Evolution of Sheared Dense Granular Flow
Jerry Gollub . Haverford College & Univ. of Pennsylvania
J.-C. Tsai
G.VothI ) Crystallization transition
-- 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
Grains
W Udriving
x
z
Light sheetCamera
Light sheet
Fluid level
(acrylic)
(aluminum)
Glass beads(diameter=d)
Position sensor
Wzx
Camera
Gaps smaller than 0.5d
Steady driving+ torque measurement
Experimental Setup --cross-sectional view
--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 )
Normal load W >> beads’ total weight & fluid’s viscous drag
I) Crystallization transition -- internal slices
Grains
W Udriving
x
z
-0.03
-0.02
-0.01
0.00
-450
-300
-150
0
0 10000 20000 300000.00
0.01 0 12 24 36 48 600.0
0.2
t=10000s
t = 0s
h(t
)-h(
0) (mm
)
(h(t
)-h(
0))
/ H0
t = 0s t=30000st=20000st=10000s
I(1)
Time: t (s)
Lb
a
c t = 30000s
f( k
x )
kx / (2p/L)
Horizontal slice (XY plane):
Vertical slice (XZ plane):
x
I) Crystallization transition -- movies
XZ slice:
(9hrs total @ ~900X)
Grains
W Udriving
x
z
XY slice (before trans.)
XY slice (after trans.)
I) Crystallization transition-- time-resolved measurements
The ordering transition
results in step changes of
granular volume (),
shear force (),
and particle speed
(stronger decay downwards).
-0.06
-0.03
0.00
-900
-450
0
0.00
0.03
0.18
0.21
0.24
0 10 20 30 400.0
0.1
0.000
0.001
0.002
0.003
0.004
0 20000 40000 60000 800000.0
1.0
Volume Change
h(t
)-h(
0) (
mm)
(h(t
)-h(
0))
/ H0
Shear Force
t(t)
L
b
a
c(a.u.)
Instantenous FFT Spectrum ( t = 60000s )
f(kx)
kx / (2p/L)
Spatial Ordering
I(1
)
b
a
c
G0
< V
x >
G0 /
Ud
rivi
ng
t = 0 t=60000s
Time: t (s)
b
a
c
ccd
x 10-4
Particle Speed(averaged over region G0)
( -3 %)
( -15 %)
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”
* Final volume: ** Degree of final ordering:
(case of thin layers)
Final states vs. Total mass (movies)
(Volume quantization is found to exist for flows as thick as 23~24 layers!)
96 100 104 108 112 116 120
0
200
400
600
800
1000
1200
< h
>fin
al- h
100g
(mm
)
Mass (gram)
incomplete ordering
incomplete ordering
13 layers
14
12
-1800
-1200
-600
0
100 1000 10000 100000
-600
0
Fluid-immersed particles 200g (24 layers)
168g (20 layers) 136g (16 layers) 120g (14 layers) 116g (14 layers) 111g (13 layers) 108g (13 layers)
h(t)
- h
i
(mm
)
Dry particles 200g (24 layers)
Time (s)
ca
b
I) Crystallization transition -- timescales & behavior of dry
particles
(ii) Dry particles:
Ordering transition occurs, but takes much longer!
(Driven at the same speed:)
(i) Dependence on layer thickness:
{Fig.5, PRL 91,064301}
II) Non-unique final states
Grains
W
Grains
W Udriving
First
Using a bumpy bottom:
§ Shearing with an oscillatory pre-treatment:
then
drive back and forthby a few cycles;(102 d each way)
continuously shear at a fixed velocity.
II)--stochastic selection of final states
100 1000 10000 100000-800
-700
-600
-500
-400
-300
0 1 2
7 7
8 9 10
3 4 5 6
h(t)
(mm
)
Time (s)
Number of oscillatory cyclesapplied prior to one-way shearing:
--- partial ordering
1
20
0
(MOVIEs)
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 ? ”
* ) 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(*)
(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.
http://www.haverford.edu/physics-astro
/Gollub/internal_imaging
Ref: PRL 91, 064301 (2003) & subsequent papers
Oscillatory driving –basic phenomena (1)
Temporary volume decrease induced by oscillatory shearing
(of sufficiently compacted packing):
14200 14400 14600 14800 15000
-900
-800
-700
-600
-500
-400
-300
-120
12
0 3000 6000 9000
h(t)
(mm
)
Time (s)
a b
10 cycles
U0(t
) (d
/s)
t - ti (s)
Disordered Ordered
30 cycles
0 200 400 600 800 1000
Vx
(t)
Time step (dt = 0.2s)
Udriving
( t ) / 10
0 200 400 600 800 1000
Vx
(t)
Time step (dt = 0.2s)
Udriving
( t ) / 10
III ) Oscillatory shear –basic phenomena (2)
Instantaneous mean velocity Vx(t), measured at the same height:
Disorderedstate
Orderedstate
(dt ~ 0.05Td)(sudden drop h ~ d/5.)
x
z
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 )
0 -5 -10 -15 -20 -25
10-6
10-5
10-4
10-3
10-2
10-1
100
bez/d
aez/5d
< V
x >
/
Ud
rivi
ng
z / d
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
Other driving speeds
Driving speed 12 d/s
0.12 d/s :
1.2 d/s :
(2) velocity profile & displacement timescales
Time-averaged grain velocity of the ordered state
(@~30X)
x
z