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Lecture IV: Dense Granular Flows
Igor Aronson
Materials Science Division Argonne National Laboratory
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Outline
• Stationary Flow Stability
• Phase Diagram
• Avalanches in Shallow Layers
• Deep Layers/Connection to BCRE
• Comparison with MD simulations
• Granular Stick-Slips Friction
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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
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Shear Temperature: Simplified Version
)tan(tan2tantan2tan
12
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• 1, 2 – dynamic/static repose angles
•for – granular solid is unstable•for – solid/liquid equilibrium(note slightly different definition of )
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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
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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
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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:
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Phase Diagram
no flowsolid only
solid & liquid
flowliquid only
2/)1( 2/1 sh
hmin
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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
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Correction to z
x
)tan(tan2tantan2tan
12
21
0
- local slope contribution0 xh
)tan(tan2
1
12
0 xh 0- chute angle, –local slope
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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
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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
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Boundary conditions
• Inlet x=0– No-flux condition: Jx=0
– Fixed flux condition Jx=Ah3=J0 (grains supplied from hopper with the fixed rate)
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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
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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
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Two types of avalanches (theory)
15.0,25.0,2.1,3 h 05.0,25.0,07.1,5.5 h
Downhill Uphill
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Two types of avalanches (experiment)
Triangular (downhill) Balloon (uphill)
Daerr & Douady, Nature, 399, 241 (1999)
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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!
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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
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Daerr & Douady:
15.332,25 02
01
mdl 240~ (particle diameter)
msgd 5)/(~ 2/10
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Phase diagram (theory & experiment)
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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
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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
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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!
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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
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• 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
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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
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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:
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Fitting free energy: fixed points of the order parameter
MD simulations OP equation
500 particles (50x10), = 0.82; = 0.3
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2 2 2 2* * *
* *
( 1) ( , )
/
( , ) 2 exp[ ( )]
0.6; 0.26; 25; 2
t
xy yy
D G
G A
A D
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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
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Newtonian Fluid + Contact Part
Kinematic viscosity in slow dense flows: ≈
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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
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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
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Slip event: MD simulations
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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
&&
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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
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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
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12.6; 326
Constit. Relation ((1 ) )
C Cm
C C
V
&
21 2 2 220 * * *(1 )A
m
&
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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
&
& =
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Comparison: Spring deflection vs time
theory: ODE
MD simulations
theory: PDE
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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.