Growing length scale in gravity-driven dense granular flow.
ABSTRACT We report simulations of a two-dimensional, dense, bidisperse system of inelastic hard disks falling down a vertical tube under the influence of gravity. We examine the approach to jamming as the average flow of particles down the tube is slowed by making the outlet narrower. Defining coarse-grained velocity and stress fields, we study two-point temporal and spatial correlation functions of these fields in a region of the tube where the time-averaged velocity is spatially uniform. We find that fluctuations in both velocity and stress become increasingly correlated as the system approaches jamming. We extract a growing length scale and time scale from these correlations.
arXiv:0806.2413v1 [cond-mat.soft] 15 Jun 2008
Growing length scale in gravity-driven dense granular flow
Shubha Tewari∗and Bidita Tithi
Department of Physics, Mount Holyoke College,
50 College Street, South Hadley, MA 01075
Department of Biochemistry, University of Toronto, Toronto, Ontario M5S 1A8
Martin Fisher School of Physics, Brandeis University,
Mailstop 057, Waltham, MA 02454-9110
(Dated: June 15, 2008)
We report simulations of a two-dimensional, dense, bidisperse system of inelastic hard disks
falling down a vertical tube under the influence of gravity. We examine the approach to jamming
as the average flow of particles down the tube is slowed by making the outlet narrower. Defining
coarse-grained velocity and stress fields, we study two-point temporal and spatial correlation func-
tions of these fields in a region of the tube where the time-averaged velocity is spatially uniform.
We find that fluctuations in both velocity and stress become increasingly correlated as the system
approaches jamming. We extract a growing length scale and time scale from these correlations.
PACS numbers: 45.70.-n, 81.05.Rm, 83.10.Pp
∗Electronic address: email@example.com
†Electronic address: firstname.lastname@example.org
Granular materials (powders, seeds, grains, sand) consist of macroscopic particles that
interact via dissipative short-ranged or contact forces . Granular systems are athermal,
since the characteristic energies needed to move a single grain are many orders of magnitude
larger than thermal energies. In the absence of external forces, there is no motion; and the
state of the system under the application of external forces varies with the magnitude of the
force as well as with the packing density of the grains. In this paper, we will describe results
from numerical simulations of the dense gravity-driven flow of grains down a vertical tube,
as in an hourglass, and the transition from a flowing state to one that is stuck, or jammed.
The questions addressed here fall within the rubric of a proposal  to unify disparate
systems under a common framework, that posits that there are some universal aspects to the
slowing down of the dynamics in many disordered systems as they move from a mobile state
to one that is frozen; such a transition is labeled a jamming transition. There is no static
structural signature of the transition from a mobile to a jammed state: unlike first order
freezing transitions, there is no discontinuous change in the density or broken translational
symmetry. Nor is there a clear signature of a diverging length scale derived from a two-
point static correlation function as in a second-order phase transition, though there are other
indications of a diverging length scale as a critical density is approached [3, 4, 5].
A dense column of grains in a vertical hopper flows down at a steady rate rather than
accelerating under gravity because the weight of the column is supported by the walls. The
rate of flow decreases as the size of the opening at the outlet of the hopper is decreased,
and ultimately jams when the opening is a few particle diameters across. It is well known
that the distribution of load in a static column of sand is spatially inhomogeneous and
organized along linear structures called force chains [6, 7]. The question remains open
as to whether these structures begin to form in the flowing state as the flow slows. In
this article, we present evidence from simulations for increasing spatial correlations in both
velocity and stress fluctuations as the flow rate decreases. We extract a length scale from
these correlations, and find that the flow rate dependence of the length scales for velocity
and stress are in exact correspondence. Our results agree very well with recent experimental
observations  of growing spatial correlations in velocity fluctuations as the flow in a vertical
hopper approaches the jamming threshold, and help clarify how these correlations arise in
a flow that is dense, continuous, and highly collisional.
There have been many efforts towards extracting a length scale in granular systems.
Inhomogeneous force chains were visualized in sheared systems using photoelastic beads 
and their spatial correlations quantified  - these observations have been primarily in
the quasistatic regime, where the beads stay in contact rather than undergoing collisions.
Force measurements using a photoelastic plate at the base of a sheared, cylindrical pack of
beads found a change in the distribution of forces as jamming was approached . Earlier
measurements in flow in a rotating drum  found evidence of clustering, but with a
power law distribution of cluster sizes and hence no chosen length scale. Growing spatial
correlations were seen at the free surface of chute flow down a plane , with a length scale
of the order of a few grain diameters. Previous simulations of flow in a vertical tube geometry
indicated an inhomogeneous distribution of stresses . Earlier results  on the same
simulations we report on in this article indicated that the most frequently colliding particles
organize into chain-like structures that form repeatedly and break up as the particles move
down the hopper. It was found that the chain direction coincided with the principal axis of
the collisional stress in the system, with the lifetime of correlations in the stress fluctuations
increasing with decreasing flow rate . There was quantitative agreement between the
simulational results and experiments in a two-dimensional hopper geometry [8, 17].
The absence of a clear structural signature of the approach to jamming has led various
groups to look for spatial inhomogeneities in the dynamics. This approach was pioneered in
structural glasses where a variety of techniques probing local response functions showed 
that the dynamics of a sample became increasingly heterogeneous near the glass transition.
Two point correlation functions did not show clear signatures of heterogeneity, however,
an analysis of four-point correlation functions in simulations of supercooled liquids [19, 20]
showed evidence for growing spatial correlations between localized density autocorrelations.
Evidence for dynamic heterogeneity was found in experiments on colloidal glasses, in which
highly mobile particles were found to cluster , with a cluster size that increased as
the glass transition was approached. Heterogeneous dynamics have now also been seen
in experiments on dense granular material under shear . In our simulations as well,
spatial heterogeneities were found in the mean-squared displacements of particles when a
method  of maximizing the difference between highly mobile and less mobile regions of the
sample was used . While the size of the spatial heterogeneity varied between five and six
particle diameters as a function of flow velocity, the ”cage-size” or length scale over which the
heterogeneity was maximized did increase as the flow velocity decreased towards jamming.
However this increase in length scale was typically smaller than a particle diameter, and the
connection with the collisional dynamics and force chains was not clear.
In the current paper, we analyze the development of spatial correlations in both kinetic
and dynamical variables in the flowing state, and show that the extent of these increases as
jamming is approached. We would like to emphasize that the changing length scale is seen
in the two-point correlation functions of the velocity and stress. We also draw qualitative
connections between these correlations and the chains of frequently colliding particles.
In the sections to follow, we first describe the simulation, and our method of defining
coarse-grained velocity and stress fields. We then discuss our results for the time-averaged
fields and the temporal and spatial correlations in both velocity and stress fluctuations, and
conclude with a discussion of our results.
II. DESCRIPTION OF SIMULATION
The results we describe here are obtained from a two-dimensional event-driven simulation
of bidisperse hard disks falling under the influence of gravity in a vertical hopper. We use
the same particle dynamics as Denniston and Li , and have described our setup in some
detail in an earlier paper . To summarize: the inter-particle collisions are instantaneous
and inelastic, and there is no friction between the particles. As a result, momentum transfer
between colliding particles always occurs along the vector separating their centers. The
relative velocity between colliding particles i and j is reduced by a coefficient of restitution
µ, defined in the usual way:
i).ˆ q = −µ(uj− ui).ˆ q (1)
iare the particle velocities after the collision, and ˆ q is a unit vector along
the line separating the centers of the particles. Frictional effects at the wall are simulated by
introducing a coefficient of restitution µwallin the tangential direction: the loss of vertical
momentum at the walls is what allows the flow to reach a steady state. In order to avoid
inelastic collapse, all collisions become elastic when the relative velocity at the collision is
below a certain threshold ucut. A particle exiting the base has a probability p of being
reflected, else it exits the system and is re-introduced at the top. The flow rate of particles
in steady state is controlled by the size of the opening at the base. The results described
here are for a simulation of 1000 particles of diameter 1 and 1.2 respectively, where grains
are chosen at random to have one or the other size. The other simulation parameters are
µ = 0.8, µwall= 0.5, p = 0.5 and ucut= 10−3, and the mass of the smaller grains is set
to 1. Lengths are expressed in units of the smaller particle diameter. In these units, the
rectangular region of the hopper has width 20 and height 76.5. The simulation is run for
a total of 1000 simulation time steps, of which the first 500 are discarded. In units of the
simulation time, the average time between collisions for a given particle is on the order of
Earlier results reported on these simulations [15, 16] were based on a particle-based
analysis of the system as the size of the opening, or the flow rate, was decreased. Over a
timescale larger than a typical collision time but shorter than the time taken for a particle
to fall through its own diameter, particles with the highest frequency of collisions appear to
repeatedly organize into linear structures that form and break. These structures were shown
to carry much of the collisional stress , and their lifetime increased with decreasing flow
rate, but no evidence was found for a growing length scale.
In this paper, we seek to go beyond the particle-level analysis of the system in order to
quantify the correlations signaled by the frequently colliding chains of particles and look for
indications of increasing order in the system as it approaches jamming. Thus in the work
described here, we have constructed coarse-grained variables, looking at the system in terms
of velocity, stress and density fields and the spatial and temporal variations of these fields.
We also view this approach as a useful first step in developing a continuum description of
A. Coarse-grained Fields
The system area is divided into square boxes of side equal to two particle diameters. We
define a box velocity