Evolution of Sheared Dense Granular Flow
z
Udriving
W
x
Jerry Gollub
.
Haverford College & Univ. of Pennsylvania
Grains
J.-C. Tsai
I ) Crystallization transition
G.Voth
-- rheological change
-- role of B.C.
-- ‘quantization’ effects
II ) Non-unique final states
-- ‘stochastic’ selection
-- stabilization of disordered state
III ) Quasi-static internal dynamics: crystallized vs. disordered states
Experimental Setup
--cross-sectional view
Normal load W >> beads’ total
weight & fluid’s viscous drag
--Glass beads:
d = 0.6mm
immersed in fluid
--Driving:
constant speed,
fixed normal load
--Fluid:
index-matched
fluorescent dye
+ laser sheet
* Volume measurement
(height of upper surface)
** Shear force
measurement
~30d,
( Circumference ~ 800d )
Movie : the initial state (with driving speed = 8 d/s)
I) Crystallization transition
-- internal slices
z
Udriving
W
x
Grains
Vertical slice (XZ plane):
a
t = 0s
t=10000s t=20000s t=30000s
t=10000s
t = 0s
(t)-h(0)) / H0
b Horizontal slice (XY plane):
0.00
0
x
-0.01
-150
-0.02
-300
h(t)-h(0) (mm)
L
I) Crystallization transition
-- movies
z
W
x
Grains
XZ slice:
XY slice (before trans.)
XY slice (after trans.)
(9hrs total @ ~900X)
Udriving
I) Crystallization transition
a
t=0
t=60000s
Spatial Ordering
0.004
-- time-resolved measurements
0.002
G0
I
The ordering transition
0.001
f(kx)
results in step changes of
0.000
0.0
0
b
10
20
30
0.00
Volume Change
-0.03
( -3 %)
(h(t)-h(0)) / H0
granular volume (),
Instantenous
FFT Spectrum
( t = 60000s )
0.1
L
40
kx / (2p/L)
0
-450
-900
-0.06
cc0.24
Shear Force
0.21 (a.u.)
t(t)
shear force (),
0.18
( -15 %)
0.03
0.00
dc
< Vx >G0 / Udriving
and particle speed
(stronger decay downwards).
h(t)-h(0) (mm)
(1)
0.003
Particle Speed
(averaged over region G0)
-4
1.0 x 10
0.0
0
20000
40000
Time: t (s)
60000
80000
I)
-- Role of boundary condition
Final states (after a long steady shearing from above) with
flat bottom or mono-layer bottom
|
bumpy bottom
I)
-- “Quantization effects”
** Degree of final ordering:
* Final volume:
< h >final- h100g (mm)
(case of thin layers)
1200
14
1000
800
13 layers
Final states vs. Total mass
(movies)
600
400
200
0
incomplete ordering
12
incomplete ordering
96 100 104 108 112 116 120
Mass (gram)
(Volume quantization is found to exist for flows
as thick as 23~24 layers!)
I) Crystallization transition
-- timescales & behavior of dry particles
(Driven at the same speed:)
(i) Dependence on
layer thickness:
(mm)
c
{Fig.5, PRL 91,064301}
a
Fluid-immersed particles
200g (24 layers)
0
168g (20 layers)
136g (16 layers)
120g (14 layers)
116g (14 layers)
111g (13 layers)
108g (13 layers)
-600
h(t) - hi
-1200
-1800
(ii) Dry particles:
0
b
Dry particles
200g (24 layers)
-600
100
1000
10000
100000
Time (s)
 Ordering transition occurs, but
takes much longer!
II) Non-unique final states
Using a bumpy bottom:
§ Shearing with an oscillatory pre-treatment:
First
then
W
W
Grains
Grains
drive back and forth
by a few cycles;
(102 d each way)
Udriving
continuously shear at a
fixed velocity.
II)
--stochastic selection of final states
Number of oscillatory cycles
applied prior to one-way shearing:
0
7
1
7
2
-300
(MOVIEs)
0
h(t) (mm)
-400
1
-500
-600
0
3
4
5
6
2
8
9
10
-700
-800
100
1000
Time (s)
--- partial ordering
10000
100000
II)
--stochastic evolution (movies)
2
1
II)
-- stabilization of disordered state
“Effectiveness” of partial ordering by oscillatory shear
before the
|
after the
long unidirectional shearing
long unidirectional shearing
II ) Non-unique final states
Facts:
* Both states can be stablized.
* Transition is possible ONLY when uncompacted;
preparation history matters.
* Reversal of crystallization transition NEVER occurs.
* Crystallized state: less shear force, stronger velocity decay,
less dissipative.  “preferred state”
 How is history ‘recorded’ in granular packing?
 “Attractors ? ”
III ) Quasi-static internal dynamics
-- comparing velocity profiles
III ) Quasi-static internal dynamics
-- particle trajectories: xi(t) & yi(t)
xi(t)
1d
time
yi(t)
III ) Quasi-static internal dynamics
-- yi(t): ordered vs. disordered states
* ) Additional information
Steady shearing of binary mixture
(The r.m.s. size dispersion in the previous experiments is about 4%.)
Binary mixture:
(d=1.0 mm vs. 0.6 mm), (25% vs. 75%) by weight,
with some of the 1.0 mm grains painted black as tracers.
(~3000X Real time)
Summary & Theoretical challenges(*)
http://www.haverford.edu/physics-astro
/Gollub/internal_imaging
(1*) Shear flows can have non-unique final states.
(2) For a nearly mono-disperse packing,
rheology of cyrstallized state and disordered state are compared.
(3*) Both boundary condition and preparation history have
profound effects on crystallization transition.
the reversal of crystallization never occurs.
Ref: PRL 91, 064301 (2003) & subsequent papers
.. More info
Oscillatory driving
–basic phenomena (1)
Temporary volume decrease induced by oscillatory shearing
U0(t) (d/s)
(of sufficiently compacted packing):
12
0
-12
h(t) (mm)
-300
-400
30 cycles
10 cycles
a
b
Disordered
Ordered
-500
-600
-700
-800
-900
14200
14400
14600
14800
Time (s)
15000
0
3000
6000
t - t (s)
9000
III ) Oscillatory shear
–basic phenomena (2)
Instantaneous mean velocity Vx(t),
measured at the same height:
Vx (t)
x
Disordered
state
Udriving( t ) / 10
0
200
400
600
Time step (dt = 0.2s)
800
1000
400
600
Time step (dt = 0.2s)
800
1000
Udriving( t ) / 10
Ordered
state
Vx (t)
z
0
200
(dt ~ 0.05Td)
(sudden drop Dh ~ d/5.)
3D structure of the velocity field
3D structure of the
disordered final state (partially ordered at sidewalls)
After 2 weeks of steady shearing at a driving speed 12d/s:
Multiple vertical slices
(y = W0/3 W0/6)
Multiple horizontal slices
(z = -H0/2  -1d )
III ) Quasi-static internal dynamics
-- comparing velocity profiles
(24 layers)
(22 layers)
(2) velocity profile & displacement timescales
Time-averaged grain velocity of the ordered state
< Vx > / Udriving
10
z
0
10
-1
10
-2
10
-3
10
-4
10
-5
10
-6
x
Other driving speeds
z/5d
ae
0.12 d/s :
Driving speed 12 d/s
sampling rates :
0.06 fps
0.6 fps
1.2 fps
2.4 fps
4.8 fps
12 fps
60 fps
0.02 fps
0.00667 fps
0
-5
-10
1.2 d/s :
z/d
be
-15
z/d
-20
-25
(@~30X)