Abstract:
The study of granular materials has received a recent upsurge of interest in physics.
This has been motivated by both the relevance of such flows to a wide range of
industrial and geological processes, and by the realization that granular materials
provide an excellent test bed for a number of fundamental questions in the context
of modern fluid dynamics and nonequlibrium statistical mechanics. Most of the
theories of granular fluid are based on the Boltzmann or Enskog-Boltzmann equation of inelastic hard spheres. In this regard particle dynamics simulations make a
bridge between theory and experiment.
An event-driven molecular dynamics code has been developed which is based
on the paper of Lubachevsky (1991) and the book by Rapaport (1995). The computational approach employed in this study mainly consists of initialization, bookkeeping and diagnostic parts. This code is very fast and can handle a large number
of particles (A'^ ~ 10^) and has been generalized to three dimensions. It can be
used to simulate various kind of plane granular flows (Couette flow, Poiseuille flow,
Chute flow, etc.) with rough, frictional particles. For the simplest model of rough,
inelastic spheres, two material parameters are needed to characterize the collision
process: the normal coefficient of restitution, e, and the tangential coefficient of
restitution, /?. For a more realistic collision model of rough particles, we have taken
into account the effect of Coulomb friction which helps to distinguish between the
sliding and the rolling contacts, resulting in a contact-angle (7) dependent tangential restitution coefficient (3{'y).
With the above code, we have probed various microscopic and macroscopic
propreties of unbounded granular shear flow for which a steady linear velocity
profile is applied with a constant velocity gradient via shifting the replica images
of top and bottom boundaries without deforming the box (Lees-Edwards boundary condition). This code has been tested for bounded shear flow too for which
physical boundaries arc required: wall-particle interactions are modeled using the
same collision dynamics of particle-particle interaction. For either case, the system
is allowed to reach a 'non-equilibrium' steady-state condition by monitoring the
temporal evolution of system's kinetic energy.
In this work we have thoroughly examined the effects of particle roughness and
rotation on the probability distributions of fluctuating translational and rotational
velocities as well as density and spatial velocity correlations in the Boltzmann
limit (for which the particle volume fraction, 0, is close to zero) of unbounded
shear flow. We found that both translational and rotational velocity distribution
functions (VDF) arc non-Gaussian with stretched exponential tails, except in the
limiting cases of perfectly smooth (/? = —1) and rough (0—1) particles with elastic
collisions (e = 1). One important finding is that the translational and rotational
velocities are correlated in direction. Particle roughness has important effects on
oricntational and velocity correlations even when the collisions are perfectly elastic
(e = 1) and the system is homogeneous. Oricntational and spatial velociy correlation are responsible for non-Gaussian distributions of translational and rotational
velocities. With increasing system density, the dissipation-induced density inhomogeneity
is observed over the whole domain. A pronounced asymmetry about the mean
value is observed for the probability distributions of local density, local shear rate
and local spanwise rotational velocity. Therefore the calculation of "local" VDF
is a proper way to study such inhomogeneous systems. For a moderately dense
system {(() — 0.3), an interesting phenomenon is observed for the local VDF of
streamwise translational velocity, its tails undergo a transition from an stretched
exponential to an under-populated Gaussian distribution with decreasing dissipation and finally to a Gaussian for no dissipation. The VDF of spanwise rotational
velocity makes a transition from stretched exponential tails to a Gaussian with
decreasing dissipation. For the dense system {4> = 0.5) with dissipation, the VDF
for streamwise translational velocity is a Gaussian with under-populated tails. The
effect of Coulomb friction on VDFs has been studied for different values of friction
coefficient ^ ior (p = 0.3 with the critical roughness being set to Po — 0. With
the incorporation of Coulomb friction, a pronounced asymmetry of the tails of the
VDFs of rotational velocities is seen and the skewness of the distribution increases
with increasing dissipation.Lastly, we have calculated some rheologoical properties of unbounded shear
flow of rough, frictional particles. Our simulations results on pressure and shear
viscosity compare well with the predictions of Lun's (1991) rheological model at
small dissipations. The model predictions deteriorate with increasing dissipation
which is tied to the inherent assumptions of the underlying model which is valid for
quasi-elastic (e ~ 1) particles in the prefectly smooth and rough (|/3| ~ 1) limits.
Our results on normal stress differences (A/i and A/'2) suggest that a non-Newtonian
constitutive model is required for moderately dissipative rough, frictional particles.