Fourier state of a fluidized granular gas

This con en has been downloaded om IOPscience. Please sc oll down o see he ull ex . Download de ails: IP Add ess: 150.214.182.194 This con en was downloaded on 15/06/2015 a 17:28 Please no e ha e ms and condi ions apply. Fou ie s a e o a luidized g anula gas View he able o con en s o his issue, o go o he jou nal homepage o mo e 2001 Eu ophys. Le . 53 432 (h p://iopscience.iop.o g/0295-5075/53/4/432) Home Sea ch Collec ions Jou nals Abou Con ac us My IOPscience 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)). 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