S eady uni o m shea low in a low densi y g anula gas
J. J. B ey, M. J. Ruiz-Mon e o, and F. Mo eno
Fı
´sica Teo
´ ica, Facul ad de Fı
´sica, Uni e sidad de Se illa, Apa ado de Co eos 1065, E-41080 Se illa, Spain
~Recei ed 8 Oc obe 1996!
The s eady uni o m shea low o a low densi y g anula sy em is s udied by means o a kine ic model
equa ion and also by di ec Mon e Ca lo simula ion o he Bol zmann equa ion. The o me can be exac ly
sol ed o a bi a y shea a e and dissipa ion. Explici exp essions o he one-pa icle dis ibu ion unc ion
and he p essu e enso a e ob ained. Compa ison o he esul s wi h hose o p e ious heo ies is p esen ed in
he app op ia e limi s. The simula ion shows ha he model ep oduces ai ly well he alues o he s esses
and, in pa icula , he phenomenon o no mal s ess di e ences. The ag eemen is also e y good o he
eloci y dis ibu ion unc ion in he he mal eloci y egion, al hough signi ican disc epancies appea o la ge
eloci ies. @S1063-651X~97!12803-3#
PACS numbe ~s!: 05.20.Dd, 47.50.1d, 47.20.2k
I. INTRODUCTION
The de elopmen o kine ic model equa ions desc ibing
he dynamics o pa icles, which collide inelas ically, seems
o be an impo an s ep owa ds he unde s anding o he
complex phenomena aking place in apid g anula lows
@1–3#. Ne e heless, he nonconse a ion o ene gy in colli-
sions in oduces qui e d as ic changes in he physics o he
p oblem, and he gene aliza ion o he exis ing heo ies o
o dina y luids is a om being an easy ask. This e e s o
bo h he own de i a ion o p oposal o a kine ic equa ion o
he dis ibu ion unc ion o he sys em and also o sol ing he
equa ion o ob ain consis en hyd odynamic-like equa ions
desc ibing he mac oscopic e olu ion. Al hough a hyd ody-
namic desc ip ion, in e ms o he densi y, he mac oscopic
low eloci y, and he g anula empe a u e is sugges ed by
analogy wi h no mal luids and also by expe imen s and
compu e simula ions, a de i a ion om a mo e undamen al
desc ip ion o he sys em is needed.
In he las yea s, se e al kine ic equa ions o he one-
pa icle dis ibu ion unc ion o apid g anula lows ha e
been p oposed, ex ending he heu is ic a gumen s leading o
he Bol zmann and Enskog equa ions o o dina y gases
@4–7#. In addi ion, a mo e basic app oach o he p oblem,
s a ing om he Liou ille equa ion, has been p esen ed @8#.
Ne e heless, e en i i is assumed ha hese equa ions a e
ele an o he desc ip ion o apid g anula lows, a unda-
men al di icul y a ises. The s anda d p ocedu e o ob aining
solu ions o kine ic equa ions is by means o he Chapman-
Enskog me hod @9#, in which an expansion in he g adien s
o he hyd odynamic ields is ca ied ou . Fo molecula lu-
ids, he e e ence s a e abou which g adien s a e conside ed
is he Maxwellian local equilib ium dis ibu ion. The co e-
sponding e e ence dis ibu ion o inelas ic pa icles is com-
plex and has no been de e mined o da e. As a consequence,
he limi o small dissipa ion ~i.e., he coe icien o es i u-
ion
a
close o 1!is usually conside ed. Ne e heless, many
o he phenomena peculia o apid g anula luids a e clea ly
associa ed o nonlinea and heological e ec s and o no
e y small dissipa ion. This includes p ope ies such as sig-
ni ican no mal s ess di e ences unde shea @2,10,11#,
spon aneous o ma ion o dense clus e s su ounded by e-
gions o low densi y @12–15#, and inelas ic collapse @13#.
The easons men ioned abo e indica e ha i is ins uc i e
o conside simple model kine ic equa ions o inelas ic gases
which allow con olled and de ailed analysis. Kine ic models
ha e p o en o be e y use ul in he s udy o a om equi-
lib ium s a es o dilu e molecula gases @16#. Fo many
physical si ua ions, exac solu ions o he models ha e been
de i ed, and compa ison wi h nume ical solu ions o he ex-
ac equa ions ob ained by compu e simula ion shows a ai ly
good ag eemen . Ve y ecen ly @8#, wo kine ic models o
sys ems o inelas ic ha d sphe es ha e been p oposed. Bo h
models a e o mula ed as app oxima ions o he e ised En-
skog equa ion, bu wi h di e en quan i a i e accu acy. He e
we will conside he limi o low densi ies and la ge leng h
scales as compa ed wi h he diame e o he pa icles. In his
limi , he wo models lead o he same equa ion, and become
a kine ic model o he Bol zmann equa ion o inelas ic ha d
pa icles.
Mos o he simples a om equilib ium physical si ua-
ions co espond o s eady s a es. In his pape , we s udy an
unbounded, s eady, and uni o m shea low ~USF!. While in
molecula luids such a s a e is no possible due o iscous
hea ing, in g anula luids his e ec can be balanced by
dissipa ion in collisions. The simplici y o he model allows
us o ob ain he exac solu ion desc ibing he s eady USF and
also all he physically ele an quan i ies. This is done wi h-
ou in oducing any expansion in he g adien s o in he in-
elas ici y o collisions. The esul s o he p essu e enso
show aniso opy, i.e., no mal s ess di e ences.
I mus be no ed ha ou app oach is qui e di e en om
o he s, in which a speci ic o m is assumed o he one-
pa icle dis ibu ion unc ion desc ibing a gi en s a e. Fo
ins ance, in he heo y by Jenkins and Richman @17#a gen-
e alized Maxwellian is conjec u ed o model he dis ibu ion
unc ion o he s eady USF. The eloci y-independen coe -
icien s a e iden i ied h ough he balance equa ion o he
second o de luc ua ing eloci y co ela ion enso . This
heo y, which in p inciple also keeps all o de s in he shea
a e and dissipa ion, leads o no mal s ess di e ences which
a e in good ag eemen wi h he esul s o molecula dynam-
ics simula ions. On he o he hand, he model we use has
been o mula ed o a bi a y condi ions, and no speci ic ap-
PHYSICAL REVIEW E MARCH 1997VOLUME 55, NUMBER 3
55
1063-651X/97/55~3!/2846~11!/$10.00 2846 © 1997 The Ame ican Physical Socie y
p oxima ion is in oduced o apply i o he s eady USF. To
ob ain he one-pa icle dis ibu ion unc ion one has o sol e
he kine ic model equa ion unde he app op ia e condi ions.
Ano he in e es ing app oach o he same p oblem @18#
has been ca ied ou by expanding he Bol zmann equa ion o
Bu ne o de . The pe u ba i e expansion me hod is speci ic
o he s eady USF, and i is based on he obse a ion ha in
his s a e he shea a e scales as
A
e
, whe e
e
512
a
2. The
heo y esul s in a p edic ion o he no mal s ess di e ences
which a e e y close o hose o Jenkins and Richman, when
he la e a e expanded o he same o de o app oxima ion.
The di ec Mon e Ca lo simula ion me hod @19#was de-
elopped o ob ain nume ical solu ions o he Bol zmann
equa ion, and has been success ully applied o a g ea a ie y
o si ua ions. Since i is o mula ed o an a bi a y sca e ing
law, he e is no di icul y in using i o s udy g anula lows
@20#. He e we p esen esul s ob ained o he s eady USF,
and compa e hem wi h he p edic ions o he model kine ic
equa ion. Fo he componen s o he p essu e enso he
ag eemen u ns ou o be e y good o e a wide ange o
alues o he coe icien o es i u ion. Rega ding he one-
pa icle dis ibu ion unc ion, he ag eemen is only semi-
quan i a i e. This is no su p ising, since he de ailed Bol z-
mann collision ope a o is eplaced in he model by a much
simple e ec i e e m, which is de e mined by he second
momen s o he one-pa icle dis ibu ion.
The s uc u e o he pape is as ollows. In Sec. II he
model is b ie ly discussed, and he basic equa ions a e p e-
sen ed. In addi ion, a u he simpli ica ion o he o mula ion
in Re . @8#in in oduced. I consis s o an e alua ion o he
dissipa ion sou ce e m in he local equilib ium app oxima-
ion. We belie e his is consis en wi h he spi i o he
model, and makes he calcula ions much simple . The model
kine ic equa ion is pa icula ized o he USF in Sec. III, and
he s eady p essu e enso is compu ed in Sec. IV. The esul s
a e compa ed o he heo ies o Jenkins and Richman @17#
and o Sela, Goldhi sch, and Noskowi z @18#. Sec ion V
deals wi h he one-pa icle dis ibu ion unc ion, which is
also compa ed wi h he abo e heo ies in he low dissipa ion
limi . In Sec. VI we discuss he Mon e Ca lo simula ion e-
sul s and, inally, Sec. VII p o ides a sho summa y and
conclusions.
II. BASIC EQUATIONS OF THE MODEL
We conside a gas o iden ical smoo h disks (d52) o
sphe es (d53) o diame e
s
and mass m, whose collisions
a e cha ac e ized by a cons an coe icien o no mal es i u-
ion
a
in he in e al (0,1#. The Bol zmann equa ion o he
one-pa icle dis ibu ion unc ion, ( , , ), is @7,8#
S
]
]
1 1•“1
D
~ 1, 1, !5JB@ 1, 1,
u
~ !#,~1!
whe e JBis he ~inelas ic!Bol zmann collision ope a o
JB@ 1, 1,
u
~ !#5
s
d21
E
d 2
E
d
s
ˆQ~g•
s
ˆ!~g•
s
ˆ!
3@
a
22 ~ 1, 1
8, ! ~ 1, 2
8, !
2 ~ 1, 1, ! ~ 1, 2, !#.~2!
In he abo e exp ession,
s
ˆis he uni ec o poin ing om
he cen e o pa icle 2 o he cen e o pa icle 1 a con ac ,
Qis he Hea iside s ep unc ion, g5 12 2, and
1
85 1211
a
2
a
~g•
s
ˆ!
s
ˆ,~3!
2
85 2111
a
2
a
~g•
s
ˆ!
s
ˆ~4!
a e p ecollisional eloci ies leading a e collision o eloci-
ies 1and 2.
The local pa icle numbe densi y n, low eloci y u, and
empe a u e Ta e de ined in he usual way,
n~ , !5
E
d ~ , , !,~5!
n~ , !u~ , !5
E
d ~ , , !,~6!
d
2n~ , !kBT~ , !5
E
d 1
2m@ 2u~ , !#2 ~ , , !.~7!
F om Eq. ~1! he ollowing e olu ion equa ions o hese
ields a e easily ob ained:
]
n
]
1“•~nu!50, ~8!
]
u
]
1u•“u1~nm!21“•P50, ~9!
d
2nkB
]
T
]
1d
2nkBu•“T52~¹u!:P2“•q2~12
a
2!
,
~10!
whe e
P~ , !5
E
d m~ 2u!~ 2u! ~ , , !~11!
is he p essu e enso ,
q~ , !5
E
d m
2~ 2u!2 ~ , , !~12!
is he hea lux, and
~ , !5m
s
d21
8
E
d 1
E
d 2
E
d
s
ˆQ~g•
s
ˆ!
3~g•
s
ˆ!3 ~ , 1, ! ~ , 2, !~13!
55 2847STEADY UNIFORM SHEAR FLOW IN A LOW DENSITY . . .
is a sou ce e m desc ibing he a e o dissipa ion in colli-
sions.
Sol ing Eq. ~1! o a gi en si ua ion is a o midable ask.
E en in he case o o dina y gases, e y li le is known abou
solu ions o he Bol zmann equa ion o a om equilib ium
s a es, and he p esence o dissipa ion complica es in a non-
i ial way he s uc u e o he Bol zmann equa ion. The sim-
ples physical si ua ion one can hink o o a g anula luid is
he so-called homogeneous cooling s a e ~HCS!, o which
he sys em is uni o m and he ime dependence occu s en-
i ely h ough he empe a u e. Al hough i has been shown
@7,21# ha Eq. ~1!admi s a solu ion desc ibing such s a e, he
exac o m o he dis ibu ion unc ion is no known. Fo
hese easons, i is wo h looking o model kine ic equa ions
simila o hose which ha e p o en o be so ui ul o o -
dina y luids @16#. The basic idea is o eplace he Bol zmann
collision ope a o wi h a simple o m, while p ese ing he
essen ial p ope ies o he exac equa ion.
Ve y ecen ly @8#, a kine ic model along he lines ske ched
abo e has been p oposed. The Bol zmann collision ope a o
is sepa a ed in o wo pa s: one in he eloci y subspace
spanned by 1, , and 2, and he o he one in he subspace
o hogonal o i . The i s con ibu ion is e ained exac ly,
while he second one is app oxima ed by a single elaxa ion
ime e m. This gua an ees ha he balance equa ions ~8!–
~10!a e p ese ed by he model and ha he luxes Pand
q, and also he sou ce e m w, a e he same unc ionals o he
dis ibu ion unc ion as in he Bol zmann equa ion, i.e.,
hey a e gi en by Eqs. ~11!–~13!. Since he de ails a e de-
sc ibed elsewhe e @8,22#, we only gi e he esul he e. The
Bol zmann collision ope a o is eplaced by
JB→2
n
~ 2 l!21
nkBT l
c
~V!~12
a
2!
,~14!
whe e we ha e in oduced he peculia eloci y V5 2u,
c
~ !5m 2
dkBT21, ~15!
and lis he local equilib ium dis ibu ion,
l~ , , !5n~ , !
F
m
2
p
kBT~ , !
G
d/2
exp
F
2mV2~ , !
2kBT~ , !
G
.
~16!
Also,
n
@n( , ),T( , )#is an e ec i e local collision e-
quency, speci ied in mo e de ail below.
The model de ined by Eq. ~14!, al hough much simple
han he o iginal Bol zmann equa ion, is s ill oo complica ed
o allow exac calcula ions in mos o he applica ions. This
is due o he nonlinea unc ional dependence o
on he
dis ibu ion unc ion. Then, we go a s ep u he in he sim-
pli ica ion, and app oxima e
by i s local equilib ium alue,
i.e.,
~ , !→
l~ , !5m
s
d21
8
E
d 1
E
d 2
E
d
s
ˆQ~g•
s
ˆ!
3~g•
s
ˆ!3 l~ , 1, ! l~ , 2, !
5~d21!
S
p
m
D
1/2
n2
s
d21~kBT!3/2.~17!
In summa y, ou model kine ic equa ion is gi en by
S
]
]
1 •“
D
52
n
~ 2 l!21
nkBT l
c
~V!~12
a
2!
l.
~18!
T i ially, his equa ion also leads o he balance equa ions
~8!–~10!, wi h he only di e ence ha in he las o hem he
sou ce e m
is eplaced by
l. In he elas ic collision
limi , he kine ic model educes o he well-known
Bha naga -G oss-K ook ~BGK!equa ion @23#.
Applica ion o Eq. ~18! o he HCS is e y simple. The
solu ion is gi en by a Maxwellian wi h a ime dependen
empe a u e. The ime e olu ion o his la e quan i y is
gi en by Eq. ~10!, which o he HCS has he solu ion
T~ !5T~0!
S
11
0
D
22
,~19!
wi h
0
215~12
a
2!d21
d
s
d21n
S
p
kBT~0!
m
D
1/2
.~20!
As we ha e al eady men ioned, he exac solu ion o he
Bol zmann equa ion o he HCS is no known and, in pa -
icula , i is easy o check ha i is no a Maxwellian ~excep
in he elas ic limi ,
a
51). Ne e heless, bo h heo e ical
s udies @7,21#and Mon e Ca lo simula ions @20#ha e shown
ha de ia ions om Maxwellian a e quan i a i ely small o
he mal eloci ies and Eq. ~19!p o ides a e y accu a e ap-
p oxima ion o he ime e olu ion o he empe a u e.
III. UNIFORM SHEAR FLOW
We wan o s udy he uni o m shea low ~USF!, which is
cha ac e ized by a cons an linea eloci y p o ile
u~ !5a• ,~21!
whe e aij5a
d
ix
d
jy ,abeing he cons an shea a e. Fo
simplici y we conside an unbounded sys em. In addi ion, he
hea lux anishes and he densi y n(0) and empe a u e
T(0) a e uni o m. Fo his low, Eqs. ~8!,~9!, and ~10!imply
ha he densi y is cons an in ime, he p essu e enso is
uni o m, and he empe a u e obeys he ene gy balance
d
2n~0!kB
]
T~0!
]
52aPxy
~0!2~12
a
2!
l
~0!.~22!
.This equa ion clea ly shows he qui e di e en beha io
o elas ic and inelas ic luids unde shea . While o molecu-
la luids he USF is always a ime-dependen s a e whose
empe a u e inc eases mono onically in ime due o iscous
2848 55J. J. BREY, M. J. RUIZ-MONTERO, AND F. MORENO
hea ing, in g anula media he a ia ion o he empe a u e
a ises om he balance o wo opposi e e ec s: iscosi y and
dissipa ion in collisions. As a consequence, a s eady s a e is
possible when bo h e ec s cancel each o he . Ou aim in he
ollowing will be o analyze such a s a e. I is wo h men-
ioning ha shea ed g anula media posses mul iple s eady
s a es, depending on he ini ial and bounda y condi ions @24#.
He e we es ic ou sel es o he USF as desc ibed abo e,
assuming ha he sys em has eached such a s a e, and wi h-
ou paying any a en ion o i s s abili y. A de ailed analysis o
hese ques ions will be published elsewhe e.
Le us in oduce posi ion and eloci ies coo dina es wi h
espec o a ame mo ing wi h he low eloci y u.We
de ine he Lag angian coo dina es by
R5L~ !• ,V5 2a• ,~23!
whe e
Lij~ !5
d
ij2aij .~24!
In e ms o hese, Eq. ~18!becomes
]
]
1LijVj
]
]
Ri
2aijVj
]
]
Vi
52
n
~ 2 l!21
nkBT l
c
~12
a
2!
l.
~25!
Since he USF is mac oscopically homogeneous in he
Lag angian ame, we expec he associa ed dis ibu ion
unc ion, (0), o ha e he same p ope y, i.e., o be indepen-
den o R. The e o e, Eq. ~25! educes o
]
~0!
]
2aijVj
]
~0!
]
Vi
52
n
~0!~ ~0!2 l
~0!!21
n~0!kBT~0! l
~0!
c
~0!
3
~12
a
2!
l
~0!.~26!
This equa ion implies ha
]
^
Vi
&
~0!
]
52a
d
ix
^
Vy
&
~0!,~27!
whe e he angula b acke s deno e a e aging wi h espec o
. The e o e, i he ini ial s a e e i ies
^
V
&
(0)50, his p op-
e y is kep in ime by Eq. ~26!. In he ollowing we will limi
ou conside a ion o ini ial condi ions o his kind. This is he
eason why, o eloci ies in he Lag angian ame, we used
he same symbol as o he peculia eloci ies in oduced in
Sec. II. Then he local equilib ium dis ibu ion appea ing in
Eq. ~26!has he o m
l
~0!~V, !5n~0!
S
m
2
p
kBT~0!~ !
D
d/2
exp
S
2mV2
2kBT~0!~ !
D
,
~28!
wi h all he possible ime dependence occu ing h ough he
empe a u e.
IV. PRESSURE TENSOR
By mul iplying Eq. ~26!by ViVjand in eg a ing o e V,i
is s aigh o wa d o ob ain he e olu ion equa ions o he
componen s o he p essu e enso in he USF,
]
]
Pij
~0!1aikPjk
~0!1ajkPik
~0!52
n
~0!~Pij
~0!2
d
ijp~0!!
22
d
d
ij~12
a
2!
l
~0!,~29!
whe e p5(iPii /d5nkBTis he hyd os a ic p essu e. Fo
a
51 we eco e he equa ions o an o dina y gas unde
USF in he BGK app oxima ion, which ha e been ex en-
si ely s udied @25–27#. F om Eq. ~29!, in pa icula , one ob-
ains
]
]
p~0!52 2
daPxy
~0!22
d~12
a
2!
l
~0!,~30!
]
]
Pxy
~0!52aPyy
~0!2
n
~0!Pxy
~0!,~31!
]
]
Pyy
~0!52
n
~0!~Pyy
~0!2p~0!!22
d~12
a
2!
l
~0!.~32!
This closed sys em o equa ions has he i ial solu ion
Pxy
(0)5Pyy
(0)5p(0)50. A second s eady solu ion, deno ed by
an as e isk, is gi en by
Pxy
*52 a
n
*
F
p*22
d~12
a
2!
l
*
n
*
G
,~33!
Pyy
*52
n
*
aPxy
*,~34!
a2
n
*
F
p*22
d~12
a
2!
l
*
n
*
G
2~12
a
2!
l
*50. ~35!
Fo gi en alues o he shea a e a, he numbe densi y
n, and he es i u ion coe icien
a
, Eq. ~35!de e mines he
alue o he p essu e ~o empe a u e!a which he s eady
s a e is possible. I is con enien a his poin o speci y he
o m o he e ec i e collision equency
n
. Dimensional
analysis equi es ha
n
5cn
s
d21
S
p
kBT
m
D
1/2
,~36!
whe e cis a dimensionless cons an . We a e going o ix his
by equi ing ha he model gi es he co ec ~Bol zmann!
alue o he Na ie -S okes shea iscosi y in he elas ic
limi
a
51. This leads o c516/5b, wi h b.1.016, o
d53~sphe es!, and c52/b, wi h b.1.022, o d52~disks!
@28#. When Eq. ~36!is used in Eqs. ~33!–~35! hey educe o
P
˜
xy52
H
~12
a
2!~d21!
c
F
122~d21!
dc ~12
a
2!
G
J
1/2
,
~37!
55 2849STEADY UNIFORM SHEAR FLOW IN A LOW DENSITY . . .
P
˜
yy52 P
˜
xy
a
˜
5122~d21!
dc ~12
a
2!,~38!
a
˜
25d~d21!~12
a
2!
dc22~d21!~12
a
2!.~39!
We ha e in oduced he dimensionless p essu e enso P
˜
ij
and shea a e a
˜
in he s eady s a e as
P
˜
ij5Pij
*
p*,a
˜
5a
n
*.~40!
Equa ion ~39!shows ha in he s eady s a e he a io o
he shea a e o he collision equency depends only on he
es i u ion coe icien . The simple dependence o he p esu e
enso on he shea a e ound he e con as s wi h he com-
plex one ound in he ime-dependen uni o m shea low o
o dina y gases @25–27#. Le us suppose we ca y ou a se ies
o expe iences, all o hem in he same sys em, i.e., wi h
gi en alues o
a
and n, bu wi h di e en alues o he
shea a e, measu ing he shea iscosi y in he s eady s a e.
Taking in o accoun ha Eq. ~39!implies ha T*}a2,i
ollows om Eq. ~37! ha he gene alized shea iscosi y
h
*is linea in he shea a e,
h
*~a![2Pxy
*
a}a.~41!
O cou se, he p opo ionali y cons an in his exp ession de-
pends on he coe icien o es i u ion.
The o he componen s o he p essu e enso a e easily
compu ed by means o Eq. ~29!, pa icula ized o he s eady
s a e. One ob ains
P
˜
xx5112~d21!2~12
a
2!
dc ,~42!
and, o a h ee-dimensional sys em,
P
˜
zz5P
˜
yy ,P
˜
xz5P
˜
yz50. ~43!
Le us no e ha he educed p essu e enso in he s eady
s a e is exp essed as a unc ion o
a
alone, being indepen-
den , in pa icula , o he applied shea a e. Then, o in-
s ance, o he no mal s ess a ios we ha e
Pxx
*
Pyy
*5dc12~d21!2~12
a
2!
dc22~d21!~12
a
2!,~44!
Pyy
*
Pzz
*51. ~45!
The e o e, he model p edic s aniso opy o he diagonal
e ms o he p essu e enso in he shea plane, bu no in he
plane pe pendicula o he low. The exis ence o no mal
s ess di e ences in shea lows o no mal luids is e y well
known. I is a iscome ic e ec which appea s beyond he
Na ie -S okes app oxima ion @29#. Fo a low densi y USF
hey ha e been calcula ed using he BGK model @27#. The e,
di e ences be ween Pxx and Pyy bu no be ween Pyy and
Pzz we e ound.
As we ha e al eady men ioned, an app oxima ed heo y
o he s eady USF o smoo h inelas ic disks has been o -
mula ed by Jenkins and Richman @17#. Thei esul s o he
second eloci y momen s in he low densi y limi ead, in ou
no a ion,
a
˜
~JR!25~11
a
!~12
a
2!~723
a
!2
c2~917
a
!,~46!
P
˜
xy
~JR!52 4
2529
a
@~917
a
!~12
a
!#1/2,~47!
P
˜
yy
~JR!5917
a
2529
a
,~48!
P
˜
xx
~JR!541225
a
2529
a
.~49!
Al hough hese exp essions may appea a he di e en
om he co esponding esul s ob ained om ou model ki-
ne ic equa ion, Figs. 1 and 2 show ha he disc epancies a e
qui e small o low dissipa ion, and end o anish when
a
app oaches uni y. Le us also men ion ha ecen molecula
dynamics simula ions @24#ha e ound ha he alues o he
s eady empe a u e o a dilu e gas o inelas ic disks can be
closely i ed by an empi ical ela ion, which in ou uni s
eads
a
˜
22516Ac2
p
S
1
12
a
21B
D
,~50!
wi h A.0.080 and B.20.54. The abo e exp ession has he
same unc ional dependence on he es i u ion coe icien as
Eq. ~39!. In addi ion, om his la e equa ion one iden i ies
A.0.100 and B.20.511, which a e su p isingly close o
he simula ion alues, aking in o accoun ha no excluded
olume e ec s a e conside ed in ou model, and also ha he
low obse ed in Re . @24#is highly nonuni o m and i ex-
hibi s ime-dependen mic os uc u es.
Sela, Goldhi sch, and Noskowi z @18#ca ied ou a s udy
o a wo-dimensional g anula USF o Bu ne o de using
he Bol zmann equa ion. They a i ed a exp essions o he
s eady empe a u e and he p essu e enso o he o ms
a
˜
25C~12
a
2!1D~12
a
2!3/21O~12
a
2!2,~51!
P
˜
xy5E~12
a
2!1/21O~12
a
2!3/2,~52!
P
˜
xx511F~12
a
2!1O~12
a
2!2,~53!
P
˜
yy511G~12
a
2!1O~12
a
2!2.~54!
He e C,D,E,F, and Ga e dimensionless cons an s. In
Table I we compa e he alues o hese cons an s gi en in
Re . @18#wi h hose ob ained by expanding he exp essions
de i ed in his sec ion. Also gi en a e he esul s ob ained
om he expansion o he Jenkins and Richman heo y. The
alues om he h ee heo ies a e e y close and, in pa icu-
2850 55J. J. BREY, M. J. RUIZ-MONTERO, AND F. MORENO

la , he ag eemen be ween ou esul s and he Bu ne ex-
pansion o he Bol zmann equa ion is excellen .
V. VELOCITY DISTRIBUTION FUNCTION
The o mal solu ion o Eq. ~26!can be w i en as
~0!~V, !5e2se aijVj
]
]
Vi ~0!~V,0!
1
E
0
d 8e2~s2s8!e~ 2 8!aijVj~
]
/
]
Vi!
F
n
~0!~ 8
!
21
n~0!kBT~0!~ 8
!
c
~0!~V, 8
!~12
a
2!
l
~0!~ 8
!
G
3 l
~0!~V, 8
!,~55!
whe e
s[
E
0
d 8
n
~0!~ 8!~56!
is a measu e o he a e age numbe o collisions pe pa icle
be ween 0 and ,s85s( 8), and exp@ aijVj(
]
/
]
Vi)‡is a shi
ope a o in eloci y space,
e aijVj~
]
/
]
Vi!h~V!5h@L
~2 !•V#.~57!
O cou se, he ime-dependen empe a u e in Eq. ~55!
mus be consis en ly calcula ed, i.e., i is gi en by he solu-
ion o Eqs. ~30!–~32!. Le us now conside he long ime
limi o (0)( ). Assuming ha he s eady s a e discussed in
Sec. IV is eached, we know ha he empe a u e ends o he
cons an alue gi en by Eq. ~39!, and sdi e ges. The con-
clusion is ha (0)(V, ) app oaches he s eady o m
FIG. 1. Reduced shea a e a
˜
as a unc ion o
he coe icien o es i u ion
a
o a wo-
dimensional s eady USF. The solid line co e-
sponds o he p esen heo y, and he dashed line
o ha o Jenkins and Richman.
FIG. 2. The p essu e enso o a wo-
dimensional s eady USF. The solid lines co e-
spond o he p esen heo y, and he dashed ones
o ha o Jenkins and Richman.
55 2851STEADY UNIFORM SHEAR FLOW IN A LOW DENSITY . . .
*~V!5
E
0
`d e2
n
* e aijVj~
]
/
]
Vi!
3
F
n
*21
n~0!kBT*
c
*~V!~12
a
2!
l
*
G
l
*~V!,
~58!
wi h
c
*~V!5mV2
dkBT*21. ~59!
I can be checked ha his exp ession ep oduces he alues
o he s eady p essu e enso ound in Sec. IV, and also ha
i gi es a anishing hea lux. Now we in oduce dimension-
less quan i ies by
V
˜
5
S
2kBT*
m
D
21/2
V,~60!
˜
51
n~0!
S
2kBT*
m
D
d/2
*.~61!
Then Eq. ~58! educes o
˜
~V
˜
!5
p
2d/2
E
0
`dse2sesa
˜
V
˜
y~
]
/
]
V
˜
x!
3
F
12d21
c
S
2
dV
˜
221
D
~12
a
2!
G
e2V
˜
2,~62!
whe e use has been made o Eqs. ~17!and ~36!. In he case
o a sys em o ha d sphe es, he ma ginal dis ibu ion o he
zcomponen o he eloci y is easily ob ained,
˜
z~V
˜
z![
E
2`
`dV
˜
x
E
2`
`dV
˜
y
˜
~V
˜
!5
p
21/2
3
F
122~12
a
2!
3c~2V
˜
z
221!
G
e2V
˜
z
2.~63!
This exp ession shows a limi a ion o ou model kine ic
equa ion, namely, ha he dis ibu ion unc ion akes un-
physical nega i e alues o la ge enough eloci ies. Ne e -
heless, he eloci y alues o which Eq. ~63!is nega i e a e
gi en by
Vz
2.
F
3c
2~12
a
2!11
G
kBT*
m,~64!
and, he e o e, he dis ibu ion unc ion is posi i e in all he
ange o he mal eloci ies, e en in he limi
a
→0. I is hen
possible ha Eq. ~62!p o ides an accu a e app oxima ion o
he solu ion o he Bol zmann equa ion in all he ele an
egion o he dis ibu ion unc ion. This poin will be ana-
lyzed in de ail in Sec. VI.
The ma ginal eloci y dis ibu ion in he shea plane, also
o d53, is
˜
xy~V
˜
x,V
˜
y![
E
2`
`dV
˜
z
˜
~V
˜
!
5
p
21
E
0
`dse2se2[~V
˜
x1sa
˜
V
˜
y
!
2
1V
˜
y
2
]
3
H
12
4
~
12
a
2
!
3c@~V
˜
x1sa
˜
V
˜
y!21V
˜
y
2#
J
.
~65!
In o de o compa e wi h p e ious heo ies, we ha e ca ied
ou a pe u ba i e expansion o
˜
o d52 in powe s o
12
a
2. The esul is
˜
~V
˜
!5
p
21e2V
˜
2
F
12V
˜
2sin2
u
A
c~12
a
2!1/2
1~122V
˜
21V
˜
2cos2
u
11
2V
˜
421
2V
˜
4cos4
u
!
312
a
2
c1O~12
a
2!3/2
G
,~66!
He e, we ha e in oduced a pola ep esen a ion o he e-
loci y, V
˜
(V
˜
,
u
), whe e
u
is measu ed wi h espec o he
s eaming di ec ion x.
Jenkins and Richman @17#assumed ha he s eady dis i-
bu ion unc ion has he o m o a gene alized Gaussian,
˜
~JR!51
p
A
u
K
u
exp~2V
˜
•K21•V
˜
!,~67!
whe e Kis a ma ix which accoun s o he aniso opy exis -
ing in he USF. When he low densi y limi o he abo e
dis ibu ion is conside ed and he esul expanded o o de
(12
a
2) one ob ains
˜
~JR!~V
˜
!5
p
21e2V
˜
2
F
12V
˜
2sin2
u
A
2~12
a
2!1/2
1~122V
˜
212V
˜
2cos2
u
11
2V
˜
421
2V
˜
4cos4
u
!
312
a
2
41O~12
a
2!3/2
G
.~68!
I we neglec he di e ence be ween c(;1.96) and 2, bo h
exp essions ag ee o o de (12
a
2)1/2. The e ms o o de
(12
a
2) ha e he same dependence on he eloci y modulus
TABLE I. Compa ison o he alues o he coe icien s in ex-
pansions gi en by Eqs. ~51!–~54!ob ained wi h di e en heo ies
and in he p esen wo k.
Model Sela, Goldhi sch,
and Noskowi z Jenkins and Richman
C0.511 0.511 0.522
D00 0
E20.714 20.714 20.707
F0.511 0.522 0.5
G0.511 0.522 0.5
2852 55
J. J. BREY, M. J. RUIZ-MONTERO, AND F. MORENO
and also on
u
, bu all he cons an p e ac o s di e . In spi e
o his di e ence, le us no ice ha bo h exp essions ~66!and
~68!make he same con ibu ion o he diagonal pa o he
p essu e enso when he cons an cis again app oxima ed by
2. This can be e i ied by a di ec calcula ion ~see also Table
I!.The dis ibu ion unc ion o he USF o a wo-
dimensional g anula gas has also been calcula ed a he
same o de by Sela, Goldhi sch, and Noskowi z in Re . @18#.
The e, a much mo e complica ed dependence on bo h pola
componen s o he eloci y is ound. By compa ing wi h Jen-
kins and Richman esul s, he au ho s conclude ha he con-
s an s appea ing as p e ac o s o he second ha monics in
u
in Eq. ~68!can be conside ed as ough a e ages o he
co esponding unc ions ob ained by hem. We e e he
eade o hei pape o de ails.
VI. DIRECT MONTE CARLO SIMULATION
The di ec simula ion Mon e Ca lo me hod @19#has
p o ed o be a e y use ul ool o ob ain nume ical solu ions
o he Bol zmann equa ion o molecula luids. Ve y e-
cen ly, i has also been applied o s udy he HCS o a low
densi y g anula low @20#. Since he de ails o he me hod
ha e been ex ensi ely discussed in Re . @19#, hey will no be
gi en he e.
We saw in Sec. III ha he dis ibu ion unc ion o he
USF becomes homogeneous in he Lag angian ame. The
conside a ion o homogeneous s a es allows a g ea simpli i-
ca ion o he simula ion. The e o e, we ha e ca ied ou he
simula ion in he Lag angian ame and es ic ed ou sel es
o solu ions o he Bol zmann equa ion which s ay homoge-
neous in ha ame. In o he wo ds, we nume ically sol ed
he equa ion @compa e wi h Eq. ~26!#
]
~0!
]
2aijVj
]
~0!
]
Vi
5J
B@V
u
~0!#.~69!
O cou se, his implies ha he possibili y o spon aneous
o ma ion o spa ial inhomogenei ies is elimina ed in he
FIG. 3. Time e olu ion o he educed shea
a e a
¯
o
a
50.8, and di e en alues o he
shea a e a. In all cases, he ini ial dis ibu ion
was homogeneous wi h a Maxwellian eloci y
dis ibu ion in he Lag angian ame. Time is
measu ed in uni s o c„2
A
pn
(0)…21.
FIG. 4. S eady educed shea a e a
˜
as a unc-
ion o he coe icien o es i u ion
a
o a dilu e
sys em o ha d disks. The solid line co esponds
o he kine ic model, and he symbols a e esul s
om he Mon e Ca lo simula ion.
55 2853STEADY UNIFORM SHEAR FLOW IN A LOW DENSITY . . .
simula ion. Ou aim he e is o s udy he p ope ies o he
homogeneous s eady s a e. I s s abili y will be analyzed else-
whe e.
Fo homogeneous sys ems he e is no need o spli he
sys em in o cells and, consequen ly, he spa ial coo dina es
o he pa icles do no play any ole in he simula ion. In
addi ion, no bounda y condi ions mus be in oduced. In ou
simula ion we ha e conside ed a h ee-dimensional sys em.
The numbe o pa icles is N51000, and he esul s ha e
been a e aged o e 500 di e en ajec o ies. The ime in e -
al D o e which i is assumed ha ee mo ion, including
he e ec o he ine ial o ce, and collisions, a e uncoupled
has been aken
d
50.025
n
0
21, whe e
n
05(2
A
p
/c)
n
( ),
wi h
n
( ) being he ins an aneous alue o he collision e-
quency gi en by Eq. ~36!. We conside ed a noncons an ime
s ep in o de o gua an ee ha i always emains much
smalle han he a e age ime be ween collisions, in spi e o
he change o he empe a u e @30#. The ini ial eloci y dis-
ibu ion in all cases is a Maxwellian.
In Fig. 3 we p esen he ime e olu ion o a
¯
[a/
n
( ) o
a
50.8 and ou di e en alues o he shea a e, namely,
a50.1, 0.25, 0.5, and 1. Time is measu ed in uni s o
c„2
A
pn
(0)…21, whe e
n
(0) is he ini ial collision e-
quency. A e an ini ial ansien pe iod, all cu es con e ge
o he same s eady alue, as p edic ed by Eq. ~39!. The same
quali a i e beha io has been ound o all he educed com-
ponen s o he p essu e enso Pij( )/p( ). The e o e, in he
ollowing we will concen a e on he dependence o he
s eady alues o he educed quan i ies on he es i u ion co-
e icien
a
, once we ha e checked hey do no depend on he
shea a e.
The esul s ob ained o he s eady educed shea a e a
˜
o di e en alues o
a
a e shown in Fig. 4. The s a is ical
e o s a e smalle han he symbols used o ep esen he
da a. Also plo ed is he p edic ion o ou model kine ic
equa ion, i.e. Eq. ~39!wi h d53. I is seen ha he ag ee-
men is ema kable a low dissipa ion, al hough he disc ep-
ancy inc eases as he es i u ion coe icien dec eases. We
FIG. 5. The same as in Fig. 4 o he educed
p essu e enso P
˜
ij.
FIG. 6. Ma ginal eloci y dis ibu ion unc-
ion in he di ec ion pe pendicula o he shea
plane. The symbols a e simula ion da a, and he
solid line co esponds o he model kine ic equa-
ion. The es i u ion coe icien is
a
50.8.
2854 55
J. J. BREY, M. J. RUIZ-MONTERO, AND F. MORENO