Jounal of Appled Mathematcs and Physcs 6 4 687-697 Publshed Onlne August 6 n ScRes http://wwwscpog/jounal/jamp http://dxdoog/436/jamp64877 Asymptotc Solutons of the Knetc Boltzmann Equaton and Multcomponent Non-Equlbum Gas Dynamcs S A Seov S S Seova Fundamental Reseaches Depatment Russan Fedeal Nuclea Cente All-Russan Scentfc Reseach Insttute of Expemental Physcs Saov Russa St Petesbug State Unvesty St Petesbug Russa Receved 6 June 6; accepted 4 August 6; publshed 3 August 6 Abstact In the atcle coect method fo the knetc Boltzmann equaton asymptotc soluton s fomulated the Hlbet s and Enskog s methods ae dscussed The equatons system of multcomponent nonequlbum gas dynamcs s deved that coesponds to the fst ode n the appoxmate asymptotc method fo soluton of the system of knetc Boltzmann equatons Keywods Knetc Boltzmann Equaton Multcomponent Non-Equlbum Gas Dynamcs Intoducton In 9 Hlbet consdeed the knetc Boltzmann equaton fo one-component gas as an example of ntegal equaton and poposed a ecpe fo ts appoxmate asymptotc soluton see [] Chapte~XXII Hlbet s ecpe was nconvenent fo pactcal use because the fve abtay functonal paametes of the fst and the followng appoxmatons of the velocty dstbuton functon had to be found by solvng the dffeental equatons n patal devatves equatons of gas dynamcs of the fst and hghe odes Fve yeas late Enskog poposed to use zeo condtons condtons wth zeo ght-hand sdes to detemne the fve abtay functonal paametes of the fst and followng appoxmatons of the velocty dstbuton functon The mposton of the zeo condtons leads n fact to usng dffeent compason scales n the asymptotc expanson of the velocty dstbuton functon and n the asymptotc expanson of the patcle numbe densty the mean mass velocty and the tempeatue that ae deved fom the asymptotc expanson of the velocty dstbuton functon by ntegaton ove veloctes wth dffeent weghtng functons As a esult of paalogsm of the method of successve appoxmatons one has to set vaable coeffcents of the same tems of the unfed compason scale equal to each othe patal tme devatves vansh n the necessay condtons of solutons exstence of ntegal equatons of hghe odes see below and wth them tems of gas-dynamc equatons coespondng to vscosty heat conducton vansh Enskog mpoved the stuaton by the ntoducng see fo example [] Chapte 7 Secton 5 of the unsubstantated expanson of patal tme devatve: How to cte ths pape: Seov SA and Seova SS 6 Asymptotc Solutons of the Knetc Boltzmann Equaton and Multcomponent Non-Equlbum Gas Dynamcs Jounal of Appled Mathematcs and Physcs 4 687-697 http://dxdoog/436/jamp64877
= θ t t The appoach of Stumnsk who had poposed n 974 n [3] hs appoxmate asymptotc method of soluton of the system of knetc Boltzmann equatons fo multcomponent gas dffes fom the appoach of Enskog to asymptotc soluton of the Boltzmann equatons system fo gas mxtue n that how the nfntesmal paamete s ntoduced n the Boltzmann equatons system fo gas mxtue e the soluton s constuctng n anothe asymptotc lmt In substance Stumnsk s method of soluton of knetc equatons system s the same as Enskog s method Stumnsk used the patal tme devatve expanson as Enskog dd In secton below wll be poposed the coect method of asymptotc soluton of the knetc Boltzmann equatons system fo multcomponent gas mxtue fo the appoach that combnes Enskog s and Stumnsk s appoaches; n patcula t wll be shown how one has to modfy Enskog s method: n addton to asymptotc expanson of the velocty dstbuton functon -component patcles of gas mxtue t s necessay to detemne and to use the expanson of the patcle numbe densty n of -component mean mass velocty u and tempeatue T of the gas mxtue Futhe n the Secton 3 the system of nfntesmal fst ode equatons of multcomponent non-equlbum gas dynamcs appeang dung the pocess of the soluton of the system of Boltzmann equatons by successve appoxmatons method n the Secton as necessay condton of the exstence of appoxmate asymptotc soluton of the ntegal equatons system s consdeed n moe detal Ths atcle s condensed veson of ou atcle axv:33675 Notatons used below ae close to notatons n []; t s assumed that all egaded functons ae contnuous and contnuously dffeentable so many tmes as t s necessay f the devatves ae consdeed and all egaded ntegals convege = Coect Method of Soluton of the Knetc Boltzmann Equatons System The Boltzmann equatons system that descbes change of dependent on t and spatal coodnates pescbed by adus-vecto the velocty dstbuton functons f t c due to collson wth patcles of othe components of mxtue of aefed monatomc gases whee c ae the veloctes of patcles of -component of the mxtue {see [] Chapte 8 Equaton ; dscusson of the devaton of the Boltzmann equatons system and ts applcablty ange see fo example n [] Chaptes 3 and 8 [4] Chapte 7 ; below the cental nteacton of molecules ae consdeed only when the foce wth whch each molecule acts on the othe s dected along the lne connectng the centes of the molecules} could be wtten as: f f X f ' ' + c + = f f j f f j gj bdbdε dcj t m c j N ' ' = f f ff kdkdc N ; j N j j j j n N s a set of ndexes that ae numbeng components of the mxtue; X s an extenal foce whch acts on the molecule of the -component; m s the mass of the molecule of the -component; g j s the modulus of the elatve velocty of colldng patcles gj = c cj ; b s the mpact dstance ε s the azmuth angle k s the unt vecto dected to the cente of mass of the colldng patcles fom the pont of closest appoach see k g k s detemned by equalty [] Chapte 3 Fgue 3; the scala functon j j g bdbdε = k dk; 3 j by pme n and below the veloctes and the functons of veloctes afte the collson ae denoted Let us ntoduce followng notatons: ' ' ' ' j J f f = ff f f kdkd c 4 J f f = ff f f kdkd c ; 5 j j j j j j to dffe veloctes of colldng molecules of the same knd n the one velocty s denoted by c j and the othe s denoted by c wthout any ndex and the ndex of the coespondng velocty dstbuton functon f s omtted 688
In Enskog s appoach the dffeental pats of the Boltzmann Equatons that ae denoted by f below ae consdeed to be small as compaed wth the ght-hand sdes of Equatons see [] Chapte 7 Secton 5 theefoe the ndcato of nfnte smallness θ s fomally ntoduced n the Boltzmann equatons system n the followng way: θ f = J f f N 6 j j j In Stumnsk s appoach to the asymptotc soluton of the Boltzmann equatons system the dffeental pats of the Boltzmann Equatons and the collson ntegals of the patcles of -component wth the patcles of the othe components ae consdeed to be small as compaed wth the collson ntegal of the patcles of -component between each othe theefoe the ndcato of nfntesmalty θ s ntoduced n the Boltzmann equatons system n anothe way: θ f = J f f θ J f f N 7 j j j It s possble to combne Enskog s appoach wth Stumnsk s appoach Fo ths pupose we dvde the set of mxtue components N nto two subsets: the subset of components that we call fomally nne components we could consde the case when thee ae some subsets of nne components but ths case does not fundamentally dffe fom the one consdeed below the only dffeence s that the notaton become moe complcated and the subset of components that we call extenal components To dffe the two goups of mxtue components we denote the subset of ndexes of nne components ˆN as well as the ndexes of nne components ˆ N ˆ and the subset of ndexes of extenal components N as well as the ndexes of extenal components N ; the ntesecton of the sets ˆN and N s the empty set ˆ N N = and the unon of these sets s the set of ndexes of all mxtue components N = N ; f an asseton concens both knds of components the specal symbols wll be omtted In new notatons the Boltzmann equatons system can be ewtten as: θ f = J f f θ J f f ˆ 8 j ˆ ˆ j ˆˆ ˆ ˆj j ˆ ˆ ˆj j N θ f = J f f θ J f f N 9 j j j Let us wte the asymptotc expanson of the velocty dstbuton functon f of patcles of -component as fomal sees of successve appoxmatons n powes of θ : f = f + θ f + θ f + The dffeental pats of the Equatons 3 ae wtten as: X f = + c + f + θ f + t m c = + θ + θ + whee f f X f + c + t m c = = cf wth [] Chapte 7 Sectons 4 5 and [3] In - the patal tme devatve expanson s not used n contast to that how t was made by Enskog and futhe by Stumnsk As esult descbed below method fo soluton of the system of knetc Boltzmann equatons dffe fundamentally fom Enskog s method and Stumnsk s method Substtutng and n 8 and equatng coeffcents at the same powes of θ to each othe we obtan the equatons system of the method of successve appoxmatons fo fndng the velocty dstbuton functons of nne components patcles of gas mxtue f ˆ ; takng ntoduced notatons 4 5 and nto account the system can be ewtten as: 689
ˆˆ ˆ ˆ ˆ ˆ j j ˆj J f f = N 3 s s ˆ + Jˆˆ fˆ fˆ + Jˆˆ fˆ fˆ + Jˆˆ fˆ f j j j j j ˆj ˆj ˆj s= ˆj s s J ˆ fˆ f = ˆ = j j j Ns= + Smlaly substtutng and n 9 and equatng coeffcents at the same powes of θ to each othe we obtan the equatons system of the method of successve appoxmatons fo fndng the velocty dstbuton functons of patcles of extenal components of gas mxtue 4 f : J f f = N 5 s s + J f f + J f f + J f f s= s s J f f = = j j N j s= + Speakng about an ode of appoxmaton below we assume the ode to be equal to the value of ndex n 4 6 Accodng to 5 3 n zeo ode appoxmaton we have the followng system of ntegal equatons to fnd the velocty dstbuton functons of patcles of nne components of gas mxtue f : ˆ ˆ ' ' ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ k c j j j j j ˆj ˆj ˆj J f f = f f f f k d d = N 7 The geneal soluton of the equatons system 7 can be wtten as a set of the Maxwell functons: 3/ mˆ cˆ b 3 kβ m ˆ fˆ = βˆ e 3 ˆ πkβ whee k s the Boltzmann constant Patcle numbe densty n of the -component mean mass velocty u and tempeatue T of nne components of mxtue ae ntoduced by ntons: î 6 8 n= fd c 9 u nm = m c fd c ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3 kt n = m f d ˆ ˆ cˆ u ˆ c ˆ ˆ ˆ n k s the Boltzmann constant Fom 9- the equalty s obtaned: 3 kt n + u n m = m c f d c ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ that s convenent to use below nstead of nton Accodng to ntons 9 n addton to the asymptotc expanson t s necessay to detemne asymptotc expansons fo patcle numbe densty n of the -component mean mass velocty u = + θ + θ + 3 n n n n 69
and tempeatue T of nne components of mxtue θ θ 4 u = u + u + u + T = T + θt + θ T + 5 Substtutng and 3-5 n 9 and equatng tems of the same nfntesmal ode we obtan Cad N ˆ + 4 scala elatons that connect asymptotc expansons and 3-5: and In 7 8 the notatons ae ntoduced ˆ cˆ ˆ ˆ ˆ f d = n N 6 s s s s ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ mc f dc = m nu = m n u = ρ u 7 ˆ ˆ ˆ s= s= 3 mc f dc = k nt + m nu ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 3 = k n T m n s s s s s q q ˆ + ˆ ˆ u u ˆ s= ˆ s= q= s 3 s s s s q q = k nˆ T + ˆ ρ u u s= s= q= s mn s ˆ ˆ ˆ 8 ˆ ρ = 9 s s ˆ ˆ n ˆ = n 3 In patcula fo = fom 6-8 we obtan expessons fo abtay functons 3 t t β ˆ b t β n 8 though the zeo ode appoxmatons to local values of the î -component numbe densty the mean mass velocty and the tempeatue of nne components of the mxtue: ˆ ˆ β t = n t 3 b t = u t 3 3 β t = T t 33 Accodng to 4 5 zeo ode ntegal equatons fom whch the velocty dstbuton functons f of patcles of oute components of the mxtue ae found: ' = ' J f f f f f f kdkdc= N 34 ae smple than Equatons 7 and dffe actually fom 7 only by lack of summaton ove components Theefoe smlaly 8 the geneal soluton of the equatons system 34 can be wtten as a set of the Maxwell functons: whee β and 3 3/ m c b 3 kβ m = f β 3 e N πkβ β ae some ndependent of c scala functons of spatal coodnates ned by the adus vecto and tme t and b s a vecto functon of and t Let s add to the nton of the numbe densty of patcles of -component ntons of mean velocty u and tempeatue T of oute component of mxtue: 35 69
fom 9 36 37 the equalty s obtaned: u nm = mc fd c 36 3 kt n = ; m c u f d c 37 3 kt = n + u n m m c c f d 38 that s convenent to use below nstead of nton 37 Let s ente smla 4-5 asymptotc expansons of oute -component mean velocty u and oute -component tempeatue T = + + + u u θu θ u 39 + + + T = T θt θ T 4 Substtutng 3 39 4 n 9 36 38 and equatng tems of the same nfntesmal ode we obtan fo each 5 scala elatons that connect asymptotc expansons 3 39 4: cf wth 6-8 In 4 43 the notaton s used Fo = 3 t c f d = n 4 s= s= s s s s m c f dc = m n u = m n u = ρ u 4 3 = mc f dc k nt + m nu 3 = k n T m n s s s s s q q + u u s= s= q= 3 = k n T ρ s s s s s q q + u u s= s= q= s s mn 43 ρ = 44 fom 4-43 we obtan expessons fo abtay functons t β b t and β n 35 though the zeo ode appoxmatons to local values of the numbe densty the mean velocty and the tempeatue of oute -component of the mxtue: β t = n t 45 b t = u t 46 3 β t = T t 47 Fo the velocty dstbuton functons of nne components of gas mxtue ntegal equatons system 4 whch takng 5 and equalty nto account can be ewtten n the fom ' ' ˆ ˆ j ˆ ˆj ˆ f ae found fom the f f f f 48 s s s s ˆ + ˆˆ ˆ ˆ ˆ ˆ = ˆˆ ˆ ˆ ˆ ˆˆ ˆ J f f + j j J f f j j J f χ f j j J f f j ˆ χ j ˆj ˆj s= j Ns= ˆj ˆj ˆj ˆˆ k cˆ ˆ ˆ j j ' ' ˆ ˆ χˆ χ j ˆ χ j ˆ χˆ j = f f + k d d N 49 69
n 49 functons ˆ f ae wtten as ˆ = ˆ ˆ f f χ whee ˆ χ ae new unknown functons The left-hand sdes of Equatons 49 nvolves functons that ae known fom the pevous step of the successve appoxmatons method Unknown functons χ ˆ ente lnealy only nto the ght-hand sdes of Equatons 49 Theefoe the geneal soluton of the system of Equatons 4 s a famly of functons of a fom ˆ Ξ ˆ + ˆ ˆ ˆ ˆ Φ ˆ ˆ ˆ f ˆ ˆ ˆ Φ ˆ ˆ ˆ ˆ ˆ { f = ξ } whee { Ξ = f } { ξ = φ } a famly of functons { } s a N patcula soluton of the system of nhomogeneous Equatons 49 and a famly of functons geneal soluton of the system of homogeneous equatons ˆj + ˆ ˆ ' ' ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ k c j j j j ˆj { φ } s the = f f φ φ φ φ k d d N 5 ˆ Multplyng Equatons 5 by φ ntegatng ove all values of c ˆ summng ove î and tansfomng ntegals we obtan k c c fˆ f ˆ ˆ + ˆ ' ˆ ' ˆ kˆˆ d d ˆ d j j j j ˆj 4 ˆ ˆj ˆ φ φ φ φ = 5 Fom 5 we conclude that φ ae lnea combnatons of the summatonal nvaants of the collson ψ l = 3 : 3 φ ˆ = α ˆ + a mˆ cˆ + α mc ˆ ˆ 5 3 whee α ˆ and α ae some ndependent of c ˆ scala functons of spatal coodnates ned by the adus vecto and tme t and a s a vecto functon of and t as well as above abtay functons a 3 and α ae dentcal fo all nne components of the mxtue and hence 3 ˆ fˆ ˆ + ˆ ˆ + ˆ ˆ ˆ ˆ a mc mc N ξ = α α 53 To smplfy futhe evaluatons accodng to the expesson fo as whee ˆ β î l f see 8 and 3-33 let us ewte 53 ˆ ˆ 3 ξˆ = fˆ βˆ + b mˆ cˆ u + β mˆ cˆ u N 54 b and the system of nhomogeneous equatons 3 β ae new functons of and t Famly of functons + ˆ ˆ ' ' ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ k c j j j j ˆj ˆj ˆ { χ } s a soluton of ˆ F = f f χ χ χ χ k d d N 55 whee F ˆ denote left-hand sdes of the Equatons 49 taken wth opposte sgn Multplyng Equatons 55 by l ψ ˆ l = 3 ntegatng ove all values of c ˆ and tansfomng ntegals as above we obtan as necessay condton fo the exstence of solutons of the system of ntegal Equatons 55 the necessty of the fulfllment of equaltes: ψ F dc = ˆ 56 ˆ ˆ ˆ ˆ l ˆ ˆ c ˆ ψ F d = l = 3 57 Among nfntesmal set of patcula solutons of the system of Equatons 55 dffeent fom each othe on some soluton of the system of homogeneous Equatons 5 unque soluton { Φ } ˆ ˆ N ˆ may be chosen such that ψ f Φ dc = ˆ 58 ˆ ˆ ˆ ˆ ˆ l ˆ ˆ Φ ˆ c ˆ ψ f d = l = 3 59 693
Havng substtuted expesson fo ˆ ˆ ˆ f N = ˆ = Ξ ˆ + ξ ˆ f f f m m 3 ˆ Φ ˆ + ˆ βˆ + b ˆ cˆ u + β ˆ cˆ u n 6-8 takng 8 9-33 and 58-59 nto account we obtan a system of ˆ equatons [constant equatons fo asymptotc expansons and 3-5]: 3 3 ˆ ˆ + ˆ ˆ ˆ ˆ 6 Cad N + 4 algebac n β n kt β = n N 6 u 3 β ρ b ρ u β = ρ u 6 3 m n ˆ ˆ ˆ + ˆ kt s s + ˆ ˆ ˆ kt s= nˆ 3 + ˆ ˆ + ˆ kt m u β ρ kt u b ˆ s= s= q= fom whch we fnd expessons fo functons β 3 3 + kt 5 ˆ ˆ 4 n kt + ρ u s 3 s s s s q q = k nˆ T + ˆ ρ u u t β ˆ b t and 3 t 63 β though vaable coe- ffcents of asymptotc expansons of the patcle numbe densty of î -component of the mean mass velocty and of the tempeatue of nne components of the mxtue β nˆ 3 s s ˆ = ˆ ˆ ˆ ˆ n T n T n n T s= s ˆ ˆ ρ u u ρ nˆ kt s= q= ρ ρ u s s q q s s + u ˆ ˆ ˆ u u n kt s= 64 s s b = ˆ ˆ ˆ ρ u ρ u ρ kt s= 65 β 3 s s = ˆ ˆ n T n T s= nˆ k kt kt kt s + ˆ ρ u u ˆ ρ 3 ˆ n s= q= ρ ρ u s s q q s s u ˆ ˆ 3 ˆ u u n s= Then the fulfllment of equaltes 56-57 can be consdeed as the dffeental equatons the -ode equatons of gas dynamcs fo fndng ˆ n u ˆ ˆ T = The patal soluton of the system of nhomogeneous Equatons 55 constucted fo example usng expanson of ˆ cˆ Φ ˆ { } ˆ N ˆ 66 satsfyng 58-59 may be Φ n sees n tems of Sonne polynomals wth expanson 694
coeffcents dependng on and t see [] o [4]; such constucton poves exstence of solutons of the system of ntegal Equatons 49 Fo the velocty dstbuton functons of oute components of gas mxtue f may be smlaly found fom the ntegal equatons system 6: whee = = 3 f Ξ + ξ f Φ + f β + b m c u + β m c u 67 β n 3 s s = n T n T n n T s= kt kt n s s ρ ρ s= s s s q q ρ u u ρ u s= q= + u u u n 68 b = u u s s ρ ρ ρ kt s= 69 β k 3 s s = n T n T n kt s= kt kt + 3 n s s ρ ρ s= s s s q q ρ u u ρ u s= q= u u u 3 n The fulfllment of analogous 56-57 equaltes ψ 7 l F dc = N l = 3 can be consdeed as the dffeental equatons the -ode equatons of gas dynamcs fo fndng T = 3 The System of Fst Ode Equatons of Multcomponent Non-Equlbum Gas Dynamcs Let us consde n moe detal the system of nfntesmal fst ode Equatons 56-57 7 = n 7 u deved above as the necessay and suffcent condton of the soluton exstence of the fst ode ntegal equatons system 4 6 = To smplfy tansfomatons accodng to the expessons fo velocty dstbuton functons of patcles of nfntesmal zeo ode 8 35 functons Ψ l ˆ Ψ l may be used n 56-57 7 = athe than l functons ψ ψ l espectvely: ˆ Ψ = m 7 m C 73 Ψ = = mc 74 3 Ψ 695
fo nne components ˆ = ˆ fo oute components C c u = C c u At tansfomaton of dffeental pats of the Equatons 56-57 and 7 we use equaltes: l l nψ l l f l Ψ Ψ dc f dc f dc n Ψ = Ψ = t t t t t l l l f l Ψ l Ψ c c c c c c c c Ψ d = Ψ f d f d = n Ψ n 76 l X X X f c l c l d f d n m c c m c m Ψ = Ψ = Ψ In 75-77 the ba above symbol wth ndex denotes the aveage of the value: V = Vf dc ; n and c ae consdeed as ndependent vaables At aveagng n 77 t s assumed that extenal foce X actng on the patcle of speces s ndependent of the patcle velocty t s assumed also that ntegals dependng on extenal foces X ae convegent and poduct Ψ l X f tends to zeo when c tends to nfnty Afte smple tansfomatons fom 56-57 and 7 = we obtan followng system of nfntesmal fst ode equatons of multcomponent non-equlbum gas dynamcs: n ˆ ˆ u t ˆ ˆ = n N u + + J ˆ nˆ Xˆ u u ˆ ˆ p j t N j N ˆ ˆ ρ p ˆ = ˆ ρ 8 ˆ E + qˆ + ˆp : + = t u J ˆ ˆ E u ˆ ˆ E j N j N 75 77 78 79 8 n t = n u N u J X u u p j t j n m + p + = n n m N u q u E j t j E + + p : + J = E N In accodance wth the geneal nton of pessue tenso of -component of gas mxtue 8 83 84 and wth the geneal nton of -component heat flux vecto cf wth [] Chapte 3 4 n 79-84 p = m c u c u fd c 85 q = m c u c u fd c 86 ˆ ˆ cˆ u cˆ u ˆ p ˆ = n m = n ˆ kt U= p ˆ U 87 696
s nne components pessue tenso of zeo ode U s the unt tenso double poduct of two second ank tensos w and w ' w:w ' = w w ' =w ' :w α β αβ βα ˆ ˆ ˆ ˆ ˆ s nne components heat flux vecto of zeo ode ˆp s nne components hydostatc pessue of zeo ode [] Chapte 3 s the scala qˆ = n m c u c u = 88 3 = ˆ ˆ cˆ u = ˆ ˆ E ˆ n m n kt 89 s zeo ode ntenal enegy of patcles of nne components pe unt volume whch s equal n ths case to enegy of the tanslatonal chaotc moton howeve the enegy tansfe equatons wtten n fom 8 and 84 can be used n moe geneal cases as well cf wth [4] Chapte 7 6 n 87-89 aveagng 78 s pefomed wth Maxwell functon f fom 8; s -component pessue tenso of zeo ode s -component heat flux vecto of zeo ode î p = n m c u c u = n kt U= p U 9 p s -component hydostatc pessue of zeo ode q = = n m c u c u 9 3 = c u = E n m n kt 9 s zeo ode ntenal enegy of patcles of -component pe unt volume n 9-9 aveagng 78 s pefomed wth Maxwell functon f fom 35 Geneal analytc expessons fo ntegals J fom 8 8 and 83 84 that depend on the J p j E j nteacton coss-secton can be deved n geneal case when sepaate components wth Maxwell velocty dstbuton functon of patcles have dffeent mean veloctes and tempeatues System of nfntesmal fst ode equatons of multcomponent non-equlbum gas dynamcs 79-84 s poposed to use fo descbng tubulent flows Refeences [] Hlbet D 9 Gundzüge ene Allgemenen Theoe de Lneaen Integalglechungen Teubne Lepzg and Beln In Geman [] Chapman S and Cowlng TG 95 The Mathematcal Theoy of Non-unfom Gases Cambdge Unvesty Pess Cambdge [3] Stumnsk VV 974 Influence of Dffuson Velocty on Flow of Gas Mxtues Pkladnaya Mathematca Mechanca [Appled Mathematcs and Mechancs] 38 3- In Russan [4] Hschfelde JO Cutss ChF and Bd RB 954 Molecula Theoy of Gases and Lquds Wley New Yok 697
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