We develop a kinetic-theory framework to investigate the steady rheology of a dilute gas interacting via a repulsive potential under uniform shear flow. Starting from the Boltzmann equation with a restitution coefficient that depends on the impact velocity and potential strength, we derive evolution equations for the stress tensor based on Grad's moment expansion. The resulting expressions for the collisional rates and transport coefficients are fitted with simple analytical functions that capture their temperature dependence over a wide range of shear rates. Comparison with direct simulation Monte Carlo (DSMC) results shows excellent quantitative agreement for the shear stress, temperature anisotropy, and shear viscosity. We also analyze the velocity distribution functions, revealing that the system remains nearly Maxwellian even under strong shear.
We develop the kinetic theory of dilute cohesive granular gases in which the attractive part is described by a square well potential. We derive the hydrodynamic equations from the kinetic theory with the microscopic expressions for the dissipation rate and the transport coefficients. We check the validity of our theory by performing the direct simulation Monte Carlo.
We develop the rheology of a dilute granular gas mixture. Motivated by the interaction of charged granular particles, we assume that the grains interact via a square shoulder and well potential. Employing kinetic theory, we compute the temperature and the shear viscosity as functions of the shear rate. Numerical simulations confirm our results above the critical shear rate. At a shear rate below a critical value, clustering of the particles occurs.
The spontaneous symmetry breaking taking place in the direction perpendicular to the energy flux in a dilute vibrofluidized granular system is investigated, using both a hydrodynamic description and simulation methods. The latter include molecular dynamics and direct Monte Carlo simulation of the Boltzmann equation. A marginal stability analysis of the hydrodynamic equations, carried out in the WKB approximation, is shown to be in good agreement with the simulation results. The shape of the hydrodynamic profiles beyond the bifurcation is discussed.
Hydrodynamic equations for granular gases driven by the Fokker-Planck operator are derived. Transport coefficients appeared in Navier-Stokes order change from the values of a free cooling state to those of a steady state.
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 …
Using particle simulations of the uniform shear flow of a rough dilute granular gas, we show that the translational and rotational velocities are strongly correlated in direction, but there is no orientational correlation-induced singularity at perfectly smooth ($β=-1$) and rough ($β=1$) limits for elastic collisions ($e=1$); both the translational and rotational velocity distribution functions remain close to a Gaussian for these two limiting cases. Away from these two limits, the orientational as well as spatial velocity correlations are responsible for the emergence of non-Gaussian high vel…
The Green-Kubo relation for two models of granular gases is discussed. In the Maxwell model in any dimension, the effective temperature obtained from the Green-Kubo relation is shown to be frequency independent, and equal to the average kinetic energy, known as the granular temperature. In the second model analyzed, a mean-field granular gas, the collision rate of a particle is taken to be proportional to its velocity. The Green-Kubo relation in the high frequency limit is calculated for this model, and the effective temperature in this limit is shown to be equal to the granular temperature. T…
We analyze the asymptotic behavior of linear Fokker-Planck equations with time-dependent coefficients. Relaxation towards a Maxwellian distribution with time-dependent temperature is shown under explicitly computable conditions. We apply this result to the study of Brownian motion in granular gases as introduced J. J. Brey, J. Dufty and A. Santos (1999), by showing that the Homogenous Cooling State attracts any solution at an algebraic rate.
The long-time behavior of the current auto-correlation 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 $τ^{-d/2}$, while the correlation function of heat flux decays as $τ^{-(d+2)/2} \exp(-ζ^* τ)$ with the dimensionless cooling rate $ζ^*$, the spatial dimension $d$ and the scaled time $τ$ in terms of the collision frequency. The result of our numerical simulation of the freely cooling granular gases is consistent with…
An impact of particles' roughness on the self-diffusion coefficient in granular gases is investigated. For a simplified collision model where the normal and tangential restitution coefficients are assumed to be constant we develop an analytical theory for the diffusion coefficient, which takes into account non-Maxwellain form of the velocity-angular velocity distribution function. We perform molecular dynamics simulations for a gas in a homogeneous cooling state and study the dependence of the self-diffusion coefficient on restitution coefficients. Our theoretical results are in a good agreeme…
Motivated by recent experiments reporting non-Gaussian velocity distributions in driven dilute granular materials, we study by numerical simulation the properties of 2D inelastic gases. We find theoretically that the form of the observed velocity distribution is governed primarily by the coefficient of restitution $η$ and $q=N_H/N_C$, the ratio between the average number of heatings and the average number of collisions in the gas. The differences in distributions we find between uniform and boundary heating can then be understood as different limits of $q$, for $q \gg 1$ and $q \lesssim 1$ res…
While memory effects have been reported for dense enough disordered systems such as glasses, we show here by a combination of analytical and simulation techniques that they are also intrinsic to the dynamics of dilute granular gases. By means of a certain driving protocol, we prepare the gas in a state where the granular temperature $T$ coincides with its long time limit. However, $T$ does not subsequently remain constant, but exhibits a non-monotonic evolution before reaching its non-equilibrium steady value. The corresponding so-called Kovacs hump displays a normal behavior for weak dissipat…
The expansion of the velocity distribution function for the homogeneous cooling state (HCS) in a Sonine polynomial series around a Maxwellian is shown to be divergent, though Borel resummable. A convergent expansion for the HCS has been devised and employed to obtain the HCS velocity distribution function and (using it) the linear transport coefficients for a three dimensional monodisperse granular gas of smooth inelastic spheres, for all physical values of the coefficient of normal restitution. The results are in very good agreement with findings of DSMC simulations.
We study velocity statistics of electrostatically driven granular gases. For two different experiments: (i) non-magnetic particles in a viscous fluid and (ii) magnetic particles in air, the velocity distribution is non-Maxwellian, and its high-energy tail is exponential, P(v) ~ exp(-|v|). This behavior is consistent with kinetic theory of driven dissipative particles. For particles immersed in a fluid, viscous damping is responsible for the exponential tail, while for magnetic particles, long-range interactions cause the exponential tail. We conclude that velocity statistics of dissipative gas…
Dry, freely evolving granular materials in a dilute gaseous state coalesce into dense clusters only due to dissipative interactions. This clustering transition is important for a number of problems ranging from geophysics to cosmology. Here we show that the evolution of a dilute, freely cooling granular gas is determined in a universal way by the ratio of inertial flow and thermal velocities, that is, the Mach number. Theoretical calculations and direct numerical simulations of the granular Navier--Stokes equations show that irrespective of the coefficient of restitution, density or initial ve…
We investigate hydrodynamic noise in a dilute granular gas during the homogeneous cooling state, by means of a proper application of the Direct Simulation Monte Carlo (DSMC) algorithm. The DSMC includes a source of randomization which is not present in Molecular Dynamics (MD) for inelastic hard disks. Notwithstanding this difference, a fair quantitative agreement is found, including a violation of the fluctuation-dissipation relation for the noise amplitude of the same order observed in MD. This study suggests that deterministic collision dynamics is not an essential ingredient to reproduce, u…
A new type of two phase coexistence in a vibrated granular material is reported, in the limit where energy transfer between the particles and the bottom wall occurs due to discrete collisions. A horizontally homogeneous bed develops inhomogeneities in which a dense and a dilute phase separate and coexist. The dilute region resembles a gas, while the dense region is like a liquid with a velocity distribution very different from a Gaussian.
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 Couet…
Two granular gases separated by an adiabatic piston and initially in the same macroscopic state are considered. It is found that a phase transition with an spontaneous symmetry breaking occurs. When the mass of the piston is increased beyond a critical value, the piston moves to a stationary position different from the middle of the system. The transition is accurately described by a simple kinetic model that takes into account the velocity fluctuations of the piston. Interestingly, the final state is not characterized by the equality of the temperatures of the subsystems but by the cooling ra…
Motivated by recent experiments reporting non-Gaussian velocity distributions in driven dilute granular materials, we study by numerical simulation the properties of inelastic gases as functions of the coefficient of restitution $η$ and concentration $ϕ$ with various heating mechanisms. We show that there are marked, qualitative differences in the behavior for uniform heating (as is frequently assumed theoretically) and for particle systems driven at the boundaries of the container (as is frequently done in experiments). In general, we find Gaussian velocity distributions for uniform heating a…
An overview of the hydrodynamic profiles derived by kinetic theory tools (Boltzmann equation and kinetic models) for the gravity-driven Poiseuille flow with particles colliding either elastically (planar and cylindrical geometries) or inelastically (planar geometry) is presented. The results obtained through second order in the gravity acceleration show that the Navier-Stokes predictions are qualitatively incorrect over distances of the order of the mean free path, especially in the case of the temperature profile
We study dynamics of freely cooling granular gases in two-dimensions using large-scale molecular dynamics simulations. We find that for dilute systems the typical kinetic energy decays algebraically with time, E(t) ~ t^{-1}, in the long time limit. Asymptotically, velocity statistics are characterized by a universal Gaussian distribution, in contrast with the exponential high-energy tails characterizing the early homogeneous regime. We show that in the late clustering regime particles move coherently as typical local velocity fluctuations, Delta v, are small compared with the typical velocity,…
We numerically and theoretically investigate how the softness of particles affects the rheology of sheared dilute granular gases. We find that the kinetic theory predicts the deviation of the flow curve from the Bagnold scaling, and it works well below a certain shear rate when we compare with the simulation results. It is also found that there is no theoretical solution above this shear rate, which is because the energy loss due to inelastic collisions cannot be balanced with the energy injection by the shear.
We study the equilibrium behavior of one-dimensional granular clusters and one-particle granular gases for a variety of velocity dependent coefficients of restitution $r$. We obtain equations describing of the long time behavior for the cluster's pressure, r.m.s. velocity and granular inter-spacing. We show that for extremely long times, clusters with velocity dependent coefficients of restitution are unstable and dissolve into homogeneous, quasi-elastic gases, but clusters with velocity independent $r$ are permanent. This is in accordance with hydrodynamic studies pointing to the transient na…
This paper investigates the lift force exerted on an elliptical obstacle immersed in a granular flow through analytical calculations and computer simulations. The results are shown as a function of the obstacle size, orientation with respect to the flow direction (tilt angle), the restitution coefficient and ellipse eccentricity. The theoretical argument, based on the force exerted on the obstacle due to inelastic, frictionless collisions of a very dilute flow, captures the qualitative features of the lift, but fails to reproduce the data quantitatively. The reason behind this disagreement is …
We evidence a Kovacs-like memory effect in a uniformly driven granular gas. A system of inelastic hard particles, in the low density limit, can reach a non-equilibrium steady state when properly forced. By following a certain protocol for the drive time dependence, we prepare the gas in a state where the granular temperature coincides with its long time value. The temperature subsequently does not remain constant, but exhibits a non-monotonic evolution with either a maximum or a minimum, depending on the dissipation, and on the protocol. We present a theoretical analysis of this memory effect,…
We report the statistical properties of spherical steel particles rolling on an inclined surface being driven by an oscillating wall. Strong dissipation occurs due to collisions between the particles and rolling and can be tuned by changing the number density. The velocities of the particles are observed to be correlated over large distances comparable to the system size. The distribution of velocities deviates strongly from a Gaussian. The degree of the deviation, as measured by the kurtosis of the distribution, is observed to be as much as four times the value corresponding to a Gaussian, si…
We report the results of an experimental investigation of segregation in a binary mixture of dry particles subjected to horizontal oscillatory excitation. The thin layer of particles was driven by the stick--slip frictional interaction with the surface of a horizontal tray. As the packing fraction of the mixture was increased the evolution of distinct phases was observed. We identified them as a binary gas, segregation liquid and segregation crystal and provide both microscopic and macroscopic measures to identify their properties. Finally, we draw some analogies between segregation in our gra…
We study the behavior of an assembly of $N$ granular particles contained in two compartments within a simple kinetic approach. The particles belonging to each compartment collide inelastically with each other and are driven by a stochastic heat bath. In addition, the fastest particles can change compartment at a rate which depends on their kinetic energy. Via a Boltzmann velocity distribution approach we first study the dynamics of the model in terms of a coupled set of equations for the populations in the containers and their granular temperatures and find a crossover from a symmetric high-te…
The temporal evolution of a dilute granular gas, both in a compressible flow (uniform longitudinal flow) and in an incompressible flow (uniform shear flow), is investigated by means of the direct simulation Monte Carlo method to solve the Boltzmann equation. Emphasis is laid on the identification of a first "kinetic" stage (where the physical properties are strongly dependent on the initial state) subsequently followed by an unsteady "hydrodynamic" stage (where the momentum fluxes are well-defined non-Newtonian functions of the rate of strain). The simulation data are seen to support this two-…
The free evolution of inelastic particles in one dimension is studied by means of Molecular Dynamics (MD), of an inelastic pseudo-Maxwell model and of a lattice model, with emphasis on the role of spatial correlations. We present an exact solution of the 1d granular pseudo-Maxwell model for the scaling distribution of velocities and discuss how this model fails to describe correctly the homogeneous cooling stage of the 1d granular gas. Embedding the pseudo-Maxwell gas on a lattice (hence allowing for the onset of spatial correlations), we find a much better agreement with the MD simulations ev…
Simulation results of dense granular gases with particles of different size are compared with theoretical predictions concerning the pair-correlation functions, the collison rate, the energy dissipation, and the mixture pressure. The effective particle-particle correlation function, which enters the equation of state in the same way as the correlation function of monodisperse granular gases, depends only on the total volume fraction and on the dimensionless width A of the size-distribution function. The ''global equation of state'' is proposed, which unifies both the dilute and the dense regim…
We report a general class of steady and transient states of granular gases. We find that the kinetic theory of inelastic gases admits stationary solutions with a power-law velocity distribution, f(v) ~ v^(-sigma). The exponent sigma is found analytically and depends on the spatial dimension, the degree of inelasticity, and the homogeneity degree of the collision rate. Driven steady-states, with the same power-law tail and a cut-off can be maintained by injecting energy at a large velocity scale, which then cascades to smaller velocities where it is dissipated. Associated with these steady-stat…
A variant of the Direct Simulation Monte Carlo method is used to study the behavior of a granular gas, in two and three dimensions, under varying density, restitution coefficient, and inelasticity regimes, for realistic vibrating wall conditions. Our results agree quite well with recent experimental data. The role of the energy injection mechanism is discussed, as well as the behavior of state-functions, such as pressure, under realistic boundary conditions. Upon a density increase, we find signals of a clustering transition.
We report an experimental study of particle kinematics in a 3-dimensional system of inelastic spheres fluidized by intense vibration. The motion of particles in the interior of the medium is tracked by high speed video imaging, yielding a spatially-resolved measurement of the velocity distribution. The distribution is wider than a Gaussian and broadens continuously with increasing volume fraction. The deviations from a Gaussian distribution for this boundary-driven system are different in sign and larger in magnitude than predictions for homogeneously driven systems. We also find correlations …
Dry, wet, dense, and dilute granular flows have been previously considered fundamentally different and thus described by distinct, and in many cases incompatible, rheologies. We carry out extensive simulations of granular flows, including wet and dry conditions, various geometries and driving mechanisms (boundary driven, fluid driven, and gravity driven), many of which are not captured by standard rheology models. For all simulated conditions, except for fluid-driven and gravity-driven flows close to the flow threshold, we find that the Mohr-Coulomb friction coefficient $μ$ scales with the squ…
We predict a p-wave Cooper pairing of the spin-polarized fermions in a binary fermion-boson mixture due to the exchange of density fluctuations of the bosonic medium. We then examine the dependence of the Cooper paring temperature on the parameters of the system. We finally estimate the effect of combining the boson-induced interaction with other pairing mechanisms, e.g the Kohn-Luttinger one, and find that the critical temperature of p-wave Cooper pairing can be realistic for experiment.
It is demonstrated that recent results on 1d granular gas in a container with a vibrating piston, which was modelled by a shock wave propagation, can be understood with a modelling using ideas coming from the "thermodynamics of a single particle". Defining e as the square root of the energetic-restitution coefficient of a single collision, and N as the total number of grains, the mean loss during a round trip of the momentum is calculated in the limit N(1-e)<<1. It is also demonstrated that the system cannot propagate sound waves nor shock waves in the limit of N(1-e)<<1 and that hydrodynamics…
Anisotropic pseudopotential relevant to collisions of two particles polarized by external field is rigorously derived and its properties are investigated. Such low-energy pseudopotential may be useful in describing collective properties of dilute quantum gases, such as molecules polarized by electric field or metastable $^3P_2$ atoms polarized by magnetic field. The pseudopotential is expressed in terms of reactance (K--) matrix and derivatives of Dirac delta-function. In most applications it may be represented as a sum of traditional spherically-symmetric contact term and anisotropic part. Th…
Granular media such as sand and sugar are ubiquitous in nature and industry but are less well understood than fluids or solids. We consider the behavior of rapid granular flows where the transfer of momenta by collisions dominates. The physics is quite different for the opposite limit of static or slowly moving grains (e.g., sand piles). To gain understanding of granular flows we consider two problems that have been investigated with experiments, particle simulations and hydrodynamic theory: vertically oscillating granular layers and flow past an obstacle. Oscillating granular layers spontaneo…
Previous experimental and theoretical evidence has shown that convective flow may appear in granular fluids if subjected to a thermal gradient and gravity (Rayleigh-Bénard-type convection). In contrast to this, we present here evidence of gravity-free thermal convection in a granular gas, with no presence of external thermal gradients either. Convection is here maintained steady by internal gradients due to dissipation and thermal sources at the same temperature. The granular gas is composed by identical disks and is enclosed in a rectangular region. Our results are obtained by means of an eve…
Granular flows and vibro-fluidised granular gases have been extensively studied recently; most of the theoretical analyses and the experimental descriptions use temperature and other thermodynamics concepts. However, taking the very simple case of a vibro-fluidised gas made of identical particles, we show the lack of efficiency of such concepts for the understanding of the physics of such systems. This results from both (i) the fact that the vibrator does not transmit the same amount of energy to each particle, but an amount which depends on its mass and/or its size and (ii) from the fact that…
We employ Navier-Stokes granular hydrodynamics to investigate the long-time behavior of clustering instability in a freely cooling dilute granular gas in two dimensions. We find that, in circular containers, the homogeneous cooling state (HCS) of the gas loses its stability via a sub-critical pitchfork bifurcation. There are no time-independent solutions for the gas density in the supercritical region, and we present analytical and numerical evidence that the gas develops thermal collapse unarrested by heat diffusion. To get more insight, we switch to a simpler geometry of a narrow-sector-shap…
It has been recently shown (Fouxon et al. 2007) that, in the framework of ideal granular hydrodynamics (IGHD), an initially smooth hydrodynamic flow of a granular gas can produce an infinite gas density in a finite time. Exact solutions that exhibit this property have been derived. Close to the singularity, the granular gas pressure is finite and almost constant. This work reports molecular dynamics (MD) simulations of a freely cooling gas of nearly elastically colliding hard disks, aimed at identifying the "attempted" density blowup regime. The initial conditions of the simulated flow mimic t…
Thermal wall is a convenient idealization of a rapidly vibrating plate used for vibrofluidization of granular materials. The objective of this work is to incorporate the Knudsen temperature jump at thermal wall in the Navier-Stokes hydrodynamic modeling of dilute granular gases of monodisperse particles that collide nearly elastically. The Knudsen temperature jump manifests itself as an additional term, proportional to the temperature gradient, in the boundary condition for the temperature. Up to a numerical pre-factor of order unity, this term is known from kinetic theory of elastic gases. We…
