Order Parameter Description of Shear Flows in Granular Media

1
Order Parameter Description of Shear Flows in
Granular Media

• Igor Aronson (Argonne)
• Lev Tsimring, Dmitri Volfson (UCSD)
• Publications
• Continuum Description of Avalanches in Granular
  Media, Phys. Rev. E 64, 020301 (2001)
• Theory of Partially Fluidized Granular Flows,
  Phys. Rev. E 65, 061303 (2002)
• Order Parameter Description of Stationary Gran
  Flows, Phys. Rev. Lett. 90, 254301 (2003)
• Partially Fluidized Granular Flows, Continuum
  theory and MD Simulations, Phys. Rev. E. 68,
  021301 (2003)
• Stick-Slip Dynamics in a Granular Layer under
  Shear, Phys. Rev. E 69, 031302 (2004)
• Supported by US DOE, Office of Basic Energy
  Sciences

SAMSI Workshop, NC 2004
2
Outline

• Introduction
• Experimental observations of partially fluidized
  granular flows
• Theoretical description order parameter model
• Examples
• Near-surface shear flows
• Stick-slips in shear flows
• Avalanches in thin chute flows
• MD simulations and fitting the OP theory
• Conclusions

3
Onset of Motion FluidizationQuest for
Constitutive Relation

• Various phenomena avalanches, slides, surface
  flows, stick-slips are related to the transition
  from granular solid to granular liquid
• Theoretical descriptions of granular solid and
  granular liquid are very different need for
  unification
• Universal description of partially fluidized
  flows requires a constitutive relation valid for
  both granular solid and granular liquid

4
Recent publications

• Experiments
• Nasuno, Kudrolli, Bak and Gollub, PRE, 58, 2161
  (1998).
• Daerr and Douady, Nature, 399, 241 (1999)
• Pouliquen, Phys. Fluids, 11, 542 (1999)
• Veje, Howell, and Behringer, PRE, 59, 739 (1999)
• Mueth, Debregeas, Karczmar, Eng, Nagel, Jaeger,
  Nature, 406, 385 (2000)
• Komatsu, Inagaki, Nakagawa, Nasuno, PRL, 86, 1757
  (2001)
• Bocquet, Losert, Schalk, Lubensky, and Gollub,
  PRE 65, 011307 (2001)
• Molecular dynamics
• Silbert et al, PRE 64, 051302 (2002)cond-mat/020
  6188
• Aharonov and Sparks, PRE 65, 051302 (2002)
• Continuum theories
• Bouchaud, Cates, Ravi Prakash, and Edwards,
  J.Phys.France, 4, 1383 (1994)
• Boutreux, Raphael, and de Gennes, PRE, 58, 4692
  (1998)
• Bocquet, Losert, Schalk, Lubensky, and Gollub,
  PRE 65, 011307 (2001)
• Rajchenbach, Phys. Rev. Lett. 88, 014301 (2002)
  89, 074301 (2002)
• Aranson and Tsimring, PRE, 64, 020301(R)
  (2001)65, 061303 (2002)

5
Near-surface granular flows
Komatsu, Inagaki, Nakagawa, Nasuno, PRL, 86, 1757
(2001)
h
exponential velocity profile
6
Driven shear flow under a heavy plate
load
Tsai, Voth, Gollub, PRL 2003
Udriving 7.2mm/s 12 d/s
Particle size d0.6 mm Channel width ? 30d,
circumference ? 750d, depth 050d.
7
Taylor-Couette granular flow (2D)
Veje, Howell, and Behringer, Phys.Rev.E, 59, 739
(1999)
Velocity profile
8
Taylor-Couette granular flow (3D)
Mueth, Debregeas, Karczmar, Eng, Nagel, Jaeger,
Nature, 406, 385 (2000)
60mm
9
Taylor-Couette flow - 2
Bocquet et al, 2001
10
Chute flows
O.Pouliquen, 1999
Daerr Douady, 1999
11
Avalanches in chute flows
Daerr Douady, Nature, 399, 241 (1999)
Triangular (down-hill)
Balloon (up-hill)
12
Avalanches phase diagram
Daerr Douady, Nature, 399, 241 (1999)
uphill avalanches
spontaneous avalanching
Bistability
downhill avalanches
No flow
13
Stick-slip motion of grains
Nasuno, Kudrolli, Bak and Gollub, PRE, 58, 2161
(1998).
sliding speed V5.67 mm/s
sliding speed V5.67 mm/s
sliding speed V11.33 mm/s
14
Stick-slip motion - 2
15
Theoretical model
Euler equation
where
- density of material ( 1) g
- gravity acceleration v - hydrodynamic
velocity D/Dt - material
derivative - stress tensor
div v0 incompressibility condition
16
Stress-stain relation for partially fluidized
granular flow
Here
- strain rate tensor
- viscosity
- quasistatic (contact) part
fluid
solid
has non-zero off-diagonal elements
17
Stress-stain relation in partially fluidized
granular matter

• the diagonal components (pressures)
  related to the components of the static stresses
  may weakly depend on the order parameter
• shear stresses are strongly dependent on the
  order parameter r
• viscosity h may also weakly depend on the order
  parameter

18
Equation for the order parameter
f
Ginzburg-Landau free energy for shear melting
phase transition
r
0
1
t0,l characteristic time length
r
1
Two stable states r rf and r 1 One
unstable state ru
d is a control parameter
d
1
0
r0 liquid r1 solid
19
Near-surface granular flows
z
no gravity
x
Order parameter equation
Boundary condition
Control parameter
or by differentiating
20
Near-surface granular flows - theory (cont.)
,
Constitutive relation
Balance of forces requires
In non-dimensional units,
21
Analogy Maxwell stress relaxation condition
In visco-elastic fluids,
In our case,
fluid
solid
22
Stationary shear flow profile at large W
23
Relaxation oscillations at small W
Bifurcation diagram
24
Asymptotic velocity profiles (2D vs 3D)
Theory
Why?
25
3D shear granular flow
Force balance
In deep granular layers,
Control parameter
are major and minor principal stresses
and
26
3D shear granular flow (cont.)
Explicit z-dependence in the control parameter!
From OP equation near
and the same scaling for v
27
Chute flow
Equilibrium conditions
Stresses
Control parameter
(j1, j2 - static/dynamic repose angles)
28
Chute flow
Boundary conditions r 1 for z -h
(sticky bottom) r z 0 for z 0 (free
surface)
Stability of the uniform solid state r 1
OPE
Perturbation
Eigenvalue
Stability limit
29
Flow existence limit
Stationary OPE
Solution exists only for
1st integral
30
Single mode approximation
Close to the stability boundary
Here
31
Constant feeding flux at the top
t
1000
0
Stationary flow solutions
40
1.0
.99
0.9
.95
0.8
.90
d
0.7
30
0.6
0.5
0
40
80
.85
J/
m
h
20
.55
.60
.65
10
.70
0
0
10
20
30
40
50
60
J/
m
500
x
periodic avalanches
continuous flow
32
Two types of avalanches (theory)
Downhill
Uphill
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Two types of avalanches
Daerr Douady, Nature, 399, 241 (1999)
Triangular (downhill)
Balloon (uphill)
34
Avalanche cross-sections
uphill
20
Uphill front speed
Secondary avalanche
h
10
0
downhill
20
h
10
1st order transition!
0
0
200
400
600
x
35
Deep chute (sandpile)
For hgtgt1,
- x
Symmetry x
No triangular avalanches in sanpiles!
36
Quantitative comparison with experiment
Model parameters
t, characteristic time l, characteristic
length j1,j2, static/dynamic repose angles h,
viscisity coefficient
Daerr Douady
(particle diameter)
Ertas et al (MD simulations of chute flow)
37
Phase diagram (theory and experiment)
38
What is the order parameter?
39
Molecular Dynamics Simulations
We consider non-cohesive, dry, disk-like grains
with three degrees of freedom. A grain p is
specified by radius Rp, position rp,
translational and angular velocities vp and wp.
Grains p and q interact whenever they overlap, Rp
Rq - rp rp gt 0
We use linear spring-dashpot model for normal
impact, and Cundall-Strack model for oblique
impact.
Stress tensor
Restitution coefficient e
2304 particles (48x48), e 0.82 m 0.3
Pext 13.45,Vx24
Friction coefficient m
All quantities are normalized using particle size
d, mass m, and gravity g
40
Order parameter for granular fluidizationstatic
contacts vs. fluid contacts Microscopic Point of
View

• Zst is the static coordination number
• the number of long-term ( gt1.1tc)
• contacts per particle.
• Z is the total coordination number
• the total number of contacts per
• particle.

Stationary profiles of coordination numbers Z,
Zst, and order parameter in a system of 4600
grains. e 0.82 m 0.3 Pext 13.45, Vx 48
41
Order parameter for granular fluidizationElastic
vs Kinetic Energy Macroscopic Point of View

• U elastic energy stored in grains
• T fluctuational kinetic energy (granular
  temperature)

42
Stick-slip granular friction
43
Stress tensor
Reynolds stress tensor part of the stress
tensor due to short-term collisional contacts (t
lt 1.1tc). part of the stress tensor due to
force chains between particles ( t gt
1.1tc). Static stress tensor
Fluid stress tensor
44
Stationary near-surface flow shear stress
45
Couette flow in a thin granular layer (no gravity)
500 particles (50x10), e 0.82 m 0.3 Pext
13.45
Adiabatic change in shear force
46
Bifurcation diagram
47
Small initial perturbation in a bistable region
48
Order parameter fixed points
MD simulations

• 500 particles (50x10),
• 0.82 m 0.3

OP equation
49
Fitting the constitutive relation
Fit qy(r) (1-r1.2)1.9
Fit q(r) (1-r)2.5 Phenomen. theory q(r)1-r
qx(r) (1-r)1.9
50
Newtonian Fluid Contact Part
Kinematic viscosity in slow dense flows h12
51
Relation to Bagnold Scaling
Bagnold relation (1954)
Silbert, Ertas, Grest, Halsey, Levine, and
Plimpton, Phys. Rev. E 64, 051302 (2001) Fitting
Bagnold scaling relation
52
Fitting the constitutive relation
Fit f(r) 1 - (1-r)3
53
Fluid stress vs. shear strain rate
2300 grains e 0.82 m 0.3 Pext13.45.
Bagnold scaling?
54
Heavy plate under external forcing no gravity
Equation of motion for the plate
Constitutive relation
Order parameter equation
55
Heavy plate under external forcing no gravity
56
Shear flow of grains with gravity
MD simulations, box 48x48
57
Shear flow with gravity continuum model
58
Shear flow under gravity continuum model
59
Slip event MD simulations
60
Slip event MD simulations
61
Slip event MD simulations
62
Shear flow of grains with gravity
MD simulations box 96x48
63
Example stick-slips thick surface driven
granular flow with gravity
m
k
V0
y
Set of equations for sand
g
Ly
x
Equations for heavy plate
5000 particles (50x100), e 0.82 m 0.3
Pext 10,50,Vtop5,50
64
Simplified theory reduction to ODE

• Stationary OP profile
• x width of fluidized layer
• (depends on shear stress), r1(4r-1)/3
• Stationary solution exists only for specific
  value of d(y) (symmetry between the roots of OP
  equations) which fixes position of the front

y
x
g
x
65
Perturbation theory

• Substituting r into OP equation and performing
  orthogonality one obtains
• Regularization for xltlt1 (l is the growth rate
  of small perturbations)

66
Resulting 3 ODE

• 2 Equations for Plate
• 1 Equation for width of fluidized layer

67
Comparison Spring deflection vs time
theory PDE
theory ODE
MD simulations
68
Constitutive relations for stress components
indeterminacy
Static stress tensor
Force balance
Veje, Howell, and Behringer, Phys.Rev.E, 59, 739
(1999)
Symmetry
(only six relations, need 3 more!)
Linear elasticity Hooks law do not apply
Linear relation between shear and normal
stresses, force chains and stress inhomogeneity
69
Conclusions

• Stress tensor in granular flows is separated into
  a fluid part and a solid part. The ratio of
  the fluid and solid parts is fitted by the
  function of the order parameter ? F (1-r)a?, ?
  S (1 -(1-r)a) ?, a ? 2.5.
• The dynamics of the order parameter is descibed
  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.

70
Future directions

• Self-consistent description of stress evolution
• Elaboration of statistical features of
  fluidization transition, effect of fluctuations.
• Extraction of order parameter from experimental
  data