Green-Kubo representation of the viscosity of granular gases

G een-Kubo ep esen a ion o he iscosi y o g anula gases J. Ja ie B ey Á ea de Física Teó ica, Uni e sidad de Se illa. Apa ado de Co eos 1065, E-41080, Se illa, Spain Abs ac . The G een-Kubo ep esen a ion o he shea iscosi y o a dilu e gas o inelas ic ha d disks de i ed om he Bol zmann equa ion is e alua ed by means o he di ec simula ion Mon e Ca lo me hod. The ela ionship be ween he one- pa icle dynamics o he o iginal exp ession and he N-pa icle dynamics needed o he simula ion is analyzed. The p esence o eloci y co ela ions in he la e is iden i ied and hei possible implica ions a e discussed. INTRODUCTION The p o o ypical model o g anula gases is an ensemble o smoo h inelas ic ha d sphe es o disks wi h a cons an coe icien o no mal es i u ion α . Fo his model, he me hods o kine ic heo y ha e been applied and, in pa icula , he Bol zmann equa ion has been ex ended [1, 2]. Then, by using a modi ica ion o he Chapmann-Enskog p ocedu e, hyd odynamicequa ions ha e been de i ed [3, 4]. As he e e ence s a e, he iso opic and homogeneouscooling s a e (HCS) o a g anula sys em is used. Due o he ene gy dissipa ion, he HCS is no s a iona y, bu i s ene gy dec eases mono onically. Hyd odynamics has been employed wi h success o desc ibe he beha io o g anula lows. As o molecula sys ems, he Chapmann-Enskogme hod leads o complica ed di e en ial equa ions ha ha e o be sol ed in some app oxima ions.I has been shown ha he a o emen ionedequa ions a e equi alen o a ep esen a ion o he anspo coe icien s ha has a simila s uc u e o he G een-Kubo ela ions o elas ic sys ems, al hough wi h signi ican di e ences [5]. Equi alen exp essions ha e been de i ed by iden i ying he hyd odynamic pa o he spec um o he linea ized Bol zmann equa ion and assuming i domina es o long long imes and wa eleng hs [6, 7]. He e, an e alua ion o he G een-Kubo ela ion o he shea iscosi y by means o N-pa icle simula ions, namely he di ec simula ion Mon e Ca lo (DSMC) me hod [8], is p esen ed. This equi es going om he one-pa icle e ec i e dynamics, as de ined by a linea ized Bol zmann ope a o , o a supposed equi alen N-pa icle dynamics. The ela ionship be ween bo h desc ip ions will u n ou o be impo an o he analysis o he esul s. The exis ence an exac mapping o he HCS on o a s eady s a e will be exploi ed as he basis o he simula ion me hod [9, 10]. THE GREEN-KUBO EXPRESSION FOR THE SHEAR VISCOSITY By using he Chapman-Enskog me hod o eigen unc ions expansions [5, 6], he shea iscosi y η o a dilu e g anula gas o Nsmoo h inelas ic ha d disks (d=2) o sphe es (d=3) o mass mand diame e σ can be exp essed in he o m η (T) = nm` 0(T)Z∞ 0dse−s ζ 0/2Zdc1 χ HCS(c1)∆xy(c1,s)Φ2,xy(c1).(1) He e nis he densi y, T he empe a u e, 0(T) = (2kBT/m)1/2wi h kB he Bol zmann cons an , `= (n σ d−1)−1, and ∆xy(c) = cxcy,Φ2,xy(c) = −cx ∂ ln χ HCS(c) ∂ cy.(2) The unc ion χ HCS(c1)is de ined om he solu ion o he Bol zmann equa ion desc ibing he HCS, HCS( 1, ), as [1] HCS( 1, ) = n −d 0[T( )] χ HCS(c1),c1= 1 0[T( )] ,(3) 815 CP762, Ra e ied Gas Dynamics: 24 h In e na ional Symposium, edi ed by M. Capi elli © 2005 Ame ican Ins i u e o Physics 0-7354-0247-7/05/$22.50 Repo Documen a ion Page Fo m App o ed OMB No. 0704-0188 Public epo ing bu den o he collec ion o in o ma ion is es ima ed o a e age 1 hou pe esponse, including he ime o e iewing ins uc ions, sea ching exis ing da a sou ces, ga he ing and main aining he da a needed, and comple ing and e iewing he collec ion o in o ma ion. Send commen s ega ding his bu den es ima e o any o he aspec o his collec ion o in o ma ion, including sugges ions o educing his bu den, o Washing on Headqua e s Se ices, Di ec o a e o In o ma ion Ope a ions and Repo s, 1215 Je e son Da is Highway, Sui e 1204, A ling on VA 22202-4302. Responden s should be awa e ha no wi hs anding any o he p o ision o law, no pe son shall be subjec o a penal y o ailing o comply wi h a collec ion o in o ma ion i i does no display a cu en ly alid OMB con ol numbe . 1. REPORT DATE 13 JUL 2005 2. REPORT TYPE N/A 3. DATES COVERED - 4. TITLE AND SUBTITLE G een-Kubo ep esen a ion o he iscosi y o g anula gases 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5 . WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Á ea de Física Teó ica, Uni e sidad de Se illa. Apa ado de Co eos 1065, E-41080, Se illa, Spain 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT App o ed o public elease, dis ibu ion unlimi ed 13. SUPPLEMENTARY NOTES See also ADM001792, In e na ional Symposium on Ra e ied Gas Dynamics (24 h) Held in Monopoli (Ba i), I aly on 10-16 July 2004. 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT UU 18. NUMBER OF PAGES 6 19a. NAME OF RESPONSIBLE PERSON a. REPORT unclassi ied b. ABSTRACT unclassi ied c. THIS PAGE unclassi ied S anda d Fo m 298 (Re . 8-98) P esc ibed by ANSI S d Z39-18 and i is an iso opic unc ion o he ec o c1. Finally, he ime dependence o ∆xy is gi en by ∆xy(c1,s) = es¯ Λc(c1)∆xy(c1),(4) ¯ Λc(c1) = Zdc2 χ HCS(c2)Zdb σ Θ(c12 ·b σ )c12 ·b σ [b σ (c1,c2)−1](1+P12)+ ζ 0 2c1· ∂ ∂ c1,(5) whe e c12 =c1−c2,Θis he Hea iside s ep unc ion, b σ is he uni ec o poin ing om he cen e o pa icle 2 o he cen e o pa icle 1 a con ac , he ope a o P12 in e changes he subindexes 1 and 2 o i s igh , and b σ (c1,c2)is an ope a o eplacing all he eloci ies c1and c2appea ing o i s igh by he pos collisional alues c0 1and c0 2gi en by c0 1≡b σ c1=c1−1+ α 2(b σ ·c12)b σ ,c0 2≡b σ c2=c1+1+ α 2(b σ ·c12)b σ .(6) The las e m on he igh hand side o Eq. (5) con ains he ime-independen educed cooling a e o he HCS, ζ 0, de ining he e olu ion o he empe a u e in ha s a e, ∂ T( ) = − ζ HCS(T)T, ζ 0=` ζ HCS 0(T).(7) Equa ion (1) di e s om he s anda d G een-Kubo exp ession o dilu e molecula sys ems in se e al aspec s: •The a e age is aken o e he eloci y dis ibu ion o he HCS and no o e he Maxwellian cha ac e izing he equilib ium s a e. Bo h di e o α <1. •The ime co ela ion unc ion couples he ans e sal momen um lux ∆xy and a “modi ied lux” Φ2,xy. •The dynamics includes an accele a ing s eaming be ween collisions. •The ime in eg al con ains, in addi ion o he co ela ion unc ion, an exponen ially dec easing in ime ac o . To ende Eq. (1) sui able o e alua ion by means o N-pa icle simula ions, some issues mus be add essed. Fi s , i p esen s he echnical di icul y ha Λcin ol es he cooling a e ζ 0 ha , he e o e, mus be known a p io i. Consequen ly, i is use ul o make a change o scale [9, 10], τ = ζ 0 2 ω 0s,w=2 ω 0` ζ 0c(8) wi h ω 0being an a bi a y ime-independen dimensionless equency. Then, Eq. (1) ans o ms in o η (T) = nm 0(T) e 0,s NZ∞ 0d e− ω 0 h∆xy( , )Φ2,xy ( /e 0,s )is .(9) He e, we ha e enamed he ime and eloci y a iables, e 0,s ≡(2kBe Ts /m)1/2whe e e Ts =2m kB ω 0` ζ 02 ,(10) and he angula b acke s deno e a e age de ined by ha( , )b( )is ≡Zd Zd e s ( )a( , )b( ),e s ( ) = ne −d 0,s χ HCS  e 0,s .(11) Mo eo e , he ime dependence is now gi en by a( , ) = e ¯ Λs ( )a( )(12) wi h ¯ Λs ( 1) = σ d−1Zd 2e s ( 2)Zdb σ Θ( 12 ·b σ ) 12 ·b σ [b σ ( 1, 2)−1](1+P12)+ ω 0 1· ∂ ∂ 1.(13) 816 N-PARTICLE REPRESENTATION In Eq. (9), he dynamics is de ined by means o he linea ope a o e Λs . Be o e compu a ing i by N-pa icle simula ion me hods, i mus be ew i en in e ms o a well de ined pa icle dynamics. The s uc u e o e Λs sugges s o in oduce an accele a ion s eaming be ween collisions, ∂ ∂ Ri( ) = Vi( ), ∂ ∂ Vi( ) = ω 0Vi( ),(14) while he e ec o a collision be ween pa icles iand jis o ins an aneously modi y hei eloci ies acco dingly wi h he same ules as in he o iginal dynamics and gi en in Eqs. (6), i.e. Vi→V0 i≡b σ Vi=Vi−1+ α 2(b σ ·Vij)b σ ,Vj→V0j≡b σ Vj=Vj+1+ α 2(b σ ·Vij)b σ .(15) Fo his dynamics, he Liou ille equa ion can be ob ained. Then, by means o any o he s anda d p ocedu es, he Bol zmann equa ion can be de i ed in he low densi y limi . I has he o m  ∂ ∂ + ω 0 ∂ ∂ · + · ∂ ∂  ( , , ) = J[ , ],(16) whe e J[ , ]is he same Bol zmann collision ope a o as in he o iginal dynamics. I is now easily e i ied ha e s ( ) is a s eady solu ion o his equa ion [10]. Le us assume ha he Liou ille equa ion has a s a iona y solu ion ρ s (Γ), whe e Γdeno es a poin in phase space. A su icien condi ion o his is ha he dis ibu ion unc ion o he HCS in he ac ual, o iginal dynamics has he scaling p ope y ρ HCS(Γ, ) = ( 0[T( )])−dN ρ ∗ HCS ({Rij,Vi/ 0[T( )]}).(17) Conside ime co ela ion unc ions o he o m CAB,s ( ) = hA( )BiN,s −hAiN,s hBiN,s ,(18) whe e hAiN,s =ZdΓ ρ s (Γ)A(Γ),hBiN,s =ZdΓ ρ s (Γ)B(Γ),(19) hA( )BiN,s =ZdΓ ρ s (Γ)A(Γ, )B(Γ),(20) wi h A(Γ) = N ∑ i=1a(Vi),B(Γ) = N ∑ i=1b(Vi).(21) The dynamical a iable A(Γ, )≡A[Γ( )] is gene a ed om A(Γ) h ough he dynamics de ined abo e.The low densi y limi o CAB,s can be ob ained by using he same p ocedu es as o molecula sys ems [10]. The hypo hesis needed a e he same as hose equi ed o de i e he Bol zmann equa ion [11], plus he addi ional assump ion ha eloci y co ela ions a e negligible in he HCS a low densi ies. Then, i is ound ha CAB,s is gi en by Eq. (11). This p o ides a di ec way o e alua ing he la e exp ession, and he e o e he shea iscosi y, by means o N-pa icle simula ions, since he dynamics de ined by Eqs. (14) and (15) is easily implemen ed. O cou se, he equali y CAB,s =ha( , )b( )is (22) only holds a low enough densi ies and as long as eloci y co ela ions e ec s a e negligible in he HCS. The DSMC me hod is a e y use ul ool o gene a e he low densi y dynamics o a sys em [8, 12]. This me hod is designed as a eal N-pa icle dynamics simula ion o a low densi y gas, consequen ly p o iding i s comple e dynamical desc ip ion. 817 012345 −6 −4 −2 0 (ln Χ)’ FIGURE 1. Plo o (lnX)0≡ ∂ ln χ HCS/ ∂ cas a unc ion o c= /e 0,s o α =0.6. The ci cles a e he nume ical de i a i e o he simula ion esul s and he solid line he i ed unc ion. The eloci y is measu ed in he uni s de ined in he ex . SIMULATION RESULTS We ha e pe o medDSMC simula ions o a sys em o N=104inelas ic ha d disks. Since we a e dealing wi h posi ion independen quan i ies in a homogeneous s a e, i is enough o conside one cell in con igu a ionspace. This allows o inc ease hes a is ics and alsoa oids ha he sys em spon aneouslyde elopsspa ial inhomogenei ies.This is impo an since he HCS is uns able wi h espec o spa ial long wa eleng h pe u ba ions [13, 14]. We in oduce a dimensionless educed shea iscosi y η ∗by scaling he iscosi y wi h i s elas ic limi in he i s Sonine app oxima ion η 0, η ∗= η (T) η 0(T), η 0=1 2 σ mkBT π 1/2 .(23) Then, Eq. (9) yields η ∗=2√2 π `e 0,s Z∞ 0d J η ( )e− ω 0 ,(24) wi h J η ( ) = 1 Nh∆xy( , )Φ2,xy( /e 0,s )is .(25) Le us emind ha wha is ac ually compu ed in he N-pa icle desc ip ion is J0 η ( ) = 1 N N ∑ i N ∑ jh∆xy( i, )Φ2,xy( j/e 0,s )iN,s .(26) The simula ions show ha he sys em eaches, a e a ansien pe iod, a s a iona y s a e wi h a empe a u e e Ts . Then, we measu ed he eloci y unc ion ∂ ln χ HCS(c)/ ∂ c,c= /e 0,s , o de e mine he modi ied lux Φ2,xy(c), de ined in Eq. (2). Fo his, he ange o was pa i ioned in o small non-o e lapping bins and he equency dis ibu ion was buil . A e wa ds, he nume ical de i a i e was compu ed. An example o he esul s, o α =0.6, is gi en in Fig. 1. He e and in he ollowing, he uni o mass is m, he uni o leng h is `= (n σ )−1, and he uni o ime is `[2kBe T(0)/m]−1/2, whe e e T(0)is he ini ial empe a u e. Mo eo e , we se kB=1, implying ha in ou uni s i is e T(0) = 1/2. Fo la ge eloci ies, ∂ ln χ HCS(c)/ ∂ c ends o a cons an , e lec ing he exponen ial decay o χ HCS(c)[15]. Once he unc ion Φ2,xy has been de e mined o each o he alues o α conside ed, he co ela ion unc ion J0 η ( ) has been measu ed. I is seen o decay exponen ially in ime, a leas o e mo e han wo decades, o all he alues o he es i u ion coe icien [16]. Then, J0 η ( )was i ed o an exponen ial, J0 η ( ) = J0 η (0)e− λη , and he i ed pa ame e s 818 0.1 0.3 0.5 0.7 0.9 α 1.0 1.2 1.4 1.6 η∗ FIGURE 2. The educed coe icien o shea iscosi y η ∗ o a sys em o inelas ic disks as a unc ion o he coe icien o no mal es i u ion α . The symbols a e om he DSMC me hod and he solid line is he i s Sonine app oxima ion. J0 η (0)and λη we e de e mined. In e ms o hem, Eq. (24) eads η ∗=2√2 π J0 η (0) λη + ω 0.(27) The alues o η ∗ob ained in his way a e shown in Fig. 2. A good ag eemen is obse ed wi h he esul s ob ained in he i s Sonine app oxima ion [17], al hough a sligh ly di e en beha io is also clea ly iden i ied. I is in e es ing o conside he ini ial alue o J0 η ( )and decompose i in he o m J0 η (0) = J0(1) η (0)+J0(2) η (0),(28) whe e J0(1) η (0) = 1 N N ∑ ih∆xy( i)Φ2,xy( i/e 0,s )iN,s ,J0(2) η (0) = 1 N N ∑ i N ∑ j6=ih∆xy( i)Φ2,xy( j/e 0,s )iN,s .(29) The i s componen , J0(1) η (0)is iden ically he same as J η (0), whe e J η ( )was de ined in Eq. (25). I is independen o he eloci y co ela ions p esen in he sys em. In ac , i s exac alue can be easily ob ained, and o he choice o ω 0used in he simula ions and he uni s we a e employing, i is J0(1) η (0) = J η (0) = 1/2 o all α [16]. The o he pa , J(2) η (0), con ains he e ec o eloci y co ela ions and was assumed o be negligible. The simula ion esul s o J0 η (0)a e plo ed in Fig. 3. Fo α <0.6, con ibu ions om J0(2) η become signi ican , indica ing he p esence o eloci y co ela ions in he sys em. This esul aises some undamen al ques ions. A e he obse ed co ela ions an a i ac o he DSMC me hod? We belie e hey a e no , since eloci y co ela ions in he HCS also ha e been ecen ly obse ed and measu ed in molecula dynamics simula ions [18]. Why he e is such a good ag eemen be ween he i s Sonine app oxima ion p edic ions and he simula ion esul s o he shea iscosi y, e en om s ong dissipa ion? A possible answe is ha he Sonine expansion is in oduced, and unca ed, bo h in he ini ial condi ions and in he dynamics o he luxes, leading o some kind o compensa ion. Fo small alues o α , which G een-Kubo exp ession is he igh one, ha wi h J η ( )o ha wi h J0 η ( )?, o a e bo h w ong? And, inally, can i be concluded om hese esul s ha he Bol zmann equa ion should be somehow e ised o s ong inelas ici y? All hese poin s clea ly dese e u he analysis. Al hough in his pape we ha e es ic ed ou sel es o he shea iscosi y, he G een-Kubo exp essions o he o he wo anspo coe icien s, associa ed wi h he hea lux, can be compu ed in a simila way. Fo no oo s ong dissipa ion, he simula ion esul s ag ee qui e well wi h he i s Sonine app oxima ion p edic ions. Ne e heless, he disc epancies g ow a he as as α dec eases below α ≈0.7. This migh be, a leas pa ially, due o he p esence 819 0 0.2 0.4 0.6 0.8 1 α 0.45 0.5 0.55 0.6 FIGURE 3. DSMC esul s o he co ela ion unc ion J0 η (0)( illed ci cles) and i s diagonal pa J0(1) η (0)(emp y ci cles). o eloci y co ela ions in he DSMC esul s. A de ailed discussion o hese anspo coe icien s will be published elsewhe e [16]. 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) REFERENCES 1. Golds hein M. and Shapi o M., J. Fluid Mech.,282, 75-114 (1995). 2. B ey J.J., Du y J.W., and San os A., Physica,87, 1051-1066 (1997). 3. B ey J.J., Du y J.W., Kim C.S., and San os A., Phys. Re . E,58, 4638-4653 (1998). 4. Sela N. and Goldhi sch I., J. Fluid Mech.,361, 41-74 (1998). 5. Du y J.W. and B ey J.J., Physica,109, 433-448 (2002). 6. B ey J.J., Du y J.W., and Ruiz-Mon e o M.J., “Linea ized Bol zmann Equa ion and Hyd odynamics o G anula gases”, in G anula Gas Dynamics, edi ed by T. Pöschel and N. B illian o , Sp inge -Ve lag, Be lin, 2003, pp. 225-246. 7. Du y J.W. and B ey J.J., Phys. Re . E,68, 030302-1-4 (2003). 8. Bi d G., Molecula gas Dynamics and he Di ec Simula ion o Gas Flows, Cla endon P ess, Ox o d (1994). 9. Lu sko J., Phys. Re . E,63, 061211-1-14 (2001) 10. B ey J.J., Ruiz Mon e o M.J., and Mo eno F., Phys. Re . E,69, 051303-1-13 (2004). 11. E ns M.H. and Cohen E.G.D., J. S a . Phys.,25, 153-180 (1981). 12. Ga cía A., Nume ical Me hods o Physics, P en ice Hall, Englewood Cli s, NJ (2000). 13. Goldhi sch I. and Zane i G. Phys. Re . Le .,70, 1619-1622 (1993). 14. B ey J.J., Ruiz-Mon e o M.J., and Cube o D. Phys. Re . E,54, 3664-3671 (1996). 15. Esipo S.E. and Pöschel T., J. S a . Phys.,86, 1385-1395 (1997). 16. B ey J.J. and Ruiz-Mon e o M.J., unpublished. 17. B ey J.J. and Cube o D., “Hyd odynamic T anpo Coe icien s o G anula Gases”, in G anula Gases, edi e by T. Pöschel and S. Luding, Sp inge -Ve lag, Be lin (2001), pp. 59-78. 18. B ey J.J., Ga cía de So ia M.I., Mayna P., and Ruiz-Mon e o M.J., Phys. Re . E, o appea (2004). 820