Hydrodynamics of an open vibrated granular system

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 061305-2 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 061305-3 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 061305-4 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 061305-5 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 HYDRODYNAMICS OF AN OPEN VIBRATED GRANULAR SYSTEM PHYSICAL REVIEW E 63 061305 061305-9