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

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

Distribution of Impulses from "spheres in 2d" simulation

fluid

glass

glass

Exponentof 4

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

Conclusions

• Power laws dependent on disorder.• Using power laws can differentiate

between granular fluid and glass.• Can differentiate between granular

glass and crystal! Is this universal?