Shea s a e o eely e ol ing g anula gases
J. Ja ie B ey, M. J. Ruiz-Mon e o, and A. Domínguez
Física Teó ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, E-41080, Se illa, Spain
共Recei ed 27 May 2008; published 1 Oc obe 2008兲
Hyd odynamic equa ions a e used o iden i y he inal s a e eached by a eely e ol ing g anula gas abo e
bu close o i s shea ins abili y. The heo y p edic s he o ma ion o a wo bands shea s a e wi h a s eady
densi y p o ile. The e is a modula ion be ween empe a u e and densi y p o iles as a consequence o he ene gy
balance, he densi y luc ua ions emaining small, wi hou p oducing clus e ing. Mo eo e , he ime depen-
dence o he eloci y ield can be scaled ou wi h he squa ed oo o he a e age empe a u e o he sys em.
The la e ollows he Ha law, bu wi h an e ec i e cooling a e ha is smalle han ha o he ee homo-
geneous s a e. The heo e ical p edic ions a e compa ed wi h nume ical esul s o inelas ic ha d disks ob ained
by using he di ec Mon e Ca lo simula ion me hod, and a good ag eemen is ob ained o low inelas ici y.
DOI: 10.1103/PhysRe E.78.041301 PACS numbe 共s兲: 45.70.Mg, 45.70.Qj, 47.20.Ky
I. INTRODUCTION
G anula gases a e ensembles o mac oscopic pa icles
whose dynamics is con olled by inelas ic bina y collisions,
sepa a ed by ballis ic mo ion 关1兴. The o ces a e usually sho
ange and epulsi e and, he e o e, in he simples o m
g anula gases a e modeled as a collec ion o smoo h inelas-
ic ha d sphe es o disks. In ecen yea s, he s udy o g anu-
la gases has a ac ed a lo o a en ion 关2,3兴, among o he
easons because hey se e as a sensi i e p o ing g ound o
kine ic heo y and nonequilib ium s a is ical mechanics.
A peculia i y o g anula gases is hei endency o spon-
aneously de elop pa e n o ming ins abili ies when e ol -
ing eely. This includes he so-called shea ing and clus e ing
ins abili ies 关4,5兴, in which he sys em de elops pa e ns bo h
in he low ield and he densi y ield. Analysis o he hyd o-
dynamic Na ie -S okes equa ions o g anula gases indi-
ca es he exis ence o se e al possible scena ios explaining
he de elopmen o hese ins abili ies and hei mu ual in lu-
ence. An illumina ing discussion o hem is gi en in Re . 关6兴.
Fo ins ance, he exis ence o non-s a iona y channel lows in
eely cooling gases leading o an a emp ed ini e- ime den-
si y blowup, which is a es ed by hea di usion, has been
shown ecen ly 关7,8兴.
In his pape , a shea low s a e o eely e ol ing dilu e
g anula gases will be desc ibed. I s o igin is ied in wi h he
shea mode ins abili y and he analysis o be p esen ed he e
is es ic ed o sys ems whose size is sligh ly la ge han he
c i ical size o his ins abili y. The la e happens o be
smalle han he c i ical size o he ins abili y o one o he
longi udinal modes associa ed wi h he densi y 关9,10兴. Con-
sequen ly, he spa ial inhomogenei ies o he hyd odynamic
ields a e expec ed o be small. Al hough he low is nons a-
iona y, all he ime dependence can be exp essed h ough he
spa ially a e aged empe a u e and, he e o e, i can be de-
sc ibed in e ms o dimensionless ime-independen quan i-
ies. Mo e p ecisely, i is an inhomogeneous wo band shea
s a e. I mus be s essed ha his s a e is exhibi ed by a
sys em wi h, o ins ance, pe iodic o elas ic bounda y con-
di ions, wi hou being d i en by he bounda ies as i happens
wi h he so-called simple o uni o m shea low 关11–13兴.
Nons a iona y s a es in which all he ime dependence occu s
h ough he empe a u e, such as he homogenous cooling
s a e 关14兴, as well as nonuni o m s a es whe e all he spa ial
dependence also occu s h ough he empe a u e a e peculia
o g anula gases 关15兴.
The exis ence o his ee shea s a e has been sugges ed
p e iously in he amewo k o a ime-dependen Ginzbu g-
Landau model o g anula gases 关10兴, and also as he esul
o a nonlinea analysis o he shea ing ins abili y 关16兴. The
heo y p esen ed in his pape di e s om hose s udies bo h
in he me hod and he esul s, whe e a he s ong disc epan-
cies occu as i will be discussed along he pape . The in e es
he e is ocussed on he comple e iden i ica ion o he hyd o-
dynamic ields cha ac e izing he ee shea s a e and he
compa ison o he heo e ical p edic ions wi h he esul s o
nume ical simula ions o he pa icles composing he sys em.
O cou se, his p o ides a e y demanding es o he alidi y
o he mac oscopic desc ip ion p o ided by he hyd ody-
namic Na ie -S okes equa ions o g anula gases.
The emainde o he pape is o ganized as ollows. In
Sec. II, he nonlinea Na ie -S okes hyd odynamic equa ions
o a dilu e g anula gas a e sho ly e iewed, and pa icula -
ized o he ee shea s a e. This is mac oscopically cha ac-
e ized by p esen ing g adien s only in one di ec ion and by a
eloci y ield pe pendicula o he g adien s. Mo eo e , i is
assumed ha he densi y inhomogenei ies a e small. By in-
oducing app op ia e dimensionless leng h and ime scales,
a closed equa ion o he eloci y low is de i ed in Sec. III.
The solu ion o his equa ion shows ha he ampli ude o he
eloci y ield decays mono onically in ime. Mo eo e , by
analyzing he empe a u e equa ion, an explici exp ession
o he ampli ude o he s eady densi y luc ua ion is ob-
ained. Also, i is seen ha he a io be ween he a e age
empe a u e o he sys em and he squa e o he ampli ude o
he eloci y ield is ime independen .
In Sec. IV, he simula ion me hod used is indica ed. I is
es ic ed o low densi y gases, whose one-pa icle dis ibu-
ion unc ion obeys he Bol zamnn equa ion, bu i mus be
s essed ha i is an N-pa icle simula ion me hod, he e o e
p o iding in o ma ion on all he space and ime co ela ions.
I is obse ed ha he sys em ac ually ends o a s eady si u-
a ion ha ing he assumed p ope ies o he ee shea s a e.
Then, he measu ed hyd odynamic quan i ies a e compa ed
wi h he analy ical heo e ical exp essions de i ed om he
PHYSICAL REVIEW E 78, 041301 共2008兲
1539-3755/2008/78共4兲/041301共9兲©2008 The Ame ican Physical Socie y041301-1
Na ie -S okes equa ions. A good ag eemen is ob ained o
small inelas ici y and sys em sizes close o he c i ical alue
o he shea ins abili y. Finally, Sec. V summa izes he ob-
ained esul s and p esen s a b ie discussion.
II. HYDRODYNAMIC EQUATIONS
The balance equa ions o he numbe densi y n共 , 兲, he
low eloci y u共 , 兲, and he g anula empe a u e T共 , 兲o a
sys em composed by smoo h inelas ic ha d sphe es 共d=3兲o
disks 共d=2兲o mass m, diame e
␴
, and coe icien o no mal
es i u ion
␣
, ha e he o m
⳵
n
⳵
+⵱·共nu兲=0, 共1兲
冉
⳵
⳵
+u·⵱
冊
u+共mn兲−1 ⵱·P=0, 共2兲
冉
⳵
⳵
+u·⵱+
␨
冊
T+2
nd共P:⵱u+⵱·q兲=0. 共3兲
Fo a dilu e gas and o Na ie -S okes o de , he p essu e
enso P, he hea lux q, and he cooling a e
␨
a e gi en by
he cons i u i e ela ions 关9,17,18兴
Pij共 , 兲=p共 , 兲
␦
ij −
␩
冉
⳵
uj
⳵
i
+
⳵
ui
⳵
j
−2
d
␦
ij ⵱·u
冊
,共4兲
q共 , 兲=−
␬
⵱T−
␮
⵱n,共5兲
␨
=
␨
共0兲+
␨
共2兲.共6兲
He e p=nT is he hyd os a ic p essu e,
␩
is he shea iscos-
i y,
␬
he 共 he mal兲hea conduc i i y, and
␮
a anspo co-
e icien peculia o g anula luids ha is e e ed o as he
di usi e hea conduc i i y. The e m
␨
共0兲deno es he con i-
bu ion o he cooling a e o ze o h o de in he g adien s o
he hyd odynamic ields, while
␨
共2兲deno es he second o de
in he g adien s con ibu ions. The la e will be neglec ed in
he ollowing, as is done in mos o he calcula ions, since i
is expec ed o gi e e y small co ec ions o he hyd ody-
namic equa ions as compa ed wi h he simila e ms coming
om he p essu e enso and he hea lux 关9兴. The anspo
coe icien s appea ing in Eqs. 共4兲and 共5兲can be exp essed as
␩
共T,
␣
兲=
␩
0共T兲
␩
*共
␣
兲,共7兲
␬
共T,
␣
兲=
␬
0共T兲
␬
*共
␣
兲,共8兲
␮
共n,T,
␣
兲=T
␬
0共T兲
n
␮
*共
␣
兲,共9兲
whe e
␩
0共T兲and
␬
0共T兲a e he low densi y 共Bol zmann兲 al-
ues o he shea iscosi y and he he mal hea conduc i i y,
espec i ely. Thei explici exp essions as well as hose o
he dimensionless unc ions
␩
*,
␬
*, and
␮
*a e gi en in he
Appendix. Finally, he exp ession o he ze o h o de cooling
a e eads
␨
共0兲共n,T,
␣
兲=nT
␩
0共T兲
␨
*共
␣
兲.共10兲
The unc ion
␨
*共
␣
兲is also gi en in he Appendix.
The hyd odynamic Na ie -S okes equa ions o he in-
elas ic gas a e ob ained by subs i u ing Eqs. 共4兲–共6兲in o Eqs.
共1兲–共3兲. They admi a simple solu ion de ined by a cons an
and uni o m densi y nH, a anishing eloci y ield uH=0, and
a ime dependen empe a u e TH共 兲obeying he equa ion
⳵
TH共 兲
⳵
+
␨
H
共0兲共 兲TH共 兲=0. 共11兲
This is he so-called homogeneous cooling s a e 共HCS兲关14兴.
This s a e is known o be uns able wi h espec o long wa e-
leng h spa ial pe u ba ions. Mo e p ecisely, linea s abili y
analysis o he hyd odynamic equa ions shows ha , o la ge
enough sys ems, he ans e sal componen o he eloci y
ela i e o he squa e oo o he empe a u e g ows in ime
leading o he ins abili y o he sys em 关4,5,9兴. This shea
ins abili y has been con i med by molecula dynamics simu-
la ions 关4,5兴and also by Mon e Ca lo simula ions o he
e ec i e dynamics associa ed o he Bol zmann equa ion
关19兴. O cou se, a e a sho ime in e al, he heo e ical
p edic ions based on he linea ized hyd odynamic equa ions
a e no alid any longe since neglec ed nonlinea e ec s
become e y impo an . Du ing he ini ial s ages o he de-
elopmen o he ins abili y, densi y inhomogenei ies occu
in he sys em. Fo dilu e sys ems whose linea size is la ge
共and compa able兲 o he c i ical sys em size o he shea
ins abili y, he spa ial inhomogenei ies seem o be domina ed
by a nonlinea coupling o he densi y wi h he ans e sal
eloci y ield 关20兴. Al hough his could indica e ha he sys-
em is going in o a clus e ing egime, in which he pa icles
end o g oup oge he o ming e y high densi y egions
coexis ing wi h e y dilu e egions, simula ion esul s indi-
ca e ha he densi y ac ually ends o a qui e smoo h p o ile
关20兴.
Because o con inuous cooling due o he inelas ici y o
collisions, s a iona y s a es a e no possible in eely e ol -
ing g anula gases 共 o ins ance, wi h pe iodic bounda y con-
di ions兲. Ne e heless, i is s ill possible ha he “ inal s a e”
eached by he sys em when he HCS is uns able, exhibi s
some scaling p ope ies so ha i can be simply iden i ied a
leas a he hyd odynamic le el o desc ip ion. He e, a en-
ion will be ocused on sys ems jus abo e he shea mode
ins abili y h eshold. Well abo e his h eshold, he physical
scena io migh be qui e di e en 关6–8兴.
Conside hyd odynamic lows e i ying he ollowing
condi ions. 共i兲The e a e g adien s in only one di ec ion aken
as he Xaxis, and 共ii兲 he hyd odynamic eloci y ield has he
o m ui共 , 兲=
␦
i,yu共x, 兲. O he speci ica ions o he s a e o
he sys em will be made when app op ia e. Use o he abo e
condi ions in o Eqs. 共1兲–共3兲shows ha he densi y p o ile
mus be s a iona y, n=n共x兲, and he componen Pxx o he
p essu e enso mus be uni o m. In he amewo k o he
Na ie -S okes app oxima ion, he la e implies ha he p es-
su e is also uni o m, p=p共 兲. Mo eo e , he equa ions o he
eloci y and empe a u e ields become
BREY, RUIZ-MONTERO, AND DOMÍNGUEZ PHYSICAL REVIEW E 78, 041301 共2008兲
041301-2
⳵
u
⳵
−共mn兲−1
⳵
⳵
x
冉
␩
⳵
u
⳵
x
冊
=0, 共12兲
⳵
T
⳵
−2共nd兲−1
再
␩
冉
⳵
u
⳵
x
冊
2
+
⳵
⳵
x
冋
冉
␬
−n
␮
T
冊
⳵
T
⳵
x
册
冎
+T
␨
共0兲=0.
共13兲
To sol e he abo e equa ions, he bounda y condi ions mus
be speci ied. A sys em o Npa icles enclosed in a cubic 共d
=3兲o squa e 共d=2兲box o side Lwill be conside ed. Pe i-
odic bounda y condi ions will be assumed a all he bound-
a ies in o de o a oid undesi able wall e ec s. I will be
con enien o he pu poses he e o in oduce a unc ion
␸
共x兲
by
n共x兲=nH关1+
␸
共x兲兴,共14兲
wi h nH⬅N/Ld. The conse a ion o he numbe o pa icles
and he pe iodic bounda y condi ions imply ha
冕
0
L
dx
␸
共x兲=0,
␸
共x+L兲=
␸
共x兲,0艋x⬍L.共15兲
Mo eo e , i will be assumed ha 兩
␸
共x兲兩Ⰶ1, i.e., he spa ial
densi y inhomogenei ies a e supposed o be small. I will be
shown in he ollowing ha his ac ually educes he ange o
applicabili y o he heo y o sys ems nea 共and abo e兲 he
h eshold o he shea ins abili y. Then, he empe a u e p o-
ile in he s a e being conside ed is gi en by
T共x, 兲⬇
␪
共 兲关1−
␸
共x兲兴,共16兲
whe e
␪
共 兲⬅p共 兲
nH
=兰0
Ldxn共x兲T共x, 兲
兰0
Ldxn共x兲共17兲
is he spa ial a e age empe a u e o he sys em a ime .I
mus be ealized ha assuming 兩
␸
兩Ⰶ1 and, consis en ly, e-
aining e ms up o i s o de in i when sol ing Eqs. 共12兲
and 共13兲, is no he same as linea izing a ound he HCS,
since no hing is in p inciple assumed abou he ampli ude o
he empe a u e
␪
共 兲o he eloci y ield u共x, 兲. This will be
discussed in de ail in he nex sec ion.
III. THE FREE SHEAR STATE
The hyd odynamic equa ions 共12兲and 共13兲can be simpli-
ied by using dimensionless leng h and ime scales de ined
by
l⬅
␯
共 兲
2
冋
m
␪
共 兲
册
1/2
x共18兲
and
s⬅1
2
冕
0
d ⬘
␯
共 ⬘兲,共19兲
espec i ely. He e
␯
共 兲⬅nH
␪
共 兲
␩
0共
␪
兲共20兲
is a cha ac e is ic equency, p opo ional o he Bol zmann
collision equency
␯
B共 兲, namely,
␯
B共 兲=共2+d兲
␯
共 兲/4. No e
ha he p e ac o in Eq. 共18兲does no depend on ime and he
leng h scaling is made wi h a cha ac e is ic leng h which is
p opo ional o he 共 ime-independen 兲a e age mean ee
pa h o he sys em. In e ms o he new scales, Eq. 共12兲can
be app oxima ed by
⳵
u共l,s兲
⳵
s−
␩
*共
␣
兲
2
⳵
2u共l,s兲
⳵
l2=0, 共21兲
whe e e ms o o de u
␸
ha e been neglec ed. The solu ion
o he abo e equa ion is a supe posi ion o monoc oma ic
wa es o he o m
uk共l,s兲=
␹
k共l兲
␻
k共s兲共22兲
wi h
␹
k共l兲= sin共kl +
␾
k兲共23兲
and
␻
k共s兲=
␻
k,0e−k2
␩
*s/2,共24兲
whe e
␾
kand
␻
k,0 a e a bi a y cons an s. The possible al-
ues o ka e es ic ed by he pe iodic bounda y condi ions
which imply
k=2
␲
q
lM
,共25兲
qbeing a posi i e in ege and lM he alue o l o x=L, i.e.,
lM=
␯
共 兲
2
冋
m
␪
共 兲
册
1/2
L.共26兲
The o m o Eq. 共24兲indica es ha in he limi o la ge ime
s, he dynamics o he sys em is go e ned by he undamen-
al mode co esponding o he lowes possible alue o k, i.e.,
k=km=2
␲
/lM. The e o e, o la ge enough imes,
u共l,s兲⬇ukm共l,s兲=
␹
共l兲
␻
共s兲共27兲
wi h
␹
共l兲=
␹
km共l兲and
␻
共s兲=
␻
km共s兲. In he same app oxima-
ion unde conside a ion, 兩
␸
共x兲兩Ⰶ1, he e olu ion equa ion
o he empe a u e 共13兲leads o
1
␪
共s兲
d
␪
共s兲
ds −m
␩
*共
␣
兲
␻
2共s兲
␪
共s兲d
冉
d
␹
共l兲
dl
冊
2
+
␨
*共
␣
兲关2+
␸
共l兲兴
+d+2
2共d−1兲关
␬
*共
␣
兲−
␮
*共
␣
兲兴d2
␸
共l兲
dl2=0. 共28兲
Using Eq. 共23兲 his becomes
SHEAR STATE OF FREELY EVOLVING GRANULAR GASES PHYSICAL REVIEW E 78, 041301 共2008兲
041301-3
1
␪
共s兲
d
␪
共s兲
ds −
␩
*共
␣
兲m
␻
2共s兲km
2
2
␪
共s兲d兵1 + cos关2共kml+
␾
km兲兴其 +
␨
*共
␣
兲
⫻关2+
␸
共l兲兴 +d+2
2共d−1兲关
␬
*共
␣
兲−
␮
*共
␣
兲兴d2
␸
共l兲
dl2=0. 共29兲
By equi ing he sum o he posi ion dependen e ms o
cancel i is ob ained ha
␸
共l兲=− 共d−1兲
␩
*共
␣
兲
4d共d+2兲
␥
共km,
␣
兲acos关2共kml+
␾
km兲兴,共30兲
whe e
␥
共
␣
,km兲⬅
␬
*共
␣
兲−
␮
*共
␣
兲−共d−1兲
␨
*共
␣
兲
2共d+2兲km
2共31兲
and
a⬅m
␻
2共s兲
␪
共s兲.共32兲
Consis ency equi es ha abe ac ually independen o sand,
mo eo e , ha i be de ined posi i e. The o me o hese
condi ions oge he wi h Eq. 共21兲leads o
1
␪
共s兲
d
␪
共s兲
ds =2
␻
共s兲
d
␻
共s兲
ds =−km
2
␩
*共
␣
兲.共33兲
This equa ion will be used la e on o de e mine he e ec i e
cooling a e o he a e age empe a u e
␪
共s兲o he sys em in
he ee shea s a e. On he o he hand, equa ing o ze o he
pa o Eq. 共29兲 ha is posi ion independen yields, a e
employing Eq. 共33兲,
a=2d关2
␨
*共
␣
兲−km
2
␩
*共
␣
兲兴
km
2
␩
*共
␣
兲.共34兲
As men ioned abo e his quan i y mus be posi i e. As a
consequence, he exis ence o he ma hema ical solu ion o
he hyd odynamic Na ie -S okes equa ions unde conside -
a ion and hence o he ee shea s a e is only posible i
km
2⬍2
␨
*
␩
*共35兲
o , equi alen ly, L⬎Lc, wi h he c i ical size Lcgi en by
Lc=共2+d兲⌫共d/2兲
2
␲
共d−3兲/2nH
␴
共d−1兲
冉
␩
*
2
␨
*
冊
1/2
.共36兲
This coincides wi h he condi ion de e mining he ins abili y
egion o he shea mode ound in he linea s abili y analysis
o he hyd odynamic equa ions a ound he HCS 关9兴.
Subs i u ion o Eq. 共34兲in o Eq. 共30兲p o ides he explici
o m o he s eady densi y p o ile in he ee shea s a e nea
he h eshold o he ins abili y
␸
共l兲=−Acos关2共kml+
␾
km兲兴,共37兲
A=共d−1兲
␩
*共
␣
兲
2共d+2兲
␥
共km,
␣
兲
冋
冉
L
Lc
冊
2
−1
册
.共38兲
The e o e, he condi ion 兩
␸
共x兲兩Ⰶ1 o mally implies ha 兩L
−Lc兩/LcⰆ1. How es ic i e his condi ion ac ually is will be
seen in he nex sec ion.
A pa icula ly simple exp ession o he ampli ude o he
eloci y ield is ob ained by scaling wi h he a e age em-
pe a u e. Equa ions 共32兲and 共34兲yield
␻
˜
共s兲⬅
␻
共s兲
冋
m
2
␪
共s兲
册
1/2
=d1/2
冋
冉
L
Lc
冊
2
−1
册
1/2
.共39兲
The e o e, he scaled mac oscopic eloci y ield is ime-
independen and all i s dependence on he a e age densi y
and inelas ici y occu s h ough he c i ical leng h Lc. A simi-
la exp ession can be ob ained o he a io be ween he e -
ec i e cooling a e o he shea s a e
␨
S
*and he cooling a e
o he HCS,
␨
*. The o me is iden i ied by w i ing Eq. 共33兲
in he o m
⳵
␪
共s兲
⳵
s+2
␨
S
*
␪
共s兲=0, 共40兲
wi h
␨
S
*=km
2
␩
*共
␣
兲
2,共41兲
ha leads o
␨
S
*
␨
*=
冉
Lc
L
冊
2
.共42兲
O cou se, in he limi L→Lc,
␪
共s兲→TH共s兲and he law o
he HCS gi en by Eq. 共11兲is eco e ed.
The a e age ene gy pe pa icle is
e
¯
T⬅m
2u2+T
¯
d
2,共43兲
wi h he ba deno ing spa ial a e age. Using Eqs. 共17兲,共22兲,
共23兲,共32兲, and 共34兲i is ound ha
e
¯
T共s兲=
␨
*共
␣
兲
␪
共s兲d
km
2
␩
*共
␣
兲=
冉
L
Lc
冊
2
␪
共s兲d
2.共44兲
Fo L→Lc, he equilib ium ela ion e
¯
T=dT/2 is eco e ed as
expec ed. This equa ion has been p e iously ob ained by Wa-
kou e al. 关10兴, in he con ex o a Landau-Ginzbu g- ype
equa ion o mo ion de i ed om he hyd odynamic equa-
ions, unde ce ain es ic ions and in he limi o nea ly
elas ic collisions 共1−
␣
Ⰶ1兲. A his poin , i is wo h men-
ioning ha he assumed pe iodic bounda y condi ions do no
play an essen ial ole in he heo y de eloped he e, al hough
hey mus be compa ible wi h he symme y o he shea
s a e. Fo ins ance, i is easily ealized ha comple ely
equi alen esul s a e ob ained i elas ic walls we e used a
he bounda ies o he sys em.
So o e al. 关16兴ha e ca ied ou a nonlinea analysis o
he hyd odynamic equa ions o a sys em o inelas ic ha d
disks close o he ins abili y h eshold. When kmis sligh ly
BREY, RUIZ-MONTERO, AND DOMÍNGUEZ PHYSICAL REVIEW E 78, 041301 共2008兲
041301-4
smalle han kc⬅2
␲
/Lc, o ici y modes wi h wa e numbe
kmexhibi a c i ical slowing down, so ha all he o he hy-
d odynamic modes can be conside ed as ensla ed by hem.
Then, i is possible o de i e closed equa ions o he ampli-
udes o he o ici y modes wi h k=kmby using he adia-
ba ic elimina ion me hod. This app oach can be expec ed o
be o mally consis en wi h he one p esen ed he e, bu quan-
i a i e and quali a i e disc epancies occu when compa ing
he esul s om bo h app oaches. Al hough he au ho s o
Re . 关16兴do no gi e much de ail o hei analysis o he
nonlinea hyd odynamic equa ions, we ha e iden i ied a wo-
old o igin o he disc epancies. Fi s ly, a nonlinea e m in-
ol ing he o ici y seems o ha e been inconsis en ly ne-
glec ed. Secondly, he linea iza ion a ound he HCS and he
adiaba ic me hod used in 关16兴is no equi alen o he ap-
p oxima ion scheme p esen ed he e, in which he eloci y
ield obeys he closed Eq. 共21兲.
IV. DIRECT MONTE CARLO SIMULATIONS
To es he heo e ical p edic ions p esen ed in he p e i-
ous sec ions and, in pa icula , he exis ence i sel o he ee
shea s a e, we ha e employed he di ec simula ion Mon e
Ca lo 共DSMC兲me hod 关21,22兴, o nume ically gene a e he
dynamics o a sys em o inelas ic ha d disks. The DSMC
me hod is a many-pa icle algo i hm designed o mimic he
e ec i e dynamics o he pa icles o a gas in he low densi y
limi . The e o e, in his limi i is expec ed o lead o he
same esul s as, o ins ance, molecula dynamics simula-
ions, wi h he ad an age o a much la ge s a is ical accu-
acy. In all he simula ions, he ini ial s a e was homogeneous
and iso opic wi h a Gaussian eloci y dis ibu ion, and a
squa e box wi h pe iodic bounda y condi ions was used. The
linea size o he sys em Lwas always la ge han, bu close
o, he c i ical one Lc, p edic ed by Eq. 共36兲, o he consid-
e ed alue o he es i u ion coe icien
␣
. Then, acco ding o
he heo y, he sys em is expec ed o gene a e he nonlinea
ee shea low desc ibed in Sec. III, i i is s able.
As usual in DSMC simula ions, he mass mo he pa -
icles will be aken as he uni o mass, he a e age mean ee
pa h ␭⬅共2冑2nH
␴
兲−1 as he uni o leng h, and 2T共0兲
⬅2
␪
共0兲, whe e T共0兲is he ini ial empe a u e, as he uni o
ene gy. The e ec o a collision be ween pa icles and sis
o ins an aneously modi y hei eloci ies acco ding o he
ule
→
⬘= −1+
␣
2共
␴
ˆ· s兲
␴
ˆ,共45兲
s→ s
⬘= s+1+
␣
2共
␴
ˆ· s兲
␴
ˆ,共46兲
whe e s⬅ − sis he ela i e eloci y and
␴
ˆis he uni
ec o poin ing om he cen e o pa icle s o he cen e o
pa icle a con ac . This co esponds o he scena io in
which he cons i u i e ela ions 共4兲–共6兲we e de i ed 关9,17兴.
One o he echnical di icul ies when nume ically simu-
la ing a eely e ol ing g anula luid is he con inuous cool-
ing o he sys em, i.e., he dec ease in magni ude o he ypi-
cal eloci y o he pa icles. As a consequence, he nume ical
inaccu acies become e y la ge a e some ime in e al. To
a oid his di icul y, a p ocedu e was in oduced based on a
change in he ime scale being used. Mo e speci ically, a
loga i hmic ime scale is in oduced 关23,24兴. Then, he dy-
namics o sys ems in s a es whe e all he ime dependence
comes h ough he a e age empe a u e because o inelas ic
cooling, can be easily mapped in o he dynamics a ound
s eady s a es. In his case, he loga i hmic ime scale is p o-
po ional o he cumula ed numbe o collisions pe pa icle.
A. T ansien dynamics
In mos o he simula ions o be epo ed, a simila ime
sequence was obse ed. The sys em emains homogeneous
wi h no mac oscopic eloci y ield o an ini ial pe iod o
ime, de eloping a e wa ds a s a e wi h wo o ices, and
inally a shea s a e cha ac e ized by wo coun e lows pa al-
lel o one o he sides o he sys em. In his s a e, he e is a
coupling be ween he eloci y and densi y ields. The spa ial
inhomogenei ies emain s a iona y and he s a e looks ime
independen in he loga i hmic ime scale used in he nume i-
cal simula ions. The pa icula loca ion o he ansien o -
ices and hei di ec ion, and hence o he bands in he shea
s a e, depends on he ini ial s a e o , in a s a is ical sense, on
he luc ua ion aking he sys em away om he HCS. An
example o he desc ibed beha io is gi en in Fig. 1, whe e
0
10
20
30
10 20 30
τ=241.2
(b)
0
10
20
30
10 20 30
τ=120.5
(a)
0
10
20
30
10 20 30
τ=363.2
(c)
0
10
20
30
10 20 30
τ=604.5
(
d
)
FIG. 1. Time e olu ion o he eloci y ield scaled wi h he
a e age he mal eloci y u共m/2
␪
兲1/2in a sys em wi h
␣
=0.95 and
L/Lc⬇1.205. The indica ed imes
␶
a e measu ed as he a e age
numbe o collisions pe pa icle. The leng hs o he a ows in he
plo s a e p opo ional o he scaled eloci ies.
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041301-5

he scaled eloci y ield u共m/2
␪
兲1/2is plo ed a ou di e -
en imes o a sys em wi h
␣
=0.95 and L=39 共L/Lc
⬇1.205兲. The o ma ion o he o ices, hei dis o ion, and
he o ma ion o shea bands a e clea ly iden i ied. The la e
emain unchanged a e hei o ma ion, and his was ob-
se ed in all cases, a leas as long as he sys em is close
enough o he shea ins abili y. The spon aneous symme y
b eaking occu ing in he inal shea s a e was obse ed in
bo h pe pendicula di ec ions and, o compa e wi h he he-
o e ical p edic ions, he xaxis was always chosen pe pen-
dicula o he eloci y ield. Mo e will be commen ed on his
issue in he inal sec ion o he pape .
To ge some addi ional insigh abou he ime e olu ion o
he sys em be o e eaching he ee shea s a e, in Fig. 2 he
a io be ween he spa ial a e age empe a u e
␪
共 兲and he
HCS empe a u e a he same ime as p edic ed by he Ha
law 共11兲is shown as a unc ion o he a e age numbe o
collisions pe pa icle
␶
. The sys em is he same as in Fig. 1
and h ee di e en simula ion ajec o ies a e plo ed. Al-
hough he ime e olu ion o he empe a u e is no exac ly
he same in all cases, a qui e simila end is obse ed. The e
is a ime in e al in which he a e age empe a u e is well
desc ibed by he Ha law 共
␶
ⱗ250兲in spi e o he sys em
ha ing al eady well de eloped o ices 共see Fig. 1兲. A e -
wa ds, o 250ⱗ
␶
ⱗ600, he a e age empe a u e decays
much slowe han in he HCS, so ha he a io g ows e y
as , un il i sa u a es a a gi en alue. The egion wi h he
la ges g ow h co esponds o he dis o ion o he eloci y
o ices. The cons an alue eached in he inal shea s a e
indica es ha he a e age empe a u e
␪
共 兲s ill obeys a Ha -
like law, bu wi h a di e en cooling a e, in quali a i e
ag eemen wi h Eq. 共40兲.
The e olu ion o he densi y is illus a ed in Fig. 3, also
o he sys em wi h
␣
=0.95 and L/Lc⬇1.205. Two Fou ie
componen s o he ela i e densi y ield
␳
⬅n/nHa e plo ed
␳
2kmand
␳
km,km. The o me co esponds o he second densi y
mode ha is he one p edic ed o su i e by he heo y, Eq.
共37兲. In he simula ions, i appea s pa allel o any o he sides
o he sys em as discussed abo e and no dis inc ion is made
when epo ing he simula ion esul s. The o he componen ,
␳
km,kmis he lowes mode along he diagonal o he sys em.
Again, he se e al cu es a e di e en simula ion ajec o-
ies. In all cases, he second ans e sal mode o he densi y
g ows om he noise le el o a cons an alue as he shea
ins abili y de elops. Mo eo e , he inal s eady alue is he
same o all ajec o ies. On he o he hand, al hough he
diagonal mode o en also g ows in he ime in e al in which
he wo o ices a e losing hei shape ans o ming in o he
shea s a e, i decays o he noise le el once his s a e is
eached.
B. Shea s a e esul s
Now he esul s ob ained o he p ope ies o he sys em
once in he ee shea s a e will be epo ed, and compa ed
wi h he heo e ical p edic ions ob ained in his pape . Con-
side i s he densi y p o ile. The simula ion da a a e e y
well i ed by a cosine unc ion as gi en in Eq. 共37兲. This has
been checked bo h by plo ing di ec ly he p o iles and by
compu ing hei Fou ie componen s. The ampli ude o he
measu ed pe u ba ion Anis plo ed in Fig. 4as a unc ion o
L/Lc o h ee alues o he es i u ion coe icien
␣
=0.97,
0.95, and 0.9. The lines a e he heo e ical p edic ions gi en
by Eq. 共38兲, he highes one co esponding o he smalles
alue o
␣
and he lowes one o he g ea es alue o
␣
.I is
seen ha he ag eemen is qui e good when L/Lcis close o
1, he disc epancies inc easing as he size o he sys em in-
c eases abo e i s c i ical alue. Also, he de ia ion is la ge
he smalle he coe icien o es i u ion. In any case, i mus
be kep in mind ha he heo y de eloped he e is by con-
s uc ion es ic ed o s a es wi h AnⰆ1.
In Fig. 5, he ampli ude ATo he s eady cosine p o ile o
1−T共x, 兲/
␪
共 兲is p esen ed o he same alues o he pa am-
e e s as conside ed in Fig. 4. Acco ding o he heo y de el-
oped he e, Eq. 共16兲, his ampli ude should be he same as o
0 200 400 600 800 100
0
τ
0
1
2
3
θ/
THCS
FIG. 2. 共Colo online兲Time e olu ion o he spa ial a e age
empe a u e no malized by he empe a u e o he e e ence HCS
o he same sys em as in Fig. 1. Time
␶
is measu ed as he a e age
numbe o collisions pe pa icle. The se e al cu es co espond o
di e en simula ion ajec o ies.
0 200 400 600 800 100
0
τ
0
0.05
0.1
ρ
FIG. 3. 共Colo online兲Time e olu ion o he second ans e sal
mode o he ela i e densi y
␳
2km共solid lines兲and he diagonal
mode
␳
km,km共dashed lines兲. The sys em is he same as in Fig. 1. The
se e al cu es co espond o di e en ajec o ies, and ime
␶
is
measu ed again as he a e age numbe o collisions pe pa icle.
BREY, RUIZ-MONTERO, AND DOMÍNGUEZ PHYSICAL REVIEW E 78, 041301 共2008兲
041301-6
he densi y p o ile, i.e., AT=An=A. Ne e heless, he simula-
ion da a indica e ha his is no he case, excep o alues
o L/Lcclose o uni y. The de ia ions o he nume ical da a
om he heo e ical p edic ions a e in opposi e di ec ions o
ATand An, being la ge o he o me . This indica es ha he
p essu e is no ac ually s ic ly uni o m as p edic ed by he
Na ie -S okes equa ions o dilu e g anula gases, bu exhib-
i s some oscilla o y p o ile. On he o he hand, he compo-
nen Pxx o he p essu e enso was ound o be uni o m, as
equi ed by he balance equa ions 共1兲–共3兲. Ne e heless, o
L/Lcsmall enough, he ag eemen be ween heo y and simu-
la ion can be quali ied as sa is ac o y. Le us men ion ha i
he elas ic alues o he anspo coe icien s 共
␩
*=
␬
*
=1,
␮
*=0兲we e used, he heo e ical p edic ion o Aas a
unc ion o L/Lcwould become independen o
␣
con a y o
wha is obse ed in he simula ions.
A qui e s ong p edic ion o he heo y is p o ided by Eq.
共39兲, whe e he ampli ude o he eloci y ield scaled wi h
he squa e oo o he a e age g anula empe a u e is ex-
p essed as a simple ime and
␣
-independen unc ion o he
a io L/Lc. This la e p ope y was seen o be e i ied in he
simula ion wi hin he s a is ical unce ain ies. Mo eo e , he
esul s displayed in Fig. 6show ha he simula ion da a a e
in good ag eemen wi h he heo e ical exp ession, al hough
again sys ema ic de ia ions a e obse ed as he alue o he
coe icien o es i u ion dec eases and/o he size o he sys-
em as compa ed wi h i s c i ical alue inc eases.
The simula ion esul s also indica e ha he decay o he
a e age empe a u e in he ee shea s a e is go e ned by a
Ha -like law 共39兲. In Fig. 7, he a io be ween he cooling
a e o he ee shea s a e,
␨
S
*, and he one o he HCS,
␨
*,is
plo ed, o he same sys ems being conside ed along his
sec ion. The heo e ical p edic ion is p o ided by Eq. 共42兲,
implying ha he ee shea s a e cools slowe han he asso-
cia ed HCS, i.e., wi h he same densi y and ini ial empe a-
1 1.2 1.4 1.6
L/Lc
0
0.1
0.2
0.3
A
n
α=0.97
α=0.95
α=0.90
FIG. 4. 共Colo online兲Dimensionless ampli ude o he cosine
densi y p o ile in he ee shea s a e as a unc ion o he a io
be ween he size o he sys em Land i s c i ical alue o he shea
ins abili y Lc. Th ee alues o he es i u ion coe icien a e consid-
e ed, as indica ed. The symbols a e simula ion esul s and he lines
a e he p edic ions o Eq. 共38兲, he lowes line co esponding o he
la ge
␣
.
1 1.2 1.4 1.6
L/L
c
0
0.1
0.2
0.3
0
.
4
A
T
α=0.97
α=0.95
α=0.90
FIG. 5. 共Colo online兲The same as in Fig. 4bu o he ampli-
ude o he cosine empe a u e p o ile.
1 1.2 1.4 1.6
L/L
c
0
0.5
1
1.5
2
2
.5
~
ω
α=0.97
α=0.95
α=0.90
FIG. 6. 共Colo online兲Dimensionless ampli ude o he mac o-
scopic low
␻
˜
⬅
␻
共 兲/关2
␪
共 兲/m兴1/2in he ee shea s a e as a unc-
ion o he leng h Lo he sys em no malized by i s c i ical alue Lc.
The solid line is he heo e ical p edic ion 共39兲, while he symbols
a e simula ion esul s o he same sys ems as in Fig. 4.
1 1.2 1.4 1.6
L/L
c
0.2
0.4
0.6
0.8
1
ζ∗
S/ζ∗α=0.97
α=0.95
α=0.90
FIG. 7. 共Colo online兲Ra io be ween he cooling a es o he
ee shea s a e and o he HCS as a unc ion o he a io L/Lc. The
symbols ha e he same meaning as in Fig. 4and he solid line is he
heo e ical p edic ion 共41兲.
SHEAR STATE OF FREELY EVOLVING GRANULAR GASES PHYSICAL REVIEW E 78, 041301 共2008兲
041301-7
u e. Again a qui e sa is ac o y ag eemen is ound be ween
heo y and simula ions, wi h he disc epancies exhibi ing he
same ends as in all he p e ious compa isons.
V. SUMMARY AND DISCUSSION
I has been shown ha he hyd odynamic Na ie -S okes
equa ions o g anula gases p edic he exis ence o a shea
s a e o eely e ol ing sys ems, whose size is sligh ly la ge
han he c i ical size o he shea mode ins abili y. Explici
analy ical exp essions o he hyd odynamic ields ha e been
de i ed. Al hough i is a nons a iona y s a e due o cooling,
all he ime dependence o he hyd odynamic ields occu s
h ough he a e age empe a u e and, he e o e, i can be
elimina ed by in oducing app op ia e dimensionless quan i-
ies. These heo e ical p edic ions ha e been ound o be in
good ag eemen wi h he nume ical esul s ob ained by he
DSMC me hod, o alues o he es i u ion coe icien
␣
close o uni y. When he sys em is mo e inelas ic, signi ican
de ia ions om he heo y a e obse ed. This was expec ed,
since he ee shea s a e is cha ac e ized, as many o he
nonequilib ium s a es o g anula gases, by a s ong coupling
be ween g adien s and inelas ici y. In he p esen case, his is
easily iden i ied h ough he dependence on
␣
o Lc.Asa
consequence, a i s o de g adien expansion like he one
leading o he Na ie -S okes equa ions also implies a limi a-
ion in he ange o alues o
␣
o which he heo y applies.
Mo eo e , i is p obably ue ha he ee shea s a e is in-
he en ly non-New onian, as is he case o he s eady uni-
o m shea s a e o a g anula gas 关25兴. A possible indica ion
o his is he inhomogenei y o he p essu e obse ed in he
simula ions as implied by he di e ence be ween he mea-
su ed alues o Anand AT.
In Re . 关6兴se e al possible hyd odynamic scena ios a e
desc ibed o he beha io o a eely e ol ing g anula luid
inside i s ins abili y egion. In ha classi ica ion, he si ua-
ion conside ed in his pape is called scena io 4. He e he
inal s a e eached by he sys em has been in es iga ed and a
quan i a i e es o he comple e scena io has been p o ided.
The ee shea s a e seems s able, in he sense ha no de ia-
ions om i ha e been obse ed in he simula ions. In his
con ex , i is wo h men ioning ha , in many cases, he wo-
o ex s a e was clea ly iden i ied o a qui e la ge pe iod o
ime be o e he sys em mo ed o he shea s a e. This may
indica e ha i is a me as able s a e wi h a la ge escape ime.
This issue clea ly dese es mo e a en ion. On he o he
hand, when Lbecomes much la ge han Lc共 ypically,
L/Lcⲏ1.7兲, o he hyd odynamic modes become ele an and
he ee shea s a e is no expec ed o occu in he sys em.
This has been con i med by he simula ion esul s.
I is also wo h compa ing in some de ail he app oach
ollowed he e wi h he nonlinea s abili y analysis ca ied ou
in Re . 关16兴. As al eady men ioned, he esul s epo ed he e
sugges ha a e m nonlinea in he eloci y has been omi ed
al hough i is o he same o de as o he s ha a e kep . This
e m d ama ically modi ies he esul s o he s abili y analy-
sis 关26兴. Mo eo e , in Re . 关16兴an adiaba ic elimina ion
me hod is used, in which he ime de i a i e o he hyd ody-
namic ields o he han he ans e se eloci y a e se equal
o ze o. In his way, hese ields can be exp essed in e ms o
u, and when he exp essions a e inse ed in o he equa ion o
u, a closed equa ion is ob ained o he la e . The app oxi-
ma ion ollowed he e is di e en . To lowes o de , he ans-
e sal eloci y ield obeys a closed equa ion by i sel , Eq.
共21兲, once he unc ion
␪
共 兲has been scaled ou by he
change o a iables. This equa ion di e s om he one ob-
ained by he adiaba ic elimina ion me hod. I is no ully
clea o us p esen ly he physical o igin o his s ong dis-
c epancy.
One ele an ques ion ha always a ises when using
smoo h inelas ic ha d pa icles o model g anula gases, is o
wha ex en he esul s would be modi ied i mo e ealis ic
models, including o ins ance o a ional deg ees o eedom
关27兴and/o eloci y dependen es i u ion coe icien s 关2兴,
we e conside ed. Al hough he hyd odynamic Na ie -S okes
equa ions including hese e ec s a e known in some limi ing
cases, hei analysis is a beyond ou p esen each. Ne e -
heless, i can be expec ed ha an ex ension o he ene gy
balance appea ing in he simple case discussed he e will hold
when mo e dissipa ion mechanisms a e included. Then i is
ou conjec u e ha a simila shea s a e bu including he
new o a ional e ec s will also show up.
ACKNOWLEDGMENT
This esea ch was suppo ed by he Minis e io de Edu-
cación y Ciencia 共Spain兲 h ough G an No. FIS2008-01339
共pa ially inanced by FEDER unds兲.
APPENDIX: NAVIER-STOKES TRANSPORT
COEFFICIENTS
In his appendix, he explici exp essions o he anspo
coe icien s and he cooling a e in oduced in Eqs. 共4兲–共10兲
a e gi en. The elas ic shea iscosi y and hea conduc i i y
a e
␩
0=2+d
8⌫
冉
d
2
冊
␲
−共d−1兲/2共mT兲1/2
␴
−共d−1兲,共A1兲
␬
0=d共d+2兲2
16共d−1兲⌫
冉
d
2
冊
␲
−共d−1兲/2
冉
T
m
冊
1/2
␴
−共d−1兲,共A2兲
espec i ely. As usual in he con ex o g anula luids, he
Bol zmann cons an has been se equal o uni y. The ac o s
accoun ing o he dependence on he es i u ion coe icien
a e gi en by
␩
*共
␣
兲=
冋
␯
1
*共
␣
兲−
␨
*共
␣
兲
2
册
−1
,共A3兲
␬
*共
␣
兲=
冋
␯
2
*共
␣
兲−2d
d−1
␨
*共
␣
兲
册
−1
关1+c*共
␣
兲兴,共A4兲
␮
*共
␣
兲=2
␨
*共
␣
兲
冋
␬
*共
␣
兲+共d−1兲c*共
␣
兲
2d
␨
*共
␣
兲
册
⫻
冋
2共d−1兲
d
␯
2
*共
␣
兲−3
␨
*共
␣
兲
册
−1
,共A5兲
BREY, RUIZ-MONTERO, AND DOMÍNGUEZ PHYSICAL REVIEW E 78, 041301 共2008兲
041301-8
␨
*共
␣
兲=2+d
4d共1−
␣
2兲
冋
1+3c*共
␣
兲
32
册
.共A6兲
In he abo e exp essions,
␯
1
*共
␣
兲=共3−3
␣
+2d兲共1+
␣
兲
4d
冋
1−c*共
␣
兲
64
册
,共A7兲
␯
2
*共
␣
兲=1+
␣
d−1
冋
d−1
2+3共d+8兲共1−
␣
兲
16
+4+5d−3共4−d兲
␣
1024 c*共
␣
兲
册
,共A8兲
c*=32共1−
␣
兲共1−2
␣
2兲
9+24d+共8d−41兲
␣
+30
␣
2共1−
␣
兲.共A9兲
I is easily e i ied ha
␬
*and
␩
* end o uni y when
␣
goes
o one, while
␮
*and
␨
* anish in his limi .
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