest.

Anomalous energy cascades in dense granular materials yielding under simple shear deformations

By using molecular dynamics (MD) simulations of dense granular particles in two dimensions, we study turbulent-like structures of their non-affine velocities under simple shear deformations. We find that the spectrum of non-affine velocities, introduced as an analog of the energy spectrum for turbulent flows, exhibits the power-law decay if the system is yielding in a quasi-static regime, where large-scale collective motions and inelastic interactions of granular particles are crucial for the anomalous cascade of kinetic energy. Based on hydrodynamic equations of dense granular materials, which include both kinetic and contact contributions in constitutive relations, we derive a theoretical expression for the spectrum, where a good agreement between the result of MD simulations and theoretical prediction is established over a wide range of length scales.

Divergence of Viscosity in Jammed Granular Materials: A Theoretical Approach

A theory for jammed granular materials is developed with the aid of a nonequilibrium steady-state distribution function. The approximate nonequilibrium steady-state distribution function is explicitly given in the weak dissipation regime by means of the relaxation time. The theory quantitatively agrees with the results of the molecular dynamics simulation on the critical behavior of the viscosity below the jamming point without introducing any fitting parameter.

Hydrodynamic instabilities in shear flows of dry cohesive granular particles

We extend the dynamic van der Waals model introduced by A. Onuki [Phys. Rev. Lett., 2005, 94, 054501] to the description of cohesive granular flows under a plane shear to study their hydrodynamic instabilities. By numerically solving the dynamic van der Waals model, we observed various heterogeneous structures of density fields in steady states, where the viscous heating is balanced with the energy dissipation caused by inelastic collisions. Based on the linear stability analysis, we found that the spatial structures are determined by the mean volume fraction, the applied shear rate, and the inelasticity, where the instability is triggered if the system is thermodynamically unstable, i.e. the pressure, p, and the volume fraction, ϕ, satisfy ∂p/∂ϕ < 0.

Power-law decay of the velocity autocorrelation function of a granular fluid in the homogeneous cooling state

The hydrodynamic part of the velocity autocorrelation function of a granular fluid in the homogeneous cooling state has been calculated by using mode-coupling theory for a finite system with periodic boundary conditions. The existence of the shearing instability, leading to a divergent behavior of the velocity flow fluctuations, is taken into account. A time region in which the velocity autocorrelation function exhibits a power-law decay, when time is measured by the number of collisions per particle, has been been identified. Also the explicit form of the exponential asymptotic long time decay has been obtained. The theoretical prediction for the power-law decay is compared with molecular dynamics simulation results, and a good agreement is found, after taking into account finite size corrections. The effects of approaching the shearing instability are also explored.

Shear viscosity of a model for confined granular media

The shear viscosity in the dilute regime of a model for confined granular matter is studied by simulations and kinetic theory. The model consists on projecting into two dimensions the motion of vibrofluidized granular matter in shallow boxes by modifying the collision rule: besides the restitution coefficient that accounts for the energy dissipation, there is a separation velocity that is added in each collision in the normal direction. The two mechanisms balance on average, producing stationary homogeneous states. Molecular dynamics simulations show that in the steady state the distribution function departs from a Maxwellian, with cumulants that remain small in the whole range of inelasticities. The shear viscosity normalized with stationary temperature presents a clear dependence with the inelasticity, taking smaller values compared to the elastic case. A Boltzmann-like equation is built and analyzed using linear response theory. It is found that the predictions show an excellent agreement with the simulations when the correct stationary distribution is used but a Maxwellian approximation fails in predicting the inelasticity dependence of the viscosity. These results confirm that transport coefficients depend strongly on the mechanisms that drive them to stationary states.

Homogeneous steady state of a confined granular gas

The nonequilibrium statistical mechanics and kinetic theory for a model of a confined quasi-two-dimensional gas of inelastic hard spheres is presented. The dynamics of the particles includes an effective mechanism to transfer the energy injected in the vertical direction to the horizontal degrees of freedom. The Enskog approximation is formulated and used as the basis to investigate the temperature and the distribution function of the steady state eventually reached by the system. An exact scaling of the distribution function of the system having implications on the form of its moments is pointed out. The theoretical predictions are compared with numerical results obtained by a particle simulation method, and a good agreement is found.

Linear hydrodynamics for driven granular gases

We study the dynamics of a granular gas heated by a stochastic thermostat. From a Boltzmann description, we derive the hydrodynamic equations for small perturbations around the stationary state that is reached in the long time limit. Transport coefficients are identified as Green-Kubo formulas obtaining explicit expressions as a function of the inelasticity and the spatial dimension.

Simulation of granular jets: is granular flow really a perfect fluid?

We perform three-dimensional simulations of the impact of a granular jet for both frictional and frictionless grains. Small shear stress observed in the experiment [X. Cheng et al., Phys. Rev. Lett. 99, 188001 (2007)] is reproduced through our simulation. However, the fluid state after the impact is far from a perfect fluid, and thus the similarity between granular jets and quark gluon plasma is superficial because the observed viscosity is finite and its value is consistent with the prediction of the kinetic theory.

Transport coefficients of driven granular fluids at moderate volume fraction

In a recent publication [Phys. Rev. E 83, 011301 (2011)], Vollmayr-Lee et al. have determined by computer simulations the thermal diffusivity and the longitudinal viscosity coefficients of a driven granular fluid of hard spheres at intermediate volume fractions. Although they compare their simulation results with the predictions of kinetic theory, they use the dilute expressions for the driven system and the modified Sonine approximations for the undriven system. The goal here is to carry out this comparison by proposing a modified Sonine approximation to the Enskog equation for driven granular fluids that leads to a better quantitative agreement.

Class of dilute granular Couette flows with uniform heat flux

In a recent paper [F. Vega Reyes et al., Phys. Rev. Lett. 104, 028001 (2010)] we presented a preliminary description of a special class of steady Couette flows in dilute granular gases. In all flows of this class the viscous heating is exactly balanced by inelastic cooling. This yields a uniform heat flux and a linear relationship between the local temperature and flow velocity. The class (referred to as the LTu class) includes the Fourier flow of ordinary gases and the simple shear flow of granular gases as special cases. In the present paper we provide further support for this class of Couette flows by following four different routes, two of them being theoretical (Grad's moment method of the Boltzmann equation and exact solution of a kinetic model) and the other two being computational (molecular dynamics and Monte Carlo simulations of the Boltzmann equation). Comparison between theory and simulations shows a very good agreement for the non-Newtonian rheological properties, even for quite strong inelasticity, and a good agreement for the heat flux coefficients in the case of Grad's method, the agreement being only qualitative in the case of the kinetic model.

Continuum simulations of shocks and patterns in vertically oscillated granular layers

We study interactions between shocks and standing-wave patterns in vertically oscillated layers of granular media using three-dimensional, time-dependent numerical solutions of continuum equations to Navier-Stokes order. We simulate a layer of grains atop a plate that oscillates sinusoidally in the direction of gravity. Standing waves form stripe patterns when the accelerational amplitude of the plate's oscillation exceeds a critical value. Shocks also form with each collision between the layer and the plate; we show that pressure gradients formed by these shocks cause the flow to reverse direction within the layer. This reversal leads to an oscillatory state of the pattern that is subharmonic with respect to the plate's oscillation. Finally, we study the relationship between shocks and patterns in layers oscillated at various frequencies and show that the pattern wavelength increases monotonically as the shock strength increases.

Segregation by thermal diffusion in moderately dense granular mixtures

A theory based on a solution of the inelastic Enskog equation that goes beyond the weak dissipation limit is used to determine the thermal diffusion factor of a binary granular mixture under gravity. The Enskog equation that aims to describe moderate densities neglects velocity correlations but retains spatial correlations arising from volume exclusion effects. As expected, the thermal diffusion factor provides a segregation criterion that shows the transition between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE) by varying the parameters of the system (masses, sizes, composition, density and coefficients of restitution). The form of the phase diagrams for the BNE/RBNE transition is illustrated in detail in the tracer limit case, showing that the phase diagrams depend sensitively on the value of gravity relative to the thermal gradient. Two specific situations are considered: i) absence of gravity, and ii) homogeneous temperature. In the latter case, after some approximations, our results are consistent with previous theoretical results derived from the Enskog equation. Our results also indicate that the influence of dissipation on thermal diffusion is more important in the absence of gravity than in the opposite limit. The present analysis, which is based on a preliminary short report of the author (Phys. Rev. E 78, 020301(R) (2008)), extends previous theoretical results derived in the dilute limit case.

Mass transport of impurities in a moderately dense granular gas

Transport coefficients associated with the mass flux of impurities immersed in a moderately dense granular gas of hard disks or spheres described by the inelastic Enskog equation are obtained by means of the Chapman-Enskog expansion. The transport coefficients are determined as the solutions of a set of coupled linear integral equations recently derived for polydisperse granular mixtures [Garzó, Phys. Rev. E 76, 031304 (2007)]. With the objective of obtaining theoretical expressions for the transport coefficients that are sufficiently accurate for highly inelastic collisions, we solve the above integral equations by using the second Sonine approximation. As a complementary route, we numerically solve by means of the direct simulation Monte Carlo method (DSMC) the inelastic Enskog equation to get the kinetic diffusion coefficient D0 for two and three dimensions. We have observed in all our simulations that the disagreement, for arbitrarily large inelasticity, in the values of both solutions (DSMC and second Sonine approximation) is less than 4%. Moreover, we show that the second Sonine approximation to D0 yields a dramatic improvement (up to 50%) over the first Sonine approximation for impurity particles lighter than the surrounding gas and in the range of large inelasticity. The results reported in this paper are of direct application in important problems in granular flows, such as segregation driven by gravity and a thermal gradient. We analyze here the segregation criteria that result from our theoretical expressions of the transport coefficients.

Spatial correlations in sheared isothermal liquids: from elastic particles to granular particles

Spatial correlations in sheared isothermal liquids for both elastic and granular cases are theoretically investigated. Using the generalized fluctuating hydrodynamics, correlation functions for both the microscopic scale and the macroscopic scale are obtained. We find the existence of long-range correlations obeying power laws. The validity of our theoretical predictions has been verified from molecular-dynamics simulation.

Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics

We study the nonequilibrium statistical mechanics of a system of freely moving particles, in which binary encounters lead either to an elastic collision or to the disappearance of the pair. Such a system of ballistic annihilation therefore constantly loses particles. The dynamics of perturbations around the free decay regime is investigated using the spectral properties of the linearized Boltzmann operator, which characterize linear excitations on all time scales. The linearized Boltzmann equation is solved in the hydrodynamic limit by a projection technique, which yields the evolution equations for the relevant coarse-grained fields and expressions for the transport coefficients. We finally present the results of molecular dynamics simulations that validate the theoretical predictions.

Transport coefficients for the hard-sphere granular fluid

In the preceding paper, linear response methods have been applied to obtain formally exact expressions for the parameters of Navier-Stokes order hydrodynamics. The analysis there is general, applying to both normal and granular fluids with a wide range of collision rules. Those results are specialized here to the case of smooth, inelastic, hard spheres with constant coefficient of normal restitution, for further elaboration. Explicit expressions for the cooling rate, pressure, and transport coefficients are given and compared with the corresponding expressions for a system of elastic hard spheres. The scope of the results for further analytical explorations and possible numerical evaluation is discussed.

Long-time tails in freely cooling granular gases

The long-time behavior of the current autocorrelation functions for the velocity, the shear stress, and the heat flux is investigated in freely cooling granular gases. It is found that the correlation functions for the velocity and the shear stress have the long-time tails obeying tau(-d/2), while the correlation function for heat flux decays as tau(-(d+2)/2) exp(-zeta*tau) with the dimensionless cooling rate zeta*, the spatial dimension d, and the scaled time tau in terms of the collision frequency. The result of our numerical simulation of the freely cooling granular gases is consistent with the theoretical prediction.

Enskog theory for polydisperse granular mixtures. II. Sonine polynomial approximation

The linear integral equations defining the Navier-Stokes (NS) transport coefficients for polydisperse granular mixtures of smooth inelastic hard disks or spheres are solved by using the leading terms in a Sonine polynomial expansion. Explicit expressions for all the NS transport coefficients are given in terms of the sizes, masses, compositions, density, and restitution coefficients. In addition, the cooling rate is also evaluated to first order in the gradients. The results hold for arbitrary degree of inelasticity and are not limited to specific values of the parameters of the mixture. Finally, a detailed comparison between the derivation of the current theory and previous theories for mixtures is made, with attention paid to the implication of the various treatments employed to date.

Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport

A hydrodynamic description for an s -component mixture of inelastic, smooth hard disks (two dimensions) or spheres (three dimensions) is derived based on the revised Enskog theory for the single-particle velocity distribution functions. In this first part of the two-part series, the macroscopic balance equations for mass, momentum, and energy are derived. Constitutive equations are calculated from exact expressions for the fluxes by a Chapman-Enskog expansion carried out to first order in spatial gradients, thereby resulting in a Navier-Stokes order theory. Within this context of small gradients, the theory is applicable to a wide range of restitution coefficients and densities. The resulting integral-differential equations for the zeroth- and first-order approximations of the distribution functions are given in exact form. An approximate solution to these equations is required for practical purposes in order to cast the constitutive quantities as algebraic functions of the macroscopic variables; this task is described in the companion paper.

Spatial force correlations in granular shear flow. II. Theoretical implications

Numerical simulations are used to test the kinetic theory constitutive relations of inertial granular shear flow. These predictions are shown to be accurate in the dilute regime, where only binary collisions are relevant, but underestimate the measured value in the dense regime, where force networks of size xi are present. The discrepancy in the dense regime is due to non-collisional forces that we measure directly in our simulations and arise from elastic deformations of the force networks. We model the non-collisional stress by summing over all paths that elastic waves travel through force networks. This results in an analytical theory that successfully predicts the stress tensor over the entire inertial regime without any adjustable parameters.

Hydrodynamics of fluid-solid coexistence in dense shear granular flow

We consider dense rapid shear flow of inelastically colliding hard disks. Navier-Stokes granular hydrodynamics is applied accounting for the recent finding that shear viscosity diverges at a lower density than the rest of the 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 enough, the uniform sheared flow gives way to a two-phase flow, where a dense "solidlike" striped cluster is surrounded by two fluid layers. The results of the analysis are verified in event-driven MD simulations, and a good agreement is observed.

Rheology of a granular gas under a plane shear

The rheology of a two-dimensional granular gas under a plane shear is investigated. From the comparison among the discrete element method, the simulation of a set of hydrodynamic equation, and the analytic solution of the steady hydrodynamic equations, it is confirmed that the fluid equations derived from the kinetic theory give us accurate results even in relatively high density cases.

Phase separation of a driven granular gas in annular geometry

This work investigates phase separation of a monodisperse gas of inelastically colliding hard disks confined in a two-dimensional annulus, the inner circle of which represents a "thermal wall." When described by granular hydrodynamic equations, the basic steady state of this system is an azimuthally symmetric state of increased particle density at the exterior circle of the annulus. When the inelastic energy loss is sufficiently large, hydrodynamics predicts spontaneous symmetry breaking of the annular state, analogous to the van der Waals-like phase separation phenomenon previously found in a driven granular gas in rectangular geometry. At a fixed aspect ratio of the annulus, the phase separation involves a "spinodal interval" of particle area fractions, where the gas has negative compressibility in the azimuthal direction. The heat conduction in the azimuthal direction tends to suppress the instability, as corroborated by a marginal stability analysis of the basic steady state with respect to small perturbations. To test and complement our theoretical predictions we performed event-driven molecular dynamics (MD) simulations of this system. We clearly identify the transition to phase separated states in the MD simulations, despite large fluctuations present, by measuring the probability distribution of the amplitude of the fundamental Fourier mode of the azimuthal spectrum of the particle density. We find that the instability region, predicted from hydrodynamics, is always located within the phase separation region observed in the MD simulations. This implies the presence of a binodal (coexistence) region, where the annular state is metastable. The phase separation persists when the driving and elastic walls are interchanged, and also when the elastic wall is replaced by weakly inelastic one.

Transport coefficients for dense hard-disk systems

A study of the transport coefficients of a system of elastic hard disks based on the use of Helfand-Einstein expressions is reported. The self-diffusion, the viscosity, and the heat conductivity are examined with averaging techniques especially appropriate for event-driven molecular dynamics algorithms with periodic boundary conditions. The density and size dependence of the results are analyzed, and comparison with the predictions from Enskog's theory is carried out. In particular, the behavior of the transport coefficients in the vicinity of the fluid-solid transition is investigated and a striking power law divergence of the viscosity with density is obtained in this region, while all other examined transport coefficients show a drop in that density range in relation to the Enskog's prediction. Finally, the deviations are related to shear band instabilities and the concept of dilatancy.

Hydrodynamics of an endothermic gas with application to bubble cavitation

The hydrodynamics for a gas of hard spheres which sometimes experience inelastic collisions resulting in the loss of a fixed, velocity-independent, amount of energy Delta is investigated with the goal of understanding the coupling between hydrodynamics and endothermic chemistry. The homogeneous cooling state of a uniform system and the modified Navier-Stokes equations are discussed and explicit expressions given for the pressure, cooling rates, and all transport coefficients for D dimensions. The Navier-Stokes equations are solved numerically for the case of a two-dimensional gas subject to a circular piston so as to illustrate the effects of the energy loss on the structure of shocks found in cavitating bubbles. It is found that the maximal temperature achieved is a sensitive function of Delta with a minimum occurring near the physically important value of Delta approximately 12,000 K approximately 1 eV.

Molecular dynamics study of the thermodynamics and transport coefficients of hard hyperspheres in six and seven dimensions

Molecular dynamics (MD) simulations are performed for six- and seven-dimensional hard-hypersphere fluids. The equation of state, velocity autocorrelation function, self-diffusion coefficient, shear viscosity, and thermal conductivity are determined as a function of density. The molecular dynamics results for the equation of state are found to be in excellent agreement with values obtained from theoretical approaches and previous MD simulations in seven dimensions. The short-time behavior of the velocity autocorrelation function is well described by the Enskog exponential approximation. The Enskog predictions for the self-diffusion coefficient and the viscosity agree fairly well with the simulation data at low densities, but underestimate these quantities at higher densities. Data for the thermal conductivity are in fine agreement with Enskog theory for all densities and dimensions studied.

Shear-induced crystallization of a dense rapid granular flow: hydrodynamics beyond the melting point

We investigate shear-induced crystallization in a very dense flow of monodisperse inelastic hard spheres. We consider a steady plane Couette flow under constant pressure and neglect gravity. We assume that the granular density is greater than the melting point of the equilibrium phase diagram of elastic hard spheres. We employ a Navier-Stokes hydrodynamics with constitutive relations all of which (except the shear viscosity) diverge at the crystal-packing density, while the shear viscosity diverges at a smaller density. The phase diagram of the steady flow is described by three parameters: an effective Mach number, a scaled energy loss parameter, and an integer number m: the number of half-oscillations in a mechanical analogy that appears in this problem. In a steady shear flow the viscous heating is balanced by energy dissipation via inelastic collisions. This balance can have different forms, producing either a uniform shear flow or a variety of more complicated, nonlinear density, velocity, and temperature profiles. In particular, the model predicts a variety of multilayer two-phase steady shear flows with sharp interphase boundaries. Such a flow may include a few zero-shear (solidlike) layers, each of which moving as a whole, separated by fluidlike regions. As we are dealing with a hard sphere model, the granulate is fluidized within the "solid" layers: the granular temperature is nonzero there, and there is energy flow through the boundaries of the solid layers. A linear stability analysis of the uniform steady shear flow is performed, and a plausible bifurcation diagram of the system, for a fixed m, is suggested. The problem of selection of m remains open.

Chapman-Enskog expansion about nonequilibrium states with application to the sheared granular fluid

The Chapman-Enskog method of solution of kinetic equations, such as the Boltzmann equation, is based on an expansion in gradients of the deviations of the hydrodynamic fields from a uniform reference state (e.g., local equilibrium). This paper presents an extension of the method so as to allow for expansions about arbitrary, far-from-equilibrium reference states. The primary result is a set of hydrodynamic equations for studying variations from the arbitrary reference state which, unlike the usual Navier-Stokes hydrodynamics, does not restrict the reference state in any way. The method is illustrated by application to a sheared granular gas which cannot be studied using the usual Navier-Stokes hydrodynamics.