Lecture III: Dense Granular Flows

1

Lecture III: Dense Granular Flows

Igor Aronson

Materials Science Division

Argonne National Laboratory

Outline

• Introduction

• Principal Experiments

• Overview of Existing Theories

• Partial Fluidization Theory (beginning)

3

large conglomerates of discrete macroscopic particles

Jaeger, Nagel, & Behringer, Rev. Mod.Phys. 1996 Kadanoff Rev. Mod. Phys.1999 de Gennes Rev. Mod. Phys.1999

Gran Mat Gran Mat

non-gas

inelastic collisions non-gas

inelastic collisions

non-liquid critical slope

non-liquid critical slope non-solid

no tensile stresses non-solid

no tensile stresses

Drop a Ball Experiment

• Granular eruption http://www.tn.utwente.nl/pof/

5

Solid Trouble

• Stress indeterminacy

• History dependence

• Hysteresis

• Plasticity and yield stress

• Is not described by granular hydrodynamics

Granular

Granular Solids at Rest

ji

ij



  Static stress tensor

 







 









zz zy

zx

yz yy

yx

xz xy

xx ij



















 Force balance:

(only six relations, need 3 more!) Symmetry:

Linear elasticity & Hook’s law do not apply

7

Constitutive Relations for Stresses

Veje, Howell, and Behringer, Phys.Rev.E, 59, 739 (1999)

Linear relation between shear and normal stresses,

Force chains and stress inhomogeneity 

 tan  

zz xx xz

    

Oriented stress linearity models

P. Claudin , J. P. Bouchaud, M. E. Cates ,

and J. P. Wittmer, Phys. Rev. E 57, 4441 (1998) -hyperbolic equation

•C. Goldenberg and I. Goldhirsch Phys. Rev. Lett. 89, 084302 (2002) -elliptic (elastic) anisotropic equation

Mohr-Coulomb failure criterion

• 

is maximum possible angle of sandpile If MCFC is saturated, the state loses its stability

g



Check the website: http://www.granular-volcano-group.org/frictional_theory.html

0

shear sterss

max tan

normal stress  

9

Mohr-Colomb criterion in 2D

• 1,2 – eigenvalues of stress tensor

• MC criterion:

If xx=yy then MC criterion |xy /yy |<tan 0

2

2 1

2

2 2

( )

2 4

( )

2 4

xx yy xx yy

xy

xx yy xx yy

xy

   

 

   

 

 

  

 

  

2 1

0

2 1

  tan 

 

 



Bistability: Bagnold hysteresis

• 

static repose angle

– sandpile cannot be built with

• 

dynamic repose angle

– an avalanche cannot be triggered if _{ }

1

 

₂

₁

11

Short Summary

• Theoretical description of granular solid and granular liquid is very different

– Granular liquid → viscous hydrodynamic

– Granular solid → plastic solid with finite yield stress

• Various phenomena: avalanches, slides, surface flows,

stick-slips are related to the transition from granular solid to granular liquid and vise versa

• Universal description of granular systems needs a

constitutive relation valid for gran solid and gran liquid

Review of Principal Experiments

• Avalanches

• Surface shear flows

• Stick-slips

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Experimental Observations

Cariboo Mountain, British Columbia, Canada http://www.avalanche.org

Benasque, Spain, Aug 16th

http://bcs2.unizar.es/activities/alternative.htm

Life Avalanche Experience

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Laboratory Avalanches

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

1.35m

0.62m Glass beads 150-300m

CCD camera winch

Two types of avalanches discovered

17

Phase diagram

No flow

spontaneous avalanching Bistability

downhill avalanches uphill avalanches

Near-surface granular flow

10 ^-6 10 ^-5 10 ^-4 10 ^-3 10 ^-2 10 ^-1 1 10 10 ² 10 ³

-10 -5 0 5 10

áv(h)ñ / áv(h0)ñ

( h - h ₀ ) / a 0

20 40

-10 0 10

Komatsu, Inagaki, Nakagawa, Nasuno, PRL, 86, 1757 (2001)

h

1 sec 1 min 1 hour

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Taylor-Couette granular flow (2D)

Veje, Howell, and Behringer, Phys.Rev.E, 59, 739 (1999)

) exp(

~ )

( r ar

V 

10 ^-1 10 ⁰ 10 ¹

r*

10 ^-5 10 ^-3 10 ^-1 10 ¹

Vq*

.777 .779 .783 .788 .790 .792 .797

0 2 4 6 8 10 12 14

r/d 10 ^-6

10 ^-5 10 ^-4 10 ^-3 10 ^-2 10 ^-1 10 ⁰

Vq/W

.777 .779 .783 .788 .790 .792 .797

Velocity profile:

Taylor-Couette granular flow (3D)

 (

)



exp

~ )

( r ar b r r

V   

Mueth, Debregeas, Karczmar, Eng, Nagel, Jaeger, Nature, 406, 385 (2000)

60mm

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Taylor-Couette flow - 2

Losert, Bocquet, Lubensky, and Gollub Phys. Rev. Lett. 85, 1428 (2000)

Review of theoretical models

• S. B. SAVAGE, Analyses of slow high-concentration flows of granular materials, J. of Fluid Mech 377, 1 (1998)

-plasticity relation between shear stress & strain rate

• Losert, Bocquet, Lubensky, & J. P. Gollub, Particle Dynamics in Sheared Granular Matter, Phys. Rev. Lett. 85, 1428 (2000)

• Bocquet, Errami, & T. C. Lubensky, Hydrodynamic Model for a Dynamical Jammed-to-Flowing Transition in Gravity Driven Granular Media, Phys. Rev.

Lett. 89, 184301 (2002)

-viscosity diverges at closed-packed density

Difficulties:

-Description of solid state and transition to motions -No hysteresis

1.75

0

(1 /

)

     

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Two-phase models

Original version:

•Bouchaud, Cates, Ravi Prakash, and Edwards, J.Phys.France, 4, 1383 (1994) - BCRE theory (see also Anita Mehta, 1993)

Modification:

•Boutreux, Raphael, and de Gennes, PRE, 58, 4692 (1995)

Highlights:

•two fractions: rolling and immobile grains

•conservation laws and phenomenological rolling/static conversion rates

Two-phase models

2 2

( ) ,

( )

r

r

R t H

v R R x R

x H

x D R

t

  

   



    



 

  





 







immobile H

_r

•H-thickness of immobile fraction

•R-thickness of rolling fraction

conversion convection diffusion

rolling R x

–local slope & _r – critical slope v- typical flow velocity

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Pierre-Gilles de Gennes, Rev Mod Phys, 1999

… the statistical physics of grains is still in its infancy. Some basic notions may emerge:

• the possibility of describing surface flows with equations coupling the two phases and reduced to a simplicity reminiscent of the Landau-Ginsburg picture of phase

transitions.

Generic theoretical description

Momentum conservation

where

- density of material ( =1) g - gravity acceleration

v - hydrodynamic velocity 

-stress tensor

div v=0 incompressibility condition

0 ₀

0 0

i ij

i j

Dv g

Dt x

  ^   



D

Dt t

   

 v

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Stress-stain relation for granular flows

const

  - viscosity, p –hydrodynamic pressure - strain-rate tensor

Stress-strain relation in Newtonian Fluid

ij ij

p

   &  

s ij f

ij

ij

 

  

Here

s



- quasistatic (contact) part

f



- fluid part

i j ij

j i

v v

x x

  ^  ^

 

&

Order parameter- degree of fluidization

1 for

0

0 for

1











 q

q

-characterizes the “state” of the granular matter:

xy s

xy xy

f

xy

q    q  

  ( ) ;  ( 1  ( ))

 0

 - pure fluid   1 - pure solid

ii i

s ii ii

i f

ii

q    q  

  ( ) ;  ( 1  ( ))

The simplest choice:



 )  1  (

q

Renormalization of stresses

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Hybrid stress-stain relation in partially fluidized granular matter

• order parameter controls balance between fluid and solid parts of stress (in fully fluidized state solid part vanishes)

•shear stresses in partially fluidized flow are reduced by the factor 1-q(  )≈ 

•liquid part is defined as

0 0

0 for 1

(1 ( )) ,

1 for 0

s

ij ij ij

q q

q

    



 

       

( )

f ij ij

p q

 

 

 & 

Orrder parameter: microscopic definition

OP characterizes the phase state of granular matter:

solid:  = 1 liquid:  = 0

Compare: Lindenmann criterion for melting of solids

Z

  Z

• Z_st -number of persistent contacts

Z≈4 in 2D and Z≈6 in 3D

2D simulations of granular shear flow between 2 plates. OP is shown by color Volfson, Tsimring & Aranson

liquid-like v

31

Time evolution of order parameter

2D MD simulations of shear flow with stick-slip, 2400 soft particles.

Supported by the National Energy Research Center for Supercomputing, DOE BES

Equation for order parameter



 



D / Dt   ^F

Ginzburg-Landau free energy for “shear melting” phase transition

Requirements: two stable states:  =  and =1 one unstable state _u

 is a control parameter (“shear temperature”)

  f ( 

) r

d l ( ) ( )]

[



f  F    

₀,l –characteristic time & length

2

(1 ) ( , )

D         G  

33

Ginzburg-Landau Free Energy

3 .

 0



5 .

 0



7 .

 0



0

1

f



1

0



1

 ( , )

G      

solid-like liquid-like

2

(1 )( )

D Dt

          

Shear Temperature: General Definition

• 

, 

– tangents of dynamic/static repose angles

•for  – granular solid is unstable

•for  – granular liquid is unstable

2

2 1

2 2

2 1

shear normal

 

 

 

 

  

 

 

35

Summary of the Model

v 2 (1 )( )

 t             

(1 ( ))

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

0

0

ij

i j

x g

 

  



Simple example: chute flow

Equilibrium conditions

, ,

, ,

cos sin

zz z xz x

xz z xx x

g g

  

  

 

   Stresses:

cos ; sin ; 0

zz g z xz g z xx xy yy yz



 

 











 / | tan

|



h

y

x

 g

z

37

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

39

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:

41

Phase Diagram

no flow solid only

solid &

liquid flow

liquid only

2 / )

1 ( 

   h

h

Theory of partial fluidization

• Igor Aranson and Lev Tsimring, Continuum description of avalanches in granular media, Phys. Rev. E 64, 020301 (2001)

• Igor Aranson and Lev Tsimring, Continuum theory of partially fluidized granular flows, Phys. Rev. E 65, 061303 (2002)

• Dmitri Volfson, Lev Tsimring, and Igor Aranson, Order Parameter Description of Stationary Partially Fluidized Shear Granular Flows Phys. Rev. Lett. 90, 254301

• Dmitri Volfson, Lev Tsimring, and Igor Aranson, Partially fluidized shear granular flows: Continuum theory and molecular dynamics simulations, Phys. Rev. E (2003)