Spontaneous Formation of Vortex in a System of Self Motorised Particles

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Spontaneous Formation of Vortex in a System of Self Motorised Particles

Y. Limon Duparcmeur, H. Herrmann, J. Troadec

To cite this version:

Y. Limon Duparcmeur, H. Herrmann, J. Troadec. Spontaneous Formation of Vortex in a Sys- tem of Self Motorised Particles. Journal de Physique I, EDP Sciences, 1995, 5 (9), pp.1119-1128.

�10.1051/jp1:1995185�. �jpa-00247123�

J. Phys. I £Fonce 5

(1995)

1119-1128 SEPTEMBER1995, PAGE 1119

Classification Physics Abstracts

64.70 05.60 46.30

Spontaneous Formation of Vortex in

_a

System of Self Motorised Particles

Y. Limon

Duparcmeur(~),

H.

Herrmann(2)

and J. P.

lhoadec(~)

(~) Groupe Matière Condensée et

Matériaux(*),

Université de Rennes I, Campus de Beaulieu,

35042 Rennes Cedex, France

(~) Physique et Mécanique des Milieux

Hétérogènes(**),

Ecole Supérieure de Physique et Chimie Industrielles de la Ville de Paris, 10 _rue Vauquelin, 75231 Paris Cedex 05, France

(Received

29 March 1995, received in final form 9 May 1995, accepted 11 May

1995)

Abstract. We study _a system of disks with inelastic collisions. Euergy _is giveu to trie system via a force of constant modulus, applied _on each particle _in trie direction of its velocity. This system shows _a dynamical phase transition from _a disordered system towards _a system orgamzed

into _a vortex when the dissipation _is increased. Trie state equation of trie disordered system is obtained by usiug kiuetic theory arguments. ^{We also show that trie transition is similar} to trie

kiuetic-cluster transition observed in the cooliug of _a gas of inelastic partiales.

1. Introduction

Some _{years ago,} the kinetic

theory

has been extended to the

study

of the

dynamics

of

granular

systems

[1-4].

The main diflerence between gases and

granular

media is energy

dissipation during

collisions. Thus either the _energy of the

granular

media decreases with

time,

_{or energy}

must be

supplied by

_{a source} to maintain the

velocity

fluctuations. This

happens

in

flowing partiales

when

potential

_energy is transformed into kinetic energy. Another _{way to} put energy in _a

granular

system ^is

through

the motion of

boundaries,

_as in the so-called Brazil nut

problem (5-7].

On _an air

table,

thin

partiales

_move and _rearrange

permanently

because of small scale

heterogeneities: they

receive energy from air

flowing upwards through

the _porous table (8].

Recently,

Vicsek et ai. (9] have introduced _a model of self-driven

partiales

_to describe

biolog- ica1ly

motivated interactions

("Molecular

motors"

(loi

In this

model,

each

partiale

is driven

with _a constant absolute

velocity

and _assumes the _average direction of motion of its

neighbours

with _some added noise. A kinetic

phase

transition from _no transport to finite net transport ^is observed when the

density

of

partiales

is increased and the noise is decreased.

The way we use here to introduce energy into _a

dry granular

material drives the system in _a

steady

state, with constant energy. The model that _we consider consists of _a two-dimensional

(*) U-R-A 804

(**) U-R-A. 857

@ Les Editions de Physique1995

assembly

of disks with hard-core interactions in _a container with fixed

boundary

conditions. It is studied

using

Molecular

Dynamic

simulations _as described

by

Herrmann

Ill].

To introduce

energy into the system, we

apply

_a constant force _on each

partiale,

such _as in schools of

fish,

or

flying

birds in formation. This force is in the direction of the

velocity

of the

partiale

and its modulus is constant. It has been observed

previously (12,13]

that _an inelastic gas exhibits

a transition from _a

homogeneous regime

towards _a

regime

with

long-range

correlation of the velocities and

positions

when the energy

dissipated during

the collisions is increased. In _our

system,

the correlation of velocities leads _{to a}

single

_vortex.

Recently, Hemmingsson

_(14] used the _same model simulated _{on a} lattice _gas and also found _vortex formation. We show that

a theoretical

approach

similar to that

presented by

Half

Iii

_can describe the system in the disordered

homogeneous phase.

The

phase diagram

and the

origin

^of the

instability

_are then discussed.

2.

Description

of trie Model

Molecular

dynamics

is _a method to

study n-body problems by solving

equations of motion.

It _was first used to

study liquids Ils].

More

recently,

it was used to

study granular

materials

Ill,16].

Let _us consider _a two-dimensional system of N

partiales

^of _mass m and radius R

= o-S, in

a square box of

length L,

with fixed

boundary

conditions. When _two

partides

and

j overlap ii-e-,

^{their distance} is smaller than the _sum of their

radii)

three forces

Ill]

act _on

partiale1:

1) an elastic restoration force

foi _"

Ym(r~j 2R)n il)

where Y is the

Young

modulus normalized

by

the mass, m the _mass of the

partiale,

_rjj the distance between the two

overlapping partiales,

and _n the

unitary

vector built _on r~j

pointing

^from to

j;

ii)

_a

dissipation

due to

inelasticity

of the collisions

fdiss

"

~'fnTllviJ.~)~

12)

where _~fn is _a

phenomenological dissipation

coefficient in the normal direction and

V~j

= V~

Vj

_is the relative

velocity

of

partiales

and

j;

iii)

_a shear force friction which is chosen _as

fshear _"

~'Îsm(Vzj.s)S (3)

where _~fs is _a

phenomenological dissipation

coefficient and _s the unitary vector obtained

by rotating

n

by

90°.

Dunng

^collisions with the

walls,

the force

applied

_on the

partiales

_is similar to that used for

interparticle

collisions. It _means that collisions with watts also

dissipate

_energy. Partiales _are not allowed to rotate, and _no Coulomb friction is considered. In the

following,

_we will take

~fs = ~fn = ~f. As Hook's law is used for the restoration

force,

the _energy restitution coefficient of

collisions,

_e, does not

depend

_on the relative velocities of the

colliding partiales.

An exact

relationship

between _e and _{~f can} be derived from the

equation

^of motion. This

relationship

does not depend on

partiale velocity

and _can be written _as follows:

~f ce

(~~

+

ln~ e)~°'~/f

_In

e

(4)

N°9 VORTEX IN A SYSTEM OF SELF MOTORISED PARTICLES l121

A first-order

expansion

^{of this} formula when _e is close to

gives

_{~f ce} 1 _e.

An externat

driving

force is

applied

at every time step on each

partiale

to put _energy ^into the system. The modulus of this force is constant and its direction is

given by

the

velocity direction;

_so the force

applied

_on the ith

partiale

_can be written _as:

~~ _'~~~

Î

~~~

where V~ is the

velocity

vector of the i~~

partiale,

l~ the modulus of this vector, and _a is the

"Force factor". The modulus of the acceleration does not

depend

_on the _{mass or} the

velocity

of the

particle. lhajectories

_are calculated

using

a third order

predictor-corrector

^method

Il

_7].

For low

packing

fraction

(lower

than

o-S),

the initial

positions

of the

partiales

_are chosen

randomly by

_an R-S-A

(Random Sequential Adsorption) algorithm

_(18]; for

packing

fraction

larger

than O.S, as the R-S-A-

algorithm

_{can no}

longer

be

used,

the _centers of the

partiales

_are

initially

situated _on the nodes of _{a square} lattice. In both _cases, the initial velocities of the

partiales

_are

randomly

chosen. In order to let trie system reach

equilibrium,

measurements

are started after about 10~ _{time iterations}

(ten seconds). During

each

simulation,

the total energy of the system

(kinetic

_energy and elastic

energy)

is recorded. In order to characterize the

global

motion of

particles

_we have used the kinetic momentum defined _as:

N

M

=

£

_{OMj A Vi}

₍₆₎

z=1

where OMj is the vector

linking

the center of the i~~

partiale

to the center of the box. If

a vortex appears in the system, ^the time average of M has _a finite value. For _a disordered system, ^the time average of M will be _zero. Thus the kinetic momentum can be used _{as an} order parameter to characterize _a

single

vortex.

3. Kinetic

Regime

Theoretical Results

One of the fundamental

assumptions

of the kinetic

theory

is that of

complete

molecular chaos.

This _means that there _{are no} correlations between the velocities of the individual

colliding

partiales.

In _our system, we assume that there _are neither

velocity

_nor position correlations.

Thus,

_{we can}

apply

_a modified kinetic

theory

to find _{a state}

equation

of the system. The

principle

is to consider that the system _is in _a

steady

state with _a balance between the

input

energy and the _energy

dissipated by

collisions. Both

energies

can be

calculated,

and this leads to _an

equation giving

the _{mean square}

velocity

_{as a} function of the

packing

fraction

C,

the force factor _a and the energy restitution coefficient _e of collisions.

The _energy given _per ^{time unit} to the ith

partiale by

the force factor is:

~)

= ma(

(7)

By averaging

_over the whole system, the _{mean energy}

given

_per

partiale

and _per time unit _can be written _as:

Î °~'~~~~

_~~~

where V is _some average over the

partiale

velocities. We have verified that the

velocity

distri- bution _{is never} far from _a Maxwell

distribution,

_so that all _averages of velocities of the

partiales

are

proportional

to each other. As in the

following,

_{we use}

proportionality

instead of strict

equality,

V _can be the _{mean square}

velocity,

_as well _as the most

probable velocity.

In the _same way, we do _not

distinguish

between _mean

velocity

of the

partiales

and _mean relative

velocity

of two

partiales.

To calculate the

dissipated

_{energy, we}

neglect

eflects of the walls and consider that _energy

is

dissipated only by

collisions between

partiales.

The _energy

dissipated

_per

partiale

and _per time unit is

proportional

to the number of collisions _per time unit and to the _{average energy}

dissipated

at each collision.

For _a two-dimensional ideal _gas, the number of collisions for _one

particle

_per unit of time is of the form:

~

~ °~

l ~~~

where is the _mean free

path

of the

particles.

To

apply

this relation to a

granular

medium,

we have first _{to assume} that the contact time is very small

compared

to the time of free

flight.

This is

clearly

true in the _case of low

packing

fraction and

high

restitution coefficient. In the

case of _our

granular

system, _as the

packing

fraction _is

high,

the size of trie

partiales

cannot be

neglected

in front of the _mean free

path

of molecules in _an ideal _gas. Thus _we write the _mean free

path

in the

following

form:

=

r(R) (10)

where is the _mean free

path

of _a molecule _{in a} ideal _gas and

r(R)

is _a

perturbation taking

into account the reduction of the _mean free

path

due to the size of the

partiales.

As the

packing

fraction C is _an

easily

^accessible

quantity

in a

granular medium,

_{we use} the

following relationsuip

for À:

À=~=$withc=~)(~ _Ill)

a = 2R is the section of collision. If _{we assume} that tuere _are

only

frontal

collisions,

_one could expect

r(R)

= 2R. Non-frontal collision leads to _a value of

TIR)

< 2R.

Assuming

tuat there _are

only binary

collisions and that tue

impact

parameter ^is a random

variable,

_an

exact calculation

gives TIR)

₌

~~.

In _{a more}

general

_case

(multiple collisions),

_{as a} first 2

approximation

_we take

r(R) proportional

_to tue radius R of the

partiales.

Then _we write:

°~

lÎ ^~

_~

~~~~

wuere k is _a dimensionless constant. When the

packing

fraction

approaches

tue value Go _"

1/k,

tends to 0.

The _energy

dissipated during

_one

binary

collision is also

proportional

to

il e)mV~,

where

e is the energy restitution coefficient.

Using

the _energy

dissipation

coefficient d ₌ 1 _{e, we} find that the

dissipated

_{energy per}

partiale

and _per time unit is

proportionalto

the cube of the _mean

velocity

divided

by

the _mean free

path.

~~'

oe

ndmv~ (13)

~$

_oe m V~

(14)

A similar result has been obtained

by

Hafl

Iii using

a similar

approach.

In the _case of correlation of motion in the system, ^{like for}

flowing particles,

the _same

computation

_can be

done, replacing

V

by

the fluctuation of

velocity

around its _mean local value. For _our system, as all

partiales

N°9 VORTEX IN A SYSTEM OF SELF MOTORISED PARTICLES l123

10~

a=30

a=10 10~

~ ₂

tf

10

~

Ui

~~i

^a=0.1

0^°

2 3 4 5 6 7 8 9

lÙ y

Fig. l. Log-Log plot of the _{energy versus ~f} for _vanous values of _a, for C

= 0.45 and L

= 15. _{7 is}

increased from 1 to 11 aud theu decreased to 1. One _con notice that ail plots _cou be put _on top of each orner by dividing trie energy by _a

(trie

force

factor).

bave the _{same mass} and size and

by considering

the balance between the

dissipated

_energy

and the

input

_{energy, we} ^find the

following

equation of state for trie disordered

steady

state:

V~ ce

Î

_{d C}

k)

^R

^ils)

4. Kinetic

Regime

Numerical Results

In

Figure

1, _we have

plotted

in

logarithmic

scale the total _energy E of the system _{versus ~f} ^for diflerent values of _a, and

given

values of L and N

ii-e-, C).

^As _{~f ce}

il e),

d is

proportional

to ~f when d is close _{to zero.} We have verified that this

approximation

is valid for the results shown below. Therefore _we will _{use ~f} instead of d in the

following.

For low values of _~f, the

variations of E _are linear with the _same

slope j-1)

for ail _a, L and C. In agreement ^{with the} theoretical state

equation ils),

E is

proportional

to

1/~f

and also to a. Above _a certain value

~f~ of _~f, the _energy is

larger

thon the theoretical _{one: we} will _see below that this is due to a

transition of the system to _a vortex

regime.

In order _{to compare} ail results, we have

plotted

in

Figure

2 _a normalized energy per

particle

versus for diflerent values of _a,

d,

N and L for

~f < ~f~. The normalized _energy

En

is defined

~_

C

En =

E~ ₍₁₆₎

N _a

A _very

good

agreement ^{is found between the} theoretical

equation

of _state

(represented by

the

straight line)

and the numencal data. The value obtained for Go ₌

1/k

is 0.82 ~ D.ol. It is not

compatible

with that calculated

by assuming

^{that there} _are

only binary

collisions

(Go

" 1, _see

above). Surprisingly,

it is the _same

density

_as that of the 2D Random Close

Packing (R.C.P.).

£

Q~

~

OE

©w

~

ÎÎ~

2

#4 fl3~

q

E Z

UC

0

UCO

2 4 6

Fig. 2. Plot of the normahzed energy versus

1/C.

Disordered systems _are ail _ou the _same fine

whathever trie values of _{a, i,} L, C.

Above this

density long-range

correlations _appear in the positions of

partiales

_(19]. So the model

predicts

that the system ^has no

velocity

fluctuations around the local _average when it reaches such _a

packing

fraction.

5. Phase Wansition Numerical Results

One _can notice _m

Figure

1 that while _~f increases above the value _~f~, the _energy becomes

larger

than that calculated

by

the kinetic

theory.

This _is due to _a

phase

transition from _a disordered system _(~f < ~f~) ^to a system

organized

into _{a vortex (~f >} ~f~). ^The kinetic momentum M is used _{as a} parameter to characterize the _presence of _a

single

vortex. One _{cari see} in

Figure

³

that

M,

which fluctuates around 0 for _{~f < ~f~,} decreases

steeply

to a

negative

^value

(constant

in time and

depending

^of

L,

_{a, ~f} and

C)

when

~f ^is increased above _~f~.

Figure

4

gives

_a

snapshot

of _a system ^below _~f~, ^with no correlation of

velocity

_or

position,

M is _a white noise of _zero average.

Figure

5 shows the _same system above _~f~:

locally,

the velocities of the

particles

ail point in the _same direction. The

global

motion is _a vortex and M fluctuates _m time around _a

constant

(non-zero)

value. The

packing

is not

homogeneous,

the

density being larger

_near the

boundaries than in the _center of the cell.

From the values of

~f~ obtained in many

simulations,

witu diflerent values of _a,

L,

and

N,

we have

plotted (Fig. 6)

in the

plane (~f,1/C)

the lines

separating

the _vortex

regime (above)

from the

homogeneous regime (below),

for three values of L. It appears that the transition fines do not

depend

_{on a.} On the contrary, the transition

strongly depends

_on the size of the system: the

larger

the

box,

the lower _~f~. For _very small values of

C,

_no transition is observed

m the studied _range of _~f; the system is

homogeneous.

Then when C increases, the transition

curves show _a minimum for _a value of C close to 0.6

increasing slightly

with L.

In _a gas of inelastic partiales, a local increase of the

packing

fraction C increases the collision

frequency

and thus the local

dissipation

of energy

(dissipation

of the relative

velocity).

The pressure is

proportional

both to C and to the

velocity

fluctuations. When the

density

of

N°9 VORTEX IN A SYSTEM OF SELF MOTORISED PARTICLES l125

DD

o

-50

-1DO

2 4 6 8 in

y

Fig. 3. Liuear plot of the kiuetic momentum M _{versus ~f} for L ₌ 15, _{a =} 5, C

= 0.45. ~fis mcreased from 1 to 11 aud then decreased to1.

Fig. 4. Snapshot of _a system of 400 particles m the disordered phase

(C

₌ 0.5, _{a =} 1, _{~f =} 1). Trie

size of trie box _is L

= 25. The circles represent ^the partiales, trie Eues represeut the velocity vectors.

~o

o

Fig. 5. Snapshot of

a system ^of 400 partiales _in the vortex regime

(C

= 0.5, _{a =} 1, ₇

" 5). The

size of the box _is L

= 25. The circles represent trie partiales, trie fines represent trie velocity vectors.

?

L=15

Vomex _L=20

L=25

o

Homogeneous

uc

1 2 3 4 5 6

Fig. 6. Transition bues from _a disordered system to _a vortex regime in trie

I?i 1/C)

plane for

difierent _sizes L, _a being between ù-1 aud 1.

partiales locally

_increases, if the

dissipation

is

large id large),

there will be _a decrease of the pressure in that

place.

The flux of

partiales

will then _go from low

packing

fraction and

high

energy areas to

high packing

fraction and low _{energy areas}

amplifying

the

density

variation and

leading

to the formation of _a "duster". This is the

origin

of the transition from the "kinetic

N°9 VORTEX IN A SYSTEM OF SELF MOTORISED PARTICLES l127

regime"

to the "cluster

regime"

in the

cooling

of _{a gas} of inelastic

partiales

_as described

by

Goldhirsch and Zanetti (12] ^{and Mc Namarra} (13]. ^The

velocity

of the center of _mass of _a cluster is not modified

by

the collisions of

particles

inside the

cluster,

and in _{our case} the center of _mass of _a cluster is accelerated

by

the force. This mechanism tends to decrease

velocity

fluctuations in _a certain

neighbourhood,

and

amplify

^the

macroscopic

motion. This

is also the mechanism

explaining

trie

dynamical phase

transition observed

by

Vicsek et ai. _(9]

(the averaging

of

velocity

directions in the Vicsek model _can be

compared locally

to the

cooling

of _a

granular medium).

The _appearance of

long-range

self

organized

motion therefore _seems to be _a

general

property ^of

dissipative

interactions, Another argument ^{in favor of} the relation to the kinetic-cluster transition is the

shape

of the transition fines. For C _<

0.6,

_we have

~f~ ^c~

1/L~C~.

This is the Iimit of the

stability

of the heat mode

(homogeneous cooling)

of _a gas of inelastic

particles

_as

given by

McNamara (20] ^{who also found} a

particular

behaviour for

high packing

fractions.

6. Conclusion

Increasing

the

dissipation

_{m a}

granular medium,

_we bave observed _a transition from _a homo- geneous

regime

to a vortex

regime.

This _appears to be similar to the kinetic-cluster transition observed

by

McNamara (13]. ^In our case, a vortex is the

only possible steady

motion with

long-range

order

compatible

with fixed

boundary conditions,

_as

compared

to the two streams found

by

McNamara (20] ^when

periodic boundary

conditions _are

applied.

Some

higher

order modes _can be obtained in _a

rectangular box; nevertheless, they

_are not stable:

depending

on the initial conditions _{we can} have two metastable vortices, but after _a

while,

_one of the vortices stretches out, and its motion becomes faster and

faster,

the other _vortex becomes smaller and

smaller,

its

packing

^fraction

increases,

and its motion slows down

(the viscosity

increases with the

packing fraction).

The system ^{then evolves towards} one

elongated

vortex

filling

the entire system. ^The energy of this vortex is

higher

^than the _energy of the two small vortices. The

instability

of

high-order

^modes and the

independence

of the transition from the force factor _a

(taking

the temperature

proportional

to the

energy)

_seem to show that this

instability

is diflerent from the

Rayleigh-Benard

or other dassical

hydrodynamic

instabilities under

gradient.

This model also

gives

information _on the kinetics of

dissipative

_gases. The R-C-P limit appears to be

important

^for

dynamical properties.

It is the maximum

packing

fraction above which uncorrelated motion is _{no more}

possible.

It is also the limit where the

viscosity

increases

steeply

because of

geometrical

eflects. This

probably explains

that d~ increases for C > 0.6.

Some diflerent forces could be studied. A force

depending

on trie local

packing

fraction must

modify

the transition.

Adding

a noise to trie force could also be

interesting.

Acknowledgments

We thank D.

Bideau,

G.

Batrouni,

I.

Ippolito

and S. Roux for useful discussions. This work

was

partly supported by

the GDR CNRS

"Physique

des Milieux

Hétérogènes Complexes".

References

iii

Hafi P.K., J. Fluid. Mech. 134

(1983)

401.

[2] ^Lun C.K.K., J. Fluid. Mech. 140

(1984)

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