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

Anomalous diffusion in polydisperse granular gases

We investigate both ensemble and time-averaged mean-squared displacements of particles in a polydisperse granular system in a homogeneous cooling state and derive rigorous analytical expressions valid both at short and long time scales. The discrepancies in ensemble and time-averaged mean-squared displacements indicate ergodicity breaking in granular systems consisting of an arbitrary number of species of different sizes and masses. We compare the results of our study with Monte Carlo simulations in terms of a powerful low-rank algorithm and find a nice agreement.

Diffusion in multicomponent granular mixtures

We investigate diffusion in polydisperse granular media. We derive the mean-squared displacement of granular particles in a polydisperse granular gas in a homogeneous cooling state, containing an arbitrary amount of species of different sizes and masses. We investigate both models of constant and time-dependent restitution coefficients and obtain a universal law for the size dependence of the mean-squared displacement for steep size distributions.

Diffusion of intruders in granular suspensions: Enskog theory and random walk interpretation

The Enskog kinetic theory is applied to compute the mean square displacement of impurities or intruders (modeled as smooth inelastic hard spheres) immersed in a granular gas of smooth inelastic hard spheres (grains). Both species (intruders and grains) are surrounded by an interstitial molecular gas (background) that plays the role of a thermal bath. The influence of the latter on the motion of intruders and grains is modeled via a standard viscous drag force supplemented by a stochastic Langevin-like force proportional to the background temperature. We solve the corresponding Enskog-Lorentz kinetic equation by means of the Chapman-Enskog expansion truncated to first order in the gradient of the intruder number density. The integral equation for the diffusion coefficient is solved by considering the first two Sonine approximations. To test these results, we also compute the diffusion coefficient from the numerical solution of the inelastic Enskog equation by means of the direct simulation Monte Carlo method. We find that the first Sonine approximation generally agrees well with the simulation results, although significant discrepancies arise when the intruders become lighter than the grains. Such discrepancies are largely mitigated by the use of the second Sonine approximation, in excellent agreement with computer simulations even for moderately strong inelasticities and/or dissimilar mass and diameter ratios. We invoke a random walk picture of the intruders' motion to shed light on the physics underlying the intricate dependence of the diffusion coefficient on the main system parameters. This approach, recently employed to study the case of an intruder immersed in a granular gas, also proves useful in the present case of a granular suspension. Finally, we discuss the applicability of our model to real systems in the self-diffusion case. We conclude that collisional effects may strongly impact the diffusion coefficient of the grains.

Enskog kinetic theory of binary granular suspensions: Heat flux and stability analysis of the homogeneous steady state

The Enskog kinetic theory of multicomponent granular suspensions employed previously [Gómez González, Khalil, and Garzó, Phys. Rev. E 101, 012904 (2020)2470-004510.1103/PhysRevE.101.012904] is considered further to determine the four transport coefficients associated with the heat flux. These transport coefficients are obtained by solving the Enskog equation by means of the application of the Chapman-Enskog method around the local version of the homogeneous state. Explicit forms of the heat flux transport coefficients are provided in steady-state conditions by considering the so-called second Sonine approximation to the distribution function of each species. Their quantitative variation on the control parameters of the mixture (masses and diameters, coefficients of restitution, concentration, volume fraction, and the background temperature) is demonstrated and the results show that in general the dependence of the heat flux transport coefficients on inelasticity is clearly different from that found in the absence of the gas phase (dry granular mixtures). As an application of the general results, the stability of the homogeneous steady state is analyzed by solving the linearized Navier-Stokes hydrodynamic equations. The linear stability analysis (which holds for wavelengths long compared with the mean free path) shows that the transversal and longitudinal modes are always stable with respect to long-enough wavelength excitations. This conclusion agrees with previous results derived for monocomponent and (dilute) bidisperse granular suspensions but contrasts with the instabilities found in previous works in dry (no gas phase) granular mixtures.

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 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.

Influence of the first-order contributions to the partial temperatures on transport properties in polydisperse dense granular mixtures

The Chapman-Enskog solution to the Enskog kinetic equation of polydisperse granular mixtures is revisited to determine the first-order contributions ϖ_{i} to the partial temperatures. As expected, these quantities (which were neglected in previous attempts) are given in terms of the solution to a set of coupled integrodifferential equations analogous to those for elastic collisions. The solubility condition for this set of equations is confirmed and the coefficients ϖ_{i} are calculated by using the leading terms in a Sonine polynomial expansion. These coefficients are given as explicit functions of the sizes, masses, composition, density, and coefficients of restitution of the mixture. Within the context of small gradients, the results apply for arbitrary degrees of inelasticity and are not restricted to specific values of the parameters of the mixture. In the case of elastic collisions, previous expressions of ϖ_{i} for ordinary binary mixtures are recovered. Finally, the impact of the first-order coefficients ϖ_{i} on the bulk viscosity η_{b} and on the first-order contribution ζ^{(1)} to the cooling rate is assessed. It is shown that the effect of ϖ_{i} on η_{b} and ζ^{(1)} is not negligible, specially for disparate mass ratios and strong inelasticity.

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.

Interplay between polydispersity, inelasticity, and roughness in the freely cooling regime of hard-disk granular gases

A polydisperse granular gas made of inelastic and rough hard disks is considered. Focus is laid on the kinetic-theory derivation of the partial energy production rates and the total cooling rate as functions of the partial densities and temperatures (both translational and rotational) and of the parameters of the mixture (masses, diameters, moments of inertia, and mutual coefficients of normal and tangential restitution). The results are applied to the homogeneous cooling state of the system and the associated nonequipartition of energy among the different components and degrees of freedom. It is found that disks typically present a stronger rotational-translational nonequipartition but a weaker component-component nonequipartition than spheres. A noteworthy "mimicry" effect is unveiled, according to which a polydisperse gas of disks having common values of the coefficient of restitution and of the reduced moment of inertia can be made indistinguishable from a monodisperse gas in what concerns the degree of rotational-translational energy nonequipartition. This effect requires the mass of a disk of component i to be approximately proportional to 2σ_{i}+〈σ〉, where σ_{i} is the diameter of the disk and 〈σ〉 is the mean diameter.

Increasing temperature of cooling granular gases

The kinetic energy of a force-free granular gas decays monotonously due to inelastic collisions of the particles. For a homogeneous granular gas of identical particles, the corresponding decay of granular temperature is quantified by Haff’s law. Here, we report that for a granular gas of aggregating particles, the granular temperature does not necessarily decay but may even increase. Surprisingly, the increase of temperature is accompanied by the continuous loss of total gas energy. This stunning effect arises from a subtle interplay between decaying kinetic energy and gradual reduction of the number of degrees of freedom associated with the particles’ dynamics. We derive a set of kinetic equations of Smoluchowski type for the concentrations of aggregates of different sizes and their energies. We find scaling solutions to these equations and a condition for the aggregation mechanism predicting growth of temperature. Numerical direct simulation Monte Carlo results confirm the theoretical predictions.

Granular gases—dilute systems composed of dissipatively colliding particles—exhibit anomalous dynamics and numerous surprising phenomena. Here, Brilliantov et al. show that the aggregation mechanism can induce increase of the gas temperature despite the fact that the total kinetic energy decreases.

Heat flux of driven granular mixtures at low density: Stability analysis of the homogeneous steady state

The Navier-Stokes order hydrodynamic equations for a low-density driven granular mixture obtained previously [Khalil and Garzó, Phys. Rev. E 88, 052201 (2013)PLEEE81539-375510.1103/PhysRevE.88.052201] from the Chapman-Enskog solution to the Boltzmann equation are considered further. The four transport coefficients associated with the heat flux are obtained in terms of the mass ratio, the size ratio, composition, coefficients of restitution, and the driven parameters of the model. Their quantitative variation on the control parameters of the system is demonstrated by considering the leading terms in a Sonine polynomial expansion to solve the exact integral equations. As an application of these results, the stability of the homogeneous steady state is studied. In contrast to the results obtained in undriven granular mixtures, the stability analysis of the linearized Navier-Stokes hydrodynamic equations shows that the transversal and longitudinal modes are (linearly) stable with respect to long enough wavelength excitations. This conclusion agrees with a previous analysis made for single granular gases.

Generalized transport coefficients for inelastic Maxwell mixtures under shear flow

The Boltzmann equation framework for inelastic Maxwell models is considered to determine the transport coefficients associated with the mass, momentum, and heat fluxes of a granular binary mixture in spatially inhomogeneous states close to the simple shear flow. The Boltzmann equation is solved by means of a Chapman-Enskog-type expansion around the (local) shear flow distributions f(r)(0) for each species that retain all the hydrodynamic orders in the shear rate. Due to the anisotropy induced by the shear flow, tensorial quantities are required to describe the transport processes instead of the conventional scalar coefficients. These tensors are given in terms of the solutions of a set of coupled equations, which can be analytically solved as functions of the shear rate a, the coefficients of restitution α(rs), and the parameters of the mixture (masses, diameters, and composition). Since the reference distribution functions f(r)(0) apply for arbitrary values of the shear rate and are not restricted to weak dissipation, the corresponding generalized coefficients turn out to be nonlinear functions of both a and α(rs). The dependence of the relevant elements of the three diffusion tensors on both the shear rate and dissipation is illustrated in the tracer limit case, the results showing that the deviation of the generalized transport coefficients from their forms for vanishing shear rates is in general significant. A comparison with the previous results obtained analytically for inelastic hard spheres by using Grad's moment method is carried out, showing a good agreement over a wide range of values for the coefficients of restitution. Finally, as an application of the theoretical expressions derived here for the transport coefficients, thermal diffusion segregation of an intruder immersed in a granular gas is also studied.

Size distribution of particles in Saturn's rings from aggregation and fragmentation

Saturn's rings consist of a huge number of water ice particles, with a tiny addition of rocky material. They form a flat disk, as the result of an interplay of angular momentum conservation and the steady loss of energy in dissipative interparticle collisions. For particles in the size range from a few centimeters to a few meters, a power-law distribution of radii, ~r(-q) with q ≈ 3, has been inferred; for larger sizes, the distribution has a steep cutoff. It has been suggested that this size distribution may arise from a balance between aggregation and fragmentation of ring particles, yet neither the power-law dependence nor the upper size cutoff have been established on theoretical grounds. Here we propose a model for the particle size distribution that quantitatively explains the observations. In accordance with data, our model predicts the exponent q to be constrained to the interval 2.75 ≤ q ≤ 3.5. Also an exponential cutoff for larger particle sizes establishes naturally with the cutoff radius being set by the relative frequency of aggregating and disruptive collisions. This cutoff is much smaller than the typical scale of microstructures seen in Saturn's rings.

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.

Transport coefficients of a granular gas of inelastic rough hard spheres

The Boltzmann equation for inelastic and rough hard spheres is considered as a model of a dilute granular gas. In this model, the collisions are characterized by constant coefficients of normal and tangential restitution, and hence the translational and rotational degrees of freedom are coupled. A normal solution to the Boltzmann equation is obtained by means of the Chapman-Enskog method for states near the homogeneous cooling state. The analysis is carried out to first order in the spatial gradients of the number density, the flow velocity, and the granular temperature. The constitutive equations for the momentum and heat fluxes and for the cooling rate are derived, and the associated transport coefficients are expressed in terms of the solutions of linear integral equations. For practical purposes, a first Sonine approximation is used to obtain explicit expressions of the transport coefficients as nonlinear functions of both coefficients of restitution and the moment of inertia. Known results for purely smooth inelastic spheres and perfectly elastic and rough spheres are recovered in the appropriate limits.

Hydrodynamic Burnett equations for inelastic Maxwell models of granular gases

The hydrodynamic Burnett equations and the associated transport coefficients are exactly evaluated for generalized inelastic Maxwell models. In those models, the one-particle distribution function obeys the inelastic Boltzmann equation, with a velocity-independent collision rate proportional to the γ power of the temperature. The pressure tensor and the heat flux are obtained to second order in the spatial gradients of the hydrodynamic fields with explicit expressions for all the Burnett transport coefficients as functions of γ, the coefficient of normal restitution, and the dimensionality of the system. Some transport coefficients that are related in a simple way in the elastic limit become decoupled in the inelastic case. As a byproduct, existing results in the literature for three-dimensional elastic systems are recovered, and a generalization to any dimension of the system is given. The structure of the present results is used to estimate the Burnett coefficients for inelastic hard spheres.

Instabilities in granular binary mixtures at moderate densities

A linear stability analysis of the Navier-Stokes (NS) granular hydrodynamic equations is performed to determine the critical length scale for the onset of vortices and clusters instabilities in granular dense binary mixtures. In contrast to previous attempts, our results (which are based on the solution to the inelastic Enskog equation to NS order) are not restricted to nearly elastic systems since they take into account the complete nonlinear dependence of the NS transport coefficients on the coefficients of restitution α(ij). The theoretical predictions for the critical length scales are compared to molecular dynamics (MD) simulations in flows of strong dissipation (α(ij) ≥ 0.7) and moderate solid volume fractions (ϕ ≤ 0.2). We find excellent agreement between MD and kinetic theory for the onset of velocity vortices, indicating the applicability of NS hydrodynamics to polydisperse flows even for strong inelasticity, finite density, and particle dissimilarity.

Transport coefficients for driven granular mixtures at low density

The transport coefficients of a granular binary mixture driven by a stochastic bath with friction are determined from the inelastic Boltzmann kinetic equation. A normal solution is obtained via the Chapman-Enskog method for states near homogeneous steady states. The mass, momentum, and heat fluxes are determined to first order in the spatial gradients of the hydrodynamic fields, and the associated transport coefficients are identified. They are given in terms of the solutions of a set of coupled linear integral equations. As in the monocomponent case, since the collisional cooling cannot be compensated for locally by the heat produced by the external driving, the reference distributions (zeroth-order approximations) f(i)((0)) (i=1,2) for each species depend on time through their dependence on the pressure and the temperature. Explicit forms for the diffusion transport coefficients and the shear viscosity coefficient are obtained by assuming the steady-state conditions and by considering the leading terms in a Sonine polynomial expansion. A comparison with previous results obtained for granular Brownian motion and by using a (local) stochastic thermostat is also carried out. The present work extends previous theoretical results derived for monocomponent dense gases [Garzó, Chamorro, and Vega Reyes, Phys. Rev. E 87, 032201 (2013)] to granular mixtures at low density.

Shearing instability of a dilute granular mixture

The shearing instability of a dilute granular mixture composed of smooth inelastic hard spheres or disks is investigated. By using the Navier-Stokes hydrodynamic equations, it is shown that the scaled transversal velocity mode exhibits a divergent behavior, similarly to what happens in one-component systems. The theoretical prediction for the critical size is compared with direct Monte Carlo simulations of the Boltzmann equations describing the system, and a good agreement is found. The total energy fluctuations in the vicinity of the transition are shown to scale with the second moment of the distribution. The scaling distribution function is the same as found in other equilibrium and nonequilibrium phase transitions, suggesting the existence of some kind of universality.

Segregation of an intruder in a heated granular dense gas

A recent segregation criterion [Phys. Rev. E 78, 020301(R) (2008)] based on the thermal diffusion factor Λ of an intruder in a heated granular gas described by the inelastic Enskog equation is revisited. The sign of Λ provides a criterion for the transition between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE). The present theory incorporates two extra ingredients not accounted for by the previous theoretical attempt. First, the theory is based upon the second Sonine approximation to the transport coefficients of the mass flux of the intruder. Second, the dependence of the temperature ratio (intruder temperature over that of the host granular gas) on the solid volume fraction is taken into account in the first and second Sonine approximations. In order to check the accuracy of the Sonine approximation considered, the Enskog equation is also numerically solved by means of the direct simulation Monte Carlo method to get the kinetic diffusion coefficient D(0). The comparison between theory and simulation shows that the second Sonine approximation to D(0) yields an improvement over the first Sonine approximation when the intruder is lighter than the gas particles in the range of large inelasticity. With respect to the form of the phase diagrams for the BNE-RBNE transition, the kinetic theory results for the factor Λ indicate that while the form of these diagrams depends sensitively on the order of the Sonine approximation considered when gravity is absent, no significant differences between both Sonine solutions appear in the opposite limit (gravity dominates the thermal gradient). In the former case (no gravity), the first Sonine approximation overestimates both the RBNE region and the influence of dissipation on thermal diffusion segregation.

Impurity in a sheared inelastic Maxwell gas

The Boltzmann equation for inelastic Maxwell models is considered in order to investigate the dynamics of an impurity (or intruder) immersed in a granular gas driven by a uniform shear flow. The analysis is based on an exact solution of the Boltzmann equation for a granular binary mixture. It applies for conditions arbitrarily far from equilibrium (arbitrary values of the shear rate a) and for arbitrary values of the parameters of the mixture (particle masses m(i), mole fractions x(i), and coefficients of restitution α(ij)). In the tracer limit where the mole fraction of the intruder species vanishes, a nonequilibrium phase transition takes place. We thereby identify ordered phases where the intruder bears a finite contribution to the properties of the mixture, in a region of parameter space that is worked out in detail. These findings extend previous results obtained for ordinary Maxwell gases, and further show that dissipation leads to new ordered phases.

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.

Rheology of dense granular mixtures: particle-size distributions, boundary conditions, and collisional time scales

We computationally investigate the dependence of the rheology of dense sheared granular mixtures on their particle size distribution. We consider the simplest case of a binary mixture of two different sized particles where the fraction of large particles is varied from one simulation to the next while the total solid mass is kept constant. We find that the variation of the rheology with the particle size distribution depends on the boundary conditions. For example, under constant pressure conditions the effective friction coefficient μ(∗) (the ratio between shear and pressure stresses at the boundary) increases mildly with the average particle size. On the other hand, under constant volume conditions, μ(∗) has a nonmonotonic dependence on the average particle size that is related to the proximity of the system solid fraction to the maximum packing fraction. Somewhat surprisingly, then, μ(∗) scales with a dimensionless shear rate (a generalized inertial number) in the same way for either boundary condition. We show that, for our system of relatively hard spheres, these relationships are governed largely by the ratio between average collision times and mean-free-path times, also independent of boundary conditions.

Clustering in rapid granular flows of binary and continuous particle size distributions

The dynamic clustering phenomenon in two-dimensional simple shear flows has been investigated using molecular dynamic simulations of systems containing binary and continuous size distributions of equal-material-density particles. Particular attention has been paid to two questions: (1) Does the presence of size nonuniformities serve to enhance or attenuate the presence of clusters? (2) Do particles of a given size preferentially segregate within the clusters? With respect to the first question, the prominence of clustered regions increases with increasing deviation from the monodisperse limit in the case of both binary and continuous size distributions. With respect to the second question, the larger particles of both binary and continuous size distributions are consistently observed to segregate within the transient clustered regions. Further investigation of granular temperatures within the clustered and dilute regions reveals that this segregation is consistent with previously observed temperature-driven segregation in steady-state systems; large particles favor the lower-temperature (clustered) regions. Moreover, observation of clustering length scales suggests that large particles may favor the center of the clustered regions, where granular temperatures are expected to reach a minimum.

Partitioning of energy in highly polydisperse granular gases

A highly polydisperse granular gas is modeled by a continuous distribution of particle sizes, a , giving rise to a corresponding continuous temperature profile, T(a) , which we compute approximately, generalizing previous results for binary or multicomponent mixtures. If the system is driven, it evolves toward a stationary temperature profile, which is discussed for several driving mechanisms in dependence on the variance of the size distribution. For a uniform distribution of sizes, the stationary temperature profile is nonuniform with either hot small particles (constant force driving) or hot large particles (constant velocity or constant energy driving). Polydispersity always gives rise to non-Gaussian velocity distributions. Depending on the driving mechanism the tails can be either overpopulated or underpopulated as compared to the molecular gas. The deviations are mainly due to small particles. In the case of free cooling the decay rate depends continuously on particle size, while all partial temperatures decay according to Haff's law. The analytical results are supported by event driven simulations for a large, but discrete number of species.

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.

Brazil-nut effect versus reverse Brazil-nut effect in a moderately dense granular fluid

A segregation criterion based on the inelastic Enskog kinetic equation is derived to show the transition between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE) by varying the different parameters of the system. In contrast to previous theoretical attempts, the approach is not limited to the near-elastic case, takes into account the influence of both thermal gradients and gravity, and applies for moderate densities. The form of the phase diagrams for the BNE-RBNE transition depends sensitively on the value of gravity relative to the thermal gradient, so that it is possible to switch between both states for given values of the mass and size ratios, the coefficients of restitution, and the solid volume fraction. In particular, the influence of collisional dissipation on segregation becomes more important when the thermal gradient dominates over gravity than in the opposite limit. The present analysis extends previous results derived in the dilute limit case and is consistent with the findings of some recent experimental results.

Does the Chapman-Enskog expansion for sheared granular gases converge?

The fundamental question addressed in this Letter is whether or not the partial Chapman-Enskog expansion P(xy)= [see text for formula] of the shear stress converges for a gas of inelastic hard spheres. By using a simple kinetic model it is shown that, in contrast to the elastic case, the above series does converge, the radius of convergence increasing with inelasticity. It is argued that this paradoxical conclusion is not an artifact of the kinetic model and can be understood in terms of the time evolution of the scaled shear rate in the uniform shear flow.

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.