Steady uniform shear flow in a low density granular gas

S eady uni o m shea low in a low densi y g anula gas J. J. B ey, M. J. Ruiz-Mon e o, and F. Mo eno Fı ´sica Teo ´ ica, Facul ad de Fı ´sica, Uni e sidad de Se illa, Apa ado de Co eos 1065, E-41080 Se illa, Spain ~Recei ed 8 Oc obe 1996! The s eady uni o m shea low o a low densi y g anula sy em is s udied by means o a kine ic model equa ion and also by di ec Mon e Ca lo simula ion o he Bol zmann equa ion. The o me can be exac ly sol ed o a bi a y shea a e and dissipa ion. Explici exp essions o he one-pa icle dis ibu ion unc ion and he p essu e enso a e ob ained. Compa ison o he esul s wi h hose o p e ious heo ies is p esen ed in he app op ia e limi s. The simula ion shows ha he model ep oduces ai ly well he alues o he s esses and, in pa icula , he phenomenon o no mal s ess di e ences. The ag eemen is also e y good o he eloci y dis ibu ion unc ion in he he mal eloci y egion, al hough signi ican disc epancies appea o la ge eloci ies. @S1063-651X~97!12803-3# PACS numbe ~s!: 05.20.Dd, 47.50.1d, 47.20.2k I. INTRODUCTION The de elopmen o kine ic model equa ions desc ibing he dynamics o pa icles, which collide inelas ically, seems o be an impo an s ep owa ds he unde s anding o he complex phenomena aking place in apid g anula lows @1–3#. Ne e heless, he nonconse a ion o ene gy in colli- sions in oduces qui e d as ic changes in he physics o he p oblem, and he gene aliza ion o he exis ing heo ies o o dina y luids is a om being an easy ask. This e e s o bo h he own de i a ion o p oposal o a kine ic equa ion o he dis ibu ion unc ion o he sys em and also o sol ing he equa ion o ob ain consis en hyd odynamic-like equa ions desc ibing he mac oscopic e olu ion. Al hough a hyd ody- namic desc ip ion, in e ms o he densi y, he mac oscopic low eloci y, and he g anula empe a u e is sugges ed by analogy wi h no mal luids and also by expe imen s and compu e simula ions, a de i a ion om a mo e undamen al desc ip ion o he sys em is needed. In he las yea s, se e al kine ic equa ions o he one- pa icle dis ibu ion unc ion o apid g anula lows ha e been p oposed, ex ending he heu is ic a gumen s leading o he Bol zmann and Enskog equa ions o o dina y gases @4–7#. In addi ion, a mo e basic app oach o he p oblem, s a ing om he Liou ille equa ion, has been p esen ed @8#. Ne e heless, e en i i is assumed ha hese equa ions a e ele an o he desc ip ion o apid g anula lows, a unda- men al di icul y a ises. The s anda d p ocedu e o ob aining solu ions o kine ic equa ions is by means o he Chapman- Enskog me hod @9#, in which an expansion in he g adien s o he hyd odynamic ields is ca ied ou . Fo molecula lu- ids, he e e ence s a e abou which g adien s a e conside ed is he Maxwellian local equilib ium dis ibu ion. The co e- sponding e e ence dis ibu ion o inelas ic pa icles is com- plex and has no been de e mined o da e. As a consequence, he limi o small dissipa ion ~i.e., he coe icien o es i u- ion a close o 1!is usually conside ed. Ne e heless, many o he phenomena peculia o apid g anula luids a e clea ly associa ed o nonlinea and heological e ec s and o no e y small dissipa ion. This includes p ope ies such as sig- ni ican no mal s ess di e ences unde shea @2,10,11#, spon aneous o ma ion o dense clus e s su ounded by e- gions o low densi y @12–15#, and inelas ic collapse @13#. The easons men ioned abo e indica e ha i is ins uc i e o conside simple model kine ic equa ions o inelas ic gases which allow con olled and de ailed analysis. Kine ic models ha e p o en o be e y use ul in he s udy o a om equi- lib ium s a es o dilu e molecula gases @16#. Fo many physical si ua ions, exac solu ions o he models ha e been de i ed, and compa ison wi h nume ical solu ions o he ex- ac equa ions ob ained by compu e simula ion shows a ai ly good ag eemen . Ve y ecen ly @8#, wo kine ic models o sys ems o inelas ic ha d sphe es ha e been p oposed. Bo h models a e o mula ed as app oxima ions o he e ised En- skog equa ion, bu wi h di e en quan i a i e accu acy. He e we will conside he limi o low densi ies and la ge leng h scales as compa ed wi h he diame e o he pa icles. In his limi , he wo models lead o he same equa ion, and become a kine ic model o he Bol zmann equa ion o inelas ic ha d pa icles. Mos o he simples a om equilib ium physical si ua- ions co espond o s eady s a es. In his pape , we s udy an unbounded, s eady, and uni o m shea low ~USF!. While in molecula luids such a s a e is no possible due o iscous hea ing, in g anula luids his e ec can be balanced by dissipa ion in collisions. The simplici y o he model allows us o ob ain he exac solu ion desc ibing he s eady USF and also all he physically ele an quan i ies. This is done wi h- ou in oducing any expansion in he g adien s o in he in- elas ici y o collisions. The esul s o he p essu e enso show aniso opy, i.e., no mal s ess di e ences. I mus be no ed ha ou app oach is qui e di e en om o he s, in which a speci ic o m is assumed o he one- pa icle dis ibu ion unc ion desc ibing a gi en s a e. Fo ins ance, in he heo y by Jenkins and Richman @17#a gen- e alized Maxwellian is conjec u ed o model he dis ibu ion unc ion o he s eady USF. The eloci y-independen coe - icien s a e iden i ied h ough he balance equa ion o he second o de luc ua ing eloci y co ela ion enso . This heo y, which in p inciple also keeps all o de s in he shea a e and dissipa ion, leads o no mal s ess di e ences which a e in good ag eemen wi h he esul s o molecula dynam- ics simula ions. On he o he hand, he model we use has been o mula ed o a bi a y condi ions, and no speci ic ap- PHYSICAL REVIEW E MARCH 1997VOLUME 55, NUMBER 3 55 1063-651X/97/55~3!/2846~11!/$10.00 2846 © 1997 The Ame ican Physical Socie y p oxima ion is in oduced o apply i o he s eady USF. To ob ain he one-pa icle dis ibu ion unc ion one has o sol e he kine ic model equa ion unde he app op ia e condi ions. Ano he in e es ing app oach o he same p oblem @18# has been ca ied ou by expanding he Bol zmann equa ion o Bu ne o de . The pe u ba i e expansion me hod is speci ic o he s eady USF, and i is based on he obse a ion ha in his s a e he shea a e scales as A e , whe e e 512 a 2. The heo y esul s in a p edic ion o he no mal s ess di e ences which a e e y close o hose o Jenkins and Richman, when he la e a e expanded o he same o de o app oxima ion. The di ec Mon e Ca lo simula ion me hod @19#was de- elopped o ob ain nume ical solu ions o he Bol zmann equa ion, and has been success ully applied o a g ea a ie y o si ua ions. Since i is o mula ed o an a bi a y sca e ing law, he e is no di icul y in using i o s udy g anula lows @20#. He e we p esen esul s ob ained o he s eady USF, and compa e hem wi h he p edic ions o he model kine ic equa ion. Fo he componen s o he p essu e enso he ag eemen u ns ou o be e y good o e a wide ange o alues o he coe icien o es i u ion. Rega ding he one- pa icle dis ibu ion unc ion, he ag eemen is only semi- quan i a i e. This is no su p ising, since he de ailed Bol z- mann collision ope a o is eplaced in he model by a much simple e ec i e e m, which is de e mined by he second momen s o he one-pa icle dis ibu ion. The s uc u e o he pape is as ollows. In Sec. II he model is b ie ly discussed, and he basic equa ions a e p e- sen ed. In addi ion, a u he simpli ica ion o he o mula ion in Re . @8#in in oduced. I consis s o an e alua ion o he dissipa ion sou ce e m in he local equilib ium app oxima- ion. We belie e his is consis en wi h he spi i o he model, and makes he calcula ions much simple . The model kine ic equa ion is pa icula ized o he USF in Sec. III, and he s eady p essu e enso is compu ed in Sec. IV. The esul s a e compa ed o he heo ies o Jenkins and Richman @17# and o Sela, Goldhi sch, and Noskowi z @18#. Sec ion V deals wi h he one-pa icle dis ibu ion unc ion, which is also compa ed wi h he abo e heo ies in he low dissipa ion limi . In Sec. VI we discuss he Mon e Ca lo simula ion e- sul s and, inally, Sec. VII p o ides a sho summa y and conclusions. II. BASIC EQUATIONS OF THE MODEL We conside a gas o iden ical smoo h disks (d52) o sphe es (d53) o diame e s and mass m, whose collisions a e cha ac e ized by a cons an coe icien o no mal es i u- ion a in he in e al (0,1#. The Bol zmann equa ion o he one-pa icle dis ibu ion unc ion, ( , , ), is @7,8# S ] ] 1 1•“1 D ~ 1, 1, !5JB@ 1, 1, u ~ !#,~1! whe e JBis he ~inelas ic!Bol zmann collision ope a o JB@ 1, 1, u ~ !#5 s d21 E d 2 E d s ˆQ~g• s ˆ!~g• s ˆ! 3@ a 22 ~ 1, 1 8, ! ~ 1, 2 8, ! 2 ~ 1, 1, ! ~ 1, 2, !#.~2! In he abo e exp ession, s ˆis 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 , Qis he Hea iside s ep unc ion, g5 12 2, and 1 85 1211 a 2 a ~g• s ˆ! s ˆ,~3! 2 85 2111 a 2 a ~g• s ˆ! s ˆ~4! a e p ecollisional eloci ies leading a e collision o eloci- ies 1and 2. The local pa icle numbe densi y n, low eloci y u, and empe a u e Ta e de ined in he usual way, n~ , !5 E d ~ , , !,~5! n~ , !u~ , !5 E d ~ , , !,~6! d 2n~ , !kBT~ , !5 E d 1 2m@ 2u~ , !#2 ~ , , !.~7! F om Eq. ~1! he ollowing e olu ion equa ions o hese ields a e easily ob ained: ] n ] 1“•~nu!50, ~8! ] u ] 1u•“u1~nm!21“•P50, ~9! d 2nkB ] T ] 1d 2nkBu•“T52~¹u!:P2“•q2~12 a 2! , ~10! whe e P~ , !5 E d m~ 2u!~ 2u! ~ , , !~11! is he p essu e enso , q~ , !5 E d m 2~ 2u!2 ~ , , !~12! is he hea lux, and ~ , !5m s d21 8 E d 1 E d 2 E d s ˆQ~g• s ˆ! 3~g• s ˆ!3 ~ , 1, ! ~ , 2, !~13! 55 2847STEADY UNIFORM SHEAR FLOW IN A LOW DENSITY . . . is a sou ce e m desc ibing he a e o dissipa ion in colli- sions. Sol ing Eq. ~1! o a gi en si ua ion is a o midable ask. E en in he case o o dina y gases, e y li le is known abou solu ions o he Bol zmann equa ion o a om equilib ium s a es, and he p esence o dissipa ion complica es in a non- i ial way he s uc u e o he Bol zmann equa ion. The sim- ples physical si ua ion one can hink o o a g anula luid is he so-called homogeneous cooling s a e ~HCS!, o which he sys em is uni o m and he ime dependence occu s en- i ely h ough he empe a u e. Al hough i has been shown @7,21# ha Eq. ~1!admi s a solu ion desc ibing such s a e, he exac o m o he dis ibu ion unc ion is no known. Fo hese easons, i is wo h looking o model kine ic equa ions simila o hose which ha e p o en o be so ui ul o o - dina y luids @16#. The basic idea is o eplace he Bol zmann collision ope a o wi h a simple o m, while p ese ing he essen ial p ope ies o he exac equa ion. Ve y ecen ly @8#, a kine ic model along he lines ske ched abo e has been p oposed. The Bol zmann collision ope a o is sepa a ed in o wo pa s: one in he eloci y subspace spanned by 1, , and 2, and he o he one in he subspace o hogonal o i . The i s con ibu ion is e ained exac ly, while he second one is app oxima ed by a single elaxa ion ime e m. This gua an ees ha he balance equa ions ~8!– ~10!a e p ese ed by he model and ha he luxes Pand q, and also he sou ce e m w, a e he same unc ionals o he dis ibu ion unc ion as in he Bol zmann equa ion, i.e., hey a e gi en by Eqs. ~11!–~13!. Since he de ails a e de- sc ibed elsewhe e @8,22#, we only gi e he esul he e. The Bol zmann collision ope a o is eplaced by JB→2 n ~ 2 l!21 nkBT l c ~V!~12 a 2! ,~14! whe e we ha e in oduced he peculia eloci y V5 2u, c ~ !5m 2 dkBT21, ~15! and lis he local equilib ium dis ibu ion, l~ , , !5n~ , ! F m 2 p kBT~ , ! G d/2 exp F 2mV2~ , ! 2kBT~ , ! G . ~16! Also, n @n( , ),T( , )#is an e ec i e local collision e- quency, speci ied in mo e de ail below. The model de ined by Eq. ~14!, al hough much simple han he o iginal Bol zmann equa ion, is s ill oo complica ed o allow exac calcula ions in mos o he applica ions. This is due o he nonlinea unc ional dependence o on he dis ibu ion unc ion. Then, we go a s ep u he in he sim- pli ica ion, and app oxima e by i s local equilib ium alue, i.e., ~ , !→ l~ , !5m s d21 8 E d 1 E d 2 E d s ˆQ~g• s ˆ! 3~g• s ˆ!3 l~ , 1, ! l~ , 2, ! 5~d21! S p m D 1/2 n2 s d21~kBT!3/2.~17! In summa y, ou model kine ic equa ion is gi en by S ] ] 1 •“ D 52 n ~ 2 l!21 nkBT l c ~V!~12 a 2! l. ~18! T i ially, his equa ion also leads o he balance equa ions ~8!–~10!, wi h he only di e ence ha in he las o hem he sou ce e m is eplaced by l. In he elas ic collision limi , he kine ic model educes o he well-known Bha naga -G oss-K ook ~BGK!equa ion @23#. Applica ion o Eq. ~18! o he HCS is e y simple. The solu ion is gi en by a Maxwellian wi h a ime dependen empe a u e. The ime e olu ion o his la e quan i y is gi en by Eq. ~10!, which o he HCS has he solu ion T~ !5T~0! S 11 0 D 22 ,~19! wi h 0 215~12 a 2!d21 d s d21n S p kBT~0! m D 1/2 .~20! As we ha e al eady men ioned, he exac solu ion o he Bol zmann equa ion o he HCS is no known and, in pa - icula , i is easy o check ha i is no a Maxwellian ~excep in he elas ic limi , a 51). Ne e heless, bo h heo e ical s udies @7,21#and Mon e Ca lo simula ions @20#ha e shown ha de ia ions om Maxwellian a e quan i a i ely small o he mal eloci ies and Eq. ~19!p o ides a e y accu a e ap- p oxima ion o he ime e olu ion o he empe a u e. III. UNIFORM SHEAR FLOW We wan o s udy he uni o m shea low ~USF!, which is cha ac e ized by a cons an linea eloci y p o ile u~ !5a• ,~21! whe e aij5a d ix d jy ,abeing he cons an shea a e. Fo simplici y we conside an unbounded sys em. In addi ion, he hea lux anishes and he densi y n(0) and empe a u e T(0) a e uni o m. Fo his low, Eqs. ~8!,~9!, and ~10!imply ha he densi y is cons an in ime, he p essu e enso is uni o m, and he empe a u e obeys he ene gy balance d 2n~0!kB ] T~0! ] 52aPxy ~0!2~12 a 2! l ~0!.~22! .This equa ion clea ly shows he qui e di e en beha io o elas ic and inelas ic luids unde shea . While o molecu- la luids he USF is always a ime-dependen s a e whose empe a u e inc eases mono onically in ime due o iscous 2848 55J. J. BREY, M. J. RUIZ-MONTERO, AND F. MORENO hea ing, in g anula media he a ia ion o he empe a u e a ises om he balance o wo opposi e e ec s: iscosi y and dissipa ion in collisions. As a consequence, a s eady s a e is possible when bo h e ec s cancel each o he . Ou aim in he ollowing will be o analyze such a s a e. I is wo h men- ioning ha shea ed g anula media posses mul iple s eady s a es, depending on he ini ial and bounda y condi ions @24#. He e we es ic ou sel es o he USF as desc ibed abo e, assuming ha he sys em has eached such a s a e, and wi h- ou paying any a en ion o i s s abili y. A de ailed analysis o hese ques ions will be published elsewhe e. Le us in oduce posi ion and eloci ies coo dina es wi h espec o a ame mo ing wi h he low eloci y u.We de ine he Lag angian coo dina es by R5L~ !• ,V5 2a• ,~23! whe e Lij~ !5 d ij2aij .~24! In e ms o hese, Eq. ~18!becomes ] ] 1LijVj ] ] Ri 2aijVj ] ] Vi 52 n ~ 2 l!21 nkBT l c ~12 a 2! l. ~25! Since he USF is mac oscopically homogeneous in he Lag angian ame, we expec he associa ed dis ibu ion unc ion, (0), o ha e he same p ope y, i.e., o be indepen- den o R. The e o e, Eq. ~25! educes o ] ~0! ] 2aijVj ] ~0! ] Vi 52 n ~0!~ ~0!2 l ~0!!21 n~0!kBT~0! l ~0! c ~0! 3 ~12 a 2! l ~0!.~26! This equa ion implies ha ] ^ Vi & ~0! ] 52a d ix ^ Vy & ~0!,~27! whe e he angula b acke s deno e a e aging wi h espec o . The e o e, i he ini ial s a e e i ies ^ V & (0)50, his p op- e y is kep in ime by Eq. ~26!. In he ollowing we will limi ou conside a ion o ini ial condi ions o his kind. This is he eason why, o eloci ies in he Lag angian ame, we used he same symbol as o he peculia eloci ies in oduced in Sec. II. Then he local equilib ium dis ibu ion appea ing in Eq. ~26!has he o m l ~0!~V, !5n~0! S m 2 p kBT~0!~ ! D d/2 exp S 2mV2 2kBT~0!~ ! D , ~28! wi h all he possible ime dependence occu ing h ough he empe a u e. IV. PRESSURE TENSOR By mul iplying Eq. ~26!by ViVjand in eg a ing o e V,i is s aigh o wa d o ob ain he e olu ion equa ions o he componen s o he p essu e enso in he USF, ] ] Pij ~0!1aikPjk ~0!1ajkPik ~0!52 n ~0!~Pij ~0!2 d ijp~0!! 22 d d ij~12 a 2! l ~0!,~29! whe e p5(iPii /d5nkBTis he hyd os a ic p essu e. Fo a 51 we eco e he equa ions o an o dina y gas unde USF in he BGK app oxima ion, which ha e been ex en- si ely s udied @25–27#. F om Eq. ~29!, in pa icula , one ob- ains ] ] p~0!52 2 daPxy ~0!22 d~12 a 2! l ~0!,~30! ] ] Pxy ~0!52aPyy ~0!2 n ~0!Pxy ~0!,~31! ] ] Pyy ~0!52 n ~0!~Pyy ~0!2p~0!!22 d~12 a 2! l ~0!.~32! This closed sys em o equa ions has he i ial solu ion Pxy (0)5Pyy (0)5p(0)50. A second s eady solu ion, deno ed by an as e isk, is gi en by Pxy *52 a n * F p*22 d~12 a 2! l * n * G ,~33! Pyy *52 n * aPxy *,~34! a2 n * F p*22 d~12 a 2! l * n * G 2~12 a 2! l *50. ~35! Fo gi en alues o he shea a e a, he numbe densi y n, and he es i u ion coe icien a , Eq. ~35!de e mines he alue o he p essu e ~o empe a u e!a which he s eady s a e is possible. I is con enien a his poin o speci y he o m o he e ec i e collision equency n . Dimensional analysis equi es ha n 5cn s d21 S p kBT m D 1/2 ,~36! whe e cis a dimensionless cons an . We a e going o ix his by equi ing ha he model gi es he co ec ~Bol zmann! alue o he Na ie -S okes shea iscosi y in he elas ic limi a 51. This leads o c516/5b, wi h b.1.016, o d53~sphe es!, and c52/b, wi h b.1.022, o d52~disks! @28#. When Eq. ~36!is used in Eqs. ~33!–~35! hey educe o P ˜ xy52 H ~12 a 2!~d21! c F 122~d21! dc ~12 a 2! G J 1/2 , ~37! 55 2849STEADY UNIFORM SHEAR FLOW IN A LOW DENSITY . . . P ˜ yy52 P ˜ xy a ˜ 5122~d21! dc ~12 a 2!,~38! a ˜ 25d~d21!~12 a 2! dc22~d21!~12 a 2!.~39! We ha e in oduced he dimensionless p essu e enso P ˜ ij and shea a e a ˜ in he s eady s a e as P ˜ ij5Pij * p*,a ˜ 5a n *.~40! Equa ion ~39!shows ha in he s eady s a e he a io o he shea a e o he collision equency depends only on he es i u ion coe icien . The simple dependence o he p esu e enso on he shea a e ound he e con as s wi h he com- plex one ound in he ime-dependen uni o m shea low o o dina y gases @25–27#. Le us suppose we ca y ou a se ies o expe iences, all o hem in he same sys em, i.e., wi h gi en alues o a and n, bu wi h di e en alues o he shea a e, measu ing he shea iscosi y in he s eady s a e. Taking in o accoun ha Eq. ~39!implies ha T*}a2,i ollows om Eq. ~37! ha he gene alized shea iscosi y h *is linea in he shea a e, h *~a![2Pxy * a}a.~41! O cou se, he p opo ionali y cons an in his exp ession de- pends on he coe icien o es i u ion. The o he componen s o he p essu e enso a e easily compu ed by means o Eq. ~29!, pa icula ized o he s eady s a e. One ob ains P ˜ xx5112~d21!2~12 a 2! dc ,~42! and, o a h ee-dimensional sys em, P ˜ zz5P ˜ yy ,P ˜ xz5P ˜ yz50. ~43! Le us no e ha he educed p essu e enso in he s eady s a e is exp essed as a unc ion o a alone, being indepen- den , in pa icula , o he applied shea a e. Then, o in- s ance, o he no mal s ess a ios we ha e Pxx * Pyy *5dc12~d21!2~12 a 2! dc22~d21!~12 a 2!,~44! Pyy * Pzz *51. ~45! The e o e, he model p edic s aniso opy o he diagonal e ms o he p essu e enso in he shea plane, bu no in he plane pe pendicula o he low. The exis ence o no mal s ess di e ences in shea lows o no mal luids is e y well known. I is a iscome ic e ec which appea s beyond he Na ie -S okes app oxima ion @29#. Fo a low densi y USF hey ha e been calcula ed using he BGK model @27#. The e, di e ences be ween Pxx and Pyy bu no be ween Pyy and Pzz we e ound. As we ha e al eady men ioned, an app oxima ed heo y o he s eady USF o smoo h inelas ic disks has been o - mula ed by Jenkins and Richman @17#. Thei esul s o he second eloci y momen s in he low densi y limi ead, in ou no a ion, a ˜ ~JR!25~11 a !~12 a 2!~723 a !2 c2~917 a !,~46! P ˜ xy ~JR!52 4 2529 a @~917 a !~12 a !#1/2,~47! P ˜ yy ~JR!5917 a 2529 a ,~48! P ˜ xx ~JR!541225 a 2529 a .~49! Al hough hese exp essions may appea a he di e en om he co esponding esul s ob ained om ou model ki- ne ic equa ion, Figs. 1 and 2 show ha he disc epancies a e qui e small o low dissipa ion, and end o anish when a app oaches uni y. Le us also men ion ha ecen molecula dynamics simula ions @24#ha e ound ha he alues o he s eady empe a u e o a dilu e gas o inelas ic disks can be closely i ed by an empi ical ela ion, which in ou uni s eads a ˜ 22516Ac2 p S 1 12 a 21B D ,~50! wi h A.0.080 and B.20.54. The abo e exp ession has he same unc ional dependence on he es i u ion coe icien as Eq. ~39!. In addi ion, om his la e equa ion one iden i ies A.0.100 and B.20.511, which a e su p isingly close o he simula ion alues, aking in o accoun ha no excluded olume e ec s a e conside ed in ou model, and also ha he low obse ed in Re . @24#is highly nonuni o m and i ex- hibi s ime-dependen mic os uc u es. Sela, Goldhi sch, and Noskowi z @18#ca ied ou a s udy o a wo-dimensional g anula USF o Bu ne o de using he Bol zmann equa ion. They a i ed a exp essions o he s eady empe a u e and he p essu e enso o he o ms a ˜ 25C~12 a 2!1D~12 a 2!3/21O~12 a 2!2,~51! P ˜ xy5E~12 a 2!1/21O~12 a 2!3/2,~52! P ˜ xx511F~12 a 2!1O~12 a 2!2,~53! P ˜ yy511G~12 a 2!1O~12 a 2!2.~54! He e C,D,E,F, and Ga e dimensionless cons an s. In Table I we compa e he alues o hese cons an s gi en in Re . @18#wi h hose ob ained by expanding he exp essions de i ed in his sec ion. Also gi en a e he esul s ob ained om he expansion o he Jenkins and Richman heo y. The alues om he h ee heo ies a e e y close and, in pa icu- 2850 55J. J. BREY, M. J. RUIZ-MONTERO, AND F. MORENO la , he ag eemen be ween ou esul s and he Bu ne ex- pansion o he Bol zmann equa ion is excellen . V. VELOCITY DISTRIBUTION FUNCTION The o mal solu ion o Eq. ~26!can be w i en as ~0!~V, !5e2se aijVj ] ] Vi ~0!~V,0! 1 E 0 d 8e2~s2s8!e~ 2 8!aijVj~ ] / ] Vi! F n ~0!~ 8 ! 21 n~0!kBT~0!~ 8 ! c ~0!~V, 8 !~12 a 2! l ~0!~ 8 ! G 3 l ~0!~V, 8 !,~55! whe e s[ E 0 d 8 n ~0!~ 8!~56! is a measu e o he a e age numbe o collisions pe pa icle be ween 0 and ,s85s( 8), and exp@ aijVj( ] / ] Vi)‡is a shi ope a o in eloci y space, e aijVj~ ] / ] Vi!h~V!5h@L ~2 !•V#.~57! O cou se, he ime-dependen empe a u e in Eq. ~55! mus be consis en ly calcula ed, i.e., i is gi en by he solu- ion o Eqs. ~30!–~32!. Le us now conside he long ime limi o (0)( ). Assuming ha he s eady s a e discussed in Sec. IV is eached, we know ha he empe a u e ends o he cons an alue gi en by Eq. ~39!, and sdi e ges. The con- clusion is ha (0)(V, ) app oaches he s eady o m FIG. 1. Reduced shea a e a ˜ as a unc ion o he coe icien o es i u ion a o a wo- dimensional s eady USF. The solid line co e- sponds o he p esen heo y, and he dashed line o ha o Jenkins and Richman. FIG. 2. The p essu e enso o a wo- dimensional s eady USF. The solid lines co e- spond o he p esen heo y, and he dashed ones o ha o Jenkins and Richman. 55 2851STEADY UNIFORM SHEAR FLOW IN A LOW DENSITY . . . *~V!5 E 0 `d e2 n * e aijVj~ ] / ] Vi! 3 F n *21 n~0!kBT* c *~V!~12 a 2! l * G l *~V!, ~58! wi h c *~V!5mV2 dkBT*21. ~59! I can be checked ha his exp ession ep oduces he alues o he s eady p essu e enso ound in Sec. IV, and also ha i gi es a anishing hea lux. Now we in oduce dimension- less quan i ies by V ˜ 5 S 2kBT* m D 21/2 V,~60! ˜ 51 n~0! S 2kBT* m D d/2 *.~61! Then Eq. ~58! educes o ˜ ~V ˜ !5 p 2d/2 E 0 `dse2sesa ˜ V ˜ y~ ] / ] V ˜ x! 3 F 12d21 c S 2 dV ˜ 221 D ~12 a 2! G e2V ˜ 2,~62! whe e use has been made o Eqs. ~17!and ~36!. In he case o a sys em o ha d sphe es, he ma ginal dis ibu ion o he zcomponen o he eloci y is easily ob ained, ˜ z~V ˜ z![ E 2` `dV ˜ x E 2` `dV ˜ y ˜ ~V ˜ !5 p 21/2 3 F 122~12 a 2! 3c~2V ˜ z 221! G e2V ˜ z 2.~63! This exp ession shows a limi a ion o ou model kine ic equa ion, namely, ha he dis ibu ion unc ion akes un- physical nega i e alues o la ge enough eloci ies. Ne e - heless, he eloci y alues o which Eq. ~63!is nega i e a e gi en by Vz 2. F 3c 2~12 a 2!11 G kBT* m,~64! and, he e o e, he dis ibu ion unc ion is posi i e in all he ange o he mal eloci ies, e en in he limi a →0. I is hen possible ha Eq. ~62!p o ides an accu a e app oxima ion o he solu ion o he Bol zmann equa ion in all he ele an egion o he dis ibu ion unc ion. This poin will be ana- lyzed in de ail in Sec. VI. The ma ginal eloci y dis ibu ion in he shea plane, also o d53, is ˜ xy~V ˜ x,V ˜ y![ E 2` `dV ˜ z ˜ ~V ˜ ! 5 p 21 E 0 `dse2se2[~V ˜ x1sa ˜ V ˜ y ! 2 1V ˜ y 2 ] 3 H 12 4 ~ 12 a 2 ! 3c@~V ˜ x1sa ˜ V ˜ y!21V ˜ y 2# J . ~65! In o de o compa e wi h p e ious heo ies, we ha e ca ied ou a pe u ba i e expansion o ˜ o d52 in powe s o 12 a 2. The esul is ˜ ~V ˜ !5 p 21e2V ˜ 2 F 12V ˜ 2sin2 u A c~12 a 2!1/2 1~122V ˜ 21V ˜ 2cos2 u 11 2V ˜ 421 2V ˜ 4cos4 u ! 312 a 2 c1O~12 a 2!3/2 G ,~66! He e, we ha e in oduced a pola ep esen a ion o he e- loci y, V ˜ (V ˜ , u ), whe e u is measu ed wi h espec o he s eaming di ec ion x. Jenkins and Richman @17#assumed ha he s eady dis i- bu ion unc ion has he o m o a gene alized Gaussian, ˜ ~JR!51 p A u K u exp~2V ˜ •K21•V ˜ !,~67! whe e Kis a ma ix which accoun s o he aniso opy exis - ing in he USF. When he low densi y limi o he abo e dis ibu ion is conside ed and he esul expanded o o de (12 a 2) one ob ains ˜ ~JR!~V ˜ !5 p 21e2V ˜ 2 F 12V ˜ 2sin2 u A 2~12 a 2!1/2 1~122V ˜ 212V ˜ 2cos2 u 11 2V ˜ 421 2V ˜ 4cos4 u ! 312 a 2 41O~12 a 2!3/2 G .~68! I we neglec he di e ence be ween c(;1.96) and 2, bo h exp essions ag ee o o de (12 a 2)1/2. The e ms o o de (12 a 2) ha e he same dependence on he eloci y modulus TABLE I. Compa ison o he alues o he coe icien s in ex- pansions gi en by Eqs. ~51!–~54!ob ained wi h di e en heo ies and in he p esen wo k. Model Sela, Goldhi sch, and Noskowi z Jenkins and Richman C0.511 0.511 0.522 D00 0 E20.714 20.714 20.707 F0.511 0.522 0.5 G0.511 0.522 0.5 2852 55 J. J. BREY, M. J. RUIZ-MONTERO, AND F. MORENO and also on u , bu all he cons an p e ac o s di e . In spi e o his di e ence, le us no ice ha bo h exp essions ~66!and ~68!make he same con ibu ion o he diagonal pa o he p essu e enso when he cons an cis again app oxima ed by 2. This can be e i ied by a di ec calcula ion ~see also Table I!.The dis ibu ion unc ion o he USF o a wo- dimensional g anula gas has also been calcula ed a he same o de by Sela, Goldhi sch, and Noskowi z in Re . @18#. The e, a much mo e complica ed dependence on bo h pola componen s o he eloci y is ound. By compa ing wi h Jen- kins and Richman esul s, he au ho s conclude ha he con- s an s appea ing as p e ac o s o he second ha monics in u in Eq. ~68!can be conside ed as ough a e ages o he co esponding unc ions ob ained by hem. We e e he eade o hei pape o de ails. VI. DIRECT MONTE CARLO SIMULATION The di ec simula ion Mon e Ca lo me hod @19#has p o ed o be a e y use ul ool o ob ain nume ical solu ions o he Bol zmann equa ion o molecula luids. Ve y e- cen ly, i has also been applied o s udy he HCS o a low densi y g anula low @20#. Since he de ails o he me hod ha e been ex ensi ely discussed in Re . @19#, hey will no be gi en he e. We saw in Sec. III ha he dis ibu ion unc ion o he USF becomes homogeneous in he Lag angian ame. The conside a ion o homogeneous s a es allows a g ea simpli i- ca ion o he simula ion. The e o e, we ha e ca ied ou he simula ion in he Lag angian ame and es ic ed ou sel es o solu ions o he Bol zmann equa ion which s ay homoge- neous in ha ame. In o he wo ds, we nume ically sol ed he equa ion @compa e wi h Eq. ~26!# ] ~0! ] 2aijVj ] ~0! ] Vi 5J B@V u ~0!#.~69! O cou se, his implies ha he possibili y o spon aneous o ma ion o spa ial inhomogenei ies is elimina ed in he FIG. 3. Time e olu ion o he educed shea a e a ¯ o a 50.8, and di e en alues o he shea a e a. In all cases, he ini ial dis ibu ion was homogeneous wi h a Maxwellian eloci y dis ibu ion in he Lag angian ame. Time is measu ed in uni s o c„2 A pn (0)…21. FIG. 4. S eady educed shea a e a ˜ as a unc- ion o he coe icien o es i u ion a o a dilu e sys em o ha d disks. The solid line co esponds o he kine ic model, and he symbols a e esul s om he Mon e Ca lo simula ion. 55 2853STEADY UNIFORM SHEAR FLOW IN A LOW DENSITY . . . simula ion. Ou aim he e is o s udy he p ope ies o he homogeneous s eady s a e. I s s abili y will be analyzed else- whe e. Fo homogeneous sys ems he e is no need o spli he sys em in o cells and, consequen ly, he spa ial coo dina es o he pa icles do no play any ole in he simula ion. In addi ion, no bounda y condi ions mus be in oduced. In ou simula ion we ha e conside ed a h ee-dimensional sys em. The numbe o pa icles is N51000, and he esul s ha e been a e aged o e 500 di e en ajec o ies. The ime in e - al D o e which i is assumed ha ee mo ion, including he e ec o he ine ial o ce, and collisions, a e uncoupled has been aken d 50.025 n 0 21, whe e n 05(2 A p /c) n ( ), wi h n ( ) being he ins an aneous alue o he collision e- quency gi en by Eq. ~36!. We conside ed a noncons an ime s ep in o de o gua an ee ha i always emains much smalle han he a e age ime be ween collisions, in spi e o he change o he empe a u e @30#. The ini ial eloci y dis- ibu ion in all cases is a Maxwellian. In Fig. 3 we p esen he ime e olu ion o a ¯ [a/ n ( ) o a 50.8 and ou di e en alues o he shea a e, namely, a50.1, 0.25, 0.5, and 1. Time is measu ed in uni s o c„2 A pn (0)…21, whe e n (0) is he ini ial collision e- quency. A e an ini ial ansien pe iod, all cu es con e ge o he same s eady alue, as p edic ed by Eq. ~39!. The same quali a i e beha io has been ound o all he educed com- ponen s o he p essu e enso Pij( )/p( ). The e o e, in he ollowing we will concen a e on he dependence o he s eady alues o he educed quan i ies on he es i u ion co- e icien a , once we ha e checked hey do no depend on he shea a e. The esul s ob ained o he s eady educed shea a e a ˜ o di e en alues o a a e shown in Fig. 4. The s a is ical e o s a e smalle han he symbols used o ep esen he da a. Also plo ed is he p edic ion o ou model kine ic equa ion, i.e. Eq. ~39!wi h d53. I is seen ha he ag ee- men is ema kable a low dissipa ion, al hough he disc ep- ancy inc eases as he es i u ion coe icien dec eases. We FIG. 5. The same as in Fig. 4 o he educed p essu e enso P ˜ ij. FIG. 6. Ma ginal eloci y dis ibu ion unc- ion in he di ec ion pe pendicula o he shea plane. The symbols a e simula ion da a, and he solid line co esponds o he model kine ic equa- ion. The es i u ion coe icien is a 50.8. 2854 55 J. J. BREY, M. J. RUIZ-MONTERO, AND F. MORENO