Transcript Granular Materials: A window to studying the Transition

Computer Simulation of Gravity-Driven
Granular Flow
Western Canadian Conference for Young Researchers in Mathematics
University of Western Ontario
Department of Applied Mathematics
John Drozd and Dr. Colin Denniston
Granular Matter
• Granular matter definition
– Small discrete particles vs. continuum
• Granular matter interest
– Biology, engineering, geology,
material science, physics.
– Mathematics and computer science.
• Granular motion
– Energy input and dissipation.
• Granular experiments
– Vibration
Small Amplitude
Surface Waves

Large Amplitude
Surface Waves

3-Node Arching

C. Wassgren et al. 1996
Simulation
Other Phenomena in Granular
Materials
•
•
•
•
•
•
Shear flow
Vertical shaking
Horizontal shaking
Conical hopper
Rotating drum
Cylindrical pan
Cylindrical Pan Oscillations
Oleh Baran et al. 2001
Vibratory Drum Grinder
Raw Feed
Vibratory Drum Grinder
Grinding Media,
Rods
Reactor
Springs
Vibrator
Motor
Isolation
Spring
Ground
Product
Goal
• Find optimum oscillation that results in a
force between the rods which achieves the
ultimate stress of a particular medium that is
to be crushed between the rods.
• Minimize the total energy required to grind
the medium.
• Mixing is also important.
Event-driven Simulation Without
Gravity
Event-driven Simulation With
Gravity
Typical Simulation
Harry
Swinney
et al. 1997
Harry Swinney et al. 1997
Granular Materials: A
window to studying the
Transition from a nonNewtonian Granular
Fluid To A "Glassy"
system
Savage and Jeffrey
J. Fluid Mech. 130, 187, 1983.
Collision rules for dry granular media as
modelled by inelastic hard spheres
Velocity
 r1'   r1  1   v n    m 2
   

 r '   r  m  m   m
1
1
2 
 2  2
r1
q
r2
m 2   r1  q 
 
 q
 m1   r2  q 
T he velocitydependentcoefficient of restit ution  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
As collisions become weaker
(relative velocity vn small),
they become more elastic.
 1 , v 0  ga , a  radiu sof particle
Energyis dissipated.
C. Bizon et. al., PRL 80, 57, 1997.
1
a
0.98
0.96
0.94
0.92
0.9
6
b
5
vn
4
3
2
1
0
50
100
150
200
250
300 (free fall region)
0.7
a
0.6
Donev et al PRL96
"Do Binary Hard Disks
Exhibit an Ideal Glass
Transition?"
250 (fluid region)
0.5

0.4
polydisperse
monodisperse
0.3
0.2
0.1
20.
b
free fall
10.
fluid
vy
vy
glass
15.
5.
20.
c
1
dvy dt
15.
dvy/dt
200 (glass region)
10
0.1
0.01
10.
0
5.
100
200
300
0.
0
P
P
0.6
2
0.4
v
P
2
0.8
150
d
y
0.2
4
0.2
6
0.4
1
0.6
c
8
0
100
200
y
300
Donev et al PRE 71 P  D k BT1  / c 1
400
y
x
z
Polydispersity means
Normal distribution
of particle radii
The density in the glassy region is a constant.
In the interface between the fluid and the glass does the
density approach the glass density exponentially?
0 = 0.9
0 = 0.95
Interface width seems to increase as 0  1
0 = 0.99
0.7
a
0.6
0.5

0.4
0.3
0.2
0.1
20.
10.
fluid
glass
vy
vy
free fall
b
15.
5.
20.
dvy dt
15.
10.
5.
0.
2.5
10
1
y
0.1
0.01
0
c
  c  A exp y
100 200 300
How does  depend on (1  0) ?
Density vs Height in Fluid-Glass
Transition
  c  A exp y
0
1
0
c
0
Log10
2
0
0
3
0
0
0
4
0
0
160
180
200
220
y
240
260
280
300
0.8
0.9
0.95
0.96
0.97
0.98
0.99
0.995
0.999
Length Scale in Transition
  c  A exp y
  1  0 
0.44
Slope = 0.42 poly
Slope = 0.46 mono
"interface width diverges"
300 (free fall region)
Y Velocity
Distribution
vy
3
2
Poiseuille flow
1
a y 250
250 (fluid region)
0 5 10 15 20 25 30
x
2
vy
1.5
1
Plug flow
snapshot
200 (glass region)
0.5 b y 200
0 5 1015202530
x
2
1.5
vy
Mono-disperse
(crystallized)
only
4
1
Mono
kink
fracture
150
y
0.5 c y 150
0 5 1015202530
x
x
z
300 (free fall region)
y 250
35
vx2
vy
20
15
vy
25
2
30
10
fluid
5
250 (fluid region)
0
5
10 15 20 25 30
x
y 235
235 (At Equilibrium
Temperature)
vx2
1.2
1
vy
0.6
vy
2
0.8
0.4
200 (glass region)
equilibrium
0.2
0
5
vx2
0.2
vy
0.3
2
0.4
vy
Granular
Temperature
10 15 20 25 30
x
y 200
glass
x
0.1
0
5
10 15 20 25 30
x
150
y
z
Fluctuating and Flow Velocity
Experiment by
N. Menon and
D. J. Durian,
Science, 275, 1997.
5
 v v
2
16 x 16
vy
1
v
2/3
flow
0.5
32 x 32
0.2
In Glassy Region !
Simulation results
0.1
0.5
1
5
10
J.J. Drozd and
C. Denniston
vf
Europhysics Letters, 76 (3), 360, 2006
"questionable" averaging over nonuniform regions gives 2/3
vy
 v  v  Tg  v y  v c 
   1 in fluid glass transition
2
x
For 0 = 0.9,0.95,0.96,0.97,0.98,0.99
Subtracting of Tg and vc and not averaging over regions of different vx2
1.5
Down centre
1.25
1
Slope  = 1.0
log10
vx2
0
0
1
0
0
0
0
2
0
0.8
0.9
0.95
0.96
0.97
0.98
0.99
0.995
Tg
Tg
0
0.75
vx2
0
0.5
log10
1
0.25
0
0.25
3
50
100
150
200
y
250
300
350
400
1.5
1
log10 vy
0.5
vc
0
0.5
"Particle Dynamics in
Sheared Granular Matter"
Physical Review Letters 85,
Number 7, p. 1428
(2000)
Experiment:
(W. Losert, L. Bocquet,
T.C. Lubensky and
J.P. Gollub)
vy
h
Velocity Fluctuations
vs. Shear Rate
240
1.7
1.6
1.5
1.4
1.3
1.2
   x v y
U
5
10
h
15 20 25
x
240 , y g 185 ,
30
0
 x y   
0.9
v2x
Tg 1 2
1.6
1.8
2
2.2
2.4
3.5
3
log 10
2.5
xv y
U
2
log 10
1.5
1
U
Slope = 0.406  0.018
From simulation
Must Subtract Tg !
Experiment Slope = 0.4
Physical Review Letters 85, Number 7, (2000)
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
q
y
x
y 100
y 100
2
0.02
1
0.01
q.x q.y
xy
3
0
1
0
0.01
2
0.02
3
5
10 15 20 25 30
x
slope 0.15554674  x xy
1
ratio of slopes   fc 1    (r1  r2 )  q
2
5 10 15 20 25 30
x
slope  0.00165986
Viscosity vs Temperature
…can do slightly better…
   x v y
 x y   
x 16
0.5
xvy
1
log10
log10
yx
1.5
Slope ~ 2  1
1.92  0.084
   xy / x vy
2
2.5
3
0
0.9
0
0.95
0
0.96
0
0.97
0
0.98
0
0.99
3.5
1.5
1
0.5
0
2
log 10 v x
0.5
1
1.5
Tg
…"anomalous" viscosity.
Is a fluid with "infinite" viscosity
a useful description of the interior phase?
Transformation from a
liquid to a glass takes
place in a continuous
manner. Relaxation
times of a liquid and its
Shear Viscosity increase
very rapidly as
Temperature is
lowered.
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
momentum
after collision
minus momentum before
collision.
Experiment 
(Longhi, Easwar)
Quasi-1d Theory 
(Coppersmith, et al)
Most frequent collisions
contributing to smallest
impulses
Power Laws for Collision Times
Collision time
= time between
collisions
10
P
0.1
0.001
0.00001
1.
10
7
1) spheres in 2d
2) 2d disks
3) 3d spheres
0.01 0.02
P   

0.05
0.1
0.2
0.5
15% polydispese
r grains
  2.81 0.06 2.75to 2.87in glassy region
Similar power laws for 2d and 3d simulations!
Comparison With Experiment
: experiment 1.5 vs. simulation 2.8
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
Probability Distribution for
Impulses vs. Collision Times (log scale)
P    
2
0
ln I
2
4
6
8
10
15
12.5
10
7.5
ln
5
2.5
0
 = 2.75
 = 1.50
random packing
at early stage
 = 2.75
Is there any difference
between this glass and
a crystal?
Answer: Look at
Monodisperse grains
crystallization 
at later stage
 = 4.3
Disorder has a universal effect on
Collision Time power law.
Summary of Power Laws
Radius
Polydispersity
0%
(monodisperse)
2d disks
Spheres in 2d
4
4.3
15 %
(polydisperse)
2.75
2.85
3d spheres
4
2.87
Conclusions
• A gravity-driven hard sphere simulation was used to study
the glass transition from a granular hard sphere fluid to a
jammed glass.
• We get the same 2/3 power law for velocity fluctuations vs.
flow velocity as found in experiment, when each data point
is averaged over a nonuniform region.
• When we look at data points averaged from a uniform
region we find a power law of 1 as expected.
• We found a diverging length scale at this jamming (glass)
to unjamming (granular fluid) transition.
• We compared our simulation to experiment on the
connection between local velocity fluctuations and shear
rate and found quantitative agreement.
• We resolved a discrepancy with experiment on the
collision time power law which we found depends on the
level of disorder (glass) or order (crystal).
Momentum Conservation
kik+gi = 0
Normal Stresses Along Height
 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