Hyd odynamics o an open ib a ed g anula sys em
J. Ja ie B ey, M. J. Ruiz-Mon e o, and F. Mo eno
Fı
´sica Teo
´ ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, 41080 Se illa, Spain
共Recei ed 17 Janua y 2001; published 21 May 2001兲
Using he hyd odynamic desc ip ion and molecula dynamics simula ions, he s eady s a e o a luidized
g anula sys em in he p esence o g a i y is s udied. Fo an open sys em, he densi y p o ile exhibi s a
maximum, while he empe a u e p o ile goes h ough a minimum a high al i ude, beyond ha he empe a u e
inc eases wi h he heigh . The exis ence o he minimum is explained by he hyd odynamic equa ions i he
p esence o a collisionless bounda y laye is aken in o accoun . The ene gy dissipa ed by in e pa icle colli-
sions is also compu ed. A good ag eemen is ound be ween heo y and simula ion. The ela ionship wi h
p e ious wo ks is discussed.
DOI: 10.1103/PhysRe E.63.061305 PACS numbe 共s兲: 45.70.Mg, 81.05.Rm, 51.10.⫹y, 05.20.Dd
I. INTRODUCTION
The aim o his pape is o in es iga e he hyd odynamic
desc ip ion o a g anula luid in a g a i a ional ield when
ene gy is con inuously p o ided o he sys em om below
h ough a ib a ing pla e. Expe imen s, compu e simula-
ions, and also heo e ical s udies ha e e ealed he exis ence
o a s eady luidized s a e unde hese condi ions 关1–8兴. One
o he main conclusions o all hese s udies is ha ib o lu-
idized g anula media show a luidlike beha io , in he sense
ha hei s a e seems o be well cha ac e ized by he p o iles
o he hyd odynamic ields. O cou se, a di e en ques ion is
whe he hyd odynamic equa ions, de i ed as an ex ension o
hose o o dina y luids, p o ide a quan i a i ely, o a leas
quali a i ely, accu a e desc ip ion o wha is obse ed.
In a p e ious wo k 关9兴, we ha e analyzed a ib a ed
g anula luid in absence o ex e nal ields, paying special
a en ion o he bulk beha io o he luid, a away om he
bounda ies. The sys em was shown o exhibi a no mal be-
ha io , independen o he de ails o he bounda ies, cha ac-
e ized by a closed cons i u i e ela ionship be ween he uni-
o m p essu e and he empe a u e g adien . The idea he e is
o ex end he abo e s udy o sys ems submi ed o a uni o m
ex e nal ield. Such an ex ension is no a all i ial, since he
ield gene a es spa ial g adien s ha couple in an in ica e
way o hose associa ed wi h he inelas ici y o he sys em.
Mos o he expe imen al s udies deal wi h open sys ems,
i.e., o mally wi h a sys em o in ini e heigh , and ha is he
si ua ion we will ocus on he e. I could be expec ed ha he
beha io o he sys em becomes simple a away om he
ib a ing su ace a he bo om. Ne e heless, he p esence o
a ee su ace in oduces some addi ional complica ions in
he desc ip ion o he luid. As he densi y dec eases, pa -
icles end o mo e in a ballis ic way es ained by he g a i-
a ional o ce. This was al eady no ed by Ha 关1兴, who con-
cluded ha a hyd odynamiclike desc ip ion canno accoun
o hose e ec s.
In 1991, Clemen and Rajchenbach 关2兴ca ied ou an in-
e es ing expe imen al s udy o a luidized wo-dimensional
e ical g anula sys em. They measu ed he hyd odynamic
p o iles and es ablished a se ies o impo an obse a ions.
The poin we wan o emphasize is p esen ed in Fig. 4 o
hei pape , whe e i is obse ed ha he g anula empe a-
u e inc eases o la ge enough heigh s. Un o una ely, he
au ho s do no commen on he o igin and ele ance o his
inding. The same e ec has been ound again in expe i-
men s and in molecula dynamics simula ions by Helal e al.
关10兴. The empe a u e p o ile p esen s a minimum a a high
al i ude, beyond ha i is ound o inc ease. In o de o ex-
plain his beha io , he au ho s sugges a model ha in ol es
wo coupled di e en ial equa ions o he packing ac ion
and he empe a u e. These equa ions mus be nume ically
sol ed by using bounda y condi ions de e mined om he
simula ions. Al hough he e is a good quali a i e ag eemen
wi h he expe imen al and simula ion esul s, he physical
o igin o he ise in empe a u e as well as i s compa ibili y
wi h a hyd odynamic desc ip ion o he sys em is no clea .
In pa icula , he con inuous app oach used by Helal e al.
does no include a densi y dependen hea lux e m, ha
plays a ele an ole in g anula sys ems 关11,12兴. A de ailed
discussion o hese ques ions will be one o he main poin s
o be add essed in he ollowing sec ions.
The mos di ec implica ion o inelas ici y in collisions is
he dissipa ion o ene gy in he sys em. The balance be ween
he ene gy dissipa ed and he ene gy supplied h ough he
ib a ing wall is o en used o de i e scaling laws o ib o -
luidized g anula ma e ials, as well as bounda y condi ions o
sol e he hyd odynamic equa ions. Wa e al. 关13兴modeled
he ib a ed g anula sys em unde g a i y as an iso he mal
luid, wi h all he pa icles ha ing he same eloci y. La e
on, Kuma an 关14兴, using a kine ic heo y desc ip ion, ound
ha he eloci y dis ibu ion unc ion is a Maxwell-
Bol zmann dis ibu ion in he limi o small inelas ici y. His
exp ession o he dissipa ed powe Donly di e s om he
esul de i ed by Wa e al. 关13兴by a cons an . Ex ensi e
molecula dynamics simula ions ca ied ou by McNama a
and Luding 关5兴showed ha he p edic ion in Re . 关14兴 i ed
be e he simula ion da a han he exp ession in Re . 关13兴,
al hough signi ican disc epancies we e ound. In pa icula ,
he simula ion alues o Dindica ed a dependence on he
numbe o pa icles in he sys em ha was no accoun ed o
by he heo y. T ying o unde s and his dependence is one o
he goals o he p esen wo k.
This pape is o ganized as ollows. In Sec. II, he Na ie -
S okes-like hyd odynamic equa ions o a g anula gas a e
PHYSICAL REVIEW E, VOLUME 63, 061305
1063-651X/2001/63共6兲/061305共10兲/$20.00 ©2001 The Ame ican Physical Socie y63 061305-1
sho ly e iewed and pa icula ized o he s eady s a e o a
ib a ed sys em unde he in luence o an homogeneous ex-
e nal o ce. By in oducing an app op ia e scaling o he
space coo dina e in he di ec ion o he ield, explici exp es-
sions o he hyd odynamic p o iles a e ob ained. They in-
ol e wo cons an s ha mus be de e mined om he bound-
a y condi ions. The limi o an in ini ely high open sys em is
conside ed in Sec. III. This limi does no imply ha any o
he wo cons an s appea ing in he gene al exp ession o he
empe a u e p o ile mus anish, con a y o wha has been
es ablished in some p e ious wo ks. This is due o he p es-
ence o a ee-molecule bounda y laye in he uppe egion o
he g anula gas, so ha he alidi y o he hyd odynamic
equa ions canno be ex apola ed up o an in ini e heigh , o
which a di e gen beha io o one o he con ibu ions o he
heo e ical p edic ion o he empe a u e p o ile would show
up.A ele an consequence o he abo e is ha he empe a-
u e p o ile exhibi s a minimum, becoming an inc easing
unc ion o la ge enough heigh s. Mo eo e , by using mo-
lecula dynamics simula ions, we ha e e i ied ha he e-
gion o inc easing empe a u e is accu a ely desc ibed by he
hyd odynamic equa ions. In ac , he alues o he minimum
o he empe a u e and i s posi ion p o ide enough in o ma-
ion o build up he hyd odynamic p o iles in he bulk o he
sys em, as discussed in he las pa o Sec. III. The o m o
he p o iles o la ge heigh s, bu s ill in he hyd odynamic
egime, is analyzed in Sec. IV.
Sec ion V is de o ed o he s udy o he dissipa ion in
collisions, again by means o he hyd odynamic desc ip ion.
The esul s a e compa ed wi h p e ious wo ks as well as
wi h molecula dynamics simula ions. The o igin o he dis-
c epancies be ween he se e al heo ies is cla i ied, showing,
o ins ance, ha he exp ession o Dde i ed by Kuma an
关14兴co esponds o he low inelas ici y and small sys em
limi s o he mo e gene al o m de i ed he e. Finally, he
main conclusions a e summa ized in Sec. VI.
II. HYDRODYNAMIC EQUATIONS
We conside a g anula gas composed by smoo h inelas ic
ha d sphe es (d⫽3) o disks (d⫽2) o mass mand diame e
␴
, in p esence o a uni o m ex e nal o ce . The pa icles
collide wi h a cons an coe icien o no mal es i u ion
␣
.In
he hyd odynamic desc ip ion, i is assumed ha he s a e o
he sys em is cha ac e ized by he local numbe densi y
n( , ), eloci y low u( , ), and empe a u e T( , )关1,15兴.
Fo a dilu e gas, he ime e olu ion o hese ields is gi en by
he equa ions 关16,17兴
⳵
n⫹“•共nu兲⫽0, 共1兲
⳵
ui⫹u•“ui⫹共mn兲⫺1ⵜip⫺共nm兲⫺1ⵜj
共2兲
⫻
冋
␩
冉
ⵜiuj⫹ⵜjui⫺2
d
␦
ij“•u
冊
册
⫺m⫺1 i⫽0,
⳵
T⫹u•“T⫹2共dnkB兲⫺1p“•u⫺2共dnkB兲⫺1ⵜiuj
⫻
冋
␩
冉
ⵜiuj⫹ⵜjui⫺2
d
␦
ij“•u
冊
册
⫺2共dnkB兲⫺1“•共
␬
“T⫹
␮
“n兲⫹T
␨
(0)⫽0. 共3兲
In he abo e equa ions, p⫽nkBTis he p essu e, kB he Bol -
zmann cons an ,
␩
he shea iscosi y coe icien ,
␬
he hea
conduc i i y coe icien , and
␮
a new anspo coe icien
ha has no analogous in he elas ic limi . Al hough kBis
aken as uni y in mos o he li e a u e o luidized g anula
sys ems, we will keep i he e jus o s ess he analogy wi h
molecula luids. Finally,
␨
(0) is he cooling a e associa ed o
he ene gy dissipa ion in collisions. These quan i ies ha e he
o m
␩
⫽
␩
*共
␣
兲
␩
0共T兲,
␬
⫽
␬
*共
␣
兲
␬
0共T兲,
␮
⫽
␮
*共
␣
兲
␮
0共T兲,
共4兲
␨
(0)⫽
␨
*共
␣
兲p
␩
0,共5兲
whe e
␩
0and
␬
0a e he Bol zmann elas ic alues o he
shea iscosi y and hea conduc i i y, espec i ely,
␮
0
⫽T
␬
0/n, and
␬
*,
␩
*,
␮
*, and
␨
*a e dimensionless unc-
ions o he coe icien o no mal es i u ion
␣
. Fo
␣
→1,
␩
*and
␬
* end o uni y, while
␮
*and
␨
* anish. The
explici exp essions o hese quan i ies a e gi en in Appen-
dix A. Le us no e ha he exis ence o a anspo coe icien
␮
gi ing a con ibu ion o he densi y g adien s o he hea
lux is a peculia i y o g anula luids ha has been con-
i med by molecula dynamics 关11兴and by Mon e Ca lo
simula ions o he Bol zmann equa ion 关12兴.
We will conside a o ce o he g a i a ional ype, namely,
⫽⫺mge
ˆz,共6兲
wi h ga posi i e cons an and e
ˆz he uni ec o in he posi-
i e di ec ion o he zaxis. The sys em we will s udy is a
g anula medium o Npa icles con ained in a box o sec ion
Sand heigh L. The quan i y Sis an a ea o d⫽3 and a
leng h o d⫽2. Ene gy is added o he sys em by he bo om
o he box which ib a es in a gi en way. Since many o he
esul s we will de i e in he ollowing a e independen o he
speci ic way in which he wall is ib a ed, we delay he
discussion o he de ails o i s mo ion un il hey a e needed.
Mo eo e , we a e no in e es ed in he bounda y e ec s as-
socia ed o he side walls and, he e o e, we will conside a
egion o he sys em a away om hem. In ac , in he
molecula dynamics simula ions o be epo ed la e on, pe-
iodic bounda y condi ions we e used in he di ec ions pe -
pendicula o he ex e nal ield. Then, because o symme y
conside a ions, in he s eady s a e only g adien s in he z
di ec ion a e expec ed and he hyd odynamic equa ions e-
duce o
⳵
p
⳵
z⫽⫺nmg,共7兲
J. JAVIER BREY, M. J. RUIZ-MONTERO, AND F. MORENO PHYSICAL REVIEW E 63 061305
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2
dnkB
⳵
⳵
z
冉
␬
⳵
T
⳵
z⫹
␮
⳵
n
⳵
z
冊
⫺T
␨
(0)⫽0. 共8兲
Equa ion 共7兲implies
T
n
⳵
n
⳵
z⫽⫺ mg
kB
⫺
⳵
T
⳵
z,共9兲
and use o his exp ession in o Eq. 共8兲 oge he wi h Eqs. 共4兲
and 共5兲, leads o,
2
dnkB关
␬
*共
␣
兲⫺
␮
*共
␣
兲兴
⳵
⳵
z
冋
␬
0共T兲
⳵
T
⳵
z
册
⫺2mg
dnkB
2
␮
*共
␣
兲
⳵
␬
0共T兲
⳵
z⫺
␨
*共
␣
兲nkBT2
␩
0
⫽0.
共10兲
In o de o analyze he abo e equa ion, i is con enien o
in oduce a dimensionless leng h scale lby
l⫽
冕
z
Ldz⬘1
␭共z⬘兲,共11兲
whe e ␭(z) is he local mean ee pa h o ha d disks o
sphe es,
␭共z兲⫽关Cn
␴
d⫺1兴⫺1,共12兲
wi h C⫽2
冑
2 o d⫽2 and C⫽
␲
冑
2 o d⫽3. The a iable
lmeasu es he numbe o mean ee pa hs om he wall
loca ed a z⫽L o he pa allel plane loca ed a heigh z. Fo
z⫽0i is
l共z⫽0兲⬅l0⫽C
␴
d⫺1Nz,共13兲
Nz⫽N/Sbeing he numbe o pa icles in he sys em pe uni
o sec ion. In e ms o l, he solu ion o Eq. 共7兲can be w i en
as
p⫽mgl
C
␴
d⫺1⫹pL,共14兲
whe e pLis he p essu e o he gas nex o he uppe wall. In
pa icula , a z⫽0,
p共z⫽0兲⬅p0⫽mgNz⫹pL.共15兲
Equa ion 共10兲is equi alen o
⳵
2T1/2
⳵
l2⫹b共
␣
兲
l⫹pL
*
⳵
T1/2
⳵
l⫺a共
␣
兲T1/2⫽0, 共16兲
wi h
pL
*⫽C
␴
d⫺1
mg pL共17兲
and
a共
␣
兲⫽32共d⫺1兲
␲
d⫺1
C2共d⫹2兲3⌫共d/2兲2
␨
*共
␣
兲
␬
*共
␣
兲⫺
␮
*共
␣
兲,共18兲
b共
␣
兲⫽2
␬
*共
␣
兲⫺
␮
*共
␣
兲
2关
␬
*共
␣
兲⫺
␮
*共
␣
兲兴.共19兲
Inspec ion o Eq. 共16兲indica es ha
冑
a(
␣
) de e mines he
coupling be ween g adien s and dissipa ion ha is in insic in
g anula lows, while b(
␣
) is a scaling ac o o he inhomo-
genei ies associa ed o he ex e nal ield.
I is s ill possible o exp ess Eq. 共16兲in a mo e amilia
way by de ining a new a iable
␰
by
␰
⫽
冑
a共
␣
兲共l⫹pL
*兲⫽C
␴
d⫺1
冑
a共
␣
兲
冋
冕
z
Ldz⬘n共z⬘兲⫹pL
mg
册
.
共20兲
This a iable, as well as l, is a dec easing unc ion o he
o iginal coo dina e z. No e ha
␰
canno be de ined in he
elas ic limi
␣
→1, in which a(
␣
) anishes. Pe o ming he
change, Eq. 共16兲becomes
␰⳵
2T1/2
⳵␰
2⫹b共
␣
兲
⳵
T1/2
⳵␰
⫺
␰
T1/2⫽0, 共21兲
whose gene al solu ion is
T1/2共
␰
兲⫽A
␰
⫺
␯
I
␯
共
␰
兲⫹B
␰
⫺
␯
K
␯
共
␰
兲,共22兲
whe e
␯
共
␣
兲⫽
␮
*共
␣
兲
4关
␬
*共
␣
兲⫺
␮
*共
␣
兲兴
⬎0, 共23兲
I
␯
and K
␯
a e he modi ied Bessel unc ions o i s and sec-
ond kind, espec i ely, and Aand Ba e cons an s ha mus
be de e mined om he bounda y condi ions. Since he be-
ha io o he unc ions
冑
a(
␣
) and
␯
(
␣
) will play an impo -
an ole in he discussions in he ollowing sec ions, we ha e
plo ed hem in Fig. 1 o d⫽2. Al hough bo h quan i ies
anish in he elas ic limi ,
冑
a(
␣
) g ows much as e han
␯
(
␣
)as
␣
dec eases in he icini y o
␣
⫽1. On he o he
hand, when
␣
becomes smalle , he beha io in e s and
␯
(
␣
) p esen s a much la ge slope.
The p essu e p o ile in he
␰
scale ollows di ec ly om
Eq. 共14兲and 共20兲,
p共
␰
兲⫽mg
␰
C
␴
d⫺1
冑
a共
␣
兲.共24兲
In his way, we ha e o mally sol ed he hyd odynamic
equa ions o he sys em unde conside a ion. The exp ession
o he densi y ollows om Eqs. 共22兲,共24兲, and he equa ion
o s a e. A e wa ds, he ela ionship be ween
␰
and he o igi-
nal coo dina e zis ob ained by sol ing he equa ion
HYDRODYNAMICS OF AN OPEN VIBRATED GRANULAR SYSTEM PHYSICAL REVIEW E 63 061305
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d
␰
n共
␰
兲⫽⫺
冑
a共
␣
兲C
␴
d⫺1dz,共25兲
ha is he di e en ial o m o he de ini ion o
␰
gi en in Eq.
共20兲. O cou se, he solu ion o his equa ion in ol es he
p essu e o he gas nex o he uppe wall pL, which is un-
known up o now.
III. OPEN SYSTEMS
In o de o pa icula ize he gene al esul s ob ained in he
p e ious sec ion o a gi en physical si ua ion, i.e., o spe-
ci ic o ms o he walls a z⫽0 and z⫽L, he c ucial and
non i ial poin is he in oduc ion o he hyd odynamic
bounda y condi ions needed o de e mine he cons an s Aand
Bappea ing in Eq. 共22兲, as well as he alue o pLen e ing in
he de ini ion o
␰
, Eq. 共20兲. We a e in e es ed in an open
sys em, i.e., in he limi o in ini e heigh L. In his limi , i is
ob ious ha pL⫽0, so ha he lowes alue o
␰
is now
␰
⫽0, co esponding o he limi z→⬁. Mo e explici ly, Eq.
共20兲becomes
␰
⫽C
␴
d⫺1
冑
a共
␣
兲
冕
z
⬁dz⬘n共z⬘兲.共26兲
The beha io o he modi ied Bessel unc ions in he limi
␰
→0is关18兴
I
␯
共
␰
兲⬃1
⌫共1⫹
␯
兲
冉
␰
2
冊
␯
,共27兲
K
␯
共
␰
兲⬃⌫共
␯
兲
2
冉
␰
2
冊
⫺
␯
.共28兲
The e o e, i ollows om Eq. 共22兲 ha in he same limi ,
T1/2共
␰
兲⬃B⌫共
␯
兲
2
冉
␰
2
2
冊
⫺
␯
,共29兲
indica ing a di e gen beha io o he empe a u e as he
limi
␰
→0(z→⬁) is app oached. Ne e heless, i canno be
concluded om he e ha he cons an Bmus iden ically
anish, con a y o wha has been in e ed in o he wo ks
关1,4兴. The e a e wo main easons o ha . Fi s , he ac ha
T o mally di e ges does no imply any hing unphysical, as
long as he densi y dec eases as enough as o gua an ee ha
he local kine ic ene gy goes o ze o as zgoes o in ini y.
Second, Eq. 共22兲is based on a con inuous hyd odynamic
desc ip ion o he g anula low, and such a desc ip ion is no
alid in he egion in which
␰
is e y small, so ha he local
Knudsen numbe , de ined as he a io o he mean ee pa h
o he leng h scale o he mac oscopic g adien s, is e y
la ge. The e, he a gumen s leading om a mic oscopic de-
sc ip ion o a con inuous app oach ail, and he gas has o be
desc ibed as a ee-molecule low, wi h a so-called ansi ion
egime be ween he hyd odynamic egion and he collision-
less one 关19,20兴. The de ailed analysis o he gas in hese
egimes is an in e es ing bu e y complex p oblem ha will
be add essed elsewhe e. So, we will keep he cons an Bin
Eq. 共22兲di e en om ze o, al hough he ques ion is s ill
whe he i s con ibu ion is ele an wi hin he hyd odynamic
egion.
The p esence o he e m p opo ional o K
␯
(
␰
) in he
exp ession o T1/2 implies ha he empe a u e p o ile exhib-
i s a minimum. Using he exp essions o he de i a i es o
he modi ied Bessel unc ions 关18兴, i is ob ained ha he
minimum is loca ed a a alue
␰
⫽
␰
Tgi en by he solu ion o
he equa ion
AI
␯
⫹1共
␰
T兲⫺BK
␯
⫹1共
␰
T兲⫽0, 共30兲
and he empe a u e Tma he minimum is
Tm
1/2⫽
␰
T
⫺
␯
关AI
␯
共
␰
T兲⫹BK
␯
共
␰
T兲兴.共31兲
The e o e, i he hyd odynamic desc ip ion is alid in he
icini y o
␰
⫽
␰
T, he alues o
␰
Tand Tmallow us o de-
e mine he cons an s Aand Bcha ac e izing he hyd ody-
namic p o iles e e ywhe e in he sys em, excep in he
bounda y laye s nex o
␰
⫽
␰
0and
␰
⫽0, whe e a mo e mi-
c oscopic desc ip ion, such as ha p o ided by kine ic
heo y, is needed.
As an example, in Fig. 2 we plo he empe a u e p o ile
in he
␰
a iable ob ained by molecula dynamics simula ion
in a wo-dimensional sys em wi h
␣
⫽0.95 and Nz⫽6. The
wall a he bo om is ib a ed wi h a saw oo h eloci y p o-
ile ha ing a eloci y W⫽6. This means ha all he pa icles
colliding wi h he wall ind i wi h ha eloci y 关5,21兴. Mo e-
o e , he ampli ude o he wall mo ion is much smalle han
he mean ee pa h o he pa icles nex o i , so ha he
posi ion o he wall can be aken as ixed a z⫽0. Pe iodic
bounda y condi ions a e employed in he di ec ion pe pen-
dicula o he ield. The uni s a e de ined by m⫽1, and
␴
⫽1. We ake kB⫽0.5, and he alue o he ex e nal ield is
g⫽1. Fo he abo e alue o
␣
,i is
␯
⯝0.021. F om he
simula ion da a i is es ima ed ha
␰
T⯝0.21 and Tm⯝162.6,
and using Eqs. 共30兲and 共31兲one ge s A⯝12.2 and B
⯝0.76. The dashed line in he igu e is he empe a u e p o-
FIG. 1. Func ions
冑
a(
␣
)共solid line兲and
␯
(
␣
)共dashed line兲
de ined in he main ex , o d⫽2.
J. JAVIER BREY, M. J. RUIZ-MONTERO, AND F. MORENO PHYSICAL REVIEW E 63 061305
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ile ob ained wi h hese alues o he cons an s. A qui e good
ag eemen is obse ed be ween he cons uc ed p o ile and
he simula ion da a ou side he bounda y laye s. In pa icula ,
he ag eemen ex ends well inside he egion o small alues
o
␰
. This con i ms ha he hyd odynamic egime includes
he minimum o he empe a u e and also a pa o he sys em
whe e he empe a u e inc eases wi h z, i.e., Tinc eases as
␰
dec eases.
In Fig. 3 he same empe a u e p o ile as in Fig. 2 is
shown as a unc ion o he o iginal a iable z. I is seen ha
he inc ease o he empe a u e wi h he heigh is no jus a
heo e ical a i ac , bu in p ac ice i is obse ed o e a wide
egion in eal space. Simila esul s ha e been ob ained o
o he alues o
␣
in he in e al 0.85⭐
␣
⭐0.99. De ails o
he p ac ical limi a ions in he molecula dynamics simula-
ions will be gi en in he las sec ion o he pape .
Since he exis ence o a egion whe e he empe a u e
inc eases is associa ed wi h he p esence o he collisionless
bounda y laye and he ansi ion egime, he loca ion o he
empe a u e minimum and, consequen ly, o he hyd ody-
namic egion beyond i , a e expec ed o co espond o small
alues o
␰
. The e o e, o s udy his egion we can app oxi-
ma e in Eq. 共22兲 he modi ied Bessel unc ions by hei ex-
p essions in he limi o small a gumen s. This yields
A
B⬃21⫺2
␯
共
␯
⫹1兲⌫共
␯
⫹1兲2
␰
T
⫺2(1⫹
␯
)共32兲
and, since
␰
Tis small, i ollows ha A/BⰇ1. The conclu-
sion eached in his way is ha he e m in ol ing K
␯
(
␰
)in
Eq. 共22兲is only ele an in he egion in which
␰
is small and
K
␯
(
␰
) is la ge. Mo e p ecisely, a de ailed asymp o ic analy-
sis o Eq. 共22兲indica es ha he ele an inc easing empe a-
u e egion co esponds o
␰
2Ⰶ1 bu
␰
2ⲏ
␯
2. This ange o
alues o
␰
exis s as long as
␯
is small enough, i.e., he
sys em be no oo inelas ic. Mo eo e , he analysis shows
ha in his
␰
window one can app oxima e
K
␯
共
␰
兲⬃⫺ln
␰
.共33兲
When
␰
akes alues o he o de o uni y, he e m B
␰
⫺
␯
ln
␰
is negligible as compa ed wi h A
␰
⫺
␯
I
␯
(
␰
). Then, we p o-
pose, as an accu a e app oxima ion o Eq. 共22兲, he exp es-
sion
T1/2共
␰
兲⯝A
␰
⫺
␯
I
␯
共
␰
兲⫺B
␰
⫺
␯
ln
␰
.共34兲
In Fig. 4 we compa e Eqs. 共22兲and 共34兲 o A⫽12.2 and
B⫽0.26, which a e he alues o he cons an s ound om
he molecula dynamics da a co esponding o he si ua ion
desc ibed in Fig. 2. The ag eemen is ai ly good in he plo -
ed in e al 10⫺2⬍
␰
⬍2. The accu acy is e en be e o
la ge alues o
␰
, as expec ed om he abo e discussion.
Le us nex analyze he densi y p o ile. Equa ions 共24兲and
共34兲yield
FIG. 2. Tempe a u e p o ile in uni s de ined in he main ex , in
he
␰
a iable o a ib a ed sys em wi h
␣
⫽0.95, Nz⫽6(
␰
0
⫽1.722) and w⫽6. The solid line is om he molecula dynamics
simula ion, while he dashed line is he heo e ical p edic ion dis-
cussed in he ex .
FIG. 3. The same as Fig. 2, bu in e ms o he eal space
a iable z, measu ed in uni s o
␴
.
FIG. 4. Compa ison o he exac heo e ical exp ession o he
empe a u e p o ile 关Eq. 共22兲兴 and he app oxima ed exp ession 关Eq.
共34兲兴 o alues o he pa ame e s co esponding o he si ua ion o
Fig. 2.
HYDRODYNAMICS OF AN OPEN VIBRATED GRANULAR SYSTEM PHYSICAL REVIEW E 63 061305
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n共
␰
兲⫽p共
␰
兲
kBT共
␰
兲⫽mg
␰
1⫹2
␯
CkB
␴
d⫺1
冑
a共
␣
兲关AI
␯
共
␰
兲⫺Bln
␰
兴2.
共35兲
The compa ison o his exp ession wi h he simula ion da a
o he same sys em as in Fig. 2 is shown in Fig. 5. The
dashed line, co esponding o he heo e ical p edic ion, has
been plo ed by using he alues o Aand Bob ained by
i ing he empe a u e minimum. Again, he ag eemen is
qui e good, ou side he bounda y laye nex o he ib a ing
wall.
The densi y p o ile exhibi s a maximum a
␰
⫽
␰
n, ha is
app oxima ely gi en by he solu ion o he equa ion
I
␯
共
␰
n兲⫺2
␰
nI
␯
⫹1共
␰
n兲⫽0, 共36兲
which can be nume ically sol ed o each alue o
␯
, i.e., o
␣
. A ske ch o he de i a ion o Eq. 共36兲is gi en in Appen-
dix B. In Fig. 6,
␰
nis shown as a unc ion o
␣
o 0.5⭐
␣
⭐0.99. Le us ema k ha con a y o he posi ion o he
empe a u e minimum
␰
T,
␰
n akes alues o he o de o
uni y. Mo eo e , he dependence o
␰
non
␣
is a he weak
since i g ows oughly om 1.07 o 1.46 as
␣
dec eases om
0.99 o 0.5. Ne e heless, when he alues o
␰
na e ans-
la ed in o he lscale by means o Eq. 共20兲wi h pL
*⫽0, he
posi ion o he maximum o he densi y u ns ou o be
s ongly in luenced by he inelas ici y o he sys em, inc eas-
ing e y as as
␣
app oaches uni y.
One o he main implica ions o Eq. 共36兲is ha he posi-
ion o he densi y maximum, measu ed in he scale
␰
, does
no depend on he bounda y condi ions, i.e., on he alues o
he cons an s Aand B, o on he o al numbe o pa icles in
he sys em, as measu ed o ins ance by Nz. O cou se, in
o de o ac ually obse e he maximum in an expe imen o
in a compu e simula ion, i mus be
␰
n⬍
␰
0⬅
冑
a共
␣
兲l0⫽C
冑
a共
␣
兲
␴
d⫺1Nz.共37兲
Taking in o accoun Eq. 共11兲, he o al numbe o pa icles
Nz
(⫹)pe uni o leng h o a ea o he ib a ing wall abo e
he posi ion o he densi y maximum is
Nz
(⫹)⫽ln
C
␴
d⫺1共38兲
whe e lndeno es he posi ion o he densi y maximum in he
lscale. This numbe inc eases as
␣
inc eases. The e o e, he
mo e elas ic he sys em is he la ge is he numbe o pa -
icles needed in o de o see he maximum o he densi y.
This explains why in some molecula dynamics simula ions
he densi y p o ile shows an almos mono onic decay wi h an
appa en maximum nex o he ib a ing wall. On he o he
hand, he posi ion o he densi y maximum in he ac ual a i-
able zdoes depend on he bounda y condi ions, since he
con e sion om
␰
o zin ol es he cons an s Aand B,asi
ollows om Eqs. 共26兲and 共35兲. This is clea ly seen, o
ins ance, in he expe imen al esul s shown in Fig. 2共b兲o
Re . 关13兴. The p edic ion o ou heo y is ha he a ea bellow
he densi y p o ile o he igh o he maximum is he same
o he se e al ib a ion ampli udes. This seems o be quali-
FIG. 5. Densi y p o ile in he scaled
␰
a iable 共a兲, and in he
eal a iable 共b兲, o he same alues o he pa ame e s as in Fig. 2.
The solid line is om he molecula dynamics simula ion, while he
dashed line is he heo e ical p edic ion discussed in he ex .
FIG. 6. Theo e ical alue o he scaled posi ion o he densi y
maximum as a unc ion o he coe icien o es i u ion,
␣
.
J. JAVIER BREY, M. J. RUIZ-MONTERO, AND F. MORENO PHYSICAL REVIEW E 63 061305
061305-6
a i ely ue in he epo ed case, al hough he ib a ion am-
pli ude can a ec he deg ee o luidiza ion o he g anula
sys em, so ha o small ampli udes he heo y de eloped
he e may no apply.
IV. THE UPPER REGION OF THE GRANULAR GAS
Fo la ge z共small
␰
), bu s ill inside he egion whe e
hyd odynamics holds, he empe a u e and densi y p o iles
can be app oxima ed by
T1/2共
␰
兲⬃A
冉
1
2
冊
␯
1
⌫共1⫹
␯
兲⫺B
␰
⫺
␯
ln
␰
,共39兲
n共
␰
兲⬃mg
␰
CkB
␴
d⫺1
冑
a共
␣
兲关A2⫺
␯
⌫共1⫹
␯
兲⫺1⫺B
␰
⫺
␯
ln
␰
兴2,
共40兲
espec i ely. No e ha he own s uc u e o Eq. 共39兲, ha
does no p esen any minimum, implies ha his app oxima-
ion only holds o alues o
␰
smalle han he posi ion
␰
To
he empe a u e minimum. On he o he hand,
␰
should be
la ge enough as o he hyd odynamic desc ip ion be accu a e.
In he p e ious sec ion we ha e discussed he exis ence o a
ele an egion e i ying bo h condi ions. Subs i u ion o
hese exp essions in o Eq. 共25兲yields
d
␰
⬃⫺mg
␰
dz
kB
冋
A
冉
1
2
冊
␯
⌫共1⫹
␯
兲⫺1⫺B
␰
⫺
␯
ln
␰
册
2共41兲
and by di e en ia ion o Eq. 共39兲one ge s
T1/2 dT
dz ⬃2mgB
kB
␰
⫺
␯
共1⫺
␯
ln
␰
兲.共42兲
Fo
␯
Ⰶ1, ha means no e y inelas ic sys ems, he abo e
equa ion can be app oxima ed by
dT3/2
dz ⬃3mgB
kB.共43兲
This is compa ible wi h wha is seen in he molecula dy-
namics simula ions, al hough due o he ela i ely small
a ia ion o he empe a u e wi h zin he egion wi h posi i e
slope, he beha io p edic ed by Eq. 共43兲is ha d o disce n
om a simple linea in zp o ile. We ha e i ed he empe a-
u e p o iles ob ained by molecula dynamics simula ions o
la ge z o he beha io p edic ed by Eq. 共43兲. The alues o B
ob ained in his way we e compa ed wi h hose de e mined
om he minimum o he empe a u e, as discussed in Sec.
III, and a good ag eemen was ound. Since he cons an Bis
small, a good es ima ion o he empe a u e p o ile in his
uppe egion o he ib a ed g anula sys em is ob ained in
some cases by conside ing ha he empe a u e eaches a
cons an pla eau 关13,14,22兴.
Also o
␯
Ⰶ1, Eq. 共40兲leads o
dlnn
dz ⬃⫺mg
kBT.共44兲
The e o e, i he empe a u e is app oxima ely cons an in he
zin e al conside ed, an appa en ly exponen ial beha io o
he densi y can be obse ed. This is equi alen o saying ha
n(
␰
)⬀
␰
, as easily seen om Eq. 共40兲. Again, he simula ions
ha e con i med hese p edic ions.
I is impo an o s ess ha he egion in which he em-
pe a u e shows a minimum ollowed by an appa en ly linea
p o ile, and he densi y seems o decay exponen ially, can
only be explained co ec ly i he con ibu ion o he e m
B
␰
⫺
␯
K
␯
(
␰
) o he empe a u e p o ile in Eq. 共22兲is aken
in o accoun . Mo eo e , he abo e discussion only explains
an app oxima ely exponen ial decay o he densi y o alues
o zlying well inside he egion o inc easing empe a u e,
bu no whe e he empe a u e dec eases wi h he heigh , due
o he di e en a ia ion a es o he empe a u e in bo h
egions.
V. DISSIPATED POWER
An exp ession o he o al powe Ddissipa ed in he sys-
em is di ec ly ob ained om he hyd odynamic equa ion o
he empe a u e, Eq. 共3兲,
D⫽
冕
d dnkB
2T
␨
(0)
⫽dkBS
␨
*共
␣
兲T1/2
2C
␴
d⫺1
冑
a共
␣
兲
␩
0
冕
0
␰
0d
␰
p共
␰
兲T共
␰
兲1/2.共45兲
Upon w i ing he abo e exp ession we ha e aken in o ac-
coun ha
␩
0is p opo ional o T1/2. Using now he exp es-
sions o he p essu e and empe a u e p o iles, Eqs. 共22兲and
共24兲, espec i ely, one ge s
D⫽dkBSmg
␨
*共
␣
兲T1/2
2关C
␴
d⫺1
冑
a共
␣
兲兴2
␩
0
冕
0
␰
0d
␰␰
1⫺
␯
关AI
␯
共
␰
兲⫹BK
␯
共
␰
兲兴.
共46兲
The in eg al on he igh hand side o his exp ession can be
easily e alua ed by employing Eqs. 共B2兲wi h he esul
D⫽dkBS
␨
*共
␣
兲mgT1/2
2关C
␴
d⫺1
冑
a共
␣
兲兴2
␩
0
再
A
冋
␰
0
1⫺
␯
I
␯
⫺1共
␰
0兲⫺21⫺
␯
⌫共
␯
兲
册
⫺B关
␰
0
1⫺
␯
K1⫺
␯
共
␰
0兲⫺2⫺
␯
⌫共1⫺
␯
兲兴
冎
.共47兲
This exp ession is an exac consequence o he hyd ody-
namic equa ions o a g anula gas. In pa icula , no assump-
ion has been made abou he alues o
␰
0,A,o B.
HYDRODYNAMICS OF AN OPEN VIBRATED GRANULAR SYSTEM PHYSICAL REVIEW E 63 061305
061305-7
Le us de ine a dimensionless quan i y Fby
F⫽D
g共NmE
¯
␰
0兲1/2 ,共48兲
whe e E
¯
is he o al kine ic ene gy o he sys em,
E
¯
⫽
冕
d d
2nkBT⫽SdkB
2C
␴
d⫺1
冑
a共
␣
兲
冕
0
␰
0d
␰␰
⫺2
␯
关AI
␯
共
␰
兲
⫹BK
␯
共
␰
兲兴2.共49兲
Subs i u ion o Eqs. 共47兲and 共49兲in o Eq. 共48兲leads o an
explici exp ession o F. I is a a he complica ed and no
e y illumina ing exp ession. The e o e, we will no w i e i
he e explici ly, al hough i will be e e ed o in he ollow-
ing as Fexac . Le us now suppose ha we neglec he pa o
he p o iles ha a e esponsible o he inc ease o he em-
pe a u e, i.e., we o mally ake B⫽0. Because o he discus-
sion in Sec. III, his could be expec ed a p io i o be a good
app oxima ion as long as
␰
0is no small. Then, using he
explici exp ession o he elas ic shea iscosi y
␩
0,i is
ob ained
Fapp ox⫽4共2d兲1/2
␲
d⫺1/2
␨
*共
␣
兲
共d⫹2兲⌫共d/2兲C
冑
a共
␣
兲
␰
0
␰
0
1⫺
␯
I
␯
⫺1共
␰
0兲⫺21⫺
␯
⌫共
␯
兲
冋
冕
0
␰
0d
␰␰
⫺2
␯
I
␯
共
␰
兲2
册
1/2 .
共50兲
Al hough his exp ession is no a all simple, i only depends
on he alues o
␣
and
␰
0, bu no on he bounda y condi-
ions ha de e mine he cons an s Aand B, hen ep esen ing
a scaling law p edic ion. In he limi o a la ge sys em, in he
sense ha
␰
0Ⰷ1, he asymp o ic beha io o Eq. 共51兲is
gi en by
Fapp ox⬃8d1/2
␲
(d⫺1)/2
␨
*共
␣
兲
共d⫹2兲C⌫共d/2兲
冑
a共
␣
兲,共51兲
while o small
␰
0i is
Fapp ox⬃2共2d兲1/2
␲
(d⫺1)/2
␨
*共
␣
兲
共d⫹2兲⌫共d/2兲C
冑
a共
␣
兲
␰
0
1/2.共52兲
Kuma an 关5,14兴modeled he ib a ed g anula media as
an iso he mal luid wi h a Maxwellian eloci y dis ibu ion
and de i ed an exp ession o he dissipa ed powe Din a
wo-dimensional sys em. His exp ession leads o a alue o
he quan i y Fgi en by
F⫽共1⫺
␣
兲
冋
␲␴
Nz
2
冑
2a共
␣
兲
册
1/2
.共53兲
In he limi o quasielas ici y, i.e., o
␣
e y close o uni y,
his esul is equi alen o Eq. 共52兲. Consequen ly, as de i ed
he e, i s applicabili y is es ic ed o small inelas ici y and
small sys ems.
In Fig. 7 we ha e plo ed he unc ion F(
␰
) in he in e al
0⭐
␰
0⭐10 o
␣
⫽0.95 (
␯
⯝0.021). The solid line is Fapp ox
as gi en by Eq. 共50兲, while he dashed line is he exp ession
de i ed by Kuma an, Eq. 共53兲. The do ed line shows he
asymp o ic cons an alue p edic ed by Eq. 共51兲,F⯝0.85.
Simila beha io s a e ob ained o o he alues o
␣
. The
symbols a e molecula dynamics simula ion esul s. While
he ci cles a e om simula ions ca ied ou by us, he squa es
a e om Fig. 2 in Re . 关5兴by aking in o accoun ha he
quan i y Cpp de ined he e is ela ed o Fby
Cpp⫽共1⫺
␣
兲⫺1
冋
C
冑
a共
␣
兲
␴
Nz
册
1/2
.共54兲
Equa ion 共53兲p edic s Cpp⫽
冑
2
␲
. All he epo ed simula-
ion da a in he igu e co espond o he dilu e luidized e-
gime, i.e., o high enough eloci ies o he ib a ing wall, so
ha Fhas al eady eached a s eady alue ha does no de-
pend any mo e on w. Fo smalle alues o he eloci y, F
is an inc easing unc ion o i 关5兴. The alue o
␰
0has been
a ied by modi ying he numbe o pa icles Nz. I is seen
ha Eq. 共50兲 ep oduces ai ly well he simula ion esul s,
p o iding a de ini ely be e app oxima ion han Eq. 共53兲.In
ac , he dependence o Cpp on he size o he sys em, mea-
su ed by Nz, was al eady ealized by McNama a and Luding
关5兴. Le us also s ess ha he asymp o ic beha io o la ge
sys ems, Eq. 共51兲, is only accu a e o qui e la ge alues o
␰
0and, he e o e, i is no e y use ul in p ac ice.
We ha e also compu ed Fexac by using he alues o A
and B esul ing om he i ing o he minimum o he em-
pe a u e p o ile ob ained in he simula ions, as discussed in
Sec. III. The esul s 共no shown兲always lie be ween he
simula ion symbols and he cu e Fapp ox . The disc epan-
cies be ween Fapp ox and he simula ion esul s inc ease as
he alue o he coe icien o no mal es i u ion
␣
dec eases.
FIG. 7. Dimensionless quan i y Fde ined in he ex o
␣
⫽0.95 as a unc ion o he pa ame e
␰
0. The con inuous line is he
app oxima ed exp ession de i ed in he ex 关Eq. 共50兲兴, he dashed
line he p edic ion by Kuma an 关Eq. 共53兲兴, and he ho izon al do ed
line, he asymp o ic alue o
␰
0→⬁. The symbols a e om mo-
lecula dynamics simula ions, as discussed in he main ex .
J. JAVIER BREY, M. J. RUIZ-MONTERO, AND F. MORENO PHYSICAL REVIEW E 63 061305
061305-8
The eason is ha he ene gy dissipa ed in he egion wi h a
posi i e empe a u e slope, which is neglec ed in Eq. 共50兲,
becomes mo e ele an as he inelas ici y o he sys em in-
c eases. This is con i med by he ac ha a much be e
ag eemen is ound i he exp ession o Fexac , wi h Aand B
ob ained om he simula ion, is used. On he o he hand, i is
s ill ue ha , o la ge w,F eaches a alue ha only de-
pends on
␰
0and
␣
, hen indica ing he exis ence o a scaling
law. This scaling is no i ially seen in he exp ession o
Fexac ha depends on bo h cons an Aand Bin a non i ial
manne . Ne e heless, i mus be ealized ha Aand Ba e
no in ac independen . They mus be de e mined om he
same bounda y condi ions speci ying he ib a ing wall, al-
hough hei calcula ion ac ually equi es conside ing he ee
pa icle egion. I he e is a p opo ionali y ela ionship be-
ween Aand B, i is easily seen ha he exp ession o Fexac
u ns ou o be independen o hem.
VI. CONCLUSIONS
In his pape , we ha e s udied a luidized g anula sys em
submi ed o an ex e nal o ce o he g a i a ional ype. The
gene al conclusion we ha e eached is ha he hyd ody-
namic desc ip ion p o ided by he 共inelas ic兲Na ie -S okes
equa ions is able o explain wha is obse ed in molecula
dynamics simula ions and also in expe imen s. We ha e o-
cused on open sys ems, and showed ha he p esence o a
collisionless egime in he e y high egion o he gas mus
be aken in o accoun when in oducing he ma ching condi-
ions be ween he bulk o he g anula medium and he
bounda ies. Now we summa ize he mos impo an esul s.
共a兲The empe a u e p o ile as a unc ion o he heigh
p esen s a minimum, inc easing mono onically a e wa ds.
The minimum lies in he hyd odynamic egion, bu when
in e p e ing his esul i mus be ealized ha he hyd ody-
namic desc ip ion is no alid when he densi y becomes oo
small.
共b兲The densi y p o ile p esen s a maximum when he sys-
em has a la ge enough numbe o pa icles. This maximum
is no associa ed, in p inciple, o any clus e ing hyd ody-
namic ins abili y, bu ollows di ec ly om he Na ie -
S okes equa ions.
共c兲The posi ion o he densi y maximum is qui e accu-
a ely only de e mined by he coe icien o es i u ion o he
sys em, being independen o he numbe o pa icles and he
way in which he sys em is being ib a ed.
共d兲An accu a e desc ip ion o he ene gy dissipa ed in
collisions equi es conside ing he nonuni o mi y o he hy-
d odynamic ields. The esul s ob ained by using he exac
hyd odynamic p o iles de i ed om he Na ie -S okes equa-
ions a e in be e ag eemen wi h molecula dynamics simu-
la ions han hose using a uni o m empe a u e and an expo-
nen ially dec easing densi y.
共e兲The app oxima ions used in some p e ious wo ks ha e
been ob ained as limi ing app oxima ions o he mo e gen-
e al esul s ob ained he e. This also applies o he scaling
beha io p edic ed by some au ho s. A sound jus i ica ion o
scaling laws can only ollow om a de ailed analysis o bo h
bounda y laye s, he one nex o he ib a ing wall and ha
associa ed wi h he ansi ion o he ee-pa icle low. Ne -
e heless, we ha e obse ed in he simula ions ha he hy-
d odynamic ields seem o scale wi h he eloci y o he i-
b a ing wall, as al eady ound in Re . 关4兴
The ange o alidi y o he analysis we ha e ca ied ou
dese es some commen s. We ha e e i ied ha he e is a
easonable good ag eemen be ween he heo e ical p edic-
ions de i ed he e and he molecula dynamics esul s o
␣
⬎0.9. Fo smalle alues o he coe icien o es i u ion,
he disc epancies become impo an and hey inc ease e y
apidly as
␣
dec eases. The e a e wo main ela ed easons
ha es ic a p io i he applicabili y o ou heo y o he
small inelas ici y ange. Fo la ge inelas ici y, he g adien s
become e y la ge and he Na ie -S okes app oxima ion
ails. Mo eo e , he densi y in he icini y o i s maximum
becomes e y high so ha he low densi y hyd odynamic
equa ions should be subs i u ed by equa ions mo e accu a e
o dense g anula luids.
ACKNOWLEDGMENT
This esea ch has been pa ially suppo ed by he Di ec-
cio
´n Gene al de In es igacio
´n Cien ı
´ icayTe
´cnica 共Spain兲
h ough G an No. PB98-1124,
APPENDIX A
In his Appendix, he explici exp essions o he quan i-
ies appea ing in Eqs. 共4兲and 共5兲a e gi en 关16,17兴. The
Bol zmann elas ic alues o he shea iscosi y and he mal
conduc i i y a e
␩
0⫽2⫹d
8⌫共d/2兲
␲
⫺(d⫺1)/2共mkBT兲1/2
␴
⫺(d⫺1),共A1兲
␬
0⫽d共d⫹2兲2
16共d⫺1兲⌫共d/2兲
␲
⫺(d⫺1)/2kB
冉
kBT
m
冊
1/2
␴
⫺(d⫺1),
共A2兲
while he dimensionless unc ions ha e he o m
␩
*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
␨
*共
␣
兲⫽2⫹d
4d共1⫺
␣
2兲
冋
1⫹3
32c*共
␣
兲
册
.共A6兲
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