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Lecture II: Granular Gases & Hydrodynamics

Supported by the U.S. Department of Energy

Igor Aronson

Materials Science Division Argonne National Laboratory

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Outline

• Definitions

• Continuum equations

• Transport coefficients: phenomenology

• Examples: cooling of granular gas

• Kinetic theory of granular gases

• Transport coefficients: kinetic theory

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large conglomerates of discrete macroscopic particles

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

Gran MatGran Mat

non-gasinelastic collisions

non-gasinelastic collisions

non-liquidcritical slope

non-liquidcritical slope

non-solidno tensile stresses

non-solidno tensile stresses

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Dropping a Ball

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

Group of Detlef Lohse, Univ. Twenta

Loose sand with deep bed (it was fluffed before dropping the ball)

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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

• 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”

Definition:

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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

are post /pre collisional relative velocities

k is the direction of line impact

' '12 12 12(1 )v v v kk

'12 12&v v

v1’ v2

’

v1 v2

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The Basic Macroscopic Fields

• Velocity V

• Mass density (or number density n)

• Granular temperature T (average fluctuation kinetic energy)

Granular temperature is very different from thermodynamic temperature

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Distribution Functions

• Single-particle distribution function

f(v,r,t) = number density of particles having velocity v at r,t

• Relation to basic fields

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( , ) ( , , )

1( , ) ( , , )

number density

velocity field

granular temperatur

( , )

1( , ) ( ) ( , , )

( , )e

n r t dvf v r t

V r t dvvf v r tn r t

T r t dv v V f v r tn 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 >> ,l

,l – mean free time/path

For simple shear flow with shear rate : Vx=y

Macroscopic time scale S=1/

Granular temperature T~2l2

/S~~O(1) – formally no scale separation

, - granular temprature l

TT

Vx

Leo Kadanoff, RMP (1999): skeptic pint of view

Restitution coefficient is in the prefactor

Restitution coefficient is a function of velocity

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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

Lasting velocity correlations between different particles

Usually particles don’t stick

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Continuum equations

• Continuity equation

• Traditional form

where is the material derivati e0 vi

i

D

Dt

VD

D t

nn

t r

V

0 n

nt

V

Flux of particles J=nV, V – velocity vector

Number of particles:

Particles balance:

N ndvndv n ds n dv

t

V VÑ

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Momentum Density Equations

• force on small volume: ∫Fdv

• acceleration: ∫nDV/dt dv

• relation between force F and stress tensor ij:

• Momentum balance:

where is stress tensor, g -gravity- ii

jji

j

DVn n

Dt r

g

iji

j

Fr

Compare in ideal fluid , p is pressureF p

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The Stress Tensor

• Compare hydrodynamic stress tensor, Landau & Lifshitz

• – first (shear) and second viscosities (blue term disappear in incompressible flow)

• p – pressure (hydrostatic)

• contact part

Strain rate tensor

2

3l l

ij ijl

jii

jij

ij

l

VV

r r rp

V V

r

14444444444444444244444444444444443

cij

cij

Appears in dense flows, in granular gases ~0

Only appears when contact duration > 0

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Granular Temperature Equation

• Detail derivation in L&L, Hydrodynamics

• ~n(1-2) – energy sink term (absent in hydrodynamics)

• granular heat flux,

• – thermal conductivity

iij

j

j

j

DTn

D

V

t

Q

rr

shear heating

energy sink

heat flux

Q T

Note: energy is not conserved, but mass and momentum are

(From inelastic collisions)

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Constitutive Relations: Phenomenology

• relate materials parameters (restitution , grain size d and separation s) and variables in conservation laws n,V,T

• Typical time of momentum transfer ~s/u

u ~T1/2 – typical (thermal) velocity

• Collision rate = u/s

d

s

d

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Equation of state

• Pressure on the wall for s<<d using n~1/d3

• Volume V=N/n, N – total number of grains

• s~V-V0; V0 – excluded volume

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1~ ~

mu Tp const n

d s

0( )p V V const NT Analog of Van der Waal’s equation of state

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Viscosity coefficient

• Two adjacent layers of grains

• shear stress from upper to lower layer

• velocity gradient V/d ~dV/dy

• viscosity

x

yVx(y)

2~xy

m

s

V

d

u collision rate

momentum transfer

2 ( )xxy

dV yconst d

u

s dy

2

0

~ ~u T

ds n n

=m/d3 – density, n0- closed packed concentration

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Thermal diffusivity

• mean energy transfer between neigh layers muu

• Mean energy flux

• Thermal diffusivity

2 2122

~ ~d

Q d ud dys

u

s

m uu u

2

0

~ ~u T

ds n n

The ratio of the two viscosities is constant, like in fluids

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Energy Sink

• energy loss per collision

• Energy loss rate per unit volume

• Energy sink coefficient

2 212~ (1 )E mu

2 212~ (1 )E mn u

u

s

3 3/ 2

0

~ ~u nT

s n n

The temperature rise from collisions is very small

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Example: Cooling of Granular Gas

• Let’s for t=0 T=T0, V=0, n=const

• temperature evolution

• asymptotic behavior T ~ 1/t2

• homogeneous cooling is unstable with respect to clustering!!!

2 3/ 2(1 ) where T

T constt

:

0221

2 (1 ) 1

TT

t

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Clustering Instability

MacNamara & Young, Phys. Fluids, 1992Goldhirsch and Zanetti, PRL, 70, 1619 (1993)Ben-Naim, Chen, Doolen, and S. Redner PRL 83, 4069 (1999)

Simulations of 40,000 discs, =0.5Init. Conditions: uniform distributionTime 500 collisions/per particle

Mechanism of instability: decrease in temperature →decrease in pressure→increase in density→increase in number of collisions →increase of dissipation→ decrease in temperature ….

Q: Does the temperature reach 0 in finite time?

R: Difficult to say, in simulations sometimes it does.

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Thermo-granular convection

• inversed temperature profiles: temperature is lower at open surface

due to inelastic collisions

• Consideration of convective instability

Shaking A=A0 sin(t)Theory:Khain and Meerson PRE 67, 021306 (2003)

Experiment: Wildman, Huntley, and Parker, PRL 86, 3304 (2001)

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Kinetic Theory

• Boltzmann Equation for inelastically colliding spherical particles of radius d

• f(v,r,t) – single-particle collision function,

( , )( , ) ( , )

f tf t B f f

t

v,r

v v,r

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Collision integral

12

22 120 2

gain te

loss ter

m

m

r

( , ) ( , )1

( ,( , ) ( ) ,) ( )k v

f tB f f d d f t ft f td

1 2 1 2v k v ,r v ,rv v ,r v rk ,144444444

644444444744444444

4424444444443

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• binary inelastic collisions

• molecular chaos

• splitting of correlations: f(v1,v2,r1,r2,t)= f(v1,r1,t) f(v2,r2,t)

• k – vector along impact line

• v’1,2 –precollisional velocities

• v1,2 –postcollisional velocities

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Macroscopic variables

• averaged quantity

• stress tensor

• heat flux

• energy sink

• approximations for f(v,r,t) in Eli’s lecture

ij i jn u u

1nA Afd v

212j jQ n u u

2 23

1 2 12 1 2

(1 )( ) ( )

8

dd d v f f

n

v v v v

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Expressions for smooth inelastic spheres

• equation of state

• shear viscosities

• bulk viscosity

1 2(1 ) ( )p nT G

25 4 121 16 5( ) (1 ) ( )

6 ( )nd T G G v

G

2

8 ( )

3

ndG T

Smooth inelestic spheres, from Jenkins & Richman, Arch. Ration. Mech. Anal. 87, 355 (1985).

Copied from Bougie et al, PRE 66, 051301 (2002)

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Expressions for smooth inelastic spheres

• heat conductivity

• energy sink

25 6 3224 5 9

5( ) (1 ) ( )

8 ( )nd T G G

G

3/ 2

28 ( )1

nG T

d

Smooth inelastic spheres, from Jenkins & Richman, Arch. Ration. Mech. Anal. 87, 355 (1985).

Copied from Bougie et al, PRE 66, 051301 (2002)

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Radial distribution function

• =(/6)nd3-packing fraction

• dilute elastic hard disks (Carnahan & Starling)

• High densities (~c =0.65 closed-packed density in 3D)

2

(1 7 /16)( )

1G

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1

1( ) 1

cv

c c

G

:

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Asymptotic behaviors

Dilute Nearly closed packed

1 120

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2 2 3/ 22 2 3/ 2 2 24

30

5 shear viscosity

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25 heat conductance

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heat sink 1 1

TT

n nd

TT

n nd

n d Tn d T

n n

:

:

:

Works pretty well for sheared granular flows

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Comparison with MD: Dynamics of Shocks

J. Bougie, Sung Joon Moon, J. B. Swift, and Harry L. Swinney Phys. Rev. E 66, 051301 (2002)

Q: Why is there not a big temperature gradient?

R: There is a slow vibration, fast vibrations have a large temperature gradient

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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)

Q: Do these equations predict oscillons, waves, etc?

R: Oscillons no, waves and bubbles yes.

Comment: These equations work well for low density and restitution coefficient near 1.