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John Drozd
Colin Denniston
Simulations of Collision Times In Gravity Driven Granular Flow
• bottom sieve• particles at bottom go to top• reflecting left and right walls• periodic or reflecting front and back walls
Snapshot of 2d simulation from paper: “Dynamics and stress in gravity-driven granular flow” Phys. Rev. E. Vol. 59, No. 3, March 1999 Colin Denniston and Hao Li
3d simulation
Velocity q
qqr
qrv
r
r
r
rn
2
1
2
1
'2
'1
11
111
2
1
.dissipated isEnergy
,1ˆ
ˆ
,
,11
:lossenergy thedetermines
n restitutio oft coefficiendependent velocity The
012
120
00
0
7.0
00
gavqvv
qvv
vv
vvv
vv
v
n
nn
n
n
1r2r
Collision rules for dry granular media asmodelled by inelastic hard spheres
As collisions become weaker(relative velocity vn small),they become more elastic.
300 (free fall region)
250 (fluid region)
200 (glass region)
150
Y VelocityDistribution
0 5 1015202530x
0.5
1
1.5
2
v y cy150
0 5 1015202530x
0.5
1
1.5
2
v y by2000 5 1015202530
x
1
2
3
4
v y ay250Plug flow
kinkfracture
Poiseuille flow
Simulation of 16 by 16 system was performed by summer student Nehal Al Tarhuni
2/3flowvv
Experiment by N. Menon and D. J. Durian, Science, 275, 1997.
Simulation results
Fluctuating and Flow Velocity
v2
vf
0.5 1 5 10vy
0.1
0.2
0.5
1
2
5
vy
In Glassy Region !
Simulation
Experiment
Quasi-1d Theory (Coppersmith, et al)
(Longhi, Easwar)
Impulse defined: Magnitude ofmomentumafter collisionminus momentum beforecollision.
Related to Forces:Impulse Distribution
Most frequent collisions contributing to smallest impulses
Power Laws for Collision Times
regionglassy in 2.87 to2.75 0.06 2.81
%15
grainssepolydisperP
Similar power laws for 2d and 3d simulations!
Collision time= time between collisions
0.01 0.02 0.05 0.1 0.2 0.5
1. 107
0.00001
0.001
0.1
10
P
1) spheres in 2d2) 2d disks3) 3d spheres
Comparison With Experiment
Figure from experimental paper:“Large Force Fluctuations in a Flowing Granular Medium”Phys. Rev. Lett. 89, 045501 (2002)E. Longhi, N. Easwar, N. Menon
: experiment 1.5 vs. simulation 2.8
Discrepancy as a result of Experimental response time and sensitivity of detector.
Experiment
“Spheres in 2d”:3d Simulation withfront and backreflecting wallsseparated onediameter apart
Pressure Transducer
P
15 12.5 10 7.5 5 2.5 0ln
10
8
6
4
2
0
2
nlI
= 2.75
= 1.50
Probability Distribution forImpulses vs. Collision Times (log scale)
random packingat early stage = 2.75
crystallization at later stage = 4.3
Is there any difference between this glass and a crystal? Answer: Look at Monodisperse grains
Disorder has a universal effect on Collsion Time power law.
Radius Polydispersity
2d disks Spheres in 2d 3d spheres
0 %
(monodisperse)
4 4.3 3.4
15 %
(polydisperse)
2.75 2.85 2.87
Summary of Power Laws