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

Energy nonequipartition in a collisional model of a confined quasi-two-dimensional granular mixture

A collisional model of a confined quasi-two-dimensional granular mixture is considered to analyze homogeneous steady states. The model includes an effective mechanism to transfer the kinetic energy injected by vibration in the vertical direction to the horizontal degrees of freedom of grains. The set of Enskog kinetic equations for the velocity distribution functions of each component is derived first to analyze the homogeneous state. As in the one-component case, an exact scaling solution is found where the time dependence of the distribution functions occurs entirely through the granular temperature T. As expected, the kinetic partial temperatures T_{i} of each component are different and, hence, energy equipartition is broken down. In the steady state, explicit expressions for the temperature T and the ratio of partial kinetic temperatures T_{i}/T_{j} are obtained by considering Maxwellian distributions defined at the partial temperatures T_{i}. The (scaled) granular temperature and the temperature ratios are given in terms of the coefficients of restitution, the solid volume fraction, the (scaled) parameters of the collisional model, and the ratios of mass, concentration, and diameters. In the case of a binary mixture, the theoretical predictions are exhaustively compared with both direct simulation Monte Carlo and molecular dynamics simulations with a good agreement. The deviations are identified to be originated in the non-Gaussianity of the velocity distributions and on microsegregation patterns, which induce spatial correlations not captured in the Enskog theory.

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, discontinuous shear thickening, and rheological phase transitions in athermally sheared frictionless disks

We report on numerical simulations of simple models of athermal, bidisperse, soft-core, massive disks in two dimensions, as a function of packing fraction ϕ, inelasticity of collisions as measured by a parameter Q, and applied uniform shear strain rate γ[over ̇]. Our particles have contact interactions consisting of normally directed elastic repulsion and viscous dissipation, as well as tangentially directed viscous dissipation, but no interparticle Coulombic friction. Mapping the phase diagram in the (ϕ,Q) plane for small γ[over ̇], we find a sharp first-order rheological phase transition from a region with Bagnoldian rheology to a region with Newtonian rheology, and show that the system is always Newtonian at jamming. We consider the rotational motion of particles and demonstrate the crucial importance that the coupling between rotational and translational degrees of freedom has on the phase structure at small Q (strongly inelastic collisions). At small Q, we show that, upon increasing γ[over ̇], the sharp Bagnoldian-to-Newtonian transition becomes a coexistence region of finite width in the (ϕ,γ[over ̇]) plane, with coexisting Bagnoldian and Newtonian shear bands. Crossing this coexistence region by increasing γ[over ̇] at fixed ϕ, we find that discontinuous shear thickening can result if γ[over ̇] is varied too rapidly for the system to relax to the shear-banded steady state corresponding to the instantaneous value of γ[over ̇].

Shear thickening regimes of dense non-Brownian suspensions

We propose a unifying rheological framework for dense suspensions of non-Brownian spheres, predicting the onsets of particle friction and particle inertia as distinct shear thickening mechanisms, while capturing quasistatic and soft particle rheology at high volume fractions and shear rates respectively. Discrete element method simulations that take suitable account of hydrodynamic and particle-contact interactions corroborate the model predictions, demonstrating both mechanisms of shear thickening, and showing that they can occur concurrently with carefully selected particle surface properties under certain flow conditions. Microstructural transitions associated with frictional shear thickening are presented. We find very distinctive divergences of both microstructural and dynamic variables with respect to volume fraction in the thickened and non-thickened states.

Flow regime transitions in dense non-Brownian suspensions: rheology, microstructural characterization, and constitutive modeling

Shear flow of dense non-Brownian suspensions is simulated using the discrete element method taking particle contact and hydrodynamic lubrication into account. The resulting flow regimes are mapped in the parametric space of the solid volume fraction, shear rate, fluid viscosity, and particle stiffness. Below a critical volume fraction ϕ(c), the rheology is governed by the Stokes number, which distinguishes between viscous and inertial flow regimes. Above ϕ(c), a quasistatic regime exists for low and moderate shear rates. At very high shear rates, the ϕ dependence is lost, and soft-particle rheology is explored. The transitions between rheological regimes are associated with the evolving contribution of lubrication to the suspension stress. Transitions in microscopic phenomena, such as interparticle force distribution, fabric, and correlation length are found to correspond to those in the macroscopic flow. Motivated by the bulk rheology, a constitutive model is proposed combining a viscous pressure term with a dry granular model presented by Chialvo et al. [Phys. Rev. E 85, 021305 (2012)]. The model is shown to successfully capture the flow regime transitions.

Internal structure of inertial granular flows

We analyze inertial granular flows and show that, for all values of the inertial number I, the effective friction coefficient μ arises from three different parameters pertaining to the contact network and force transmission: (1) contact anisotropy, (2) force chain anisotropy, and (3) friction mobilization. Our extensive 3D numerical simulations reveal that μ increases with I mainly due to an increasing contact anisotropy and partially by friction mobilization whereas the anisotropy of force chains declines as a result of the destabilizing effect of particle inertia. The contact network undergoes topological transitions, and beyond I≃0.1 the force chains break into clusters immersed in a background "soup" of floating particles. We show that this transition coincides with the divergence of the size of fluidized zones characterized from the local environments of floating particles and a slower increase of μ with I.

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.

Relationship between interparticle contact lifetimes and rheology in gravity-driven granular flows

The validity of the Bagnold constitutive relation in gravity-driven granular flow down an inclined plane is studied by discrete element (DEM) simulations. In the limit of infinitely hard particles, the Bagnold relation is known to hold exactly. We determine deviations from this relation as a function of all parameters governing interparticle interactions. These include elastic compliance, inelastic dissipation, friction coefficient, and interparticle cohesion. We find significant deviations from Bagnold rheology in some regions of this parameter space and propose a generalized Bagnold relation to account for this effect. Moreover, we note a significant correlation between the breakdown of Bagnold rheology in the bulk and the appearance of a long-time tail in the two-particle contact time distributions.

Spatial force correlations in granular shear flow. I. Numerical evidence

We investigate the emergence of correlations in granular shear flow. By increasing the density of a simulated granular flow, we observe a transition from a dilute regime, where interactions are dominated by binary collisions, to a dense regime characterized by large force networks and collective motions. With increasing density, interacting grains tend to form networks of simultaneous contacts due to the dissipative nature of collisions. We quantify the size of these networks by measuring two-point force correlations and find dramatic changes in the statistics of contact forces as the size of the networks increases.