Self-diffusion in dense granular shear flows

Diffusivity is a key quantity in describing velocity fluctuations in granular materials. These fluctuations are the basis of many thermodynamic and hydrodynamic models which aim to provide a statistical description of granular systems. We present experimental results on diffusivity in dense, granular shear in a 2D Couette geometry. We find that self-diffusivities are proportional to the local shear rate with diffusivities along the mean flow approximately twice as large as those in the perpendicular direction. The magnitude of the diffusivity is D \approx \dot\gamma a^2 where a is the particle radius. However, the gradient in shear rate, coupling to the mean flow, and drag at the moving boundary lead to particle displacements that can appear sub- or super-diffusive. In particular, diffusion appears superdiffusive along the mean flow direction due to Taylor dispersion effects and subdiffusive along the perpendicular direction due to the gradient in shear rate. The anisotropic force network leads to an additional anisotropy in the diffusivity that is a property of dense systems with no obvious analog in rapid flows. Specifically, the diffusivity is supressed along the direction of the strong force network. A simple random walk simulation reproduces the key features of the data, such as the apparent superdiffusive and subdiffusive behavior arising from the mean flow, confirming the underlying diffusive motion. The additional anisotropy is not observed in the simulation since the strong force network is not included. Examples of correlated motion, such as transient vortices, and Levy flights are also observed. Although correlated motion creates velocity fields qualitatively different from Brownian motion and can introduce non-diffusive effects, on average the system appears simply diffusive.Comment: 13 pages, 20 figures (accepted to Phys. Rev. E

Investigation of the effect of a bumpy base on granular segregation and transport properties under vertical vibration

This study experimentally investigates the effect of a bumpy base on the Brazil-nut phenomenon in a vertically vibrated granular bed. The rise dynamics of an intruder is determined by the particle tracking method. The results indicate that the rise time increases with an increase in the base roughness, and the variation of the rise time with different base factors is more pronounced with smaller vibration acceleration and higher vibration frequency. A theoretical model is employed to measure the penetration length of the intruder and the drag force between the intruder and the immersed beads. The penetration length is reduced and the drag force is enhanced with surface roughness of the base. Additionally, the transport properties of the vibrated glass beads are also measured and discussed. With greater base roughness, the strength of the diffusive and convective motion is reduced leading to a weaker Brazil-nut effect

Shear-induced particle diffusion and longitudinal velocity fluctuations in a granular-flow mixing layer

In flows of granular material, collisions between individual particles result in the movement of particles in directions transverse to the bulk motion. If the particles were distinguishable, a macroscopic overview of the transverse motions of the particles would resemble a self-diffusion of molecules as occurs in a gas. The present granular- flow study includes measurements of the self-diffusion process, and of the corresponding profiles of the average velocity and of the streamwise component of the fluctuating velocity. The experimental facility consists of a vertical channel fed by an entrance hopper that is divided by a splitter plate. Using differently-coloured but otherwise identical glass spheres to visualize the diffusion process, the flow resembles a classic mixing-layer experiment. Unlike molecular motions, the local particle movements result from shearing of the flow; hence, the diffusion experiments were performed for different shear rates by changing the sidewall conditions of the test section, and by varying the flow rate and the channel width. In addition, experiments were also conducted using different sizes of glass beads to examine the scaling of the diffusion process. A simple analysis based on the diffusion equation shows that the thickness of the mixing layer increases with the square-root of downstream distance and depends on the magnitude of the velocity fluctuations relative to the mean velocity. The results are also consistent with other studies that suggest that the diffusion coefficient is proportional to the particle diameter and the square-root of the granular temperature

Diffusion and mixing in gravity-driven dense granular flows

We study the transport properties of particles draining from a silo using imaging and direct particle tracking. The particle displacements show a universal transition from super-diffusion to normal diffusion, as a function of the distance fallen, independent of the flow speed. In the super-diffusive (but sub-ballistic) regime, which occurs before a particle falls through its diameter, the displacements have fat-tailed and anisotropic distributions. In the diffusive regime, we observe very slow cage breaking and Peclet numbers of order 100, contrary to the only previous microscopic model (based on diffusing voids). Overall, our experiments show that diffusion and mixing are dominated by geometry, consistent with fluctuating contact networks but not thermal collisions, as in normal fluids

Velocity profile of granular flows inside silos and hoppers

We measure the flow of granular materials inside a quasi-two dimensional silo as it drains and compare the data with some existing models. The particles inside the silo are imaged and tracked with unprecedented resolution in both space and time to obtain their velocity and diffusion properties. The data obtained by varying the orifice width and the hopper angle allows us to thoroughly test models of gravity driven flows inside these geometries. All of our measured velocity profiles are smooth and free of the shock-like discontinuities ("rupture zones") predicted by critical state soil mechanics. On the other hand, we find that the simple Kinematic Model accurately captures the mean velocity profile near the orifice, although it fails to describe the rapid transition to plug flow far away from the orifice. The measured diffusion length $b$, the only free parameter in the model, is not constant as usually assumed, but increases with both the height above the orifice and the angle of the hopper. We discuss improvements to the model to account for the differences. From our data, we also directly measure the diffusion of the particles and find it to be significantly less than predicted by the Void Model, which provides the classical microscopic derivation of the Kinematic Model in terms of diffusing voids in the packing. However, the experimental data is consistent with the recently proposed Spot Model, based on a simple mechanism for cooperative diffusion. Finally, we discuss the flow rate as a function of the orifice width and hopper angles. We find that the flow rate scales with the orifice size to the power of 1.5, consistent with dimensional analysis. Interestingly, the flow rate increases when the funnel angle is increased.Comment: 17 pages, 8 figure

Tracer diffusion in granular shear flows

Tracer diffusion in a granular gas in simple shear flow is analyzed. The analysis is made from a perturbation solution of the Boltzmann kinetic equation through first order in the gradient of the mole fraction of tracer particles. The reference state (zeroth-order approximation) corresponds to a Sonine solution of the Boltzmann equation, which holds for arbitrary values of the restitution coefficients. Due to the anisotropy induced in the system by the shear flow, the mass flux defines a diffusion tensor $D_{ij}$ instead of a scalar diffusion coefficient. The elements of this tensor are given in terms of the restitution coefficients and mass and size ratios. The dependence of the diffusion tensor on the parameters of the problem is illustrated in the three-dimensional case. The results show that the influence of dissipation on the elements $D_{ij}$ is in general quite important, even for moderate values of the restitution coefficients. In the case of self-diffusion (mechanically equivalent particles), the trends observed in recent molecular dynamics simulations are similar to those obtained here from the Boltzmann kinetic theory.Comment: 5 figure

Diffusion in a Granular Fluid - Theory

Many important properties of granular fluids can be represented by a system of hard spheres with inelastic collisions. Traditional methods of nonequilibrium statistical mechanics are effective for analysis and description of the inelastic case as well. This is illustrated here for diffusion of an impurity particle in a fluid undergoing homogeneous cooling. An appropriate scaling of the Liouville equation is described such that the homogeneous cooling ensemble and associated time correlation functions map to those of a stationary state. In this form the familiar methods of linear response can be applied, leading to Green - Kubo and Einstein representations of diffusion in terms of the velocity and mean square displacement correlation functions. These correlation functions are evaluated approximately using a cumulant expansion and from kinetic theory, providing the diffusion coefficient as a function of the density and the restitution coefficients. Comparisons with results from molecular dynamics simulation are given in the following companion paper

Diffusion of impurities in a granular gas

Diffusion of impurities in a granular gas undergoing homogeneous cooling state is studied. The results are obtained by solving the Boltzmann--Lorentz equation by means of the Chapman--Enskog method. In the first order in the density gradient of impurities, the diffusion coefficient $D$ is determined as the solution of a linear integral equation which is approximately solved by making an expansion in Sonine polynomials. In this paper, we evaluate $D$ up to the second order in the Sonine expansion and get explicit expressions for $D$ in terms of the restitution coefficients for the impurity--gas and gas--gas collisions as well as the ratios of mass and particle sizes. To check the reliability of the Sonine polynomial solution, analytical results are compared with those obtained from numerical solutions of the Boltzmann equation by means of the direct simulation Monte Carlo (DSMC) method. In the simulations, the diffusion coefficient is measured via the mean square displacement of impurities. The comparison between theory and simulation shows in general an excellent agreement, except for the cases in which the gas particles are much heavier and/or much larger than impurities. In theses cases, the second Sonine approximation to $D$ improves significantly the qualitative predictions made from the first Sonine approximation. A discussion on the convergence of the Sonine polynomial expansion is also carried out.Comment: 9 figures. to appear in Phys. Rev.