A KINETIC RELAXATION MODEL FOR BIMOLECULAR CHEMICAL REACTIONS

Transcription

1 Bullen of he Inue of Mahemac Academa Snca (New Sere) Vol. (7), No., pp A KINETIC RELAXATION MODEL FOR BIMOLECULAR CHEMICAL REACTIONS BY M. GROPPI AND G. SPIGA Dedcaed o he memory of Franceco Premuda Abrac A recenly propoed conen BGK ype approach for chemcally reacng ga mxure dcued, whch accoun for he correc rae of ranfer for ma, momenum and energy, and recover he exac conervaon equaon and collon equlbra, ncludng ma acon law. In parcular, he hydrodynamc lm derved by a Chapman Enkog procedure, and compared o exng reul for he reacve and non reacve cae. In addon, numercal reul are preened for non oropc pace homogeneou problem n whch phycal condon allow reducon of he negraon over he hree dmenonal velocy pace o only one dmenon.. Inroducon Knec approache o chemcally reacng ga mxure conue he proper ool of nvegaon n everal crcumance, and allow a rgorou dervaon and jufcaon for he mo common macrocopc decrpon ued n he hydrodynamc regme [, ]. On he oher hand, nonlnear chemcal collon negral of Bolzmann ype are defnely no eay o deal wh Receved December, and n reved form March,. AMS Subjec Clafcaon: 8C, 76P. Key word and phrae: Knec heory, chemcal reacon, BGK model. Th work wa performed n he frame of he acve ponored by MIUR (Projec Mahemacal Problem of Knec Theore ), by INdAM, by GNFM, and by he Unvery of Parma (Ialy), and by he European TMR Nework Hyperbolc and Knec Equaon: Aympoc, Numerc, Analy. 69

2 6 M. GROPPI AND G. SPIGA [June [] and mpler approxmae model would be convenen for praccal applcaon. Followng he experence of ga knec heory, relaxaon me approxmaon of he ype propoed by Bhanagar, Gro, and Krook [] and by Welander [] (uually denoed a BGK model) eem o be he fr canddae n ha drecon. Typcally, he fne rucure of he collon operaor replaced by a blurred mage whch rean only qualave and average propere, and precrbe relaxaon oward a local equlbrum wh a rengh deermned by a uable characerc me. The procedure mu be carefully deved n order o avod well known drawback whch are for a mul pece ga [6, 7], a unavodable when dealng wh a bmolecular chemcal reacon A +A A +A () a we hall do n h work. Some relaxaon model for he chemcal collon operaor have been nroduced que recenly n he leraure [8, 9]. In parcular, he laer paper follow he conen BGK raegy propoed n [7] for ner mxure, whch preerve povy and ndfferenably prncple, and reor o a ngle BGK collon erm for each pece ( =,,, ), decrbng boh mechancal (elac) and chemcal encouner, and drfng he drbuon funcon f oward a uable local Maxwellan M. Indeed, he macrocopc parameer relevan o uch equlbrum are no he acual feld, momen of f (number deny n, drf velocy u, emperaure T ), bu ome oher fcou feld n,u,t, conruced ad hoc n order o recover he exac exchange rae for ma, momenum and energy, a gven by he whole Bolzmann-lke collon operaor. The machnery of coure heaver han for chemcally ner mxure, nce ha o accoun for ranfer of ma and for energe of chemcal lnk, bu made poble by he explc knowledge (for Maxwell ype neracon) of he momen of he chemcal collon negral []. I ha been hown n [9] ha uch relaxaon model of BGK ype a conen approxmaon of he Bolzmann knec decrpon, whch reproduce n parcular he correc macrocopc conervaon equaon and he correc collon equlbra, a Maxwellan a a common ma velocy and emperaure, and wh number dene relaed by he well known ma acon law. The preen paper amed a proceedng furher along he lne propoed n [9] by nvegang ome apec ha were lef here a a fuure work. We hall focu n parcular on wo pon here. The fr he hydrodynamc lm for mall collon me of he preen BGK equaon

3 7] A KINETIC RELAXATION MODEL 6 (va an aympoc Chapman Enkog expanon) n he collon domnaed regme, whenhereacvecollon meareofheameorderahemechancal collon me, and boh hermal and chemcal equlbrum are reached evenually. Th regme ha been already nroduced by Ludwg and Hel [], and nvegaed n parcular by Ern and Govanggl [] for arbrary reacve ga mxure of polyaomc pece ung Enkog expanon and Bolzmann-ype collon erm. The econd pon he exenon of llurave numercal calculaon o non oropc uaon, lke hoe whch are ymmerc around one pace drecon (of nere for nance n he clacal evaporaon condenaon problem). A h ll prelmnary age we conder only a ngle bmolecular reacon beween monoaomc pece and wh Maxwell ype neracon, bu he BGK raegy can be exended o more general uaon, where he advanage relaed o he avalably of explc formula for coeffcen and o he much eaer numercal approach become even more gnfcan. I mgh be noced ha reacon () nclude a a pecal cae hree aom reacon lke AB +C B +AC, () for whch a lghly mpler reamen would be n order []. The arcle organzed a follow. Afer preenng n Secon he man feaure of he condered BGK model, Secon devoed o a dealed aympoc analy of he Chapman Enkog ype leadng o cloed macrocopc equaon a he Naver Soke level, whch are compared o analogou reul obaned boh n he chemcally ner cae from BGK equaon [7] and n he chemcally reacve cae va Grad expanon echnque []. Secon deal nead wh he pecalzaon of he relaxaon model o non-oropc n velocy and one-dmenonal n pace phycal problem, n whch, followng a procedure propoed n [], he compuaonal machnery become lgher hank o he reducon of he hree fold velocy negral o negraon on he real lne. Fnally numercal reul for ome llurave pace-homogeneou e cae are preened and brefly dcued.. BGK Equaon We recall and dcu here he man feaure of he relaxaon me approxmaon nroduced n [9] for he chemcal reacon model worked ou

4 6 M. GROPPI AND G. SPIGA [June n [, ]. Model knec equaon read a f f +v x = ν ( M f ) =,...,, () where M a local Maxwellan wh dpoable calar parameer M (v) = n ( m πkt ) ] exp [ m (v u ) KT =,..., () andν henvereofhehrelaxaon me, poblydependngon macrocopc feld, bu ndependen of v. The above auxlary feld n, u, T are deermned by requrng ha he exchange rae for ma, momenum and oal (knec plu chemcal) energy followng from () concde wh hoe deduced from he correpondng Bolzmann equaon. The laer rae are known analycally for Maxwell molecule [], even n he chemcal frame, and may be expreed n erm of ome phycal parameer lke mae m (wh m +m = m +m = M), reduced mae µ r, energe of chemcal lnk E, and energy dfference beween reacan and produc E = = λ E (wh λ =λ = λ = λ = ), convenonally aumed o be pove. Oher eenal parameer are he mcrocopc collon frequence (conan wh repec o he mpac peed g n our aumpon) and π νk r (g) = νr k (g) = πg σ r (g,χ)( coχ) k nχdχ k =,, () ν (g) = πg π where σ and for dfferenal cro econ, and ν r conran on he exchange rae reul n n = n + λ ν Q m n u = m n u + ν r= n KT = n KT m [n u n (u ) ]+ ν σ (g,χ)nχdχ, (6) νr. The above φ r u r + λ ν m uq (7) ψ r T r r=

5 7] A KINETIC RELAXATION MODEL 6 + ν r= ν r µ r m +m rn n r (m u +m r u r ) (u r u ) [ + λ Q ν m u + M m KT + M KT λ ] M m M E, where Γ an ncomplee gamma funcon [6] and Q = ν ( Γ π, E KT ) [ n n ( m m m m ( E KT Γ ) e E KT (, E KT ) ) e E KT n n ]. (8) In addon, he ymmerc ngular marce Φ and Ψ are defned by φ r = ν r µ r n n r δ r ψ r = Kν r µ r l= m +m rn n r δ r K ν l µ l n n l (9) l= ν l µ l m +m ln n l, () and global macrocopc parameer (ncludng ma deny ρ, vcoy enorpandhea fluxq) are expreedn erm of ngle componen parameer by n = n, ρ = m n, u = m n u, ρ = = = nkt = n KT + ρ (u k u k u ku k ), = = [ ] (u p = p + ρ u ) j u u j δ (u k u k u ku k ), = = q = q + p ( ) u j u j + n KT (u u ) = = = + ρ (u k u k)(u k u k)(u u ). = () Acual conerved quane n he Bolzmann collon proce are n number

6 6 M. GROPPI AND G. SPIGA [June of even, and may be choen a hree combnaon of number dene lke n +n, n +n, n +n, he hree componen of he ma velocy u, and he oal nernal (hermal + chemcal) energy nkt + = E n. Th yeld a e of 7 calar exac non-cloed macrocopc conervaon equaon (n +n r )+ x (n u +n r u r ) = (,r) = (,), (,), (,) (ρu)+ x (ρu u+p) = ( ) [( ρu + nkt + E n + x ρu + nkt+ E n )u = = ] +P u+q+ E n (u u) = () = where P = nkti + p he preure enor. Noce ha he hree ndependen deny combnaon conerved by collon would repreen he aomc dene n he cae of reacon (). The e () of exac conervaon equaon correcly reproduced by he relaxaon model () [9], and anoher ndcaon of he robune of h approxmaon he fac, proved agan n [9], ha collon equlbra alo concde wh he acual one, and are provded by a even parameer famly of Maxwellan M (v) = n ( m πkt ) ] exp [ m KT (v u), =,...,, () wh equlbrum dene relaed by he well known ma acon law of chemry n n n n = ( µ µ ) ( ) E exp. () KT Oher nereng feaure are n order for hee BGK equaon, bu are le crucal for he preen purpoe, and wll no be dcued here; he nereed reader referred o [9] for deal. Anoher pon ha wa lef open n [9] he mo convenen choce of he nvere relaxaon me ν, whch n fac wll be maer of fuure nvegaon. In h paper we wll ck o he opon of reproducng he acual average number of collon (regardle f mechancal or chemcal) akng place for each pece, whch

7 7] A KINETIC RELAXATION MODEL 6 lead o ν = ν = ν = ν = r= r= r= r= ν r n r + ( Γ π, E ) ν KT n ν r nr + ( Γ π, E KT ν r n r + ( Γ π, E KT ν r nr + ( Γ π, E KT ) ν n () )( µ µ )( µ µ ) e E KT ν n ) e E KT ν n.. Hydrodynamc Regme Equaon () may be caled, meaurng all quane n erm of ome ypcal value n order o make hem dmenonle. Ung macrocopc cale for pace and me varable, and meaurng relaxaon me n un of a ypcal mcrocopc value, lead o equaon whch look exacly he ame a (), f he ame ymbol reaned for dmenonle varable, wh only he appearance of he Knuden number ε, rao of he mcrocopc o he macrocopc me cale, downar on he rgh hand de f f +v x = ε ν ( M f ) =,...,. (6) The parameer ε mall n collon domnaed regme, and end o zero n he connuum lm we are nereed n. We hall perform a formal Chapman Enkog aympoc analy wh repec o uch mall parameer, o fr order accuracy, n order o acheve Naver Soke lke hydrodynamc equaon a a cloure of he conervaon equaon (). To h end he drbuon funcon f are expanded a f = f () +εf (), (7) and conequenly mlar expanon hold for n, u, T. However, hydrodynamc varable mu reman unexpanded [7], namely n + n r = n () +n r() (,r) = (,),(,),(,)

8 66 M. GROPPI AND G. SPIGA [June u = ρ m n () u () (8) = nt + E n = nt() + = E n () (wh n = = n() and ρ = = m n () no expanded eher), yeldng he conran = n () = n () = n () = n () = n m n () u () + m n () u () =. = = E T() (9) Noce ha nernal energy, a uual, an hydrodynamc feld, bu, conrary o clacal (non reacve) ga dynamc for monoaomc parcle, no amenable only o he ga emperaure T, o ha he laer mu be alo expanded, a T () +εt (). Thee expanon nduce of coure mlar expanon for all varable, n parcular for he auxlary feld and for he Maxwellan M. Themacrocopc collon frequence ν, dependng on he dene n, mu be expanded a well. Equang fnally equal power of ε n (6) yeld o leadng order (ε ) M () (v) f () (v) =, () and o he nex order (ε ) [ ] ν () M () (v) f() (v) = f () +v f() x () where he fr erm of a formal expanon of he me dervave operaor, o be condered a an unknown of he problem... Zero order oluon Equaon (), () are uneay, hghly nonlnear, negro funconal equaon n he unknown f (), nce her negral momen are needed n he defnon of he parameer deermnng he auxlary Maxwellan M (). However, equaon () yeld, n cacade, n () = n (), u () = u (),

9 7] A KINETIC RELAXATION MODEL 67 T () = T (), hen Q () = ν ( Γ π, E T () ) [ n () n () ( m m m m from whch he zero order ma acon law follow We have furher on n () n () n () n () = ( µ µ ) e E T () n () n () ] =, () ) ( ) E exp T (). () φ r() u r() =, =,..., () = from whch, due o he propere of marx φ r() [], u () = u for all and m n () u () =, () and fnally = ψ r() T r() =, =,..., (6) = mplyng, agan for he propere of marx ψ r() [], T () = T () for all. Noce ha all pece hare, o leadng order, he ame drf velocy, equal o he global ma velocy u, and he ame emperaure, equal o he leadng erm of he global emperaure T, n agreemen wh he fac ha u an hydrodynamc varable, wherea T no, and mu be expreed a T () +εt (). In concluon we have f () (v) = M () (v) = n () ( m πt () ) exp [ m T ()(v u) ], (7) for =,...,, wh even free parameer, nce he n () and T () mu be bound ogeher by (). Equaon (7) yeld mmedaely p () = and q () = for all, from whch alo p () = and q () = for he leadng erm of vcoy enor and hea flux. Before gong on o he nex ep, we can elec a unknown for he

10 68 M. GROPPI AND G. SPIGA [June ough Naver Soke ype equaon he even calar varable n (), =,...,, and u, and expre T (), wherever needed, by mean of (). Conervaon equaon may be rewren a ( n () +n r()) + [ (n x () +n r()) u] +ε x ( n () u () +n r() u r()) = (,r) = (,), (,), (,) (ρu)+ x (ρu u)+ x (nt() )+ε x (nt() )+ε x p() = ( ) ρu + nt() + E n () [( = ] + x ρu + nt() + E n )u () +ε (nt () u ) x = +ε ( p () u ) ( ) +ε x x q() +ε x E n () u () = (8) and her cloure acheved f we are able o deermne, reorng o (), = conuve equaon for he quane u (), T (), p () we have furher nt () = n () T (), p () = = = p (), q () = = q (), and q(), for whch + T() = n () u (). (9).. Fr order correcon Sandard manpulaon allow o evaluae he me and pace dervave of f (), and o expre M () a he dervave of M wh repec o ε a ε = ; n h way we oban a formal oluon of () a { f () = f () n ()n() + m T ()u() (v u)+ [ m T ()T() T ()(v u) ] } { ν () f () n () n () + m u T () (v u)+ T () [ m T () T ()(v u) ] } { ν () f () n () m u n () v+ x T () x :v (v u) } + T () [ m T () x T ()(v u) ] v ()

11 7] A KINETIC RELAXATION MODEL 69 where n () = n () + λ ν () Q () m n () u () = m n () u () + ν () n () T() = n () T() + ν () + λ ν () ( [ E Q () M m T () M T() Γ and Q () = n () Q, wh ( Q = ν Γ π, E ) T () {[ n () +n () + n n () n () φ r() u r() () r= ψ r() T r() r= ) e E T () (, E T () ) λ ] M m M E, ( E ) ]( m m ) e E T () m m T () +n () +n ()}. () A paen algebra allow now o recompue he auxlary feld n (), u (), from he drbuon funcon () and o make h oluon effecve T () by ung () and (). Skppng echncal deal, deny feld provde he compably condon n () + x (n() u) = λ n () Q =,...,. () Velocy feld yeld he compably condon u and he algebrac equaon = u +u x = ρ x (nt() ) () φ r() u r() = x (n() T () ) ρ() ρ x (nt() ), =,...,. () Th e of equaon he ame whch are when he Chapman Enkog algorhm appled o he reacve Grad momen equaon [], and, a he ame me, concde wh he reul obaned n [7] for a chemcally

12 6 M. GROPPI AND G. SPIGA [June ner mxure (apar from he upercrp (), no neceary here). The marx φ r() ngular, bu () unquely olvable when coupled o he conran (). The oluon, mua muand, goe hrough he ame ep of eher [7] or [], and may be ca a u () = r= L r() ρ () ρ r() x (nr() T () ) (6) where L r() a uable marx, dependng on he n (), whch can be proved o be ymmerc [7], reproducng hu he Onager relaon [8]. The expreon gven by (6) alo concde wh clacal expreon for dffuon veloce of ner mxure [9, ], he only dfference beng he preence of n () nead of he acual dene n. Indeed, marx L r() conrbued only by elac caerng and ndependen from he chemcal reacon and from E, and number dene would be hydrodynamc feld n he non reacve cae. Fnally, emperaure feld recompued from () yeld, afer ome algebra, a e of lnear algebrac equaon for he T () wh marx coeffcen ψ r() (no hown here for brevy, nce uele for our purpoe) plu he compably condon T () +u T() x + T() x u = n Q E n (). (7) Now T () can be evaluaed from () and T () he dervave n (), a T () = (T() ) E may be ca n erm of λ n () n (), (8) = whch mple an addonal compably condon beween () and (7), deermnng n () n erm of he choen unknown and of her paal graden. Bearng he fr of (9) n mnd, and reorng o () n order o elmnae unneceary paal graden, he calculaon yeld T () = [+ T(){ Q ( T ()) n ]} E n () u, (9) x and nereng o remark ha concde wh he emperaure correcon found n [] for he ame phycal problem by reorng o he Grad =

13 7] A KINETIC RELAXATION MODEL 6 momen mehod. Of coure h correcon pecfc for he chemcal reacon () and would no appear n any non reacve ga mxure. In parcular, T () would vanh n he lm E... Vcou re and hea flux In prncple, he aympoc problem o fr order for he drbuon funcon olved, nce eay o ee ha he algebrac yem for he T () unquely olvable a well, and ha alo all zero order me dervave can be made explc. However, n order o acheve he dered hydrodynamc equaon from (8), uffce o compue p () and q () by uable negraon of he drbuon funcon. More precely we have P () =m R (v u )(v j u j )f () (v)d v, p () =P () δ rp() () and q () = T() n () u () + R m (v u )(v u) f () d v () o be ued hen n (9). When compung P (), no dffcul o check ha he addend of f () nvolvng he fr order correcon yeld a enor proporonal o he deny, whch conrbue nohng o he devaorc par p (), and he ame occur o he econd addend, nvolvng he operaor. For he hrd addend, nvolvng paal graden, he ame feaure n order for he graden of n () and T (), wherea he rae of ran enor conrbue a erm n() T () ν () [δ ( u x + x u ) ( u +( δ ) + u )] j x j x () where he quare bracke he um of u x j + u j x and of an oropc enor. Gong on and compung p () and p (), one end up wh p () = T () = n () ν () ( u + u j x j x x uδ ). () Th Newonan conuve equaon correpond o a vcoy coeffcen µ = T () = n () ν (), ()

14 6 M. GROPPI AND G. SPIGA [June exacly he ame obaned n [7] from he BGK equaon for a chemcally ner ga mxure. An expreon of he ame ype wa obaned for he reacve cae n [] by he Grad mehod, bu wh a dfferen vcoy coeffcen, ha wa provded here by formal nveron of uable marce. Pang o (), we may pl agan f () n hree dfferen addend and evaluae eparaely he relevan conrbuon. The fr one, wh fr order correcon, leave afer negraon only he erm T() n () u (). In a mlar fahon, from he econd addend we have only T()n() ν () u. More conrbuon come from he paal graden, and hey can be pu ogeher a T () T () m ν () o ha here reul n () q () = n () T () ν () n () T () x T()n() m ν () [ u ν () u u n() T () T () x m ν (), x +u u x + ] ρ () (n () T () ) x T () x + T() n () (u () u() ) () and, upon ung he econd of () for u (), and () and () for he quare bracke, we end up mply wh In concluon, from (9) q () = T() q () = = n() T () T() m ν (). (6) x n () m ν () T () x + T() = a Fourer conducon law wh a hermal conducvy λ = T() = n () m ν () n () u (), (7). (8) Once more, h reul concde wh he correpondng one for he ame BGK raegy appled o a non reacve mxure [7], and reproduce he rucure of hea flux for he reacve cae a obaned n [], where agan conducvy wa gven by he nveron of ceran marce.

15 7] A KINETIC RELAXATION MODEL 6.. Concluon Summarzng our reul, hydrodynamc equaon of Naver Soke ype for he preen relaxaon model of he chemcal reacon () n a ga mxure provded by he e (8) of even paral dfferenal equaon for he even calar unknown n () and u, coupled o he rancendenal algebrac equaon () for T (), and o he conuve equaon (6) for u (), (9) for T (), () for p (), and (7) for q(). All ogeher, hey read a ( n () +n r()) + [ (n () +n r()) u] +ε ( n () u () +n r() u r()) = x x (,r) = (,), (,), (,) (ρu)+ x (ρu u)+ x (nt() )+ε x (nt() )+ε x p() = ( ρu + nt() + E n ()) + [( x ρu + nt() + E n ()) ] u = +ε x (nt () u ) +ε x (p () u ) +ε x q() +ε x n () n () n () n () = u () = r= ( µ µ ) ( ) E exp T () L r() ρ () ρ r() x (nr() T () ) = ( E n () u ()) = = T () = [+ T(){ Q ( T ()) n ]} E n () x u = p () = T () n () ( u + u j ) x j x x uδ = q () = T() ν () = n () m ν () T () x + T() = n () u (). (9) Euler equaon correpond o he lmng cae ε =. Th BGK aympoc lm ha he ame form of he Bolzmann aympoc lm worked ou n [] va a Grad momen expanon, and only he defnon of vcoy coeffcen and hermal conducvy dffer n he wo approache. The preen reul dffer nead ubanally from he BGK aympoc lm ha would be n order f he chemcal reacon were wched off, ha wa

16 6 M. GROPPI AND G. SPIGA [June horoughly derved and dcued n [7]. In uch a cae n fac he kernel of he collon operaor egh dmenonal, and all dene, a well a he emperaure, are hydrodynamc varable, o ha Naver Soke equaon are made up by egh paral dfferenal equaon, ncludng connuy equaon for each pece. In oher word, alo wh Bolzmann model, he lnearzed collon operaor n he preen regme conan erm accounng for reacve collon, a oppoed o he one ha would be obaned f chemcal reacon were aben, or were condered a much lower proce. I remarkable however ha all fr order correcon needed for he cloure (excep, of coure, he emperaure correcon, or addonal reacve calar preure, whch pecular of he chemcal reacon [, ]) are expreed by conuve equaon whch are, mua muand, he ame n he reacve and n he non reacve cae. Th fac reproduce known reul eablhed a he Bolzmann level []. A he BGK level, here are addonally reacve conrbuon n he ranpor coeffcen hrough he facor ν n (), whch are affeced by he reacve collon frequency ν. Thee conrbuon however affec only vcoy and hermal conducvy hrough ν () n () and (8).. Axally Symmerc Problem A aed n he Inroducon, we pecalze now our BGK equaon (), (), (7) o problem wh axal ymmery wh repec o an ax (ay, x x), n he ene ha all ranvere paal graden vanh, and he ga drfng only n he axal drecon. In oher word, drbuon funcon f depend on he full velocy vecor v (molecular rajecore are hree dmenonal) bu dependence on he azmuhal drecon around he ymmery ax uch ha all ranvere componen of he macrocopc veloce u vanh. Th occur, for nance, when he drbuon funcon depend on v only hrough modulu and laudnal angle wh repec o ha ax. A well known n he leraure [], h allow a enble mplfcaon of he calculaonal apparau, and a praccal reducon o a fully one dmenonal problem, hough decrbng ll a hree dmenonal velocy pace. On he oher hand, h knd of problem no only mporan for heorecal nvegaon, bu alo que frequen n praccal applcaon: uffce o recall here he clacal evaporaon condenaon problem []. I mmedaely een ha he prevou aumpon mply u = u =,

17 7] A KINETIC RELAXATION MODEL 6 and hen u = u = and u = u = (for all ). I convenen o nroduce new unknown φ = f dv dv, ψ = (v +v)f dv dv, () (negraon range from o + ), each dependngonly on one pace and one velocy varable. Of coure, φ and ψ provde a reduced decrpon of he velocy drbuon, f compared o f, bu hey uffce for everal purpoe, a hown below. Droppng he ubcrp alo from veloce, he fundamenal macrocopc parameer are n fac deermned by φ and ψ a n = φ dv, u = n vφ dv, Seng KT m = n [ M (v) = n ( m πkt (v u ) φ dv + ] ψ dv. () ) / ] exp[ m (v u ), () KT mulplcaon of () by and (v + v ) and negraon wh repec o (v,v ) R yeld hen he par of BGK equaon φ +v φ x = ν (M φ ) ψ +v ψ x = ν ( KT m M ψ ), () whch are acually coupled, nce parameer n, u, T appearng n () follow from (7) and from (). Collon equlbra are gven by φ = M, ψ = KT m M, where M (v) = n ( m πkt ) / ] exp[ m (v u). () KT I now a fve parameer famly of Maxwellan, wh dene relaed by he ma acon law (). We have proceed numercally equaon () n ome mple pace homogeneou e cae, manly for llurave purpoe, whou havng n mnd a pecfc reacve mxure, all quane beng meaured n u-

18 66 M. GROPPI AND G. SPIGA [June able cale. Momen of he drbuon funcon φ and ψ appearng n () have been numercally evaluaed by mean of he compoe rapezodal quadraure rule on a uffcenly large bounded velocy nerval [ R, R]. A reference problem (Problem A) we have aken an oropcally caerng mxure (hen ν r = νr ) where he mcrocopc par collon frequence ν r are gven by Table. Table. Elac collon frequence ν r. ν r The reacve endohermc chemcal collon frequency nead ν =., namely he ypcal chemcal collon me abou one order of magnude larger han he ypcal caerng collon me. Th done merely n order o pobly eparae n he numercal oupu he effec due o mechancal encouner from hoe due o chemcal reacon, and underood ha we reman n he phycal regme decrbed n he Inroducon [, ]. Mae ake he value m =.7, m =.6, m = 8, m = 7., () choen ju a an llurave example, and he energy dfference n he chemcal bond E =. Inal condon have been eleced a Maxwellan hape characerzed by he parameer n () = 9 n () = n () = n () = u () = u () = u () = u () = 6 T () = T () = T () = T () =. (6) Thee value deermne unquely, va he fve dpoable ndependen fr negral (n +n, n +n, n +n, u, and nt + = E n ) and he ma acon law (), he unque Maxwellan equlbrum, decrbed by (), wh parameer whch urn ou o be n =.688 n =.687 n =. n = 9. u =.96 T =.. (7)

19 7] A KINETIC RELAXATION MODEL 67 Evoluon of all macrocopc parameer can be deduced from () upon negraon of he numercally compued φ and ψ. A dcued n [9], uch evoluon correcly reproduced by he preen BGK equaon, and ndeed our reul concde o compuer accuracy wh he one of []. Typcal rend are hown n Fgure and. Equalzaon of veloce and emperaure due o elac caerng, and acually occur on he hor (mechancal) cale. Relaxaon of dene n and emperaure T (non conerved quane) due nead o chemcal neracon, and ruled n 7 6 u 9.. Fgure. Trend of dene n (lef) and of veloce u (rgh) veru me for problem A... T T Fgure. Tme evoluon of emperaure T on he hor (lef) and on he long (rgh) me cale for Problem A.

20 68 M. GROPPI AND G. SPIGA [June by he longer reacve characerc me. In he condered problem we oberve an overall ranformaon of produc no reacan, and a correpondng emperaure ncreae due o ranformaon of energy from chemcal o hermal. In general, overhoo or underhoo of ome pece emperaure can occur, a ndcaed by Fgure. 8 6 φ - - v 7 Fgure. Funcon φ veru and v for Problem A. Relaxaon of he drbuon funcon φ oward he equlbra () lluraedbyfgure, relevano =, onhe(v,)plane. Aexpeced, he nal hape ge rongly modfed n a fr hor nal me layer, domnaed by mechancal encouner, hen a reored, bu dfferen, Maxwellan evolve moohly, a he chemcal pace, oward he equlbrum M, followng he low me varaon of n and T, and wh a praccally conan u. The approach of φ o he aocaed local Maxwellan ( m M = n ) / ] πkt exp[ m KT (v u ), (8) where n, u, T are he acual me dependen macrocopc parameer, well depced by he devaon φ M veru v and, hown n Fgure for =. The devaon zero nally, becaue of he nal Gauan hape, hen undergoe pove and negave varaon, whch reman confned however boh n amplude (order one enh) and n doman (a neghborhood of

21 7] A KINETIC RELAXATION MODEL 69 he varyng macrocopc velocy). In addon, he devaon vanhe dencally n a very hor me, of he order of he mechancal relaxaon me, and even maller han he velocy or emperaure equalzaon me.. φ M v 7 Fgure. Zoom of he devaon φ M veru and v for Problem A. 6.. u u u.,u u.,u.. u u Fgure. Dfference beween acual and auxlary mean veloce (u and u repecvely) veru me: =, (lef), =, (rgh) for Problem A. Fnally, a meaure of he approach o equlbrum provded alo by he dfference beween he acual and he auxlary macrocopc feld, whch mu relax o zero for ncreang me. We exhb here he rend of he veloce u and u n Fgure. Dfference are agan raher mode, and end quckly o vanh, on he fa mechancal cale. A predced by he

22 6 M. GROPPI AND G. SPIGA [June BGK equaon hemelve, we have u < ( u > ) for u < u (u > u ); n parcular, we can oberve ha crong of he acual and fcou curve may occur only for he pece wh a non monoonc rend for u (.e., =,), and acually occur when u =. φ 8 φ 8 - v 7 - v 6 6 φ φ - - v v Fgure 6. Inal and fnal velocy profle for φ n Problem B. We nexexamne hereponeof ourrelaxaon model o achange of nal hape from a Gauan o a bmodal drbuon, a depced n Fgure 6 (Problem B). Such drbuon are made up by wo dencal ymmercally dplaced Maxwellan, choen n uch a way ha he macrocopc nal condon reman he ame of Problem A, a gven by (6). Th requremen deermne hem unquely, once he peak eparaon u agned. We have aken here u =.8 u = u = u =. (9) Snce equaon (6) are unchanged wh repec o Problem A, evoluon of all macrocopc varable reman he ame, a well a he fnal equlbrum drbuon M, whch are alo gven n Fgure 6. The hree dmenonal

23 7] A KINETIC RELAXATION MODEL 6 plooffgure7howφ veruv and; aferarapdranonoaunmodal hape, evolve eenally n he ame way a for Problem A (ee Fgure ). Fgure 8, o becompared o Fgure, repor on he devaon of φ from he local Maxwellan M for ProblemBon hehormecale. Ican benoced ha relaxaon o local hermodynamcal equlbrum occur eenally on he ame me cale a for Problem A, bu dfference are larger by almo wo order of magnude, due o rong devaon of he nal hape from a Gauan ( almo a double ream drbuon for h pece). 7 6 φ - v 8 Fgure 7. Funcon φ veru and v for Problem B. φ M v 8 Fgure 8. Zoom of he devaon φ M veru and v for Problem B.

24 6 M. GROPPI AND G. SPIGA [June u 7 6 T Fgure 9. Trendofveloce u (lef)andofemperauret (rgh) veru me for problem C. Fnally, we have condered a hrd e cae (Problem C) n order o ee he repone of our model o anoropy of caerng. For h purpoe we change, wh repec o Problem A, only he mcrocopc collon frequence for momenum ranfer ν r, by akng a unform reducon facor,.e. ν r = νr, (,r), whch replace he equaly νr = ν r relevan o oropc caerng. In h way, caerng que forwardly peaked (manly grazng collon), whch mean ha, hough mechancal encouner are ll much more frequen han chemcal reacon, momenum and energy ranfer n elac caerng become much le effecve, and hen all relaxaon phenomena drven by mechancal collon become lower and almo comparable o chemcal relaxaon. Th ndeed wha we acually verfed numercally, and lluraed by Fgure 9, where we plo veloce and emperaure veru me, and realze ha equalzaon me have ncreaed, roughly peakng, by a facor of wh repec o he Problem A (ee Fgure and ). The lower rae of varaon of macrocopc parameer ha hen nereng mplcaon a he knec level. In fac, n agreemen wh a lower magnude of me dervave, we oberve a enble decreae of he dfference beween acual and auxlary macrocopc feld. An example provded by Fgure, where we plo for comparon he emperaure T and T for Problem A and for Problem C (noce agan he dfferen me cale for he wo problem). A a conequence of he much maller devaon of fcou feld from he acual one, he auxlary Maxwellan M oward whch our collon relaxaon model force parcle drbuon are much cloer now o he acual local Maxwellan M han for he reference

25 7] A KINETIC RELAXATION MODEL 6 Problem A Problem A T T T T.. Problem C Problem C T T T T Fgure. Comparon of acual and auxlary emperaure (T and T repecvely) veru me for problem A (above) and Problem C (below).. φ M v 6 8 Fgure. Zoom of he devaon φ M veru and v for Problem C.

26 6 M. GROPPI AND G. SPIGA [June problem. Devaon of drbuon funcon φ from he correpondng localmaxwellanarehenmallerhanforproblema.thhown, for =, n Fgure, where he dfference, nally zero becaue of he Gauan nal hape, end agan o zero (correpondng o local equlbrum) on a hor me cale whch lghly longer han for Problem A, bu undergo flucuaon ha reman a lea one order of magnude maller han n Fgure. Reference. C. Cercgnan, Rarefed Ga Dynamc. From Bac Concep o Acual Calculaon, Cambrdge Unvery Pre, Cambrdge,.. V. Govanggl, Mulcomponen Flow Modelng, Brkhäuer Verlag, Boon, M. Gropp and G. Spga, Knec approach o chemcal reacon and nelac ranon n a rarefed ga, J. Mah. Chem., 6(999), P. L. Bhanagar, E. P. Gro and K. Krook, A model for collon procee n gae, Phy. Rev., 9(9), -.. P. Welander, On he emperaure jump n a rarefed ga, Ark. Fy., 7(9), V. Garzó, A. Sano, J. J. Brey, A knec model for a mulcomponen ga, Phy. Flud A: Flud Dynamc, (989), P. Andre, K. Aok and B. Perhame, A conen BGK-ype model for ga mxure. J. Sa. Phy., 6(), R. Monaco and M. Pandolf Banch, A BGK ype model for a ga mxure wh reverble reacon, In New Trend n Mahemacal Phyc, World Scenfc, Sngapore,, M. Gropp and G. Spga, A Bhanagar Gro Krook ype approach for chemcally reacng ga mxure, Phy. Flud, 6 (), M. B, M. Gropp and G. Spga, Grad drbuon funcon n he knec equaon for a chemcal reacon. Connuum Mech Thermodyn., (), 7-.. G. Ludwg and M. Hel, Boundary layer heory wh docaon and recombnaon. In Advance n Appled Mechanc, vol. 6, Academc Pre, New York, 96, A. Ern and V. Govanggl, The knec chemcal equlbrum regme, Phy. A, 6(998), M. B, M. Gropp and G. Spga, Flud dynamc equaon for reacng ga mxure, Appl. Mah., (), -6.. K. Aok, Y. Sone and T. Yamada, Numercal analy of ga flow condenng on plane condened phae on he ba of knec heory, Phy. Flud A: Flud Dynamc, (99), A. Roan and G. Spga, A noe on he knec heory of chemcally reacng gae, Phy. A, 7(999), 6-7.

27 7] A KINETIC RELAXATION MODEL 6 6. M. Abramowz and I. A. Segun, Ed. Handbook of Mahemacal Funcon, Dover, New York, C. Cercgnan, The Bolzmann Equaon and Applcaon, Sprnger Verlag, New York, S. R. De Groo and P. Mazur, Non-equlbrum Thermodynamc, Norh Holland, Amerdam, S. Chapman and T. G. Cowlng, The Mahemacal Theory of Non unform Gae, Unvery Pre, Cambrdge, 97.. J. H. Ferzger and H. G. Kaper, Mahemacal Theory of Tranpor Procee n Gae, Norh Holland, Amerdam, 97.. I. Samohyl, Comparon of clacal and raonal hermodynamc of reacng flud mxure wh lnear ranpor propere, Collecon Czecholov. Chem. Commun., (97), -.. Y. Sone, Knec Theory and Flud Dynamc, Brkhäuer Verlag, Boon,. Dparmeno d Maemaca, Unverà d Parma, Vale G. P. Uber /A, Parma, Ialy. E-mal: mara.gropp@unpr. Dparmeno d Maemaca, Unverà d Parma, Vale G. P. Uber /A, Parma, Ialy. E-mal: gampero.pga@unpr.