Hea lux in a ib a ed g anula gas: he di usi e hea
conduc i i y coe icien
J. Ja ie B ey∗and M.J. Ruiz-Mon e o∗
∗Física Teó ica. Facul ad de Física. Apdo. de Co eos 1065. 41080-Se illa. Spain
Abs ac . The anspo coe icien coupling hea lux and densi y g adien in a g anula gas is measu ed by aking ad an age
o he exis ence o a minimum in he empe a u e p o ile o an open ib a ed g anula medium. This empe a u e in e sion is
closely ela ed o he exis ence o he new anspo coe icien . Pa icle simula ions using he Di ec Simula ion Mon e Ca lo
me hod will be used o compu e he anspo coe icien , and he esul s will be compa ed wi h heo e ical p edic ions de i ed
om he Bol zmann equa ion. Finally, he accu acy o a bounda y condi ion equi ing he hyd odynamic hea lux o anish
o in ini e heigh s will be discussed.
INTRODUCTION
S eady s a es o g anula ma e ials a e ob ained when ene gy is supplied o he sys em in o de o compensa e he
ene gy loss in collisions. One o he easies ways o add ene gy o a sys em in expe imen s is o do so h ough a
ib a ing wall. I he ib a ion is s ong enough, he sys em will be luidized, and one can expec i s beha io o
be desc ibed by he ex ension o he Na ie -S okes equa ions o g anula sys ems. In his wo k, we will use he
hyd odynamical desc ip ion o s udy he s eady s a e o an open, ib a ed, g anula sys em in p esence o g a i y.
The g anula luid will be modelled as a sys em o inelas ic ha d pa icles, whose collisions a e cha ac e ized by a
cons an coe icien o no mal es i u ion
α
. Besides, we will be in e es ed in he dilu e limi , when he Bol zmann
equa ion applies.
A dis inc i e ea u e o g anula gases as compa ed o molecula (elas ic) ones, is ha he exp ession o he hea
lux has o be gene alized by including a new e m coupling he hea lux and he densi y g adien . This implies he
in oduc ion o a new coe icien , he di usi e hea conduc i i y
µ
, ha anishes in he elas ic limi . This coupling has
been de i ed by kine ic heo y me hods [1, 2, 3], and i s consequences con i med in compu e simula ions [4, 5, 6].
Fo he pa icula case o a ib a ed g anula gas in he p esence o g a i y, his e m implies a peculia beha io o
he empe a u e p o ile, ha inc eases wi h heigh a e a minimum. The exis ence o he minimum was i s de i ed
om he hyd odynamic equa ions [5], and was con i med by compu e simula ions [5, 7, 8], and also in expe imen s
[9]. He e, we will show ha he alue o he di usi e hea conduc i i y
µ
can be ob ained om he beha io o he
sys em a he empe a u e minimum. Then, he alue o he anspo coe icien will be compa ed wi h he heo e ical
p edic ion om he Bol zmann equa ion de i ed in [2]. Finally, he bounda y condi ions o be used when sol ing he
hyd odynamic equa ions will be also discussed.
HYDRODYNAMIC DESCRIPTION
Le us conside a sys em o Ninelas ic ha d sphe es (d=3) o disks (d=2) o mass mand diame e
σ
, in p esence o a
g a i a ional ield g=−gez, whe e gis a posi i e cons an and eza uni ec o in he Zdi ec ion. In he hyd odynamic
desc ip ion, he s a e o he sys em is comple ely speci ied by he local numbe o pa icles densi y, n( , ), he eloci y
low u( , ), and he empe a u e, T( , ). Fo a dilu e gas, he e olu ion o hese quan i ies is gi en by he ex ension o
he inelas ic case o he Na ie -S okes equa ions [2, 10],
∂
n
∂
+∇·(nu) = 0,(1)
809
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S anda d Fo m 298 (Re . 8-98)
P esc ibed by ANSI S d Z39-18
∂
u
∂
+u·∇u+1
nm∇·P−g=0,(2)
∂
T
∂
+u·∇T+2
dnkB
[P:∇u+∇·q]+T
ζ
=0.(3)
He e, kBis he Bol zmann cons an , Pis he p essu e enso ,
P=pI−
η
(∇u)+(∇u)+−2
dI∇·u,(4)
whe e p=nkBTis he hyd os a ic p essu e, I he iden i y enso , and
η
he shea iscosi y coe icien . qappea ing
in Eq. (3) is he hea lux,
q=−
κ
∇T−
µ
∇n,(5)
wi h
κ
he hea conduc i i y coe icien , and
µ
he di usi e hea conduc i i y. Finally,
ζ
is he cooling a e associa ed
o he ene gy dissipa ion in collisions. The exp ession o hese coe icien s eads:
η
=
η
∗(
α
)
η
0(T),
κ
=
κ
∗(
α
)
κ
0(T),
µ
=
µ
∗(
α
)
µ
0(T),(6)
ζ
≃
ζ
(0)=
ζ
∗(
α
)p
η
0
,(7)
wi h
η
0,
κ
0 he Bol zmann elas ic alues o he iscosi y and hea conduc i i y,
µ
0=T
κ
0/n, while
η
∗(
α
),
κ
∗(
α
),
µ
∗(
α
), and
ζ
∗(
α
)a e dimensionless unc ions o he coe icien o es i u ion. Al hough hei exp essions will no be
gi en he e ( hey can be ound in [2, 10]), i is impo an o emembe ha in he elas ic limi
α
→1,
η
∗and
κ
∗ end o
uni y, while
µ
∗and
ζ
∗ anish. Besides, bo h
η
0and
κ
0a e p opo ional o T1/2.
Le us conside ha ene gy is supplied o he sys em h ough a ib a ing wall o size Sloca ed a z=0. Al hough
he de ails o he wall mo emen a e no e y impo an he e, we will conside ha he ib a ion ampli ude is small
enough as o app oxima e he posi ion o he wall as ixed. Also, he wall mo es wi h a saw oo h p o ile, so pa icles
ind i mo ing upwa ds wi h a cha ac e is ic eloci y. When he ene gy supplied by he wall compensa es he one los
in collisions, he sys em eaches a s eady s a e. Besides, and because o he symme y o he p oblem, we can expec
ha , once in his s a e, he e will be g adien s only in he Zdi ec ion. When Eqs. (1)–(3) a e pa icula ized o his
s a e, hey ake he o m
∂
p
∂
z=−nmg,(8)
2
dnkB
dqz
dz +T
ζ
(0)=0.(9)
Besides, Eq. (8) implies ha he hea lux in his case is
qz(z) = −(
κ
∗−
µ
∗)
κ
0
dT
dz +
µ
∗
κ
0
mg
kB
.(10)
In o de o sol e hese equa ions, i is con enien o in oduce he new dimensionless leng h scale
ξ
as:
ξ
=√aZ∞
z
dz01
λ
(z0),(11)
whe e
λ
(z)is he local mean ee pa h o ha d disks o sphe es,
λ
(z) = [Cn
σ
d−1]−1, wi h C=2√2 o d=2 and
C=
π
√2 o d=3. Le us no ice ha , when z→∞,
ξ
→0, while
ξ
akes i s maximum alue,
ξ
0=√aC
σ
d−1Nz, wi h
Nz=N/S, a z=0. The coe icien aappea ing in Eq. (11) is a unc ion o he coe icien o es i u ion and eads
a(
α
) = 32(d−1)
π
d−1
C2(d+2)3Γ(d/2)2
ζ
∗(
α
)
κ
∗(
α
)−
µ
∗(
α
),(12)
anishing, he e o e, in he elas ic limi . In e ms o he new scale, he solu ion o he Na ie -S okes equa ions is [5]:
T1/2(
ξ
) =
ξ
−
ν
[AI
ν
(
ξ
)+BK
ν
(
ξ
)],(13)
810
0 20 40 60
z
0.0
0.5
1.0
1.5
T/TA
n/nh
FIGURE 1. Tempe a u e and densi y p o iles o a sys em o ha d sphe es wi h
α
=0.925,
ξ
0=1.92. The symbols a e om
he simula ions, while he lines a e he solu ion o he hyd odynamic equa ions, wi h he a bi a y cons an s de e mined om he
empe a u e minimum.
n(
ξ
) = mg
ξ
1+2
ν
CkB
σ
d−1pa(
α
)[AI
ν
(
ξ
)+BK
ν
(
ξ
)]2,(14)
whe e I
ν
and K
ν
a e he modi ied Bessel unc ions o i s and second kind, and Aand Ba e cons an s o be de e mined
om he bounda y condi ions. The pa ame e
ν
is
ν
(
α
) =
µ
∗(
α
)
4[
κ
∗(
α
)−
µ
∗(
α
)] >0,(15)
In he limi
ξ
→0 (z→∞)I
ν
→0, while K
ν
→∞[11]. Ne e heless, his does no imply ha he cons an Bhas o be
aken iden ically equal o ze o, as he e is no hing unphysical in he ac ha Tdi e ges as a as he densi y dec eases
as enough as o gua an ee ha he local kine ic ene gy densi y goes o ze o in ha limi . Besides, i is clea ha
he hyd odynamic desc ip ion will no be alid o e y la ge heigh s, as he densi y will ha e decayed o e y small
alues, and he local Knudsen numbe will be e y small. Then, we will keep he Bcons an di e en om ze o.
The p esence o he e m p opo ional o K
ν
in he empe a u e p o ile implies ha i has a minimum loca ed a
ξ
=
ξ
mgi en by
AI
ν
+1(
ξ
m)−BK
ν
+1(
ξ
m) = 0,(16)
and he empe a u e Tma he minimum is
T1/2
m=
ξ
−
ν
m[AI
ν
(
ξ
m)+BK
ν
(
ξ
m)].(17)
Then, i hyd odynamics is alid in he icini y o he empe a u e minimum, he cons an s Aand Bcan be de e mined
om he measu ed empe a u e p o iles. I has been also shown [5] ha he densi y has a maximum a
ξ
=
ξ
n ha is
app oxima ely gi en by he solu ion o he equa ion:
I
ν
(
ξ
n)−2
ξ
nI
ν
+1(
ξ
n) = 0,(18)
which can be sol ed nume ically o each alue o
α
. Le us jus commen ha
ξ
n akes alues o he o de o uni y.
O cou se, in o de o obse e he densi y maximum in an expe imen he numbe o pa icles in he sys em has o be
la ge enough so ha
ξ
0>
ξ
n. I his condi ion is no ul illed, he densi y will decay mono onically wi h heigh .
In o de o check he abo e hyd odynamic desc ip ion, compu e simula ions o wo and h ee dimensional sys ems
by using he Di ec Simula ion Mon e Ca lo me hod (DSMC) [12] ha e been pe o med. This me hod is pa icula ly
sui ed o his p oblem as we a e in e es ed in he low densi y limi , and as i allows o exploi he symme y o he
sys em in he ans e sal di ec ion. In Fig. 1 he s eady densi y and empe a u e p o iles o a sys em o ha d sphe es
ha e been plo ed. The densi y has been scaled wi h he ini ial, homogeneous densi y, while he empe a u e is scaled
wi h some a bi a y alue. Space is measu ed in uni s o he ini ial, homogeneous, mean ee pa h. The alues o he
811
0.70 0.80 0.90 1.00
α
0.0
0.2
0.4
0.6
µ∗
FIGURE 2. Reduced anspo coe icien
µ
∗as a unc ion o
α
o a sys em o ha d sphe es. The solid line is he heo e ical
p edic ion de i ed in [2], and he ci cles he esul s om he simula ion o a ib a ed sys em. The squa es a e also simula ion esul s
bu using a G een-Kubo exp ession o
µ
.
pa ame e s o he simula ions we e
α
=0.925,
ξ
0=1.92 . The symbols a e he esul s o he simula ions: as p edic ed,
he densi y shows a maximum, while he empe a u e exhibi s a minimum, inc easing om he e on. The con inuous
lines a e he heo e ical p edic ion, wi h Aand Bde e mined om he empe a u e minimum. The ag eemen is e y
good, con i ming he alidi y o hyd odynamics in he empe a u e minimum egion. The same quali a i e esul s a e
achie ed o o he alues o he pa ame e s bo h in he wo and h ee dimensional cases, as a as he coe icien o
es i u ion is no oo low. Due o an in insic coupling be ween g adien s and inelas ici y in his s eady s a e, e aining
only up o he Na ie -S okes o de in he Chapman-Enskog expansion may be no enough o small alues o
α
.
THE HEAT FLUX
In he s eady s a e o a g anula ma e ial, he hea lux is gi en by Eq. (11). A he empe a u e minimum, as he
de i a i e o T anishes, we ha e
qz(zm) =
µ
∗
κ
0(Tm)mg
kB
.(19)
Then, he compu a ion o he hea lux a he empe a u e minimum p o ides a di ec measu emen o he scaled
di usi e hea conduc i i y coe icien ,
µ
∗. I mus be poin ed ou ha Eq. (19) only equi es he alidi y o he
hyd odynamic desc ip ion, and no addi ional condi ion has o be in oduced. In Fig. 2 we ha e plo ed he coe icien
µ
∗as a unc ion o
α
o a sys em o ha d sphe es. The solid line is he heo e ical p edic ion de i ed in [2] by using he
Chpaman-Enskog expansion om he Bol zmann equa ion. The ci cles a e he esul s o he simula ion using Eq. (19).
The e o ba s a e ob ained by gi ing di e en alues o he pa ame e
ξ
0and he eloci y o he ib a ing wall. The
ag eemen be ween heo y and simula ion is qui e good, e en a he lowes alues o
α
in es iga ed. Ne e heless,
i mus be said ha , o hose alues, he shape o he p o iles begin o show disc epancies om he heo e ical
p edic ion de i ed he e. This is he eason o he la ge e o ba s o he lowes alues o
α
, and o no ha ing
conside ed smalle alues o his pa ame e . Finally, we ha e also included in he igu e he esul s o independen
DSMC simula ions whe e
µ
∗was compu ed by means o G een-Kubo exp essions de i ed in Re . [13] (squa es). The
ag eemen is again e y good. I could be a gued ha i is no su p ising o ob ain a good ag eemen be ween he esul s
o he DSMC simula ion and a heo e ical desc ip ion de i ed om he Bol zmann equa ion. Ne e heless, i mus be
emembe ed he he heo e ical de i a ion equi es gi en hypo hesis and app oxima ions ( alidi y o hyd odynamics,
g adien expansions, Sonine expansion) ha a e no assumed a all by he simula ion me hod.
I is impo an o s ess ha he ac ha he hea lux does no anish a he empe a u e minimum is a di ec p oo
o he exis ence o he coupling be ween hea lux and densi y g adien . Besides, Fig. 2 shows ha , al hough
µ
∗→0
in he elas ic limi , i s con ibu ion canno be neglec ed as
α
goes beyond he quasi-elas ic limi .
812

0.70 0.80 0.90 1.00
α
0.0
0.5
1.0
1.5
ξm,n
FIGURE 3. Posi ion o he empe a u e minimum
ξ
mand o he densi y maximum
ξ
nin he h ee dimensional case. The lines a e
he heo e ical p edic ion (solid line o
ξ
m, do -dashed line o
ξ
n. The symbols a e he esul s o he DSMC simula ions.
Eq. (19) p o ides he hea lux a he empe a u e minimum. A gene al exp ession o qz alid a any posi ion can
be ob ained when he solu ion gi en by Eq. (14) is subs i u ed in Eq. (11), and i eads
qz=A(
κ
∗−
µ
∗)2mg
κ
0
kBT1/2I
ν
−1(
ξ
)−B
AK
ν
−1(
ξ
)
ξ
1−
ν
.(20)
Le us conside now he limi z→∞o , equi alen ly,
ξ
→0. Taking in o accoun he asymp o ic beha io o he
modi ied Bessel unc ions, i ollows ha , o e y la ge heigh s is gi en by:
qz=A(
κ
∗−
µ
∗)2mg
κ
0
kBT1/22−
ν
2
Γ(
ν
)−Γ(1−
ν
)B
A.(21)
I we equi e ha he hea lux anishes o e y la ge heigh s, we ge he ela ion
B
A=2
Γ(
ν
)Γ(1−
ν
),(22)
i.e. he a io o he wo cons an s is gi en by a unc ion o
α
alone. Taking in o accoun he beha io o he Γ unc ion,
µ
∗=0 (i.e.
ν
=0) implies B=0, and he empe a u e would be cons an o la ge heigh s. I we in oduce he abo e
ela ion in o he equa ion ha de e mines he posi ion o he empe a u e minimum, Eq. (16), we ge a closed equa ion
o
ξ
m:
I
ν
+1(
ξ
m)−2
Γ(
ν
)Γ(1−
ν
)K
ν
+1(
ξ
m) = 0.(23)
Then, he posi ion o he empe a u e minimum in a ib a ed sys em in he scaled space a iable is a unc ion only o
he coe icien o no mal es i u ion, independen o he o he ele an pa ame e s o he sys em (numbe o pa icles,
g a i y and ib a ion eloci y).
In Fig. 3 he posi ion o he empe a u e minimum
ξ
min he h ee dimensional case is plo ed as a unc ion o
α
.
The symbols a e he esul s o he DSMC simula ion, while he solid line is he solu ion o Eq. (23). The ag eemen
is e y good, suppo ing he use o he abo e men ioned bounda y condi ion o sol e he hyd odynamic equa ions. I
mus be no iced ha he alidi y o his bounda y condi ion is no a all clea , as i is imposed in he e y la ge heigh s
egion, i.e., once he densi y has decayed o e y small alues and he hyd odynamic desc ip ion is no alid. O cou se,
he hea lux mus anish in ha egion, bu he ques ion is whe he his can be ansla ed in o an e ec i e bounda y
condi ion o he hyd odynamic equa ions. Fo ins ance, a anishing hea lux implies a cons an empe a u e g adien
whose alue is di ec ly ela ed o
ν
. Bu iden i ying he asymp o ic egion whe e his linea beha io is achie ed is
813
e y di icul in p ac ical applica ions, because o he ailu e o hyd odynamics o desc ibe he uppe egion o he
sys em [6]. He e we ha e shown ha he condi ion o anishing lux a la ge heigh s ansla es in o a condi ion inside
he hyd odynamic egion, so i can be clea ly es ed.
We ha e also included in Fig. 3 he posi ion o he densi y maximum,
ξ
n. The do -dashed line is he heo e ical
p edic ion gi en by he solu ion o Eq. (18), while he iangles a e he esul s o he simula ions. The ag eemen is
again qui e good, al hough some disc epancies appea o he lowes alues o
α
s udied. The eason o his migh
be ha Eq. (18) is no exac [5], and is in ac ob ained o
ν
1, while o
α
=0.7,
ν
∼0.14. Ne e heless, i mus
be no iced he weak dependence on
α
o
ξ
nas compa ed o
ξ
m. In ac , o he alues o he coe icien o es i u ion
conside ed in he simula ions
ξ
n∼1.
In conclusion, hyd odynamics p o ides a e y use ul ool o s udy non-homogeneous s eady s a es o g anula
sys ems. Ne e heless, his hyd odynamic desc ip ion has dis inc i e cha ac e is ics ha canno be guessed om he
one o molecula luids. In pa icula , a new anspo coe icien , he di usi e hea conduc i i y, has o be in oduced.
In his wo k we ha e shown ha his new anspo coe icien has ele an consequences in he hyd odynamic p o iles,
so i canno be neglec ed in he desc ip ion o g anula lows.
ACKNOWLEDGMENTS
We acknowledge inancial suppo om he Minis e io de Ciencia y Tecnología (Spain) h ough G an No. BFM2002-
00303 (pa ially inanced by FEDER unds).
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