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

Movie : the initial state (with driving speed = 8 d/s)

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)--stochastic evolution (movies)

12

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)

time

xi(t)

yi(t)

1d

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(*)

(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

.. More info

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 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)

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