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Fou ie s a e o a luidized g anula gas
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2001 Eu ophys. Le . 53 432
(h p://iopscience.iop.o g/0295-5075/53/4/432)
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Eu ophys. Le .,53 (4), pp. 432–437 (2001)
EUROPHYSICS LETTERS 15 Feb ua y 2001
Fou ie s a e o a fluidized g anula gas
J. J. B ey, D. Cube o, F. Mo eno and M. J. Ruiz-Mon e o
F´ısica Te´o ica, Uni e sidad de Se illa - Apa ado de Co eos 1065
E-41080, Se illa, Spain
( ecei ed 22 Sep embe 2000; accep ed in final o m 1 Decembe 2000)
PACS. 05.20.-y – Classical s a is ical mechanics.
PACS. 05.20.Dd – Kine ic heo y.
PACS. 81.05.Rm – Po ous ma e ials; g anula ma e ials.
Abs ac . – I is shown ha he Bol zmann equa ion o smoo h inelas ic ha d disks o
sphe es admi s a solu ion desc ibing a s eady s a e cha ac e ized by uni o m p essu e and linea
empe a u e p ofile. Such a s a e has been obse ed p e iously bo h in nume ical solu ions o
he Bol zmann equa ion and in molecula dynamics simula ions. Qui e peculia ly, p essu e
and empe a u e g adien a e no independen bu hei a io is a unc ion o he coefficien o
es i u ion. Se e al p ope ies o he solu ion a e discussed. In pa icula , i is shown ha a
linea Fou ie -like law is e ified o a bi a y empe a u e g adien .
The Bol zmann equa ion, modified o accoun o inelas ic wo-pa icle collisions, is o en
used as an idealized desc ip ion o dilu e apid g anula flows [1]. Ne e heless, e y li le
is known abou i s solu ions. E en o he simples possible s a e, co esponding o a eely
e ol ing homogeneous sys em, he eloci y dis ibu ion is only known in some app oxima-
ion [2]. Fo inhomogeneous si ua ions, he a ailable solu ions a e es ic ed o sys ems wi h
small spa ial g adien s o o he quasielas ic limi [3–5].
Mos o he expe imen al si ua ions deal wi h a g anula sys em o which ene gy is con-
inuously supplied in o de o keep i fluidized and, e en ually, in a s eady s a e. O en, his
is done by means o a ib a ing wall. The s eady s a e eached by he sys em is a om
equilib ium and also om he ime-dependen homogeneous cooling s a e (HCS) desc ibing
he ee e olu ion o a g anula sys em. Excep in he limi o e y small inelas ici y, a di ec
ex ension o he hyd odynamic desc ip ion ob ained by expanding a ound he HCS is no ex-
pec ed o apply. In ac , i is known ha inelas ici y and g adien s a e coupled o a g anula
gas in he p esence o a wall p o iding ene gy o he sys em [6,7].
Al hough he bounda y laye nex o he ib a ing wall mus be aken in o accoun o
a comple e desc ip ion o he sys em, a away om he wall he s a e o he g anula flow
may be cha ac e ized in a simple way, wi hou explici e e ence o he wall. This s a e is he
so-called no mal s a e and co esponds o he he modynamic limi o he sys em [8]. S a ing
om a hyd odynamic desc ip ion, i has ecen ly been shown [9] ha his mac oscopic s a e
exis s in he bulk o a sys em confined be ween a ib a ing wall and a s eady one, a leas
in he limi o small g adien s (and inelas ici y). This was al eady implici in he molecula
dynamics esul s p esen ed in e . [6]. In his pape , we in es iga e he ex ension o he abo e
s a e o highe inelas ici ies by means o he inelas ic Bol zmann equa ion.
c
EDP Sciences
J. J. B ey e al.:Fou ie s a e o a g anula gas 433
The no mal s eady s a e we will conside is qui e peculia . A fi s poin o ealize is ha i
has no analogous in molecula sys ems. In he elas ic limi , he g adien s anish and he s a e
o he sys em becomes he usual he modynamic equilib ium. Fo non- anishing dissipa ion,
he p essu e o he gas and he empe a u e g adien a e uni o m, bu hey canno ake
a bi a y alues. Thei a io is de e mined by he deg ee o inelas ici y o he collisions. This
coupling be ween he g adien o a hyd odynamic field and he alue o ano he (uni o m)
field is cha ac e is ic o g anula gases, and a simila beha iou has been ound o shea ed
sys ems [10, 11]. The in e es o his s a e is no pu ely o mal, since i is he one obse ed
in molecula dynamics simula ions [6], in nume ical solu ions o he Bol zmann equa ion by
means o he di ec Mon e Ca lo simula ion me hod [9], and quali a i ely in expe imen s [12].
Al hough we ha e no been able o explici ly cons uc he solu ion o he Bol zmann
equa ion o he s a e, some gene al ea u es can be discussed. In pa icula , i will be shown
o exhibi an algeb aical ail o la ge eloci ies in he di ec ion o dec easing empe a u e,
indica ing an o e popula ion o pa icles wi h la ge eloci ies, as compa ed wi h he Gaussian
dis ibu ion. This seems o be a qui e gene al ea u e o g anula sys ems [2].
The Bol zmann equa ion o he one-pa icle dis ibu ion unc ion ( , , ) o a dilu e gas
o smoo h inelas ic ha d sphe es (d=3)o disks(d= 2) o mass mand diame e σis
(∂ + ·∇) =σd−1d 1d
σθ(g·
σ)(g·
σ)(α−2b−1−1) ( , , ) ( , 1, ),(1)
whe e g= − 1,
σis a uni ec o along he line o cen e s o he wo colliding pa icles,
θis he Hea iside s ep unc ion, and b−1is an ope a o ans o ming all he eloci ies
and 1 o i s igh in o hei p ecollisional alues defined by momen um conse a ion and
g=g−(1+α)(g·
σ)
σ. Inelas ici y in collisions is cha ac e ized h ough a cons an coefficien
o no mal es i u ion α, in he ange 0 <α≤1. The local numbe densi y n( , ), flow eloci y
u( , ), and g anula empe a u e T( , ) a e gi en by
n=d , nu=d , d
2nT =m
2d V2 , (2)
whe e V( , )= −u. Balance equa ions o hese quan i ies a e ob ained by aking eloci y
momen s in he Bol zmann equa ion (1),
∂ n+∇·(nu)=0,(3)
∂ u+u·∇u+(mn)−1∇·P=0,(4)
∂ T+u·∇T+2(dn)−1(P:∇u+∇·q)+Tζ =0.(5)
In he abo e equa ions he p essu e enso Pand he hea flux qa e he same unc ional o
as o molecula fluids, P=md VV ,q=m
2d V2V , while he cooling a e ζ,which
anishes in he elas ic limi , is gi en by
ζ( , )=mπ d−1
2σd−1(1 −α2)
4dΓd+3
2nT d 1d 2| 1− 2|3 ( , 1, ) ( , 2, ).(6)
We will sea ch o a solu ion s o he Bol zmann equa ion e i ying he ollowing condi-
ions: a) i is ime independen ; b) he e is no mac oscopic flow eloci y; c) he e a e g adien s
only in he di ec ion o he x-axis; d) i scales in he o m
s(x, )=n(x)m
2T(x)d/2
ϕ(ξ),ξ=m
2T(x)1/2
.(7)
434 EUROPHYSICS LETTERS
This solu ion is simila o he one ound by Goldsh ein and Shapi o [13], desc ibing he
e olu ion o a homogeneous eely e ol ing g anula fluid. Because o eq. (2), he unc ion ϕ
mus e i y he condi ions
dξϕ(ξ)=1,dξξϕ(ξ)=0,dξξ2ϕ(ξ)=d
2.(8)
Fo he s a e we a e conside ing, eq. (3) becomes an iden i y, while eq. (4) oge he wi h
eq. (7) imply ha he p essu e enso is uni o m. Finally, om he ene gy balance equa ion
(5) we ge
1
2
dT
dxdξξ2ξxϕ(ξ)=−(1 −α2)πd−1
2σd−1p
4Γ d+3
2dξ1dξ2|ξ1−ξ2|3ϕ(ξ1)ϕ(ξ2),(9)
whe e p=d−1iPii =nT is he cons an p essu e. Since he igh -hand side o his equa ion
does no depend on x, consis ency equi es ha dT(x)
dx=a=cons ,i.e. he empe a u e p ofile
mus be linea in x.
Equa ion (9) es ablishes a ela ionship be ween he empe a u e g adien and p essu e
ha canbeexp essedin he o m
a
pσd−1=I[ϕ]≡−
(1 −α2)πd−1
2
2Γ d+3
2dξ1dξ2|ξ1−ξ2|3ϕ(ξ1)ϕ(ξ2)
dξξ2ξxϕ(ξ).(10)
Fo symme y conside a ions, he dis ibu ion unc ion smus be an e en unc ion o he
eloci y componen s pe pendicula o he x-di ec ion. This p ope y, o cou se, is p ese ed
by he Bol zmann equa ion. Then, in pa icula , he e is no hea flux pe pendicula o he
empe a u e g adien . Subs i u ion o eq. (7) in o eq. (1) and use o eq. (10) leads o
−I[ϕ]ξxϕ(ξ)+1
2ξx
∂
∂ξ·[ξϕ(ξ)]=dξ1d
σθ(χ·
σ)|χ·
σ|(α−2b−1−1)ϕ(ξ)ϕ(ξ1),(11)
whe e χ=ξ−ξ1. The s uc u e o his equa ion is consis en wi h ou assump ion on he
exis ence o a solu ion o he Bol zmann equa ion o he o m (7). Then, ϕand also Idepend
only on he coefficien o es i u ion αand eq. (10) implies ha , o gi en α, p essu e and he
g adien o empe a u e a e no independen , bu hei a io has a fixed alue in he s eady
s a e we a e conside ing. A simila beha iou has been ound o he uni o m shea flow s a e,
o which he empe a u e and he shea a e a e ela ed [10,11].
The componen o he hea flux in he x-di ec ion is
qx=2T(x)
m1/2
pdξξ2ξxϕ(ξ).(12)
Taking in o accoun eq. (10) i ollows ha his flux is p opo ional o he empe a u e g adien
a, namely i can be w i en in he o m
qx=−Φ(α)κ0a, (13)
whe e Φ is an unknown unc ion o he coefficien o es i u ion and κ0is he Bol zmann
elas ic hea conduc i i y. The e o e, we ha e ob ained ha he hea flux is linea ly coupled
o he empe a u e g adien . An app oxima ed exp ession o Φ(α) can be ob ained by means
J. J. B ey e al.:Fou ie s a e o a g anula gas 435
o he Chapman-Enskog expansion. Using he esul s ob ained in e s. [4] and [14] i is ound
ha
Φ(α)=κ∗(α)−µ∗(α),(14)
whe e κ∗(α)andµ∗(α) a e unc ions only o he es i u ion coefficien and hei explici
exp essions can be ound in [9,14].
I is wo h men ioning ha o molecula sys ems (α= 1) he Bol zmann equa ion also
has an exac solu ion o Maxwell molecules, ep esen ing a s eady s a e in which he Fou ie
law is obeyed o a bi a y empe a u e g adien [15].
The e is no eason o expec ha he dis ibu ion unc ion shas a simple dependence on
he empe a u e g adien . E en mo e, calcula ions based on model kine ic equa ions o he
nonlinea Bol zmann equa ion and applied o analogous s a es [16] seem o indica e ha he
dis ibu ion unc ion has a qui e complica ed o m. Ne e heless, om he esul s ob ained in
e s. [4] and [14], he exp ession o slinea ized in a,i.e. in he fi s Chapman-Enskog o de ,
and e alua ed in he fi s Sonine app oxima ion can be w i en down. We will concen a e on
he ma ginal eloci y dis ibu ion ϕx(ξx)=dξ⊥ϕ(ξ), whe e he in eg a ion is ca ied ou
o e he d−1 componen s o ξpe pendicula o he x-axis. Fo his unc ion i is ound ha
ϕx(ξx)=π−1/2e−ξ2
x1−4d2ζ∗(κ∗−µ∗)
(d+ 2)(d−1) 1/2ξ2
x−3
2ξx.(15)
Upon de i ing his exp ession, i has been assumed, wi hou loss o gene ali y, ha ∂T/∂x =
a>0. O he wise, ξxmus be changed in o −ξx. A ac o o a/pσd−1has been elimina ed
by using he Na ie -S okes equa ion o he empe a u e (see also eq. (10)), so ha eq. (15)
is w i en in a o m which is consis en wi h eq. (11). We s ess again ha eq. (15) is, in
p inciple, use ul only in he small-g adien limi , ha o he s a e we a e dealing wi h is
equi alen o small inelas ici y. Mo eo e , i is es ic ed o alues o |ξx|o he o de o
uni y. In fig. 1 we compa e eq. (15) wi h nume ical esul s ob ained om di ec Mon e Ca lo
simula ion o he Bol zmann equa ion o a ib a ed sys em [9, 17] o d= 2. Da a om wo
diffe en posi ions in he bulk o he sys em ha e been plo ed (ci cles and squa es). The
ela i e diffe ence o empe a u es be ween bo h posi ions is abou 30%. The da a o e lap
consis en ly wi h he scaling law in eq. (7). The e is a qui e good ag eemen wi h eq. (15) o
small eloci ies and low dissipa ion, he de ia ions becoming mo e significan as he alue o
αdec eases.
I is possible o ob ain some in o ma ion abou he asymp o ic o m o high eloci ies
o he solu ion o he Bol zmann equa ion by means o a me hod in oduced by K ook and
Wu [18] in he elas ic case, and employed o homogeneous g anula sys ems by Esipo and
P¨oschel [19] and by an Noije and E ns [2]. F om eq. (11) one ob ains
−I[ϕ]3
2ξxϕx(ξx)+1
2ξ2
x
∂
∂ξx
ϕx(ξx)=J(+)
x[ϕ]−J(−)
x[ϕ].(16)
He e J(+)
xand J(−)
xdeno e he con ibu ions om he gain and loss e ms o he Bol zmann
collision ope a o , espec i ely, i.e.
J(−)
x=dξ⊥dξ1d
σθ(χ·
σ)|χ·
σ|ϕ(ξ)ϕ(ξ1) (17)
and simila ly J(+)
x. Again, we assume ha a>0, so ha pa icles a i ing a a gi en posi ion
wi h posi i e ξxcome om coole egions. The e o e, pa icles wi h la ge and posi i e ξx

436 EUROPHYSICS LETTERS
321 0123
ξx
0.0
0.2
0.4
0.6
0.8
ϕx
α=0.99
321 0123
ξx
0.0
0.2
0.4
0.6
0.8
ϕx
α=0.975
Fig. 1 – Ma ginal eloci y dis ibu ion in he di ec ion o he empe a u e g adien o α=0.99 and
α=0.975 in a wo-dimensional ib a ed sys em. The con inuous line is he heo e ical p edic ion
gi en in he main ex , he symbols a e om di ec Mon e Ca lo simula ion o he Bol zmann equa ion
and co espond o wo diffe en posi ions (squa es and ci cles) in he bulk, and he do ed line is he
Gaussian, plo ed o e e ence.
a a gi en loca ion a e mos ly gene a ed as an effec o collisions wi h o he pa icles in he
same egion. On he o he hand, pa icles wi h la ge |ξx|bu ξx<0 migh come om egions
co esponding o highe empe a u es and he p obabili y hey a e gene a ed as an effec o
collisions a he gi en loca ion is low. Fo he la e pa icles, i.e. hose wi h ξx<0and
|ξx|1, he gain e m is negligible as compa ed wi h he loss one. The dominan con ibu ion
o he collision in eg al co esponds o collisions whe e he eloci y o pa icle 1 is ypically
in he he mal egion, so ha |χ|can be eplaced by |ξ|. Since he same kind o easoning
can be applied o he in eg al o e he no mal componen ξ⊥,|ξ|can in u n be eplaced by
|ξx|. Equa ion (17) can be app oxima ed by
J(−)
x≈πd−1
2
Γd+1
2|ξx|ϕx(ξx),(18)
so ha eq. (16) can be simplified o
−I[ϕ]3
2ξxϕx(ξx)+1
2ξ2
x
∂
∂ξx
ϕx(ξx)=πd−1
2
Γd+1
2ξxϕx(ξx),(19)
whose solu ion has he o m
ϕx(ξx)=A|ξx|−b(α),(20)
whe e Ais an unde e mined in eg a ion cons an and
b(α)= 2πd−1
2
Γd+1
2I+3.(21)
No e ha o a>0, i is I[ϕ]>0, as is seen om eq. (10). In he same fi s Sonine
app oxima ion used o de i e eq. (14), one ob ains
I=32(d−1)πd−1(1 −α2)
d(d+2)
2Γ(d/2)2(κ∗−µ∗)1/2
.(22)
J. J. B ey e al.:Fou ie s a e o a g anula gas 437
Then, we ha e ob ained ha o la ge componen o he eloci y in he opposi e di ec ion o
he inc ease o he empe a u e, he e is an algeb aic ail in he ma ginal eloci y dis ibu ion.
This co esponds o an o e popula ion o hose eloci ies when compa ed wi h he Gaussian
dis ibu ion. As al eady men ioned, his seems o be a qui e gene al ea u e o sys ems o
pa icles wi h dissipa i e in e ac ions.
Since in mos o he p e ious heo e ical analysis o eloci y dis ibu ion unc ions o
g anula gases, an exponen ial a he han algeb aical decay has been ound [2], i could be
a gued whe he he esul p esen ed he e is an a i ac o he asymp o ic analysis we ha e
ca ied ou . Ne e heless, a ca e ul inspec ion o each o he s eps leading o eq. (20) shows
ha i ep esen s in any case he as es possible decay o he ma ginal eloci y dis ibu ion
o la ge nega i e alues o ξx. The e o e, in pa icula , such decay canno be exponen ial. O
cou se, he e is no con adic ion he e, as he p e ious s udies co espond o diffe en physical
si ua ions, in which he sys em was ei he e ol ing eely o d i en by a s ochas ic o ce ac ing
on all he pa icles. Mo eo e , he algeb aical decay is compa ible wi h he exis ence o he
fi s momen s o he eloci y dis ibu ion, since he exponen o he powe law is always la ge
ha h ee and i is expec ed o g ow e y as as αapp oaches uni y (see eq. (10)). Fo la ge
componen o he eloci y in he di ec ion o posi i e g adien o he empe a u e, i.e. ξx1,
he dominan pa o he collision in eg al is he gain e m J(+)
x, and he easoning de eloped
abo e does no apply.
∗∗∗
This esea ch was pa ially suppo ed by G an No. PB98-1124 om he Di ecci´on Gene al
de In es igaci´on Cien ´ıfica y T´ecnica (Spain).
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