1

Lecture IV: Dense Granular Flows

Igor Aronson

Materials Science Division Argonne National Laboratory

2

Outline

• Stationary Flow Stability

• Phase Diagram

• Avalanches in Shallow Layers

• Deep Layers/Connection to BCRE

• Comparison with MD simulations

• Granular Stick-Slips Friction

3

Summary of Lecture III

2v (1 )( )t

(1 ( ))jiij ij

j i

vvq

x x

144424443

div v=0

Order parameter equation

Constitutive relation for shear stress

Mass conservation + Momentum conservation

0 0iji

j

gx

4

Shear Temperature: Simplified Version

)tan(tan2tantan2tan

12

21

• 1, 2 – dynamic/static repose angles

•for – granular solid is unstable•for – solid/liquid equilibrium(note slightly different definition of )

5

Chute: stability of solid state Boundary conditions:

= 1 for z h (rough bottom)

z = 0 for z free surface

OPE: ))(1(2 t

Perturbation:

Eigenvalue:

1),2/cos(1 constAhzAe t

22 4/1 h

Stability limit: 2/)1( 2/1 sh

h

y

x

g

z

6

Stationary Flow

Stationary OPE: 0))(1( zz

1st integral:

constcz 232 3/)1(22/

Velocity profile – from stress constitutive relations:

(1 ) 0xxz

v

z

Boundary conditions:

z

(rough bottom)0 =1 for

0 =0 for 0 (open sur f ace)x

z x

v z h

v z

h

y

x

g

z

7

Stationary Flow Existence Limit

Solution exists only for 1 )(min hh

1

032min

0)(3/)1(22/

min

c

dh

z

2/1~for )2/1log(2min h

1for 2/12/min h

Solving the first integral:

8

Phase Diagram

no flowsolid only

solid & liquid

flowliquid only

2/)1( 2/1 sh

hmin

9

Single mode approximation:depth averaging

Close to the stability boundary )2/cos(),,(1 hztyxA

2 23

2

8(2 ) 31

4 3 4t xx yy

AA A A A A

h

Order parameter equation

A(x,y,t) – slowly varying amplitude

Orthogonality/Solvability condition 0

cos2h

zB B dz

h

10

Correction to z

x

)tan(tan2tantan2tan

12

21

0

- local slope contribution0 xh

)tan(tan2

1

12

0 xh 0- chute angle, –local slope

11

Mass Conservation Law

t x x y yh J J

3

2 sin)8(2 g

3

0( )

h

x xJ v z dz Ah down-hill flux of grains

“dimensionless” mobility

Transverse flux Jy is neglected since Jx>> Jy

12

Model

• Order parameter equation

• Local “shear temperature”

• Evolution of layer depth

2 23

2

8(2 ) 31

4 3 4t xx yy

AA A A A A

h

0 xh

3t x x xh J Ah

13

Boundary conditions

• Inlet x=0– No-flux condition: Jx=0

– Fixed flux condition Jx=Ah3=J0 (grains supplied from hopper with the fixed rate)

14

Numerical Methods

• Implicit Crank-Nicholson code for A

• Number of mesh point in x – 600-2400

• Number of mesh points in y – 600

• Time of integration op to 2000 units

• h – 2-16 =0.025, =3, Lx=400, Ly=200

• Unit of length is about grain diameter

15

Fixed Flux at the Inlet of the Chute

x

t

4.0J 6.0J

0 1000

500

continuous flow periodic avalanches

•Large flux – steady flow, h is adjusted according to J0

•Small flux – periodic sequence of avalanches, Period T ~1/J0

Space-time plot of the height hRed – max, blue - min

16

Two types of avalanches (theory)

15.0,25.0,2.1,3 h 05.0,25.0,07.1,5.5 h

Downhill Uphill

17

Two types of avalanches (experiment)

Triangular (downhill) Balloon (uphill)

Daerr & Douady, Nature, 399, 241 (1999)

18

Transition from down-hill to up-hill:1D analysis of avalanche cross-sections

h

0

10

20

h

0 200 400 600x

0

10

20

uphill

downhill

07.1

02.1

Secondary avalanche

1.05 1.06 1.07 1.08 1.09 1.10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

V

Uphill front speed

discontinuous transition!

19

Quantitative comparison with experiment

Model parameters

, characteristic timel, characteristic length1,2, static/dynamic repose anglesviscosity coefficient

3

2 sin)8(2

g

)tan(tan2

1

12

Daerr & Douady:

15.332,25 02

01

mdl 240~ (particle diameter)

msgd 5)/(~ 2/10

20

Phase diagram (theory & experiment)

27

Infinite Layers: Exact front solution for =1/2

•does not satisfy boundary conditions•non-stationary for ≠1/2

01

1 tanh82

z z

New variable z0=const - position of the front

28

Avalanches in deep chuteUniversal approximation for exact for small and large z)

0 01 tanh tanh8 8

z z z z New variable z0 : depth of fluidized layer

0

0 1z z dz

20 0 0 0 0( ) ( )t x xz z F z G z z

Evolution of z0

0 00

0

( 1) for 10.502;

for 13.292( 1/ 2)

z zzF G

z

=

?

Bi-stable function F

29

Deep chute (cont)

200

300

for 12

for 1

zzf

zz

=

?

Expression for flux

xh 0

xt Jh xxt J 0

20(1 ) ( )

3J z dz f z

Symmetry x x

No triangular avalanches in sandpiles!

30

Connection with BCRE theoryBCRE (Bouchaud, Cates, Rave Prakash & Edwards, 1995) operates with:

•H-thickness of immobile fraction, R-thickness of rolling fraction

2

2( )

( ) ,

r

r

R R RR v D

t x xH H

Rt x

Boutreux, Raphael & de Gennes modified instability term

2

2( )r up

R R Rv D

t x xv

Our theory: •reproduces BCRE for small R (or z0)•reproduces Boutreux et al for large R •has hysteresis missing in both theories

31

• For flow with finite granular temperature

Control parameter

• T-granular temperature, 0-shear temperature• T1,2-critical temperatures for instability of overheated solid/overcooled

liquidResulting equations

1. momentum conservation2. order parameter3. granular temperature evolution

Transition to conventional granular hydrodynamics for T>>0

Connection with hydrodynamics & kinetic theory

10

2 1

T T

T T

32

Validation of Theory by MD simulations•non-cohesive, dry, disk-like grainsthree degrees of freedom. •A grain p is specified by radius Rp, position rp, translational and angular velocities vp and p. •Grains p and q interact whenever they overlap, Rp + Rq rp –rp| > 0

•linear spring-dashpot model for normal impact•Cundall-Strack model for oblique impact.•Detail: Silbert et al, Phys Rev E, 64, 051302 (2001)All quantities are normalized using particle size d,

mass m, and gravity g 2304 particles (48x48), = 0.82; = 0.3; Pext = 13.45,Vx=24

Simulations: IBM SP2 at NERSC, fastest unclassified computer in the world

Restitution coefficientFriction coefficient

33

Testbed system:Couette flow in a thin granular layer without gravity

500 particles (50x10), e= 0.82; = 0.3; P = 13.45, no gravity

Adiabatic change in shear force:

34

Fitting free energy: fixed points of the order parameter

MD simulations OP equation

500 particles (50x10), = 0.82; = 0.3

20

2 2 2 2* * *

* *

( 1) ( , )

/

( , ) 2 exp[ ( )]

0.6; 0.26; 25; 2

t

xy yy

D G

G A

A D

35

Fitting the constitutive relation

yyxxyxs

yyxxyyxxyxf

yyxxxysxyxy

fxy

sij

fijij

qqqq ,,,,,, ))(1(;)(;))(1(;)(

where

Fit: q() = (1)2.5 Phenomen. theory: q()=1-

fluid (collisional) stress solid (contact) stress f sij ij

/fxy xy

/f fyy yy

/fxx xx

Fit: qy() = (1)1.9

qx() = (1)1.9

36

Newtonian Fluid + Contact Part

Kinematic viscosity in slow dense flows: ≈

37

Relation to Bagnold Scaling

Bagnold relation (1954): 2xy &:

Silbert, Ertas, Grest, Halsey, Levine, and Plimpton, Phys. Rev. E 64, 051302 (2001) Fitting Bagnold scaling relation

38

Shear granular flows and stick-slips

Nasuno, Kudrolli, Bak and Gollub, PRE, 58, 2161 (1998).

sliding speed V=11.33 mm/s

sliding speed V=5.67 mm/s

sliding speed V=5.67 m/s

39

Slip event: MD simulations

40

Example: stick-slips thick surface driven granular flow with gravity

5000 particles (50x100), = 0.82; = 0.3; Pext =

10,50,Vtop=5,50

x

y

)(x yV

g

Set of equations for sand

2 2 2 2 20 * * *

* * 0

y

2.5

( 1)( 2 exp[ ( )])

/ / ( ) /( )

0.6; 0.26; 25; 2; 0.02

. (0)= (L )=0

(1- ) =- (

B.C

Constit. Relati )on

t

xy yy

D A

p y m y

A D

V y

Ly

V0m

Equations for heavy plate

0( )

x V

mV x V t

&&

41

Simplified theory: reduction to ODE

• Stationary OP profile:

–width of fluidized layer

(depends on shear stress), 1=(4*-1)/3

- Stationary solution exists only for specific value of (y) (symmetry between the roots of OP equations) which fixes position of the front

1 11

1 ( ( ))(1 )( ) 1 tanh

2 8

y tf y

D

x

y)(x yV

topV

g

42

Perturbation theory

• Substituting into OP equation and performing orthogonality one obtains

• Regularization for <<1 ( –is the growth rate of small perturbations)

0

F F dy

20 1

0 1

2.51

21

12.6; 326

Constit. Relation ((1 ) )

C Cm

C C

V

&

21 2 2 220 * * *(1 )A

m

&

43

Resulting 3 ODE

• 2 Equations for Plate

• 1 Equation for width of fluidized layer

0( )

x V

mV x V t

&&

20 1

2.51

or

Constit. Relati

21

on

for 1

((1 ) )

V

C Cm

&

& =

44

Comparison: Spring deflection vs time

theory: ODE

MD simulations

theory: PDE

45

Conclusions

• We introduced a theoretical description for partially fluidized granular flows based on the order parameter which is defined as a fraction of persistent contacts among the grains.

• Stress tensor in granular flows is separated into a “fluid” part and a “solid” part. The ratio of the fluid and solid parts is controlled the order parameter

• The dynamics of the order parameter is described by the Ginzburg-Landau equation with a bistable free energy functional.

• The free energy controlling the dynamics of the order parameter, can be extracted from molecular dynamics simulations.

• The viscosity coefficient calculated as a ratio of the fluid shear stress to the strain rate does not diverge at small strain rates.

• The model successfully describes dynamics of various shear granular flow: from avalanches to stick slips.