Quantitative Study of a Freely Cooling Granular Medium

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Quantitative Study of

_a

Freely Cooling Granular Medium

Pierre Deltour

(*)

^and Jean-Louis Barrat

D6partement

de

Physique

des Mat4riaux

(**),

Universit4 Claude Bernard

Lyon

_1,

69622 Villeurbanne

Cedex,

France

(Receiied

9

April

1996,

accepted

17

September 1996)

PACS.47.50.+d Non-Newtonian fluid flows PACS.05.20.Dd Kinetic

theory

PACS.47.55.Kf Multiphase ^and

particle-laden

flows

Abstract. We present _a numerical

study

of _a two-dimensional

granular

medium

consisting

of hard inelastic disks. The evolution of the medium

throughout

_a

cooling

_process is monitored.

Two different types of instabilities

(shearing

and

clustering instability)

_are found _to

develop

in the system. ^The

development

of these instabilities is shown to be in

qualitative

and quantitative

agreement ^with the

predictions

of linearized

hydrodynamic theory.

R4sum4. Nous

prdsentons

_une 4tude d'un fluide

granulaire

modAle h deux dimensions

compos4 de disques durs

in61astiques.

On s'int6resse h l'Avolution du milieu pendant le _processus de refroidissement. Dans

un tel

systAme,

deux types d'instabilit6 peuvent apparaitre spontan4-

ment. La caract6risation de leur croissance est quantitativement en accord

avec la prddiction ^de

l'hydrodynamique

lin6aris6e.

1. Introduction

Granular fluids _are dense niedia

coniposed

of

elenientary

elenients of

niacroscopic size, undergo- ing

collisions in which their

macroscopic

_energy is not conserved. In the last

decade,

the flow of

these

granular

fluids has received _a

great

attention front the

physics coniniunity,

^{both because} these media offer _a

"siniple" exaniple

of

dissipative systenis

^{and because} of their _nunierous

industrial

applications.

Two factors niake the behaviour of

granular

fluids _very different front that of niolecular fluids.

Firstly,

the

niacroscopic

size of the

particles iniplies

that external fields

(boundaries

_or

gravity)

have _a niuch

stronger

effect _on

granular

fluids.

Secondly,

the

energy of the

granular

fluid is not _a conserved variable, since the heat

dissipated

in collisions

can be considered _as lost _as far _as the flow is concerned. These _two effects _are often difficult to

disentangle

in

experinients,

and also in the nunierical siniulations that aini at a realistic

niodelling

of these

experinients ill.

A different

type

^of nunierical siniulations _was initiated

by

several groups

[2-6].

^{In this}

^approach

external fields _are

ignored,

and

only

the

dissipation

is taken into account. This

dissipation

is _nioreover niodelled in _a very

siniplistic

_way,

by

de-

scribing

the

particles

as

nionodisperse rigid

hard

spheres undergoing

inelastic collisions. The

(*)

Author for

correspondence (e-mail: [email protected]) (**)

UMR CNRS 5586

©

Les

#ditions

_de

_Physique

₁₉₉₇

"cooling

these authors start front _an

equilibriuni configuration

of _an elastic

jr

=

I)

hard disc fluid.

The behaviour of this fluid after

introducing

_a nonzero restitution coefficient is followed

using

the standard niolecular

dynaniics

niethod for hard bodies

[10],

^with a collision rule that takes the

dissipation

into account. The kinetic

teniperature (average

kinetic energy per

particle)

decreases due to the

inelasticity

of the

collisions,

_so that the fluid cools down _as tinie increases.

It _was shown in

[2,5] that, depending

_on the

systeni size,

on the restitution coefficient and

density,

this

cooling

follows different routes. In the

siniplest

_case,

(sniall _systenis

_or sniall

dissipation)

the fluid reniains

honiogeneous

at all teniperatures. ^In

larger systenis

or for

larger dissipations,

either the

velocity

field _or the

density

field in the fluid

develop

instabilities and beconie

inhoniogeneous.

^It was also found

by

the _saute authors that the _occurrence of such instabilities is in

qualitative agreenient

^with the

predictions

of _a linear

stability analysis

of the

hydrodynaniic equations

for

granular

fluids

iii. Finally,

it _was discovered in

[3,

5] ^{that in} sortie cases, the

cooling

ends at _a finite tinie due to _a

singularity

in the systeni

dynaniics,

which _was shown to

correspond

to _an infinite nuniber of collisions within _a finite tinie. This

singularity,

observed both in I and 2

diniensions,

_was described _{as an} "inelastic

collapse"

of the

systeni.

In this _{paper, a} detailed and

quantitative analysis

of the

instability

^of

honiogeneous cooling

of

granular

fluids is

attenipted.

Our alai is to conipare

quantitatively

the

predictions

of

granular

kinetic

theory

and

granular hydrodynamics

to the results of molecular

dynaniics

simulations of the

cooling

of _an inelastic hard disk fluid. The paper is

organized

as follows. The niain

predictions

of

granular hydrodynaniics concerning

^the

cooling probleni

_are

briefly

recalled.

Coniputational

details

concerning

the siniulation _are

given

in Section 3. The different

reginies occurring during

the

cooling

_are

analyzed

in Section

4,

and

conipared

with the theoretical

predictions.

Our niain focus will be _on the

growth

rate of the

density instability,

that _can be

coniputed by nionitoring

the structure factor of the systeni as a function of tinie.

Finally,

the

probleni

of the "inelastic

collapse"

is addressed in Section 5, where _a

possible

niethod for

avoiding

this

singularity

in the

systeni dynaniics

is

proposed.

2.

Hydrodynamic Analysis

of the

Cooling

Problem

The

hydrodynaniic equations

that have been

proposed

to describe

granular

flow _{[7] are} based _on

niass and nionientuni

conservation,

and _{are very} siniilar _to the usual Navier-Stokes

equations.

The

only

modification is the _appearance of _{a new term} in the _energy

(or teniperature) equation, accounting

for the loss of energy in the collisions. These

equations

_can be

conipactly

written in the forni

j)

=

-PV.v, (1)

Dv

Pm

^{" -V}

^P,

=

⁽²⁾

p~~

= -V

Q

tr

(fib) ~fT~/~, (3)

where

D/Dt

is the

hydrodynaniic

derivative, _p is the

density, b

_the

symnietrized velocity gradient

tensor, ^P the stress tensor,

Q

the heat flux and

~f represents ^the rate of _energy lost due to inelastic collisions. For _a hard disk

fluid,

the

equation

of state and the

expression

of the various

transport

coefficients _can be obtained from Jenkins and Richnian kinetic

theory

_[9].

These

expressions

are recalled in

Appendix

A. The energy sink term,

~fT~/~,

has also been written in the forni

appropriate

for hard disks. _~f in that _case is _a function of the

density

and the restitution

coefficient,

which at least in the low

density

liniit niust be

proportional

to p and

(I r).

This _can be understood front the

following reasoning:

the kinetic _energy loss per

particle

_per unit tinie is

proportional

to the collision

frequency (i.e.

to

pT~/~)

and to the

energy loss _per collision

(I r)T.

A trivial solution of the

cooling probleni

forniulated above

corresponds

to _an

honiogeneously cooling fluid,

with _a uniforni

density,

_a

vanishing velocity field,

and _a uniforni

teniperature

with _an

algebraic

tinie

decay

T(tl

= To

1

+

)) ₍₄₁

Here to _"

2po/l'foTo~/~)

sets the tinie scale for

teniperature decay

in the fluid. The linear

stability

of this

honiogeneous

solution has been

investigated

in references

[2,11].

For _coni-

pleteness,

the niain steps ^{of this}

analysis

will be

repeated

here. The linearized

equations

^that describe the evolution of _a sinusoidal

perturbation

6p

=

6p~

_exp

(ik r), (5)

bv =

bvk

_exp

(ik r) (6)

bT

=

bTk

_exp

(ik r)

,

(7)

around the

honiogeneous

solution are

~~Pk

-i

(k

. vk) > ~~~

ok .

~°

at (9)

~~k

(10) -

bki ' ^{jk~ (k fi}

Po ~~ _~°

~

~~ '

T~

b6Tkj

_ 2~~~

~

(ik

~~~~ °~ ~

~°

°~ ~°j~oToi/2[T)

-

^{3/2u Ill}

~

l~

~~ ~~

As for

^usual

fluids,

the _{transverse part}

of

theelocity

field

conipletely

decouples _front

the

longitudinal

part,

and decays ^with tinie

as

_ii

+

t/to ~~°~~~~°~~° The part of the

A B C

',

_',,

~~'~0.0

_{O-1 0.2}

ko

Fig.

I. The

decay

rate of the

velocity

field disturbances

computed by solving equation (4)

for p = 0.2, _{r =} 0.9 is shown in this

figure

_as

a function of wavevector. The dashed line indicates the

decay

rate of the _transverse modes. The black lines

correspond

to the real part of the

decay

rate of the three

longitudinal

modes.

have _an

algebraic

tinie

dependence:

6p~

=

b)~[l+t/to)~, ₍₁₂₎

6vk

"

6Pk

_{(1 + t}

/to)~~~, (13)

bTk

#

bTk _(I

_{+ t}

/to)~~~ (14)

The

exponents ((k)

for the three triodes _are the three roots of the deterniinant of the

following

set of

equations

b>k ~~~ti

_~~

+

k~To P'(Uo)

+ Uo

Ill

l~l

⁺

ill]11

⁺

^6fk ik~PoP'(Uo)1

- °

6p~ (~

_to

(-2

⁺

^{T)/~u ~~}

_au ^~°

^'(uo)()

⁺

^{6Tk ((}

⁺

¹⁾

^{~° +}

^ok~j

^{= 0.}

opo to

A

typical plot

of the _wavevector

dependence

of these three roots,

together

with the

growth

rate of the

velocity perturbations,

is shown in

Figure

I. It niust be

eniphasized

that the

stability

of

velocity

disturbances is determined

by

the

comparison

between the

growth

exponent of the disturbance with the value -I that characterizes the

decay

of the thermal

velocity.

Hence _a

growth exponent larger

than -I for the transverse _or

longitudinal velocity

fields is indicative of _an

instability

of the

macroscopic velocity.

If the

growth exponent

^{of the}

longitudinal

_ve-

locity

field is

larger

than

-I,

_a

corresponding instability

in the

density

field will follow front

equation (12).

This

analysis yields

to the

prediction

of three different

possible

behaviours of the

systeni, depending

on the value of the

paranieters

^and on the

systeni size,

that introduces _a lower wavevector cutoff. If this lower cutoff is in the

region

C of

Figure I,

the

homogeneous

solution will be

linearly

stable. This

reginie

will be described _as the

honiogeneous

^kinetic

regime.

If the lower cutoff _{moves to} the

region

^B in

Figure I,

the transverse

velocity

field will beconie

unstable while the

systeni

reniains

honiogeneous.

In this

"shearing" reginie.

first observed in reference

[2],

a

shearing

flow will

develop

in the

system. Finally,

for _a lower cutoff

corresponding

to the

region A,

_an

instability

of the

longitudinal velocity

^{field and} the

corresponding instability

in the

density

field will take

place together

with the

shearing instability.

^In this

"clustering"

regime,

the

growth

^of

density

disturbances will

yield

to the formation of dense clusters of

particles,

_as first observed in reference

[12].

All three situations have

already

been observed in numerical siniulations of the

cooling

in two-diniensional

granular

fluids. The aini of the next sections will be to attenipt a

quantita-

tive

analysis

of the behaviour of _a

cooling granular fluid,

and to conipare the results to the

predictions

suniniarized above.

3.

Computational

Details

The niodel siniulated in this work is in all

respects

siniilar to that studied in

[2, 5].

^The systeni is niade up of N hard inelastic disks of dianieter _a, in _a square cell of size L with

periodic

boundary

conditions. The cell size L _sets the lower cutoff in _wavevector space,

kmin

= 27r

IL.

A standard cell-linked Molecular

Dynaniics algorithni

for hard bodies

[10]

^{is used. In} a first step, ^the

systeni

^is

equilibrated

with _a coefficient _r

equal

to

unity.

^At tinie t ₌

0, inelasticity

is switched _on and

cooling

starts, with _an initial

teniperature To-

The restitution coefficient

enters

through

_a

simple

niodification of the standard collision rule between hard

disks,

the

velocities of the two disks after _a collision

being given by

u(

= ui

(l

₊

r)[fi (ui u2))fi, (15)

2

~~

~~ ~ ~~ ~ ~~~~ ~~~

~~~~~'

~~~~

where the

prinies

^denote the

quantities

after collision and fi is _a unit vector

along

^the centers line from

particle

1 towards

particle

2. The natural units in this

probleni

_are the

particle

mass m and dianieter, and the thernial energy ^{at t} = 0, I.e.

To-

The

corresponding

tinie unit is

T =

(m/T)I/~a.

The state of the

systeni

^{is defined}

by

three diniensionless

nunibers,

which _are the reduced size L

la (or equivalently

the reduced cutoff

k[~~

₌

kmina),

the reduced

density p*

₌

a~N/L~,

and the restitution coefficient _r.

The state of the fluid

during

^the

cooling

_was nionitored

by

_a

systeniatic coniputation

of

coarse-grained (hydrodynaniic) density

and

velocity

fields. The _coarse

graining

is

obtaining

here front _a division of the

systeni

into 100 _square subcells.

Besides,

statistical

quantities characterizing

the state of the

systeni

^{have also been}

systeniatically coniputed.

These

quantities

are the nionienta of the

velocity

distribution of individual

particles,

the

pair

correlation function

g(r)

for

interpartide distance,

and the _structure factor

S(k)

=

6p~6p_~. (17)

This structure factor _can be

coniputed

for all wavevectors

conipatible

with the

periodic

bound- ary condition. of the forni

(n~, ny)kmjn.

^{As the}

systeni

is not in _a

stationary state,

these

quanti-

ties _are tinie

dependant.

^A

large enough systeni

^{is thus} necessary to obtain reasonable statistics

i-o

O-O

O-O 1.0 2.0 3.O 4.O

r/ a

Fig.

2. Pair correlation functions

girl

computed ^{for the initial}

(in

black shifted up

by 0.5)

and final

(in grey) configurations

in _an

homogeneously cooling

system

(N

= 1600. p*

= 0.5. _{r =}

0.99).

The temperature

drops by

_a factor of170000 without

significant change~

in the pair correlation function.

without tinie

averaging.

The values of N

investigated

in this work vary front N

= 1600 _to

,l/ =10000.

4. Results

4.I. KINETIC REGIME.

According

to the

analysis

of Section

I,

the kinetic

regime

corre-

sponding

to _a stable

honiogeneous cooling

will be observed

(at

_a

given density

^and restitution

coefficient)

for sniall

enough systenis.

Such _a situation allows _a clear

testing

of _sortie of the

hypothesis

of the kinetic

theory description

of the

granular

fluid. In

particular,

the

pair

_corre~

lation function and

velocity

distribution _can be

conipared

to that of _an elastic hard disk fluid

throughout

the

cooling

_process. The

teniperature decay

can be nionitored and

compared

to the theoretical

prediction (Eq. (4)),

and the

decay

tinie to

(or equivalently

the coefficient _~

(pi conipared

to the

prediction

^of kinetic

theory.

The

pair

correlation of _an

honiogeneously cooling granular

fluid after the

teniperature

has

dropped by

_a factor of170000 is shown in

Figure

2. This

coniparison

shows that the local structure of the

cooling granular

niediuni

(which

deterniines its

gquation

^of

state)

reniains

essentially

identical to that of _an

equilibriuni

fluid. The

study

of the

velocity

distribution function shows that this distribution reniains Maxwellian

throughout

^the

cooling.

This-

siniilarity

between the _structure and

velocity

distribution of the

granular

fluid and the usual hard disk fluid

suggests

^{that the kinetic}

theory

of Jenkins [9] is

applicable.

This

expectation

is borne out

by

the

study

of the tinie

dependence

of the fluid

temperature.

^As

shown in

Figure

3, the temperature

decay

is

perfectly

described

by equation (4).

The

density

600.0

soo.o

400.0

~

~~

300.0

~

200.0

ioo.o

~~0.0

_{100000.0 200000.0 300000.0}

t/1

Fig.

3. Evolution of the square ^root of the inverse temperature _versus t in _an

homogeneously cooling

system

jr

₌ 0.99, p* = 0.1, N

=

1600).

The solid line

corresponds

to the

hydrodynamic prediction

in the kinetic

regime.

dependence

of the

decay

tinie to ^is

conipared

in

Figure

4 to the

prediction

^{of kinetic}

theory (see appendix B).

The

agreenient

^is

extreniely good,

and

suggests

^{that all the}

transport

coefficients

appearing

in the

hydrodynaniic equations

_can ^{be estiniated}

using

^this kinetic

theory.

4.2. SHEARING REGIME. if the restitution coefficient _r decreases _or if the size of the

systeni increases,

the

hydrodynamic theory predicts

_a

reginie

in which transverse fluctuations of the

velocity

field _are unstable. This

reginie

is indeed observed in the

siniulations,

as shown in

Figure

5. A shear flow that

corresponds

to the sniallest wavevector

conipatible

with the

periodic boundary

conditions

develops

in the

systeni.

^{In this}

reginie,

the total kinetic energy of the

systeni (which

in that _case is not the

teniperature,

since the

systeni

has

developed

_an

ordered flow

pattern) appreciably

deviates front

equation (4),

_as shown in

Figure

6.

4.3. CLUSTERED REGIME. For _even

larger

_{systems or} smaller restitution

coefficients,

the

cooling granular

fluid beconies

inhoniogeneous,

_as shown in

Figure

7. This

spontaneous

fornia- tion of

density inhoniogeneities (or clusters)

was first observed in the simulations of the

cooling probleni by

Goldhirsch and Zanetti and

Young

and McNaniara

[2, 5].

^{Two different}

explana-

tions have been

put

^forward to

explain

this cluster forniation. The first _one, found in [2], ^is to consider this cluster forniation _{as a}

secondary instability

of the

shearing reginie,

due _to the

developnient

of teniperature ^and pressure

gradients.

The second

possible explanation

is that cluster formation is

directly

related to the linear

instability

of the

density

modes

predicted by hydrodynaniic theory.

In order to characterize

quantitatively

this

clustering reginie,

the structure factor

S(k, t)

of the systeni ^{has been}

coniputed

_{as a} function of tinie and wavevector. The

corresponding

data

ioo.o

O-O

O-O 0.2 0.4 0.6 0.8

No~/L~

Fig.

4.

Enskog

corrected value of the

decay

time calculated

as a function of

density

for _r

= 0.98,

compared

to the values obtained in the simulations

(N

=

1600).

is shown in

Figure

8. The

growth

of the

density inhoniogeneities

results in the _appearance of

a low wavevector

peak

in the structure

factor,

that

rapidly

increases with tinie.

According

to

hydrodynaniics,

the tinie

dependence

of

S(k,t)

should be

algebraic,

I.e.

2t(k)

S(k,t)

=

S(k, 0) 1

+

-) ₍₁₈₎

to

so that the ratio

In(S(k,t)) In(S(k,°))

~

~~~~j ~~~j

~~

~

~

)j

should be

independent

of tinie. This ratio is

plotted

in

Figure

9 _{as a} function of wavevector for different tinies.

2((k)

_{seenis to} be

reasonably independent

of

tinie,

and its low wavevector

value _{appears to} be consistent with the

prediction

of linearized

hydrod»namics.

-The

high

wavevector deviation _can be

explained by

the contaniination of these triodes

by

the

growth

of the

long wavelengths.

Hence the

density instability

_can be

interpreted

_as

resulting

front _a linear

instability

of the

honiogeneous

solution of the

hydrodynaniic equations.

Note that it

was

recently

observed

by

McNaniara and

I"oung

that the

"clustering"

fluid

eventually develops

for

long

times into _an ordered flow

pattern

^{of the}

"shearing" type.

This is also consistent

with

hydrodynamics,

since the

growth

rate of the transverse

velocity

^triodes is

positive.

The

description

of the forniation of this

shearing

flow in _an

inhomogeneous _systeni, however,

is

beyond

the

possibilities

of linearized

hydrodynaniics.

~ J~

i~ ~ _{~o CR~ c~7 #O ~i}

_c~>

~~m

~

~

~ ~ ~

_xr

ti~ ~m ~m ~ _~ 4#~lJ _~5~

_{t~ ~}

/~*

©#4#©#4#©m4m4m~n~4#

©#zb~~©~~fim Q~~©~©iffi

£F~4m

_rS

~l~ ~ i~> ~ 4jm~

_~

t~

~

iY1~ ~ / t/ ~

##

~ ~

_C~S

#o ~

_~~/

iJ

c~~

~

~fi~fi~4>#omfi~fimfi#fi#fiCtfi fiofisfio#fi#fic47#fic#2~fi#fi

Fig.

5. Velocity field in the

shearing

regime ^{after the} granular medium has

spontaneously

developed

a flow pattern

corresponding

to the lowest _wave vector

compatible

with the

boundary

conditions.

(N

₌ 1600, p* ₌ 0.I, _{r =}

0.92).

5. Inelastic

Collapse

and How to Avoid it

The inelastic

collapse singularity

_was first observed

by

_{[3, 6]} in siniulations of unidiniensional inelastic

systenis.

^This

collapse

_can be described _as the _appearance of _an infinite nuniber of _cor- related collisions between _a few

particles, taking place

in _a finite tinie. The _saute

phenonienon

was observed in two diniensions

by _[5].

It _was shown that in that _case the correlated collisions take

place

between _a sniall nuniber of

essentially aligned particles,

so that the unidiniensional

situation is

practically reproduced.

In order to avoid this inelastic

collapse,

_a

slightly

niodified collision rule between the

particles

can be introduced. At each

collision,

the relative

velocity

of the two

particles

is first

coniputed according

to the usual rule

(Eqs. (is, 16)),

then rotated

by

_a sniall

(less

than 5

degrees)

randoni

angle.

This _can be

justified by invoking

the unavoidable

roughness

of actual solid

particles,

conservation of

angular

nionientuni

being (virtually)

ensured

by

a transfer to the internal

degrees

of freedom of the

particles.

As to inelastic

collapse,

the alai of this niodified collision rule is to hinder the forniation of correlated

particle

lines that _cause this

singularity.

Indeed,

inelastic

collapse

_was not observed in the siniulations where this "randoni" collision rule _was used, while under the _saute conditions _a

systeni following

the "deterniinistic" collision rule

always

under1N.ent inelastic

collapse (Fig. 10).

Hence inelastic

collapse

_appears to be _a

pathology

related _to the _use of

purely specular

collision rule between

particles,

rather than _a characteristic of inelastic fluids.

o o o

200.0

~ o

°

~ o

ioo.o

~~0.0

_{10000.0 20000.0 30000.0 40000.0}

t /1

Fig.

6.

Square

root of the inverse of the temperature versus t in the

shearing regime.

The solid line

extrapolates

towards the first moments of the _run. There is _a substantial deviation from this kinetic

regime

fit.

(N

= 1600, p* ₌ 0.1, _{r =}

0.92).

6. Conclusion and

Perspectives

The niain

objective

of this work _was to _assess the

validity

of the

hydrodynaniic description

of

granular

fluids

originally proposed by [7],

^{and of the kinetic}

theory

calculation of the associated

transport

coefficients. The

study

of the

particularly siniple "cooling

fluid" _case and of the associated instabilities

provides

_an ideal benchniark for this

description.

The

coniparison

between nunierical siniulations and theoretical

predictions

in this

siniple

_case shows that the

theory

is

quantitatively

accurate. A siniilar conclusion _was also reached in _a recent

study by

McNaniara and

Young [13],

^{who showed that the} transitions between the different

cooling

reginies

were

correctly predicted by

the

theory.

The

description

of the inelastic

collapse phenonienon

observed

by

McNaniara and

Young

is

obviously beyond

the

possibilities

of kinetic

theory

_or

hydrodynaniics.

It _was shown that this

phenonienon

_can

easily

be avoided

by introducing

_a small amount of randoniness in the

collisions between

particles,

siniilar _to what would be caused

by

the natural

roughness

of

granular particles.

Obviously,

_a correct

description

of

granular

fluids cannot be achieved without _a

knowledge

of the

boundary

conditions that must be used for the

hydrodynamic equations.

These

conditions,

and in

particular

those that

correspond

to the very

important

_case of

vibrating

solid

walls,

_are

not known. Their deterniination,

through

the

quantitative coniparison

of nunierical siniulation and

theory,

will be the

subject

^{of future} work.

o

o o

o

o o

o

o

o

o o

oo o

o o

~ o

° o~

o ~

o o ^o ~

~~ o o°

~ co

o o o

~ °

o o

~

°

o o~

o°

~

~

o o

o

~ o

° o

o ° ~

o o

~

o o

o

~ o o

Fig.

7. Final

configuration (141

collisions _per

particle)

of _a simulation in the cluster

regime. (N

₌ 1600, p* ₌ 0.25, _{r =}

0.6).

Acknowledgments

This ~v"ork _was

supported by

the Pole

Scientifique

de ModAlisation

Nuni4rique

at

ENS-Lyon

and the Centre National d'Etudes

Spatiales (Aide 96/CNES /0367).

Appendix

A

Expressions

for the

Transport

Coefficients and the

Equation

of State The Navier-Stokes like

equations describing

_a

granular

fluid _are:

Dp

$

^{" ~P} ~ ^'V.

Dv

~

p

~ Dt ^'

p~~

= -V

Q

tr

(P _D) ^~T~/~

D

/Dt

is the

hydrodynaniic derivative,

D the

syninietrized velocity gradient

tensor, ^P the stress tensor,

Q

the heat flux and _~

represents

^the rate of energy lost due to inelastic collisions. The

definition for these

quantities

is:

~

=

~+jv.vj,

S-O

~'~o.o

~~~~

5.o ko

Fig.

8. Evolution of the structure factor

during

cluster formation

IN

= 1600, p* = 0.5, r =

0.4).

The _curves from the bottom _are separated

by

10 collisions _per particle. Note the

large

increase of the

structure factor in the

long wavelength

limit

(k

_-

0).

~ l

bv~

bvjj

" 2

bxj

^~

bz~

^'

fl

=

phi

_2~t

(b ^{(V vi )}

,

Q

" -~VT.

The various

transport

coefficients _and

equation

of state _are

Ph =

P'(UlpT,

~t =

p'(u)T~/~pa,

K =

K'(u)T~/~pa,

~ =

~i'(U)~,

where

p', ~t', K',

_{~f' are} functions

only

of the solid fraction _u

=

p/ps.

These functions _are:

~lj~j

^~~+~*

s~

~~~~~

~j~

^~

~

~~ ~

*~~~ ~«'

~~~~~

~j~

^~

~ ^~~

~

*~~~ ^~«'

2.0

1-s

j

~l

4

^{f ~}

~'°

~

^~ _V

j

g ^o

(

~

c ~

fi (

o.5 ^{° ~ #~}

(

~ ~

~

fl ^~

o

o-o

n

n -0.5

O-O 0.1 0.2

k a

Fig.

9. Growth exponent ^{of the}

density

field disturbance, obtained from the simulation using

equation (19) (N

= 10000, p* ₌

0.5,r

=

0.9)

The different

symbols correspond

to different times

(t /to

" 3, 10, 25, 50, 77, 105, 135,

173).

The

prediction

of the

hydrodynamic description

of the

instability

is the solid line.

~i'(Ul

=

)li-rl),

«

where _s~ is defined _as

~

j~j il-U)~

*

ii 7u/161'

Appendix

B

Enskog Expansion

In the _case of _a hard _core

fluid,

_a

senii-enipirical

niodification of Boltzniann

equation

introduced

by Enskog,

widens the range of

applicability

of the kinetic

approach

to

higher

densities

[14].

The

Enskog approxiniation

accounts for the finite size of the disks in the collision terni of the Boltzniann

equation.

When two

particles collide,

their centers are

separated by

the dianieter of the disks _a. The collision terni of Boltzniann

equation

^should thus be

niultiplied by

the

probability

of

finding

two

particles separated by

_a which is

proportional

to the

pair

correlation function evaluated at _a. This correction will have _an influence _on all the

transport

coefficients.

The

validity

of this

approach

_was checked for the

cooling

rate ~f in the kinetic

reginie.

Figure

2 shows successive

snapshots

of the

pair

correlation function in the kinetic

regime.

This function is

essentially

the _{saute as} in _a hard _core fluid at

therniodynaniic equilibriuni

o o o ~q~ ^cp p

)[j j[°ljj[)oh[iii j@f[j

~

~~°

g

f($~j~~O )/ ~o

o

Oi

_°°o ^~

$~j°°f)°j)ig

~ ~

~iil)ll~°1 l101[)I°I _~i(1~°

~

^g _~

)

_o

°w~°° _{~° ~§o}

^°°° O

o~~f

O[ ~o°oo

°l

° o _~

~o°o

DOD

o[°

p ~p

+°o

~oj~/$

^~

~ o

Fig.

10. A system with N

= 1600, p*

= 0.25, r = 0.25

obeying

the

specular

collision rule

collapses

after 3. ii collisions _per

particle.

The grey

particles

_are those involved in the last two hundred collisions.

More than 99% of these collisions _occur between the

aligned particles.

Under the _same conditions. _a system

obeying

the modified collision rule does not undergo

collapse

after 125 collisions _per

particle.

even

though

the

teniperature

has

dropped by

_a factor 170000 between the first and the last

snapshot.

Hence

g(a)

is assunied to be

given by

the usual virial

expression

~~

= 1 +

2ug(a) PT

Introducing

the

equation

of state of _an 2d hard disks fluid

[15],

^the

Enskog

corrected

cooling

rate to beconies:

1

t0 "

t0Boitz,»a,>nj

₉

°

a

Ii

_U)~

4

(1 ~j~ ^To~/~ (l 7U/16)

V~

The values of the

cooling

rate found in the siniulation _are

conipared

in

Figure

4 with this

prediction.

References

Ill

Herniann

H.J., Physica

^A 191

(1992)

263.

[2] Goldhirsch I. and Zanetti

G., Phys.

Rev. Lett. 70

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1619.

[3] McNaniara S. and

Young W.R., Phys.

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[4] McNaniara S. et

Young W-R-, Phys.

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5(1) (1993).

[5] McNaniara S. and

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Haff

P.K.,

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M.,

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