Lecture II: Granular Gases & Hydrodynamics
Igor Aronson
Materials Science Division
Argonne National Laboratory
Supported by the U.S. Department of Energy
1
Outline
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Definitions
Continuum equations
Transport coefficients: phenomenology
Examples: cooling of granular gas
Kinetic theory of granular gases
Transport coefficients: kinetic theory
2
large conglomerates of discrete macroscopic particles
Jaeger, Nagel, & Behringer, Rev. Mod.Phys. 1996
Kadanoff
Rev. Mod. Phys.1999
de Gennes
Rev. Mod. Phys.1999
non-gas
inelastic collisions
non-solid
no tensile stresses
Gran Mat
non-liquid
critical slope
3
Dropping a Ball
• Granular eruption
http://www.tn.utwente.nl/pof/
Loose sand with deep
bed (it was fluffed
before dropping the ball)
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Group of Detlef Lohse, Univ. Twenta
Granular Hydrodynamics
• Let’s live in a perfect world
-continuum coarse-grained description
-ignore intrinsic discrete nature of granular
liquid
-ignore absence of scale separation
• But, we include inelasticity of particles
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Granular gases
Definition:
• Collections of interacting discrete solid
particles. Under influence of gravity
particles can be fluidized by sufficiently
strong forcing: vibration, shear or electric
field.
• Granular gas is also called “rapid granular
flow”
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Comparison with Molecular Gases
• Main difference – inelasticity of collisions and dissipation
of energy
• Common paradigm – granular gas is collection of smooth
hard spheres with fixed normal restitution coefficient e
v12  v12'  (1  e )v12'  kk
v12 & v12'
k
are post /pre collisional relative velocities
is the direction of line impact
v1’
v2’
v1
v2
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The Basic Macroscopic Fields
• Velocity V
• Mass density r (or number density n)
• Granular temperature T (average
fluctuation kinetic energy)
Granular temperature is very different from
thermodynamic temperature
8
Distribution Functions
• Single-particle distribution function
f(v,r,t) = number density of particles having velocity v at r,t
• Relation to basic fields
n(r , t )   dvf (v, r , t )
number density
1
V (r , t ) 
dvvf (v, r , t )
velocity field

n( r , t )
1
2
T (r , t ) 
dv
(
v

V
)
f (v, r , t ) granular temperature

n( r , t )
V, v, and r are vectors
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Applicability of continuum
hydrodynamics
• Absence of scale separations between macroscopic and microscopic
scales:
Hydrodynamics is applicable for time/length scale S,L >> t,l
t,l – mean free time/path
l
, T - granular temprature
T
For simple shear flow with shear rate g : Vx=gy
Macroscopic time scale S=1/g
Granular temperature T~g2l2 Restitution coefficient is in the
t
Vx
prefactor
t/S~tg~O(1) – formally no scale separation
Restitution coefficient is a function of velocity
Leo Kadanoff, RMP (1999): skeptic pint of view
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Long-Range Correlations and Aging of
Granular Gas
• Inelasticity of collisions leads to long-range correlations
• Example: fast particle chases slow particle
elastic case – no correlation
inelastic case (sticking) – correlations
Usually particles don’t
stick
Lasting velocity correlations between different particles
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Continuum equations
• Continuity equation
Vi
Dn
n
0
Dt
ri
where
• Traditional form
D 
  V  is the material derivative
Dt t
n
 nV  0
t
Flux of particles J=nV, V – velocity vector
Number of particles:
Particles balance:
N   ndv

ndv   Ñ
nVds    nVdv


t
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Momentum Density Equations
• force on small volume: ∫Fdv
• acceleration: ∫nDV/dt dv
• relation between force F and stress tensor sij:
Fi 
s ij
r j
Compare in ideal fluid
F  p , p is pressure
• Momentum balance:
DVi s ij
n

- ng
Dt
rj
where s ij is stress tensor, g -gravity
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The Stress Tensor
• Compare hydrodynamic stress tensor, Landau & Lifshitz
 Vi V j 2 Vl 
Vl
s ij  h 

  ij
 x ij
 p ij

 rj ri 3 rl 
rl


144444444444444442 44444444444444443
Strain rate tensor
•
s
Only appears
c when contact
ij duration > 0
Appears in dense
flows, in granular
gases ~0
h,x – first (shear) and second viscosities (blue term
disappear in incompressible flow)
• p – pressure (hydrostatic)
• s ijc contact part
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Note: energy is not conserved, but mass and momentum are
Granular Temperature Equation
• Detail derivation in L&L, Hydrodynamics
Q j
DT Vi
n

s ij 
 G
Dt rj
rj
energy sink
(From inelastic collisions)
heat flux
shear heating
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•
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G ~n(1-e2) – energy sink term (absent in hydrodynamics)
Q  kT granular heat flux,
k – thermal conductivity
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Constitutive Relations: Phenomenology
• relate h,k,G materials parameters (restitution e, grain size d
and separation s) and variables in conservation laws n,V,T
s
d
d
• Typical time of momentum transfer t~s/u
u ~T1/2 – typical (thermal) velocity
• Collision rate = u/s
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Equation of state
• Pressure on the wall for s<<d using n~1/d3
mu 1
T
p~
~ const  n
2
t d
s
• Volume V=N/n, N – total number of grains
• s~V-V0; V0 – excluded volume
p(V  V0 )  const  NT
Analog of Van der Waal’s equation of state
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Viscosity coefficient
Vx(y)
y
• Two adjacent layers of grains
x
• shear stress from upper to lower layer s xy
momentum transfer
m DV u
~
d2 s
collision
rate
u dVx ( y )
• velocity gradient DV/d ~dV/dy s xy  const  d r
s dy
u
T
2
h~d r ~
• viscosity
s n  n0
2
r=m/d3 – density, n0- closed packed concentration
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Thermal diffusivity
• mean energy transfer between neigh layers
The ratio of the two viscosities is
constant, like in fluids
• Mean energy flux
muDu u
2 u d
Q~
~d
2
d
s
s dy
• Thermal diffusivity
muDu

1
2
ru 2 
u
T
k ~d ~
s n  n0
2
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The temperature rise from
collisions is very small
Energy Sink
DE ~ (1  e 2 ) 12 mu 2
• energy loss per collision
• Energy loss rate per unit volume
u
DE ~ (1  e ) mn  u 
s
2
1
2
2
• Energy sink coefficient
3
u
nT 3/ 2
G~r
~
s
n  n0
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Example: Cooling of Granular Gas
• Let’s for t=0 T=T0, V=0, n=const
T
 G : g (1  e 2 )T 3/ 2
t
• temperature evolution
T
where g  const
T0
 g (1  e )t  1
1
2
2
2
• asymptotic behavior T ~ 1/t2
• homogeneous cooling is unstable with respect to clustering!!!
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Q: Does the temperature
reach 0 in finite time?
R: Difficult to say, in
simulations sometimes it
does.
Clustering Instability
Simulations of 40,000 discs, e=0.5
Init. Conditions: uniform distribution
Time 500 collisions/per particle
MacNamara & Young, Phys. Fluids, 1992
Goldhirsch and Zanetti, PRL, 70, 1619 (1993)
Ben-Naim, Chen, Doolen, and S. Redner
PRL 83, 4069 (1999)
Mechanism of instability:
decrease in temperature →
decrease in pressure→
increase in density→
increase in number of collisions →
increase of dissipation→
decrease in temperature ….
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Thermo-granular convection
• inversed temperature profiles:
temperature is lower at open surface
due to inelastic collisions
• Consideration of convective instability
Theory:Khain and Meerson PRE 67, 021306 (2003)
Experiment: Wildman, Huntley, and Parker, PRL 86, 3304 (2001)
Shaking
A=A0 sin(wt)
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Kinetic Theory
• Boltzmann Equation for inelastically colliding spherical
particles of radius d
f ( v, r, t )
 v  f ( v, r , t )  B ( f , f )
t
• f(v,r,t) – single-particle collision function,
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Collision integral
• binary inelastic collisions
• molecular chaos
• splitting of correlations: f(v1,v2,r1,r2,t)= f(v1,r1,t) f(v2,r2,t)


loss
ter
m
64444444
47
4444444
4
8
 1

B( f , f )  d 2 
dv 2 dk (k  v12 )  2 f ( v1 ,r, t ) f ( v2 ,r, t )  f ( v1 ,r, t ) f ( v 2 , r, t ) 
k v12 0
e
442 44444444443
 144444444

gain
te
r
m


• k – vector along impact line
• v’1,2 –precollisional velocities
• v1,2 –postcollisional velocities
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Macroscopic variables
 Afdv
• averaged quantity
A 
• stress tensor
s ij  n ui u j
• heat flux
Q j  12 n u 2u j
• energy sink
G
1
n
 (1  e 2 )d 2
8n
3
d
v
d
v
v
 1 2 12 f ( v1 ) f ( v 2 )
• approximations for f(v,r,t) in Eli’s lecture
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Expressions for smooth inelastic spheres
Copied from Bougie et al, PRE 66, 051301 (2002)
• equation of state
• shear viscosities
• bulk viscosity
p  nT 1  2(1  e )G( )
h1 

nd T  165  G ( )  54 (1  12 )G (v) 2 
6G ( )
h2 
8ndG ( ) T
3 
Smooth inelestic spheres, from Jenkins & Richman, Arch. Ration. Mech. Anal. 87, 355 (1985).
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Expressions for smooth inelastic spheres
Copied from Bougie et al, PRE 66, 051301 (2002)
• heat conductivity
• energy sink
5 
k
nd T  245  G ( )  56 (1  932 )G( ) 2 
8G( )
8nG( )T 3/ 2
G
1 e 2
d


Smooth inelastic spheres, from Jenkins & Richman, Arch. Ration. Mech. Anal. 87, 355 (1985).
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Radial distribution function
• =(/6)nd3-packing fraction
• dilute elastic hard disks (Carnahan & Starling)
 (1  7 /16)
G ( ) 
2
1  
• High densities (~c =0.65 closed-packed density in 3D)
  
G ( )   1   
  c 

4v
3 c
1

1
 :
 c 


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Asymptotic behaviors
Works pretty well for
sheared granular flows
Dilute
shear viscosity
Nearly closed packed
h1 
heat conductance k 
heat sink
G
5
16  d 2
25
32  d
4
3
2
T
T
 n 2 d 2T 3/ 2 1  e 2 
h1 :
T
n0  n
T
k:
n0  n
n 2 d 2T 3/ 2
G:
1 e 2
n0  n


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Comparison with MD: Dynamics of Shocks
Q: Why is there not a
big temperature
gradient?
R: There is a slow
vibration, fast vibrations
have a large temperature
gradient
J. Bougie, Sung Joon Moon, J. B. Swift, and Harry L. Swinney Phys. Rev. E 66, 051301 (2002)
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Q: Do these equations predict oscillons,
waves, etc?
Comment: These equations work well
for low density and restitution
coefficient near 1.
R: Oscillons no, waves and bubbles yes.
References
•
•
•
•
Review:
-I. Goldhirsch, Annu. Rev. Fluid Mech 35,267 (2003)
Phenomenological Hydrodynamics:
-P.K. Haff, J. Fluid Mech 134, 401 (1983)
Derivation from kinetic theory:
-J. Jenkins and M. Richman, Arch. Ration. Mech. Anal. 87, 355 (1985).
-J.T. Jenkins and M.W. Richman, Phys. Fluids 28, 3485 (1985)
-N. Sela, I. Goldhirsch, J. Fluid Mech 361, 41 (1998)
Comparison with simulations:
-J. Bougie, Sung Joon Moon, J. B. Swift, and Harry L. Swinney Phys. Rev. E
66, 051301 (2002)
-S. Luding, Phys. Rev. E 63, 042201 (2001)
-B. Meerson, T. Pöschel, and Y. Bromberg Phys. Rev. Lett. 91, 024301 (2003)
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