Transcript Collision Times and Stress in Gravity Driven Granular Flow

Simulations of Collision Times and Stress
In Gravity Driven Granular Flow
John Drozd
Colin Denniston
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bottom sieve
particles at bottom go to top
reflecting left and right walls
periodic or reflecting front
and back walls
3d simulation 
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
Outline
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Profiles
Mechanics
Stress
Momentum
Impulse
Collision times
Velocity
Collision rules for dry granular media as
modelled by inelastic hard spheres
r1
q
 r1'   r1  1
 1 1   r1  q 

      1   v n 
 1  1  r  q  q
 r '   r  2

 2 
 2  2
T he velocit ydependentcoefficient of restitution  v n 
determinestheenergy loss :
0.7

v 
1  1   0  n  , v n  v 0
 v n   
 v0 

0 ,
vn  v0


v2  v1   qˆ
0 
v 2  v1   qˆ
 1 , v 0  ga
Energyis dissipated.
r2
300 (free fall region)
250 (fluid region)
200 (glass region)
150
300 (free fall region)
Y Velocity
Distribution
vy
4
3
2
Poiseuille flow
1
a y 250
0 5 10 15 20 25 30
x
250 (fluid region)
2
vy
1.5
Plug flow
1
0.5 b y 200
200 (glass region)
0 5 1015202530
x
2
vy
1.5
1
kink
fracture
0.5 c y 150
0 5 1015202530
x
150
300 (free fall region)
y 250
Granular
Temperature
vx2
28
vy
30
vy 2
32
fluid
26
250 (fluid region)
24
5
10
15
20
x
y 235
2
25
30
235 (At Equilibrium
Temperature)
vx2
1.75
1.5
vy 2
1.25
1
0.75
200 (glass region)
vy
equilibrium
0.5
0.25
0
15
20
x
y 200
25
30
3
2
1
vx2
7
6
5
4
0.35
vy 2
0.4
0.3
vy
12
vx2
2
vy
10
x 16
8
vy
5
0.25
150
glass
0.2
0
100
200
y
300
400
5
10
15
20
x
25
30
Fluctuating and Flow Velocity
Experiment by
N. Menon and D. J. Durian,
Science, 275, 1997.
 v v
1
0.5
2/3
flow
v2
vy2
0.2
0.1
Simulation results
by summer student
0.05
0.02
Nehal Al Tarhuni
0.01
0.5
1
2
vy
vf
5
10
Momentum Conservation
kik+gi = t(vi)+k(vivk)
Normal Stresses Along Height
kik+gi = 0, vv
 x yx   y yy   g
0
2
yy
4
6
8
10
12
0
Weight not supported
by a pressure gradient.
100
 y yy
200
y
x
300
0
400
Momentum Conservation
3
y y y,
2
1
0
1
x yx
xy
g
y 100
 x yx   y yy   g
2
3
5
 x xy
10 15 20 25 30
x
y
y 100
0.25
0.2
0.15
0.1
0.05
5
10 15 20 25
x
 0 Weightsupportedby shear stress
Shear Stress
 xy
1
1
ˆ q
ˆ  xˆ q
ˆ  yˆ 

 1    r1  r2   q

t collisions 2
1
ˆ q
ˆ  xˆ q
ˆ  yˆ 
  f c 1   r1  r2   q
2
y 100
y 100
3
0.02
2
0.01
q.x q.y
xy
1
0
0
1
0.01
2
0.02
3
5
10 15 20 25 30
x
slope 0.15554674  x xy
5 10 15 20 25 30
x
slope  0.00165986
1
ratio of slopes   fc 1    (r1  r2 )  q
2
Experimental data from the book:
“Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials”
By Jacques Duran.
Related to Forces:
Impulse Distribution
Simulation 
Impulse defined:
Magnitude of
velocity
after collision
minus velocity before
collision.
Experiment 
(Longhi, Easwar)
Quasi-1d Theory 
(Coppersmith, et al)
Most frequent collisions
contributing to smallest
impulses
Distribution of Impulses from "spheres in 2d" simulation
1
glassy region
0.5
0
Log10 P I
0.5
1
fluid region
1.5
Upturn in glassy region
Downturn in fluid region
2
2.5
0.2
0.4
0.6
Log10 I
0.8
1
Collision time distribution
P(t) = probability that a molecule survives a
time t without collision
P(t)
1
?
0
t
dt
w dt = the probability that a molecule suffers a collision between time t and t+dt
w is independent of past history (assumption of molecular chaos)
Hence
P(t+dt) = P(t) [probability that molecule
avoids collision in interval dt]
= P(t) (1- w dt)
Thus
P(t ) 
dP
dt  P(t )  P(t ) w dt
dt
P t
1
or
1 dP
 w
P dt
0.8
0.6
0.4
So that
0.2
P(t )  e
 wt
The “collision time” or “relaxation time” is
1

w
0.5
1
1.5
2
2.5
3
t
N
Power Laws for Collision Times
1.
10 8
1.
10 6
10000
100
Collision time
= time between
collisions
2d disks
spheres
1) 2d3ddisks
2) spheres in 2d
3) 3d spheres
1
0.005 0.01
0.05
0.1
0.5
1
N   
polydispese
r grains
 (2d disks / spheresin 2d)  2.75/2.85
 (3d spheres) 3.4

Similar power laws for 2d and 3d simulations!
Comparison With Experiment
: experiment 1.5 vs. simulation 2.7
Discrepancy as a result of
Experimental response time and
sensitivity of detector.
 Experiment
Pressure Transducer
“Spheres in 2d”:
3d Simulation with
front and back
reflecting walls
separated one
diameter apart 

Figure from experimental paper:
“Large Force Fluctuations in a Flowing Granular Medium”
Phys. Rev. Lett. 89, 045501 (2002)
E. Longhi, N. Easwar, N. Menon
N    
5
ln I
10
15
20
20
15
10
ln
5
0
5
1. 108
 = 2.9 0.2  2.7
100
0.01
0.0001
10
1. 10
Wall collisions only
N
PI
1
1.
6
1) grain grain
2) grain wall
10000
100
 = 1.7  0.2  1.5
6
1
10
100
I
1000
10000
1
0.0001
0.001
0.01
0.1
1
random packing
at early stage
 = 2.75
Is there any
difference between
this glass and a solid?
Answer: Look at
Monodisperse grains
crystallization 
at later stage
 = 4.3
Disorder has a universal effect on
Collsion Time power law.
Summary of Power Laws
Radius
Polydispersity
0%
(monodisperse)
2d disks
Spheres in 2d
3d spheres
4
4.3
3.4
7.5 %
(polydisperse)
2.75
2.85
3.4
15 %
(polydisperse)
2.75
2.85
3.4
Conclusions
• Power laws dependent on disorder.
• Using power laws can differentiate
between granular fluid and glass.
• Can differentiate between granular
glass and solid! Is this universal?
Not done before!
• Dynamics (collision directions) lead to
static stresses.
Future Work
• Plot impulse distributions along height for 3d
simulation to fully compare to 2D results
• Power law for distances between collisions
• Viscosity vs. Temperature power law
(Lubensky)
• Constitutive stress relations (Cates)
• Chaos and diffusion
(3D Tetrahedral Voronoi cells)
• Heat flux analysis for 3D simulation
THE END