Lecture IV: Dense Granular Flows

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

Summary of Lecture III

v 2 (1 )( )

 t             

(1 ( ))

i j

ij ij

j i

v v

x x q

   



   

            144424443

div v=0

Order parameter equation

Constitutive relation for shear stress

Mass conservation + Momentum conservation

ij

0

 g

 

 

4

Shear Temperature: Simplified Version

) tan

(tan 2

tan tan

2 tan

1 2

2

1 







   ^ _ ^

• 

, 

– dynamic/static repose angles

•for 

 – granular solid is unstable

•for 

 – solid/liquid equilibrium

(note slightly different definition of _)

Chute: stability of solid state 

Boundary conditions:



z

h



z

OPE: ^

^{ }

^{   ( 1   )(    )}

Perturbation:

Eigenvalue:

1 ),

2 / cos(

1   

 Ae

 z h A const



2 2

/ 4

1  h



   

h

y

x

 g

z

6

Stationary Flow

Stationary OPE: ^

^{  ( 1   )(    )  0}

1st integral:

const

z²

/ 2  2 (   1 ) / 3 

 

 c 



Velocity profile – from stress constitutive relations:

(1 ) 0

x

xz

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

Stationary Flow Existence Limit

Solution exists only for   1 h  h

⁽  ⁾

 _{  }



1

0 3

min 2

0

/ 2 2 ( 1 ) / 3 ( )

min



   



c h d

z

2 / 1

~ for

) 2 / 1 log(

min

  2   

h

1 for

2 / 1 2

min

  /    

h

Solving the first integral:

8

Phase Diagram

no flow solid only

solid &

liquid flow

liquid only

2 / )

1 ( 

   h

h

Single mode approximation:

depth averaging

Close to the stability boundary   1  A ( x , y , t ) cos(  z / 2 h )

2 2

3 2

8(2 ) 3

1 4 3 4

t xx yy

A A A A A A

h

 

 

  

        

 

Order parameter equation

A(x,y,t) – slowly varying amplitude

Orthogonality/Solvability condition

cos 2

h

B B z dz

h





 

     

10

Correction to 

z

x

) tan

(tan 2

tan tan

2 tan

1 2

2

1









   ^ _ ^

₀



- local slope contribution   

  h

) tan (tan

2

1

1

2



 

 

0 x

h

     ^{ }

- chute angle,  –local slope

Mass Conservation Law

t x x y y

h   J   J







  ^{2 (} 

^{ 8}

^{) g sin}

3

0^h

( )

x x

J   v z dz   Ah

“dimensionless” mobility

Transverse flux

J

J

>> J

12

Model

• Order parameter equation

• Local “shear temperature”

• Evolution of layer depth

2 2

3 2

8(2 ) 3

1 4 3 4

t xx yy

A A A A A A

h

 

 

  

        

 

0

h

    

3

t x x x

h   J     Ah

Boundary conditions

• Inlet x=0

– No-flux condition: J

=0

– Fixed flux condition J

=  Ah

=J

(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, L

=400, L

=200

• Unit of length is about grain diameter

Fixed Flux at the Inlet of the Chute

x

t



0 1000

500

•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 h Red – max, blue - min

16

Two types of avalanches (theory)

15 . 0 ,

25 . 0 ,

2 . 1 ,

3   

   

h h  5 . 5 ,   1 . 07 ,   0 . 25 ,   0 . 05

Downhill Uphill

Two types of avalanches (experiment)

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

0 200 400 600

x

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!

Quantitative comparison with experiment

Model parameters

, characteristic time l, characteristic length



,

, static/dynamic repose angles

viscosity coefficient

3

2 8) sin

( 2





   ^{ g}

) tan (tan

2

1

1

2 

 

 

Daerr & Douady:

15 . 3 32

,

25

1

     



m d

l ~  240 

ms g

d / ) 5 (

~





20

Phase diagram (theory & experiment)

Infinite Layers: Exact front solution for  =1/2

•does not satisfy boundary conditions

•non-stationary for ≠1/2

1 0

1 tanh 2 8

  ^    ^   z z ^{ }   ^  

New variable z₀=const - position of the front

28

Avalanches in deep chute

Universal approximation for exact for small and large z)

0 0

1 tanh tanh

8 8

z z z z

   ^   ^   ^{ }    ^   ^{ }   ^  

New variable z₀ : depth of fluidized layer

 

0

0

1

z z  dz

  



2

0 0

( )

( )

t

z

z F z  G z

z

     

Evolution of z₀

0 0 0

0

( 1) 0.502 for 1

; 3.29 for 1 2( 1/ 2)

z z z

F G

z





  

    

  



=

?

Bi-stable function F







Deep chute (cont)

2 0 0

3 0

0

for 1 2

for 1 z z

f z z

 

  

=

? Expression for flux

h





 



x

t

J

h    ^ 

  J

0 2

(1 ) ( )

J  z  dz  3 f z



   

Symmetry x  x

30

Connection with BCRE theory

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

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

2

( ) 2

( ) ,

r

r

R R R

R v D

t x x

H H

t R x

  

   

  

    

  

     

 

Boutreux, Raphael & de Gennes modified instability term

2

( _r ) _up 2

R R R

v D

t   v x x

      

   

Our theory:

•reproduces BCRE for small R (or z₀)

•reproduces Boutreux et al for large R

•has hysteresis missing in both theories

• For flow with finite granular temperature

Control parameter

• T-granular temperature, ₀-shear temperature

• T1,2-critical temperatures for instability of overheated solid/overcooled liquid

Resulting equations

1. momentum conservation 2. order parameter

3. granular temperature evolution

Transition to conventional granular hydrodynamics for T>>₀

Connection with hydrodynamics &

kinetic theory

1 0

2 1

T T

T T

    ^



32

Validation of Theory by MD simulations

•non-cohesive, dry, disk-like grains three degrees of freedom.

•A grain p is specified by radius R_p, position r_p, translational and angular velocities v_p and _p.

•Grains p and q interact whenever they overlap, R_p + R_q r_p –r_p| > 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; P_ext = 13.45,V_x=24

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

Restitution coefficient

Friction coefficient

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

2 0

2 2 2 2

* * *

* *

( 1) ( , ) /

( , ) 2 exp[ ( )]

0.6; 0.26; 25; 2

t

xy yy

D G

G A

A D

     

  

       

 

    



    

   

Fitting the constitutive relation

yy xx y

x s

yy xx yy

xx y

x f

yy xx xy

s xy xy

f xy

s ij f

ij ij

q q

q

q( ) ; (1 ( )) ; _{, ,} ( ) _, ; _, (1 _, ( )) _, where















































Fit: q() = (1)^2.5

fluid (collisional) stress solid (contact) stress

f s

ij ij

   

f

/

xy xy

 

f

/

yy yy

 

f

/

xx xx

 

Fit: q_y() = (1^)^1.9

36

Newtonian Fluid + Contact Part

Kinematic viscosity in slow dense flows:  ≈ 

Relation to Bagnold Scaling

Bagnold relation (1954): 

:  ^&

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

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

Slip event: MD simulations

40

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

5000 particles (50x100), = 0.82; = 0.3; P_ext =

10,50,V_top=5,50 x y

)

x(y V

g

Set of equations for sand

2 2 2 2 2

0 * * *

* * 0

y

2.5

( 1)( 2 exp[ ( )])

/ / ( ) /( )

0.6; 0.26; 25; 2; 0.02

. (0)= (L )=0

(1- ) =- ( B.C

Constit. Relation )

t

xy yy

D A

p y m y

A D

V y

         

    

  

 

  

        

     

    

 

 L_y

V₀ m 

Equations for heavy plate

(

)

x V

mV   x V t



  

&

&

Simplified theory: reduction to ODE

• Stationary OP profile:

  –width of fluidized layer

(depends on shear stress), 

=(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 1

1

1 ( ( ))(1 )

( ) 1 tanh

2 8

y t

f y D

  

    ^{ }  ^ ^{  }^ 

x

y V_x(y)

Vtop

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







 

2

0 1

0 1

2.5 1

1 2

12.6; 326

Constit. Relation

((1 ) )

C m C

C C

V

  

 

 

  

 

  

&

 





1 2 2 2

0 *

A

(1

)

m

 

 

   



   

&

Resulting 3 ODE

• 2 Equations for Plate

• 1 Equation for width of fluidized layer

(

)

x V

mV   x V t



  

&

&

 

2

0 1

or

1 2

for 1

V

C m C

 

  

  



  





&

& =

44

Comparison: Spring deflection vs time

theory: ODE MD simulations

theory: PDE

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.