HYDRODYNAMIC LIMIT FOR A GAS WITH CHEMICAL REACTIONS

Transcription

1 October 3, 003 8:48 WSPC/Trm Sze: 9n x 6n for Proceedng bp HYDRODYNAMIC LIMIT FOR A GAS WITH CHEMICAL REACTIONS M. BISI Dpartmento d Matematca, Unvertà d Mlano, Va Saldn 50, 0133 Mlano, Italy, E-mal: b@mat.unm.t G. SPIGA Dpartmento d Matematca, Unvertà d Parma, Va D Azeglo 85, Parma, Italy, E-mal: gampero.pga@unpr.t A frt approach to the hydrodynamc lmt for a ga mxture undergong moderately fat bmolecular reacton preented by a Chapman-Enkog aymptotc procedure appled to Grad 13 moment equaton. 1. Introducton Knetc formulaton of chemcal reacton n rarefed ga mxture a cumberome but eental tak whch beng gven ome attenton n the lterature. 1, One of the man output expected from the knetc approach dervaton of macrocopc equaton for the man obervable feld. Typcal tool ued for uch a purpoe nclude aymptotc method (Chapman- Enkog algorthm and moment equaton (utable cloure of a ere expanon. Analytcal manpulaton are qute awkward for a mxture even n the abence of chemcal reacton. However, t ha been poble to determne the general tructure of the Newtonan conttutve equaton and of the reultng hydrodynamc (Naver-Stoke equaton va aymptotc analy veru the mall parameter conttuted by the dmenonle molecular mean free path (Knuden number. 3 On the other hand, the Grad 13-moment approach 4 can be extended to deal wth ga mxture, 5 though explct expreon for coeffcent are avalable, to our knowledge, only for Maxwellan molecule. 6 To uch a cae we hall confne for mplcty alo n the preent paper, whch dealng wth the knetc model for a b- 1

2 October 3, 003 8:48 WSPC/Trm Sze: 9n x 6n for Proceedng bp molecular chemcal reacton propoed n Ref. 7. After calng the relevant Boltzmann-type ntegrodfferental equaton, two Knuden number pontaneouly are, repreentng the rato of a typcal elatc and, repectvely, chemcal relaxaton tme to the choen macrocopc tme cale. Mot of the pertnent lterature devoted to the cae of low chemcal reacton, n whch the elatc relaxaton tme much horter than the chemcal one, and the elatc collon operator play thu the domnant role n the evoluton proce. Th the cenaro n whch a mple cloure at the Euler level provde macrocopc equaton decrbng the chemcal reacton on t proper (low cale, after the ntal fat tranent n whch mechancal equlbrum ha been reached (ee for ntance Ref. 8. Here we am at conderng phycal tuaton n whch the chemcal reacton alo become fat wth repect to the macrocopc cale, but yet the proce manly drven by elatc catterng. Th can be modeled wthout lo of generalty, by takng elatc and chemcal relaxaton tme to be O(ɛ and O(ɛ, repectvely, wth repect to the typcal macrocopc tme, where ɛ a utable mall parameter. In th prelmnary approach we are ntereted only n the frt order correcton (namely O(ɛ to the reactve Euler equaton. Next order (O(ɛ correcton wll hopefully be condered n a future paper. Our tartng pont the et of 5 Grad equaton a worked out recently n Ref. 9. It known n knetc theory that the Chapman-Enkog procedure much eaer to apply to the Grad equaton rather than to the knetc equaton, and yeld the ame reult, at leat for Maxwellan molecule. 6 It reaonable to expect the ame n the preent chemcal frame, though the nvetgaton of the cumberome algorthm leadng from the Boltzmann to the hydrodynamc level alo left a future work. Formulaton of the problem and governng equaton are preented n Sec.. The aymptotc lmt for ɛ 0 then performed n Sec. 3 accordng to Chapman and Enkog after dcung the proper hydrodynamc varable to be ued.. Chemcal reacton model We conder a four ga mxture whoe molecule A, = 1,..., 4 nteract by bnary elatc collon and by the bmolecular reacton A 1 + A A 3 + A 4 (1 where, f E the nternal energy of chemcal lnk, we may alway aume = 4 λ E > 0, where λ 1 = λ = λ 3 = λ 4 = 1. The ntereted reader referred to Ref. 9 for all detal that wll have

3 October 3, 003 8:48 WSPC/Trm Sze: 9n x 6n for Proceedng bp 3 to be kpped here for brevty. Unknown for Grad equaton to be tuded are number dente n, mean velocte u, temperature T, vcou tree p j and heat fluxe q. Global quantte for the mxture are labeled by the ame ymbol wthout any upercrpt. Mae are denoted by m, wth M = m 1 + m = m 3 + m 4, and µ r for the reduced ma of the (, r par. For the phycal tuaton decrbed n the Introducton, caled Grad equaton read a 9 n + (n u = 1 x ɛ Q, (a ( u m n + u u j + (n + p j = 1 x j x x j ɛ R + 1 ɛ R, (b ( 3 ( n + u p j ( + n u + p u j j + q = 1 x x x x ɛ S + 1 ɛ S, (c + ( u (u k p x j + n + u j k x j u x k 3 δ j p u l kl + x k 5 ( q x j + q j u k x 3 δ j x k q k + p u j k x k + p jk x 3 δ j x k (d q + (u j q + 7 u x j 5 q j + u x j 5 q j j + u x 5 q j + 5 x j n ( x m + 5 ( p j x j m + n ( p j x j m n p j p kj m n = 1 x k ɛ W + 1 ɛ W, (e where the elatc collon term are gven by R = n ν1 r µ r n r (u r u, (3a S = n V j = ν r 1 µ r m + m r nr ν1 r 1 ]} 3 δ j(u r u + m n m r n r [ = 1 ɛ V j + 1 ɛ V j, [ 3 K(T r T + 1 ] mr (u r u, (3b µ r [ n m + m r p rj n r p j + n n r m r 3 ν r m r (m + m r (u r u (u r j u j 1 ]} 3 δ j(u r u (u r u (u r j u j m n p r j + m r n r p j, (3c

4 October 3, 003 8:48 WSPC/Trm Sze: 9n x 6n for Proceedng bp 4 n W = 5 m + m r n r (m + m r 3 ν1 r µ r n r (u r u 1 m p j [ 1 (ur u (β r + 3 β3 r n + 5 m n ] m r n r βr 4 n r r + m n β4 r (u r u + β r 1 q + m n m r n r βr 4 q r + (u r j u j ν r 1 µ r n r (u r j u j ( 1 βr p j + m n m r n r βr 4 pj } r. (3d Under our aumpton and for our purpoe, t tll content to ue, for the chemcal collon term, the cloe to mechancal equlbrum expreon derved n Ref. 9, namely [ ( m Q = λ Q 1 = λ n 3 n 4 1 m S = λ Q 1 3 m ( m 3 m 4 3 e n 1 n ] ν 34 1 R = λ m (u u Q 1, M 1 (1 λ M m M 3 ( + 1 m (u u + ( M m M W = λ Q 1 ( 5 Γ, ( 3 Γ π,, (4a [ Γ (4b ( 3, ] } 1, (4c V j = λ m Q 1 [ (u u (u j u j 1 3 δ j(u u ], (4d 5 (u u ( m M (1 λ (u u M m M + 5 ( (u u + p j n (u j u j 1 m (u u (u u 5 3 (u u ( M m M Γ ( 5, [ Γ ( 3, ] } 1. (4e

5 October 3, 003 8:48 WSPC/Trm Sze: 9n x 6n for Proceedng bp 5 The coeffcent ν1 r, ν r, βk r and ν1 34 denote utable angle ntegrated mcrocopc collon frequence for elatc (, r catterng or for the drect (endothermc reacton, whch turn out to be ndependent of molecular velocte n our hypothe. 9 Γ tand fnally for ncomplete gamma functon. Other ymbol to be ued later are the preure tenor P j = n δ j + p j and the total (thermal plu chemcal energy denty U = 3 n + 4 E n. The model ( (4 hare everal crucal properte wth the actual knetc equaton. In partcular, the ame even macrocopc conervaton equaton (correpondng to the even collon nvarant, whch read a (n + n r + x (n u + n r u r = 0 (, r = (1, 3, (1, 4, (, 4 (ρ u + (ρ u u + P = 0 x ( [ (1 1 ρu + U + x ρu + U u + P u + q + ] E n (u u =0. (5 Of coure, they do not keep trace of collon term (thu, are ɛ ndependent, and are exact but not cloed. The Euler cloure acheved, a uual, by replacng undered moment by ther equlbrum value. Now thermodynamc equlbrum characterzed (ee Ref. 9 by all equal mean velocte and temperature (u = u, T = T, all vanhng vcou tree and heat fluxe (p j = 0, q = 0, and dente related by the ma acton law 7 n 1 n n 3 n 4 = ( µ 1 µ 34 3 e. (6 Choong to play wth the 8 macrocopc feld n, u, T, reactve Euler equaton are then made up by the 7 partal dfferental equaton followng from (5 (n + n r + x (ρ u + x ( 1 ρ u + U [ ] (n + n r u = 0 (, r = (1, 3, (1, 4, (, 4 (ρ u u + x (n = 0 + x [( 1 ρ u + U + n ] u = 0, (7

6 October 3, 003 8:48 WSPC/Trm Sze: 9n x 6n for Proceedng bp 6 to be coupled to the algebrac equaton (6. Th dffer gnfcantly from the cae of low chemcal reacton, where the above feld actually conttute the 8 elatc collon nvarant, and, correpondngly, reactve Euler equaton are 8 partal dfferental equaton ncludng chemcal contrbuton. 8 The latter fact ha bearng alo on the Chapman-Enkog procedure to be developed n the next Secton. There are ndeed only 7 hydrodynamc varable, to be ued a unknown feld and to be kept unexpanded. They are gven by n 1 + n 3, n 1 + n 4, n + n 4, u, and U. 9 Th wll put ome contrant on the frt order aymptotc expanon n = n (0 + ɛ n (1 p j = p (0 j u = u (0 + ɛ p (1 j to be ntroduced nto the Grad equaton. + ɛ u (1 q = q (0 T = T (0 + ɛ T (1 + ɛ q (1 (8 3. Hydrodynamc lmt Proceedng tep by tep, the O(ɛ term from ( yeld R (0 = 0, S (0 = 0, V (0 j = 0, W (0 = 0. (9 The frt, for fxed, a lnear homogeneou algebrac ytem for u (0, wth matrx element Φ r = ν r 1 µ r n (0 n r(0 δ r l=1 ν l 1 µ l n (0 n l(0. (10 Due to the properte of the matrx Φ r etablhed n Ref. 10, there ext exactly 1 oluton, namely u (0 = u r(0, r. Snce u mut reman unexpanded, 6 we get then u (0 = u. The econd equaton n (9 agan a lnear homogeneou algebrac ytem for T (0, wth matrx element Ψ r = 3K ν1 r µ r m + m r n(0 n r(0 δ r 3K ν1 l µ l m + m l n(0 n l(0. (11 Matrx Ψ r ha the ame properte of Φ r, 10 and then there are 1 oluton, namely T (0 = T r(0, r, from whch T (0 = T (0. The thrd of (9, for fxed (, j, a lnear homogeneou algebrac ytem for p (0 j, where the matrx of coeffcent Υ r (omtted here for brevty ha non zero determnant, 10 o that only the trval oluton p (0 j = 0 left. The ame occur to the fourth of (9, for fxed, veru the q (0, wth a regular matrx of coeffcent Ξ r, 10 and then wth unque oluton q (0 = 0. l=1

7 October 3, 003 8:48 WSPC/Trm Sze: 9n x 6n for Proceedng bp 7 Untl now, the tuaton eentally the ame a for low chemcal reacton. 10 The dfference are n the O(ɛ 1 term, whch yeld Q (0 = 0, V (0 j R (0 The frt provde at once + R (1 = 0, S (0 + S (1 = 0, + V (1 j = 0, W (0 + W (1 = 0. n 1(0 n (0 n 3(0 n 4(0 = ( µ 1 µ 34 (1 3 e (0 (13 whch jut the zero-th order approxmaton of the ma acton law (6. From (4, t follow then mmedately R (0 = S (0 = V (0 j = W (0 = 0, o that the other four equaton n (1 nvolve merely the mechancal term and turn out to be the ame a (9, only wth dfferent unknown. Bearng n mnd prevou reult, we may deduce then that all drft velocte u (1 mut take, for fxed, the ame common value, whch necearly zero. For temperature we get analogouly T (1 = T 1, but, contrary to the low reacton cae, T not an hydrodynamc varable, and T 1 mut concde wth the frt order correcton T (1 to the ga temperature T = T (0 +ɛ T (1. A for p (1 j and q (1, they mut all vanh. Fnally, nvokng the fact that the quantte n 1 + n 3, n 1 + n 4, n + n 4, U mut be kept unexpanded, t readly obtaned n (1 = λ n 1(1, n 1(1 = 3 n K T (1. (14 Then, to frt order n our calng, nether Onager relaton nor Newtonan conttutve equaton appear. Dfference n the drft velocte, vcoty coeffcent, thermal conductvty would are a econd order effect. In addton to u, one may chooe a unknown feld the quantte n (0, = 1,..., 4 (whch determne alo T (0 by vrtue of (13, nce they are n a one to one relatonhp wth the hydrodynamc varable va n r + n = n r(0 + n (0 (, r = (1, 3, (1, 4, (, 4 U = 3 n (0 + E n (0. (15 The ought aymptotc cloure of (5 acheved thu by mply expreng T (1 n term of the prevou varable, whch make the tenor P explct a nk(t (0 + ɛ T (1 tme the dentty tenor, wth q = 0 and u = u. The only correcton to the Euler equaton an addtonal calar preure, a ponted out already on phenomenologcal ground. 11

8 October 3, 003 8:48 WSPC/Trm Sze: 9n x 6n for Proceedng bp 8 We pa then to the next tep, namely to O(1 term, and t uffce to conder the equaton for number dente and temperature,.e. Q (1 = n(0 S ( + S (1 = 3 n(0 ( ( (0 It eay to check that S ( = 0, and + (n(0 u x + u ( (0 x Q (1 = λ n 1(1 Q = λ n 1(1 ν 34 1 (16a + n (0 (0 u x. ( 3 Γ π, (0 (16b [( n 3(0 + n 4(0 + n 3(0 n 4(0 ( (µ ] n (0 µ 34 e (0 + n 1(0 + n (0 (17a S (1 = λ n 1(1 S = λ n 1(1 M m Q 3 M (0 1 (1 λ ( [ ( ] } (0 Γ, 3 Γ (0,. (0 Summng up Eq. (16b wth repect to, a lttle algebra yeld Q n 1(1 = 3 n 0( (0 + 3 n u ( (0 x (17b + n (0 u x. (18 Equaton (18 and (16a, whoe unknown are the functon n 1(1 and the operator 0, mut be compatble. The relaton between the zero-th order tme dervatve of T (0 and of the n (0 provded by (13 0 ( (0 = ( (0 λ 1 0 n (0. (19 n (0 In th way the 0 operator may be elmnated, yeldng a lnear algebrac equaton for n 1(1, whoe unque oluton provde the ought unknown T (1, 3 n u T (1 = 3 n K ( (0 Q [ n ( (0 λ n (0 + 3 n (0 x n u 1 n (0 ( (0 + n x ]} 1 (0 u x }. (0

9 October 3, 003 8:48 WSPC/Trm Sze: 9n x 6n for Proceedng bp 9 The O(ɛ correcton term to the preure are then affected by the patal gradent of dente and temperature, and by the dvergence of the ma velocty. Explctly, our hydrodynamc lmt read a ( n (0 + n r(0 + x (ρ u + x ( 1 ρ u + U (ρ u u + x + x [( n (0 + n r(0 u ] = 0 (, r = (1, 3, (1, 4, (, 4 [ ( nk T (0 + ɛ T (1] = 0 [( 1 ρ u + U + n (0 + ɛ n (1 ] u = 0 (1 wth U, T (0, and T (1 gven repectvely by (15, (13, and (0. Euler equaton are recovered for ɛ = 0. Second order pace dervatve do appear n (1 becaue of the conttutve equaton (0 for the temperature correcton T (1 ; uch term are all affected by the multplcatve mall parameter ɛ. Acknowledgment Th work wa performed n the frame of the actvte ponored by MIUR, by GNFM, and by the Unverty of Parma (Italy, and by the European TMR Network HYKE. Reference 1. V. Govanggl, Multcomponent Flow Modelng, Brkhäuer, Boton C. Cercgnan, Rarefed Ga Dynamc. From Bac Concept to Actual Calculaton, Cambrdge Unverty Pre, Cambrdge (U.K J. H. Ferzger and H. G. Kaper, Mathematcal Theory of Tranport Procee n Gae, North Holland, Amterdam H. Grad, Comm. Pure Appl. Math., 331 ( T. I. Gombo, Gaknetc theory, Cambrdge Unverty Pre, Cambrdge (U.K M. N. Kogan, Rarefed Ga Dynamc, Plenum Pre, New York A. Roan and G. Spga, Phyca A 7, 563 ( M. Gropp and G. Spga, Tran. Th. Stat. Phy. 30, 305 ( M. B, M. Gropp and G. Spga, Contn. Mech. Thermodyn. 14, 07 ( M. B, M. Gropp and G Spga, preprnt No. 3, Dept. Math., Unv. Parma ( I. Samohyl, Collecton Czecholov. Chem. Commun. 40, 341 (1975.