Diffusion and Mixing in Gravity-Driven Dense Granular Flows

Abstract

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 superdiffusive (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 Péclet 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.PACS numbers: PACS number(s): 45.70.Mg, Granular flow is an attractively simple and yet surprisingly complex subject [1]. Fast, dilute flows are known to obey classical hydrodynamics (with inelastic collisions), but slow, dense flows pose a considerable challenge to theorists, due to many-body interactions and non-thermal fluctuations. Beyond their fundamental scientific interest, such flows have important engineering applications [2], e.g. to new pebble-bed nuclear reactors [3], whose efficiency and safety depend on the degree of mixing in very slow granular drainage (< 1 pebble/min).Although dense granular drainage is very familiar (e.g. sand in an hourglass), it is far from fully understood. Over the past forty years, a number of theoretical approaches have been proposed for steady state flow [4,5,6,7,8]. Continuum approaches are based on the critical-state theory of soil mechanics and yield only mean velocity fields [7,8]. On the other hand, the diffusive void model [4, 5] takes a particle approach, in which 'voids' injected at the orifice cause drainage by diffusing upward and exchanging position with particles along the way. Averaging over the void trajectories yields the same continuum velocity field for particles as the 'kinematic model ' [6, 8], which provides a reasonable fit to experimental data with only one fitting parameter (the diffusion length, b) [5,6,9,10], although the void model on a regular lattice (as in Ref. [11]) underpredicts its value [12]. Remarkably, these models depend only on geometry and not on momentum, energy, etc.In spite of the success of the kinematic model, however, its only microscopic basis, the void model, greatly overpredicts diffusion. To see this, consider the Péclet number, Pe x = v z d/D x , the dimensionless ratio of advection in uniform downward flow of speed, v z , to diffusion with a horizontal diffusivity, D x , at the scale of a particle diameter, d. In the void model, when a particle falls a distance ∆h, Pe x = (∆h/∆t)d/( ∆x 2 /2∆t) = ∆h 2d/ ∆x 2 , is of order one for any conceivable packing since ∆h ≈ ∆x ≈ d, and therefore it diffuses horizontally by roughly √ ∆h. This prediction is contradicted by ev- eryday experience and our experiments below, which exhibit far less mixing. ...

Velocity profile of granular flows inside silos and hoppers

“…Recent experiments, however, have firmly rejected the void hypothesis. On the other hand, an alternative stochastic description, the Spot Model [31,32], which starts from a cooperative mechanism for random-packing rearrangements, roughly preserves the mean flow profile of the Kinematic Model, with much less diffusion and slow cage breaking, consistent with experiments [30].…”

Resolving a paradox of anomalous scalings in the diffusion of granular materials

“…In a sense, the two approaches were comparing "apples to oranges" because, as we have now demonstrated, the system could have been evolving towards another intermediate asymptotic state, thus the expected diffusive scaling of the concentration profiles had not been reached. ** In other granular systems, evidence has been presented for subdiffusive processes arising from granular force-chain networks (52), dynamic heterogeneity (53), enduring contacts (14) and caging effects (17). The conclusion of the present work is that evaluating anomalous diffusion in the tumbler system through macroscopic measurements requires careful consideration and, possibly, further experiments to establish definitively because the original experiments of ref.…”

“…A common mathematical approach to such processes is the framework of fractional partial differential equations that feature noninteger order derivatives (12). In the last decade, anomalous diffusion has been observed or suggested to be present in phenomena as diverse as tracer motion in chaotic laminar flows (13), drainage of particulates from a silo (14), protein and chromosome transport through cellular membranes (15,16), shaken granular media (17), motion of colloids in entangled actin networks (18), flow through disordered porous media (19) and across human transportation networks (20). Here, we focus on an apparent paradox of anomalous diffusion in agitated granular materials, and propose an explanation for the experimental observations without resorting to the framework of fractional partial differential equations.…”

Statistical properties of gravity-driven granular discharge flow under the influence of an obstacle

“…e-mail: katsurag@eps.nagoya-u.ac.jp Statistics of granular discharge flow has been studied by several research groups [9][10][11]. According to these works, velocity distribution of particle motion is not Gaussian and diffusive motion can be observed in horizontal direction.…”