A Study Of The Kinetic Theory Of The Steady Spherical Source Expansion Into A Vacuum

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Content This dissertation has been microfilmed exactly as received 68*1675 CHEN, T su -y e e , 1934- A STUDY OF THE KINETIC THEORY O F THE STEADY SPHERICAL SOURCE EXPANSION INTO A VACUUM. U niversity of Southern C alifornia, PtuD ., 1967 Engineering, aeronautical University Microfilms, Inc., Ann Arbor, Michigan A STU D Y O N THE KINETIC TH EO RY O F THE STEA D Y SPHERICAL SO U RCE EXPANSION INTO A V A C U U M by TSU-YEE C H E N A D is s e rta tio n P resen ted to th e FA CU LTY OF TH E G R A D U A T E SC H O O L UNIVERSITY O F SO U TH ERN CALIFORNIA In P a r tia l F u lfillm e n t o f th e Requirem ents fo r th e Degree D O C TO R OF PHILOSOPHY (Aerospace E ngineering) August 1967 \ UNIVERSITY O F S O U T H E R N CA LIFO RN IA T H E GRADUATE SC H O O L U N IV ERSITY PARK LO S A N G ELES, C A LIFO R N IA S 0 0 O 7 This dissertation, written by Tsu-Yee Chen under the direction of h.Xs...Dissertation Com mittee, and approved by all its members, has been presented to and accepted by the Graduate School, in partial fulfillment of requirements for the degree of D O C T O R OF P H I L O S O P H Y Z f t e s * e _ _ Date S e p t e ^ e r ,1 9 6 7 DISSERTATION COMMITTEE ACKNOWLEDGMENTS The a u th o r w ishes t o e x p re ss h is d ee p est a p p re c ia tio n to P ro fe s s o r H. K. Cheng f o r s u g g e stin g t h i s work and f o r h is continuous guidance and c o n s ta n t encouragem ent th ro u g h o u t th e c o u rse o f t h i s in v e s tig a tio n . He i s in d e b te d as w e ll to P ro fe s s o r R. H. Edwards fo r h is generous d is c u s s io n s and h e lp fu l s u g g e s tio n s . He i s v ery g r a te f u l to P ro fe s s o r T. S. P itc h e r f o r h is guidance d u rin g th e whole academ ic y e a rs . The a u th o r would a ls o li k e to th an k th e A erospace E n g in eerin g D epartm ent o f th e U n iv e rs ity o f S o u th ern C a lif o r n ia f o r th e f in a n c ia l su p p o rt so th a t t h i s stu d y co u ld be com pleted. The re s e a rc h d e s c rib e d h e re was su p p o rte d by N atio n a l S cien ce F oundation G ra n t. TABLE OF CONTENTS Page ACKNOW LEDGM ENTS......................................................................................................... i i LIST OF FIGURES......................................................................................................... v LIST OF SYMBOLS......................................................................................................... v i Chapter I . INTRODUCTION...............................................................,........................... 1 I I . FORMULATION................................................................................................ 6 2 .1 . Maxwell T ra n sfe r Equation in S p h erical C oordinates 2 .2 . The C o llisio n In te g ra ls 2 .2 .1 . C o llis io n In te g ra ls o f Boltzmann Equation w ith Maxwellian M olecules 2 .2 .2 . C o llis io n In te g ra ls o f B-G-K Model 2 .3 . Hypersonic Approximation 2 .3 .1 . D is trib u tio n Function and I t s Moments fo r S p h erical Source Flow Under th e H ypersonic Approximation 2 .3 .2 . T runcation o f Moment Equations 2 .4 . T h ird -o rd e r Moment Equations 2 .5 . D if fe r e n tia l Equations and Boundary C onditions o f S p h erical Source Expansion in N on-dimensional Forms iii C h ap ter Page I I I . SOLUTIONS......................................................................................................... 34 3 .1 . Z e ro th -o rd e r S o lu tio n s 3 .1 .1 . U and N 0 o 3 .1 .2 . P and P ro ± o 3 .1 .3 . S and S, ro -L O A. B-G-K Model B. M axw ellian M olecules 3 .2 . F i r s t - o r d e r S o lu tio n s 3 .2 .1 . U, and N 1 1 3 .2 .2 . P and P r i n IV. OTHER ASPECTS O N THE ZEROTH-ORDER SOLUTIONS .............................. 60 4 .1 . Z e ro th -o rd e r S o lu tio n s w ith N o n -eq u ilib riu m I n i t i a l C o n d itio n s 4 .2 . Z e ro th -o rd e r E quations o f Non-M axwellian Gases and I t s A sym ptotic B ehaviors 4 .2 .1 . Sm all End p 0 4 .2 .2 . Large Ehd p ■ + ■ “ V. CONCLUSIONS A N D REMARKS......................................................................... 75 APPENDIX A. E v a lu a tio n o f C o llis io n I n te g r a ls .......................................... 78 A .I. C o llis io n I n te g r a ls o f Non-M axwellian M olecules A .2. P e rtu rb e d M axw ellian M olecules APPENDIX B. H ypersonic Source E xpansion o f Gases w ith P e rtu rb e d M axw ellian M olecules ............................................. 91 LIST OF REFERENCES..................................................................................................... 97 iv LIST OF TABLES F igure Page 1. The Z ero th -o rd er T em peratures, Y, Y and Y ^ ................. 44 2. Z ero th -o rd er P a r a lle l and T ransverse Energy Fluxes Per U nit S o lid A ngle, s 2S and s 2S ........................... 47' ro 10 3. F ir s t- o r d e r V elo city and Number D en sity , U and - N /NQ ...................................................................... 1............................ 54 4. F ir s t- o r d e r P ressu re C o rre c tio n s, W/Y, W /Y and W 2/Y2 ..........................................................i .......................... 59 5. S o lu tio n Curves Near p a 0, Which Corresponds to s = 0, (fo r u ® 3/4 c a s e ) . The sad d le p o in t a t p = 0 in d ic a te s th a t th e eq u ilib riu m s o lu tio n near s = 0 i s s ta b le . .............................................................. 70 6. S o lu tio n Curves Near p = 0, Which Corresponds to s » (fo r w = 5/2 c a s e ) . The node p o in t a t p = 0 in d ic a te s th a t th e eq u ilib riu m s o lu tio n a t s « i s s ta b l e ........................................................................... 70 7. S o lu tio n Curves Near p • + ■ , Which Corresponds to s - * ■ < p The node p o in t a t p ■ + • ® shows th e e x iste n c e o f a "Frozen s o lu tio n " a t i n f i n i t y fo r a l l .......................................................................................................... 73 8. M olecular Encounter ............................................................................... 79 9. V ector Diagram fo r an E n c o u n te r..................................................... 81 10. Tem perature C o rre ctio n s Due to th e Weak Far D istance A ttra c tiv e F ie ld .............................................................. 94 v LIST OF SYMBOLS A = a c o n s ta n t d e fin e d by E quation (4 .1 3 ); a ls o a c o n sta n t in tro d u c e d in th e B-G-K c o llis io n term A2 = 0.436 a = /yRT , speed o f sound B 2 = - 0.4829 b = im pact p aram eter c = v e c to r p o s itio n in v e lo c ity space E = av erag e k in e tic energy o f th e p a r t i c l e s ; see E quation (2 F - re p u ls iv e fo rc e betw een two m olecules F^ * » lo c a l M axwellian d i s t r i b u t i o n fu n c tio n ; see E quation (2 F j, F2 = fu n c tio n s d e fin e d in E quations (3 .4 ) f = v e lo c ity d i s t r i b u t i o n fu n c tio n f = e l l ip s o id a l d i s t r i b u t i o n fu n c tio n ; see E quation (2 .1 6 ) © G j, G2 = fu n c tio n s d e fin e d in E quations (3 .2 ) g = r e l a t i v e speed o f two m olecules g* = r e l a t i v e speed o f two m olecules a f t e r en co u n ter H1* 1 = g e n e ra l moment o f f ; see E quation (2 .2 4 ) h = A 2 + Y2 + Z2 I j , I 2 = in te g r a ls d e fin e d in E quation (3 .4 5 ) J = in te g r a l d e fin e d in E quation (A-2) K = fo rc e c o n sta n t d e fin e d by Equation (2 .1 3 ) VI . 2) . 16) K' = fo rc e c o n s ta n t d e fin e d by E quation (A -ll) Kn = Knudsen number k = Boltzmann gas c o n s ta n t kr » k^ = c o n s ta n ts ; see E quations (2 .3 6 .5 ) and (2 .3 6 .6 ) L ., L = fu n c tio n s o f S and S , d e fin e d in E quation (2 .4 0 ) r i ~ fu n c tio n s o f d and d » d e fin e d i n E quation (2.20) r 1 = lo c a l mean f r e e p a th 1 = mean f r e e p a th a t r * r J 1Q * mean f r e e p a th a t r e s e r v o ir c o n d itio n M = Mach number M 1 = i n i t i a l Mach number M = te rm in a l Mach number m = m o le cu la r mass N = n / S j 2 n , d im e n sio n le ss number d e n s ity in th e o u te r re g io n N , Nj = z e ro th and f i r s t o rd e r d im e n sio n le ss number d e n s ity ° d e fin e d in E q u atio n s (3 .1 ) n * number d e n s ity n = n / n ^ d im e n sio n le ss number d e n s ity P = k in e tic s t r e s s te n s o r ; see E quation (2 .2 ) P = p / s , 10/3 n .R T ,, d im e n sio n le ss second moment in th e r vx 1 1 1 * o u te r re g io n P j_ = Pj_/Si 10^ 3 Hj RTj , d im e n sio n le ss second moment in th e o u te r re g io n P , P , P , P = z e ro th and f i r s t o rd e r second moments r x0 d e fin e d in E q u atio n (3 .1 ) vii p^, = second moments o f f ; se e Equation (2.26) p" , p^ = d im en sio n less second moments, pr = pr / n 1RT1> Px * V n i RTi Q = fu n c tio n o f th e p a r t i c l e v e lo c ity components Q = mean v alu e o f Q; see E quation (2 .3 ) A Q = change in Q due to c o llis io n ; see E quation (2 .9 ) Q_ » Q_ » Q. . » Q . = fo u rth moments o f f ; see S : r r r ’ E quation (2 .6 ) q = h eat flu x te n s o r; see Equation (2 .3 ) R = gas c o n sta n t r = r a d ia l d ista n c e r = v e c to r p o s itio n in o rd in a ry space r j = i n i t i a l ra d iu s r* = so n ic ra d iu s r S = — . . > ■ ------------------ , d im en sio n less t h i r d moment in th e r „ 1 0 / 3 —~ „ Sj e U jn jR tj o u te r re g io n J x d im en sio n less th i r d moment in th e S 1 e u’ 1n 1RT1 o u te r re g io n S , S = z e ro th o rd e r d im en sio n less t h i r d moment d efin ed r0 0 in Equation (3.1) s = d im en sio n less d is ta n c e d efin ed by E quations (2.22) and (2.33) Sj = d im en sio n less d is ta n c e corresponding to r = r t d T > ^ = t h i r d moments o f f ; see E quations (1 .3 ) and (1 .4 ) viii _ 0 ^ = — —------- , d im en sio n le ss t h i r d moment r u ,n , RT 1 1 1 J J t - — ----- , d im en sio n less t h i r d moment ■^J- u i n i RT1 T = average tem p eratu re Tr = p a r a l l e l te m p e ra tu re ; see E quation (1 .1 ) = tra n s v e r s e te m p e ra tu re ; see E quation (1 .2 ) Tro> T^o (T)o = z e ro th approxim ation te m p eratu res t = tim e a ls o re p re s e n tin g U = U, d im en sio n less mean r a d i a l v e lo c ity in th e o u te r re g io n U ■ u /u x , d im e n sio n less mean r a d i a l v e lo c ity U Q, U = z e ro th and f i r s t o rd e r d im en sio n less mean r a d ia l v e lo c ity d e fin e d in E quation (3 .1 ) u = p a r t i c l e v e lo c ity component i n th e r a d i a l d ir e c tio n u 1 = p a r t i c l e v e lo c ity component in th e r a d i a l d ir e c tio n a f t e r c o l l i s i o n u ^ u^ = in te g r a ls d e fin e d in E quation (3.31) ^ = mean p a r t i c l e v e lo c ity ; see E quation (2 .2 ) v = p a r t i c l e v e lo c ity component i n th e m e rid io n a l d ir e c tio n v = p a r t i c l e v e lo c ity component in th e m e rid io n a l d ir e c tio n a f t e r c o l l i s i o n W » Wj, W 2 = v a r ia b le s d e fin e d by E quations (3.42) w = p a r t i c l e v e lo c ity component in th e azim u th al d ir e c tio n W j = p a r t i c l e v e lo c ity component in th e azim u th al d ir e c tio n a f t e r c o l l i s i o n X ■ (Ty - )/T y , d e fin e d in E quation (A-18) ix Y, Y , Y = te m p eratu res o f z e ro th approxim ation, d e fin e d by E quations ( 3 .9 .1 ) - ( 3 .9 .3 ) Z j, Z2 = v a r ia b le s d e fin e d by E quation (3.19) Ani r i a * —— — ; see E quation (2.32) U1 Y = ratio of specific heats Aj = degree o f i n i t i a l n o n -eq u ilib riu m in T, d efin e d by E quation (4 .2 ) 6 = angle o f d e f le c tio n due to c o llis io n € = polar angle RTj e = ; see Equation (2.32) u 2 l mKf 2 £3 ' E2 A 2 2 j y < p = ^ - / y ; see Equation (4 .1 1 .2 ) y = v is c o s ity v = exponent index o f m o lecu lar fo rc e f ie ld p ■ sY^"1 ; see E quation (4 .1 1 .1 ) $ = c o n flu e n t hypergeom etric fu n c tio n x ^ = co n flu en t hypergeom etric fu n ctio n a ) = exponent index in th e v is c o s ity and tem p eratu re r e la tio n The symbols which a re in fre q u e n tly used and not l i s t e d here a re d efin ed where th ey appear. xi CHAPTER I ! INTRODUCTION i The stead y expansion o£ a s p h e ric a lly sym m etrical source in to a : vacuum may be considered an u sefu l model fo r se v e ra l p h y sic a l problem s o f c u rre n t i n t e r e s t . I t can be used to approxim ate th e flow along th e c e n te r lin e o f a n o zzle j e t expansion [1 ], which is o f fundam ental in t e r e s t to u n derstand th e p ro p e rty o f a m olecular beam w ith nozzle so u rces [1, 2 ], as w ell as th e perform ance o f high speed wind tu n n e ls and j e t s a t low d e n sity [3 ], The study i s a lso b a s ic to problem s as th e in te ra c tio n o f ro ck et exhaust plumes w ith surrounding s tr u c tu r e in a high vacuum environm ent and th e d is s ip a tio n o f gas sources in th e space [4, 5 ]. In a d d itio n to th e se more o r le s s p r a c tic a l a p p lic a tio n s , th e source expansion re p re s e n ts an in te r e s tin g problem o f r a r i f i e d gas dynamics from th e th e o r e tic a l p o in t o f view. The flow undergoes a I tr a n s itio n from th e continuum in th e v ic in ity o f th e source to a fre e : m olecular flow f a r downstream in th e absence o f p h y sic a l bo u n d aries. i ; Thus, in t h i s problem , one may c o n c e n tra te on th e stu d y o f th e e f f e c t o f in te rm o le c u la r c o llis io n s , avoiding th e u n c e rta in e f f e c ts o f th e ■ in te ra c tio n w ith th e s o lid s u rfa c e . C o llis io n le s s expansion o f gases has been s tu d ie d by Molmud [6 ], Narasimha [7] and Bienkowski [8]. The continuum in v is c id a n a ly s is o f ! t h i s problem was in v e s tig a te d by Owen and T h o rn h ill [9 ]. An a n a ly s is 1 o f th e source flow problem which was based on a k in e tic model was f i r s t j I made by Brock and Oman [1 0 ]. In an a ly z in g th e stea d y expansion problem j a t h ig h speed r a t io by an i t e r a ti o n m ethod, th e y used: th e B-G-K m odel, : ; but o m itte d s e v e ra l term s a ris e n from th e c u rv a tu re o f c o o rd in a te s . The r e s u l t which th ey o b ta in ed shows a ra p id t r a n s i t i o n from a c o n ti nuum s o lu tio n t o one corresp o n d in g to c o lli s i o n - f r e e flow c h a ra c te riz e d ; by a fro ze n tem p eratu re f a r downstream. T h e ir r e s u lt q u a l i t a t i v e l y ag rees w ith th e o b se rv a tio n s o f ex p erim en tal s tu d ie s [1, 2 , 11]. R ec en tly , th e s p h e ric a l and c y lin d r ic a l su p erso n ic cource flow s f o r monatomic gases were examined by Edwards and Cheng [12] and Hamel and W illis [1 3 ]. Both an aly ses to o k in to account th e term s o m itte d in : R eference [11] and used h y p erso n ic approxim ation to tru n c a te th e moment e q u a tio n s. In R eference [1 2 ], th e B-G-K model was used to compute th e ' c o llis io n mements, w h ile in R eference [1 3 ], th e M axwellian gas which is c h a ra c te riz e d by an in v e rs e f i f t h power in te r-m o le c u la r r e p u ls iv e fo rc e law was used. I f th e c o llis io n freq u en cy An in th e B-G-K model is • tak en to be p /y , th e two methods g iv e th e same moment e q u a tio n s up to second o r d e r, and h en ce, under th e h y p erso n ic ap p ro x im atio n , t h e i r r e s u l t s a re e q u iv a le n t. In both s tu d ie s , f o r th e s p h e ric a l c o u rc e , a fro zen tem p eratu re and a h ig h degree o f a n iso tro p y w ith a tra n s v e r s e i tem p eratu re v a n ish in g lik e th e f i r s t r e c ip ro c a l power were found f a r : downstream. The same tru n c a te d moment te c h n iq u e s have been a p p lie d by , Edwards and Roger [14] to study th e s tr u c tu r e o f an axisym m etric j e t . | More r e c e n tly , Edwards and Cheng [15] determ ined th e v e lo c ity I d is tr ib u tio n fu n c tio n fo r th e s p h e ric a l so u rce by in te g r a tin g th e B-G-K jmodel eq u atio n w ith th e lo c a l average q u a n titie s o b tain ed by th e moment i I : eq u atio n s under hypersonic approxim ation. T h eir a n a ly s is in d ic a te s I th a t, as a r e s u l t o f c o llis io n in th e f a r f i e l d , th e tra n s v e rs e tem per- ; a tu re and h ig h er moments a re m anifested in a th ick en in g o f th e t a i l o f th e d is tr ib u tio n fu n c tio n r a th e r th an a broadening o f th e sp ik e . i T h e refo re , th e tra n s v e rs e tem p eratu re cannot be re p re se n te d by th e c o llis io n le s s s o lu tio n even in th e f a r f i e l d . T his e x p la in s th e d iscrep an cy between th e r e s u l t s o f R eferences [12] and [13] and th a t based on a c o llis io n le s s m odel. . . The su b je c t to be in v e s tig a te d in t h i s work i s on th e stead y s p h e ric a l source expansion o f a monatomic gas in to a vacuum. The problem considered can be posed as fo llo w s: th e flow c o n d itio n s are given a t an i n i t i a l ra d iu s which i s a t f i n i t e d is ta n c e from th e o rig in and a t which th e Mach number M = u/ZyRT i s assumed to be v ery high and th e flow f ie ld can be v ery n e a rly d escrib ed by a continuum model; th e problem i s th en to fin d th e flow f ie ld outward from th e i n i t i a l ' ra d iu s as a fu n c tio n o f d is ta n c e , given i n i t i a l c o n d itio n s and gas p r o p e r tie s . The work re p re s e n ts a f u rth e r development o f th e p rev io u s study i [12], in clu d in g a g en eral study o f th e asym ptotic th eo ry fo r a I M axwellian gas o r B-G-K model under a high M, and s o lu tio n s to th e lead in g and f i r s t o rd e r approxim ations. In a d d itio n , th e e f f e c t o f I n o n -eq u ilib riu m in th e i n i t i a l c o n d itio n i s analy zed . Included a lso is a study o f th e asym ptotic b ehavior o f th e B-G-K s o lu tio n a t upstream as jw ell as f a r downstream and i t s dependence on th e m olecular fo rc e law. i ! F in a lly , th e e f fe c t o f th e d e p a rtu re from th e M axwellian fo rc e law on th e c o llis io n in te g ra l i s stu d ie d . The h ig h e r-o rd e r s o lu tio n o b tain ed , not only may serve to en larg e th e range o f th e a p p lic a b ility o f th e e x is tin g th e o ry , but provides a case wherein th e d iffe re n c e between th e Maxwellian gas and th e B-G-K model can be a n a ly tic a lly stu d ie d . The g en eral study re v e a ls t h a t , u n lik e th e a n tic ip a tio n by previous au th o rs [12, 13], Mach number M e n te rs in to th e asym ptotic expansion only as in te g ra l power o f 1/M2 in ste a d o f 1/M. As a r e s u l t , th e c o rre c tio n to th e lead in g term in th e asym ptotic expansion re q u ire s equations beyond th e th ir d o rd er moment. An assumed form o f v e lo c ity d is tr ib u tio n fu n c tio n i s th e re fo re introduced to make tru n c a tio n p o s sib le fo r th e use o f th ir d o rd er moment eq u atio n . The so lu tio n d isp la y s a n e a r-eq u ilib riu m re g io n and a non-equilibrium re g io n , w ith th e (mean) a c c e le ra tio n o f flow tak in g p lace only in th e form er re g io n . In a d d itio n to a c o rre c tio n to th e p a r a lle l and tra n sv e rse tem peratures Tr = ^ / £ u2 dudvdw - (1 .1 ) Tj_ * ^ (y 2 * wZ) dudvdw (1 .2 ) one has in th e f ir s t- o r d e r th eo ry th e two h e a t flu x e s “ / f u 3 dudvdw (1 .3 ) - J f u (v 2 + w 2) dudvdw (1.4) I 5 | and fin d s th a t th e t o t a l h eat flu x p er u n it s o lid angle in th e form o f I r 2< / fre e z e s a t i n f i n i t y and th a t o f r 2< ^ v an ish es lik e 1 /r , where r i s th e r a d ia l d is ta n c e . The study o f th e dependence o f th e s o lu tio n on in term o lecu lar ! ; fo rce law based on a m odified B-G-K model could have s ig n ific a n c e in th e m athem atical th e o ry o f s o la r wind [5 ]. I t i s found th a t fo r u s < 7 /4 (where u s c o n tro ls th e fo rc e law and e n te rs in th e v is c o s ity i m tem perature r e la tio n y * T ) th e so lu tio n c h a ra c te riz e d by a fro zen tem perature i s th e only s ta b le one a t i n f i n i t y , but fo r u s > _ 1/A, both eq u ilib riu m and fro zen so lu tio n s a re s ta b le . In C hapter I I , a d isc u ssio n o f th e hypersonic tru n c a tio n o f moment equation a t any o rd er i s p rese n ted . The th ird o rd er moment equations in s p h e ric a l symmetry a re a lso d e riv e d . The zero th and f i r s t o rd er s o lu tio n s to th e problem o f source expansion in to a vacuum, based on th e th ir d moment e q u atio n s, a re given in C hapter I I I . In C hapter IV, th e e f fe c ts o f n o n -eq u ilib riu m i n i t i a l co n d itio n as w ell as th e asym ptotic behavior o f th e s o lu tio n o f non-M axwellian gases based on B-G-K model are examined. In Chapter V, conclusions a re drawn and su g g estio n s fo r fu rth e r j work a re made. CHAPTER II FO RM U LA TIO N A b a s ic assum ption o f th e k in e tic th e o ry o f gases i s th a t a l l p h y sic a l p ro p e rtie s o f a gas can be deduced from th e m otion o f th e m olecules under c e r ta in m olecualr fo rc e in te r a c tin g law s. In th e s t a t i s t i c a l sen se, th e knowledge o f m otion o f th e m olecules, and hence th e m acroscopic o b serv ab le p ro p e rtie s o f a g as, can be d escrib ed by a d is tr ib u tio n fu n c tio n which, fo r a d ilu te g a s, i s governed by th e Boltzmann eq u atio n . The p re se n t study on th e gas expansion in to a vacuum w ill be based on t h i s eq u atio n under sp h e ric a l symmetry. Since th e s o lu tio n o f th e d is tr ib u tio n fu n c tio n i s u s u a lly hard to g e t, in s te a d o f so lv in g th e Boltzmann e q u a tio n , a s e t o f tr a n s f e r eq u a tio n s, o b tain ed by ta k in g d if f e r e n t moments on th e Boltzmann eq u atio n , w ill be d eriv ed in th e p re se n t c h a p te r as th e governing e q u atio n s. 2 .1 . Maxwell T ra n sfer Equation in S p h eric a l C oordinates The Boltzmann eq u atio n , which has been tr e a te d in most te x ts d e a lin g w ith th e k in e tic th e o ry o f g a se s, (fo r example, see Chapman and Cowling [16] o r Jean [17] i s w ritte n as f * * • V * ? • V ■ C2.1) where f ( r , c , t ) i s th e d is tr ib u tio n fu n c tio n o f a g a s, c is th e v e c to r p o s itio n in v e lo c ity sp ace, r i s th e v e c to r p o s itio n in 6 o rd in a ry space and y i s th e a c c e le r a tio n o f th e p a r t i c l e due to e x te rn a l fo rc e s and th e r o ta tio n a l v e lo c ity i f a c u r v ilin e a r co o rd in a te system i s being used. C e rta in m eaningful averages based on f a re d e fin e d by ta k in g d if f e r e n t mements: (a) The number d e n s ity n = / fd c (b) The mean v e lo c ity ^ = n " 1 / cfdc (c) The k in e tic s tr e s s te n so r 1 P = m / (c - ^ ) ( c - \f) fdc (2 .2 ) (d) The average k in e tic energy o f th e p a r t ic le s E = n * J h mc2fdc _ (e) The h eat flu x te n s o r q - m / (c - tf) (c - $) (c - ^ ) fdc I f Q (u,v,w ) re p re s e n ts any fu n c tio n o f th e v e lo c ity components (u,v,w ) o f each m olecule, th en i t s mean v alu e Q i s d e fin e d as Q = n _1 / Qfdc (2 .3 ) C onsidering th e ste a d y s p h e r ic a lly sym m etrical c a s e , w ith o u t e x te rn a l fo rc e , eq u atio n (2 .1 ) reduces to 8 v 2 » w2 3 _ r 3 " 3 uv 3 uw 3 , - r6 f. r, 3u r 3v r 3wJ * " t3 F c o l l . L J where r i s th e d is ta n c e from th e o r ig in and u ,v ,w a re th e o rth o g o n al components o f p a r t i c l e v e lo c ity in th e d ir e c tio n s o f in c re a s in g r , 6 and $ r e s p e c tiv e ly . (0 and $ a re th e m e rid io n a l and azim u th al a n g les r e s p e c tiv e ly .) M u ltip ly in g eq u a tio n (2 .4 ) by Q and in te g r a tin g o v er th e v e lo c ity sp ac e, th e Maxwell t r a n s f e r e q u a tio n i n s p h e ric a l symmetry i s o b ta in e d , where A Q i s th e change in Q due to c o l l i s i o n s . In d e riv in g e q u a tio n (2 .5 ) one assum es th a t th e p ro d u ct o f th e d i s t r i b u t i o n fu n c tio n and any polynom ial o f p a r t i c l e v e lo c ity components v a n ish e s a t both i n f i n i t i e s o f v e lo c ity sp ace. I f one d e fin e s th e fo llo w in g q u a n t i t i e s : gj- [ r 2 / uQfdc] - £ J (v2 + w2) fdc + |- J uv fdc + r / uw lw fd ^ = A Q (2 .5 ) n = J fdc u = n _1 / u fd c / (u -u )2 fdc px = / v 2fd c » / w2fdc = f ( u -u )3 fdc (2 . 6) = J (u-u) v 2fdc - j (u-u) w2fdc Qrr r r = / fd ? C2.6) O iT X1 = j (u -u )2 v 2fdc = / (u -u )2 w2fdc Qj_liX = / v^fdc = / w**fdc Q1X = / v ^ ^ d c and s u b s titu te s Q = 1, u , u 2, v2 + w2, u 3 and u (v 2 + w2) in to th e Maxwell tr a n s f e r eq uation ( 2 .5 ) , th e re s u ltin g eq u atio n s a re , ^ (nur2) = 0 (2 .7 .1 ) — b 2 (pr + nil2)] - 2rp^} = 0 (2 .7 .2 ) "T {3 r ^r 2 ^ r + 3“ pr + " 4 r(“Pj. + )> = AC «2) (2 .7 .3 ) r 2 T *dr + uP j.) + 2 r (uPj. ■ A(v2+w2) (2 .7 .4 ) r 2 ~ [r 2 C Q j.r r r + 4u J T + 6u2pr + nu7 *)] - 6r(Qr r ^ x + 2 u ^ . r 2 + u 2px )} *A (u3) (2 .7 .5 ) T {3? [l2(%-r+S S&L * * r< 2V x x - <ixxix- * r 4n<Jj_ + 2u2pA )} ■ A(u(v2+w2)) (2 .7 .6 ) One observes th a t th e f i r s t and second q u a n titie s d efin ed in equation (2 .6 ) a re th e number d e n sity and m acroscopic r a d ia l v e lo c ity r e s p e c tiv e ly , as a lre a d y m entioned in eq u atio n ( 2 .2 ) . The n ex t fo u r q u a n titie s may a ls o be r e la te d to p h y sic a l p r o p e rtie s o f a g a s. By d e f in itio n o f eq u atio n ( 2 .2 ) , one im m ediately re c o g n iz e s th a t mpr and m px a re k in e tic s tr e s s e s from which p a r a l l e l and tra n s v e rs e tem pera tu r e s can be d e fin e d by, d ir e c tio n . A ll th e se m acroscopic q u a n titie s , in t h i s c a s e , a re fu n c tio n s o f r o n ly . F u rth e r com bining and re a rra n g in g th e moment eq u atio n s ( 2 .7 .1 ) - ( 2 .7 .6 ) , th ey can be put in th e fo llo w in g form: nRT r pr , nRT p. and 3T » T + 2T. j. r -L and th a t re p re s e n ts th e h eat flu x in th e r a d ia l n u r2 c o n sta n t (2 . 8 . 1) (2 . 8 . 2) = A(u2) (2 .8 .3 ) = a(v2+w2) (2 .8 .4 ) Til « A(u3) - 3uA(u2) (2 .8 .5 ) 11 J. JL-LX r 3Q ± J + 2 ( r if ^fx )ig ^ } = A (u (v 2+w2)) - uA(v2+w2) (2 .8 .6 ) u I f th e c o l l i s i o n in te g r a l AQ, can be ex p ressed in term s o f moments o f th e d i s t r i b u t i o n fu n c tio n , one s t i l l can n o t so lv e th e s e moment e q u atio n s ( 2 .8 .1 ) - ( 2 .8 .6 ) , as th e r e a re more unknowns th a n th e re a re e q u a tio n s. In g e n e ra l, moment e q u a tio n s n ev er form a c lo se d system u n le s s a c e r ta in form o f d i s t r i b u t i o n fu n c tio n i s chosen [1 8 ], o r c e r ta in asy m p to tic lim it such as in th e h y p erso n ic approxim ation r e c e n tly used by Edwards and Cheng [12] and Hamel and W illis [13] a re c o n sid e re d . They w ill be d isc u sse d in g r e a te r d e t a i l a f t e r th e c o l l i s i o n in te g r a ls a re d isc u s s e d . 2 .2 . The C o llis io n I n te g r a ls The c o l l i s i o n i n t e g r a l, AQ, which ap p ears in th e moment e q u a tio n s ( 2 .8 .3 ) - ( 2 .8 .6 ) r e p re s e n ts th e r a t e o f change o f t o t a l amount o f Q due to c o l l i s i o n s , and i s w r itte n a s , Two s e ts o f c o l l i s i o n in te g r a ls w ill be p re se n te d in t h i s s e c tio n , namely th e c o l l i s i o n in te g r a ls o f th e Boltzmann e q u atio n w ith M axw ellian m olecules and th o se o f B-G-K m odel. 2 .2 .1 . C o llis io n I n te g r a ls o f Boltzmann E quation w ith ■ M axw ellian M olecules Upon m u ltip ly in g th e c o l l i s i o n term o f th e Boltzmann eq u a tio n A Q - / Q(- dc (2 .9 ) 12 by Q and in te g ra tin g over th e v e lo c ity space, th e r e r e s u l t s th e c o ll i s i o n in te g r a l: A Q = / Q (£’f{ - £ £ ,) gbdbd d cjd c (2.10) Equation (2.10) may a ls o be w ritte n in a sym m etrical form [1 9 ], A Q = J (Qx - Q) ffjg b d b d dCjdc (2.11) The e v a lu a tio n o f A Q through th e dynamics o f c o l l i s i o n , f o r th e g e n eral c e n tr a l re p u ls iv e m o lecu lar fo rc e law has been tr e a te d in d e t a i l in many te x ts [16,18] w ith an assumed form o f d i s tr ib u tio n fu n c tio n which d e v ia te s s l i g h t l y from M axwellian d i s t r i b u t i o n . Maxwell f i r s t found th a t th e c o llis io n in te g r a l A Q o f a m olecule w ith an in v e rse f i f t h power fo rc e law can be c a r rie d out w ithout in tro d u c in g th e ex p ressio n o f th e d is tr ib u tio n fu n c tio n . The c o llis io n in te g r a ls ap p earin g in th e moment eq u atio n s (2 .8 ) a re , A(u2) = [A2Tr/2lto] n (p^ - pr ) (2 .1 2 .1 ) A(v2+w2) = - [A2t t / 2K5] n (p^ - pr ) (2 .1 2 .2 ) A ( u 3 ) = [A2*/2]to] n [3u(p^ - pr ) + | ( J ± - J r)] (2 .1 2 .3 ) A(u(v2+w2)) = [A2w/2Km] n [- u(p± - pr ) " I - J r )1 (2 .1 2 .4 ) In th e above c o l l i s i o n moments, m i s th e mass o f th e m olecule, K is a fo rc e c o n sta n t and i s d e fin e d by where P i s th e r e p e llin g fo rc e betw een two m o lecu les a t a d is ta n c e r a p a r t , and A2 i s a c o n s ta n t which was e v a lu a te d f i r s t by Maxwell f o r th e m olecule o f in v e rs e f i f t h power fo rc e law and has a v a lu e o f 0.4 3 6 . The fo rc e c o n s ta n t K may be r e l a t e d to th e gas v i s c o s i t y , U, by th e r e l a t i o n [1 9 ], 2 .2 .2 . C o llis io n I n te g r a ls o f B-G-K Model A c o l l i s i o n term , w hich re p la c e s th e co m p licated in te g r a l form found in th e Boltzmann e q u a tio n by a sim ple r e la x a tio n form , was f i r s t proposed by B h atn ag er, e t a l . [2 0 ]. I t can be w r itte n a s : where Fw r e p r e s e n ts th e lo c a l M axw ellian d i s t r i b u t i o n fu n c tio n , N and n , u and T a r e , r e s p e c tiv e ly , th e lo c a l number d e n s ity , gas mean v e lo c ity and te m p e ra tu re ; R = k/m i s th e gas c o n s ta n t. "A" i s a f r e e p aram eter which in g e n e ra l can be r e l a t e d to th e s t a t e o f th e g a s. The p ro d u c t "An" in e q u a tio n (2 .1 5 ) s ig n i f i e s th e c o l l i s i o n freq u e n c y . A ccording to t h i s m odel, th e number o f m o lecu les l o s t p e r u n it volume p e r u n it tim e due to c o l l i s i o n i s p ro p o rtio n a l to th e kT (2 .1 4 ) ^ fit^ c o ll. An (Fm - f ) (2 .1 5 ) nam ely, F , M n i - E - ] (2 .1 6 ) (2t t RT) 14 number d e n s ity , as is expected fo r M axwellian m olecules w ith a c u t- o f f approaching d is ta n c e [2 1 ]; whereas th e r a t e o f m olecules gained p e r u n it volume due to c o llis io n equals th e number o f m olecules em itted a t th a t p o in t w ith a M axwellian d is tr ib u tio n (2 .1 6 ). The e v a lu a tio n o f th e c o llis io n in te g r a ls o f B-G-K model i s s tra ig h tfo rw a rd . The B-G-K c o llis io n moments appearing in th e same eq u atio n s (2 .8 ) a r e , A(u2) = J-An (p^ - pr ) (2 .1 7 .1 ) A (v2+w2) = - | A n ( p ± - p r ) (2 .1 7 .2 ) A(u3) = f An [3u (Pj- p r ) + | (~JT)] (2 .1 7 .3 ) A(u(v2+w2) ) = j A n [- u (p^ - pr ) - 3,J± ] (2 .1 7 .4 ) Comparing th e c o llis io n moments fo r both c o llis io n m odels, upon u sin g th e r e la tio n (2 .1 4 ), one fin d s th a t t h e i r second o rd e r c o llis io n moments a re th e same, p ro v id in g , . nmRT An = —— (2.18) However, th e th ir d o rd e r c o llis io n moments a re d if f e r e n t f o r th e s e two c o llis io n m odels. 2 .3 . H ypersonic Approxim ation Upon s u b s titu tin g th e c o llis io n in te g ra ls o b tain ed in s e c tio n 2.2 in to th e moment eq u atio n s ( 2 .8 .1 ) - ( 2 .8 .6 ) , one o b ta in s , nur 2 = c o n s t. (2 .1 9 .1 ) 15 [2n>a - a ? cr*pr )] (2.19.2) nr* u rd ,_ 9_ > . r3 ^ r V du . 1 d ,_ 2 P ^ 4r / T * — {d r (r P r5 + [ - d r = d r (r ^ r > “ = ^ < l ] } = j An(px - pr ) (2 .1 9 .3 ) * < 3 F <r V > - f * = J F ( M ) ] > r H u u = - j A n ( p x - p r ) (2 .1 9 .4 ) m x S j t a F ( r V r ) * 3 ( r 2Pr ) § * i | f ( r ^ r r r ) - & ^ r* u u - = ^ C r 2</r )> - Am£t (2.19.5) ‘l l i l * 4 0 * 4 . ) * C r V ) 3 F * - t | r 0 * W ’ ^ r H u - r3Q j_J ♦ 2 ( A 4 ) i g - J . A n /., (2 .1 9 .6 ) where, o £ j ^ Maxwell m olecules “< / r B-G' K 1 9 9 C2*20) T ■ 7-< 2 L ) Maxwell m olecules ‘ 3 -r 2 A B-G-K A - | [A2ir/2lto] = £ 1 16 As mentioned p re v io u sly , as a n a tu re o f Maxwell tr a n s f e r e q u atio n , th e moment equations (2 .1 9 .1 )-(2 .1 9 .6 ) do not form a clo sed system . The convection term s o f th e Boltzmann eq uation always in t r o duce higher o rd er moments in to th e moment eq u atio n . T h erefo re, n a tu r a lly , th e moment equations are never c lo se d . However, in th e reg io n where flow i s hy p erso n ic, i . e . , th e flow speed i s much hig h er than th e therm al speed, th e number o f v a ria b le s can be reduced by 2 dropping th e term s o f h ig h er order (Thermal speed/Flow speed) and hence a clo sed system a t c e r ta in le v e ls can be o b tain ed . Hypersonic approxim ation up to second o rd er moment eq uations has been analyzed in References [12, 13]. In th e second o rd er moment eq u atio n , th e o rd er o f m agnitude o f th e r a tio J i j k up cannot be determ ined u n le ss th e a n a ly s is o f th e th ir d o rd e r moment equations i s made. In view ing equations (2 .1 9 .5 ) and (2 .1 9 .6 ), one / ... w ill fin d th a t ■ ■ i s o f o rd er 1/M2, in ste a d o f 1/M as up a n tic ip a te d by previous workers [12, 13]. T h erefo re, th e moment eq u atio n s a re indeed clo sed a t second o rd er le v e l. But in th e th ir d o rd er moment e q u a tio n s, i t i s q u estio n ab le w hether one i s allow ed to do th e same th in g , i . e . , dropping th e h ig h e r-o rd e r moments, QijkA; sin c e th e r a t i o o f QijkA u * 4 'j * k ' may be o f o rd er o f u n ity . 17 In f a c t , as w ill be shown, th e hypersonic tru n c a tio n works only a t c e r ta in moments but not a l l o f them. In t h i s s e c tio n , a study w ill be made on th e e stim a tio n o f th e r e la tiv e o rd er o f m agnitude o f d if f e r e n t moments, from which a ru le o f n a tu ra l tru n c a tio n due to hypersonic approxim ation w ill be e s ta b lis h e d . 2 .3 .1 . D is trib u tio n Function and I t s Moments fo r S p h erical In t h i s s e c tio n , follow ing Reference [1 5 ], a g en eral moment o f th e d is tr ib u tio n fu n c tio n i s g iv en ; from which th e r e la tiv e o rd er o f m agnitude o f d if f e r e n t moments can be examined. In R eference [1 5 ], based on B-G-K model eq u atio n , under hypersonic approxim ation, th e d is tr ib u tio n fu n c tio n fo r th e sp h e ric a l source can be put in to an e x p lic it in te g ra l form as a fu n c tio n o f tem p eratu re, Source Flow under th e Hypersonic Approximation f (2 . 21) f 0(a,B) exp [- / An d r ' (2irRT) d r ' - 02/r'2 where 18 a - + v^ + w2 0 = r / v 2 + and £Q(a ,0 ) re p re s e n ts th e d is tr ib u tio n fu n c tio n a t r = r x . For r » r x th e f 0 ' term damps out ra p id ly , and th e d is tr ib u tio n fu n c tio n i s approxim ately given by f j . I f one in tro d u ce s th e dim ensionless q u a n titie s , ® = I - , s - r(— -) (2.22) Tl A n ^ 2 where T j, n j a r e , re s p e c tiv e ly , th e tem p eratu re and number d e n s ity a t th e i n i t i a l ra d iu s r ^ $ and d e fin e s th e d im en sio n less q u a n tity Mx = — — ✓yRTj upon u t i l i z i n g th e method o f Laplace, and changing th e o rd er o f in te g ra tio n , a g en eral moment o f th e v e lo c ity d is tr ib u tio n fu n c tio n can be found: 1 = J j j (u -u )* (v 2+w2)^ f dudvdw v ^ “\ i +j+3 r" J + l -b 2 . s , 2y . 1+2 = 2irK (u) J I bJ e db • e (— 1 — ) ° M j 2s 2 s l 4 j+2 ^ f ® * e x P. dx (2.24) 0 1 where, 19 n r — n l r i , u ,2 K “ ~TT7 c r J (2irRT ) 3 / z An r L2 > = ( 2n - l ) r @ p 2n M 2 2n-2 P2 n .l * P2n - [1 4 - ^ P2n+2 (2 ’25> M j S *0 M , © P0 i s given in R eference [14]. Thus, H1^ is r e la te d to M x by, H1^ = Hl j Cs)Cu)1+;j+3 (jj~)3+:5+k and k = i fo r even i (i+1) fo r odd i . 2 .3 .2 . T ru n catio n o f Moment Equations The r e s u lts ob tain ed in s e c tio n (2 .3 .1 ) w ill be used to estim ate th e r e la tiv e o rd er o f m agnitude o f d if f e r e n t moments appearing in eq u atio n s (2 .1 9 ). In eq uation (2 .1 9 .2 ), th e term s on th e r ig h t hand sid e a re o f o rd er 1/Mj2 in comparison w ith th e term on th e l e f t hand s id e . Under hypersonic approxim ation, M j » 1, th e form er may be dropped, and eq u atio n s (2 .1 9 .1 ) and (2 .1 9 .2 ) form a clo sed system to solve fo r n and u , su b jec te d to an e rro r o f o rd er 1/Mj2. 20 Exam ination on eq u atio n s (2 .19.2) , (2.24) and (2.25) in d ic a te s th a t th e term s in sid e th e square b ra c k e t o f eq u atio n s (2 .1 9 .3 ) and (2 .1 9 .4 ) a re o f o rd er 1/MX 2 in com parison w ith (o r p ) . C onsequently, th e energy flu x term s can be decoupled from th e second o rd e r moment eq u atio n s under hy p erso n ic tru n c a tio n . T his c lo se d system o f moment equations up to th e seco n d -o rd er le v e l, has been analyzed by Edwards and Cheng, and Hamel and W illis [12, 1 3 ]. The r e s u lts o f eq u atio n s (2.24) and (2 .2 5 ) a ls o show th a t a n d / u u u u a re o f same o rd e r o f m agnitude. T h e re fo re , one w ill n o t expect th a t th e h y personic tru n c a tio n can be done a t th e th ir d - o r d e r moment le v e l. However, th e term s in v o lv in g th e f if t h - o r d e r moment in th e fo u rth - o rd e r moment eq u atio n s a re o f o rd e r 1/Mj2 as compared w ith th e fo u rth -o rd e r moment, hence h y p erso n ic tru n c a tio n i s a g a in p o s s ib le . In g e n e ra l, in th e s p h e ric a lly sym m etrical c a s e , th e moment e q u atio n s c o n ta in o n ly th e moments having even-power tra n s v e rs e v e lo c ity , namely even j . In view o f th e g e n e ra l moment (2 .2 4 ), one fin d s th a t th e moments o f o d d -o rd er a re o f th e same o rd e r o f m agnitude as th o se o f n ex t h ig h e r-o rd e r, and a re o f o rd e r 1/M j2 as compared w ith th o se o f one o rd e r low er. Because o f th e f a c t th a t th e ev en -o rd er moment eq u a tio n c o n ta in s h ig h e r o d d -o rd er moments, which w ill be o f h ig h e r o rd er 1/M^2 compared w ith even moments, h y p erso n ic tru n c a tio n can be done f o r a l l ev en -o rd er except th e z e ro th -o rd e r moment e q u a tio n . As to th e odd-order moment eq u atio n s (a sid e from th e f i r s t o r d e r ) , th e 21 h ig h er even-order moment w ill be o f th e same o rd er o f magnitude as th o se od d -o rd er, hence no tru n c a tio n can be done. 2 .4 . T hird-O rder Moment Equations In has been shown in s e c tio n (2.3) th a t th e system o f moment equations (2 .1 9 .1 )-(2 .1 9 .6 ) can not be clo se d by "hypersonic tru n c a t i o n ." In o rd er to g e t a clo sed system , one way i s to r a is e th e moment equation to th e fo u rth -o rd e r le v e l. In doing t h i s , fo u r a d d itio n a l equations w ill be added to th e system w ith a corresponding in c re a se in degree o f com plexity. I f one is only in te re s te d in th ird -o rd e r moments and in ten d s to d eal w ith th e moment eq u atio n s o f o rd er no h ig h er than th re e , then some o th e r means must be used to make th e system clo se d . In t h i s stu d y , a tru n c a tio n by choosing an assumed form o f v e lo c ity d is tr ib u tio n fu n c tio n i s used to c lo se th e system a t th ird -o rd e r le v e l. Let th e d is tr ib u tio n fu n ctio n f be, f = f e ( l +3>) (2.26) fe = n[- 1 2vRTj. 2RTj. 2rrRT 2RT r r (2R T± ) Z(2RTJ -] (u-u) (v2+w2) ) . 22 which d e s c rib e s c o r r e c tly th e mean av erag es o f th e d i s t r i b u t i o n fu n c tio n up to th ir d - o r d e r moment in th e s p h e r ic a lly sym m etrical c a s e . The h ig h e r moments can be re p re s e n te d in term o f low er moments Note t h a t , a e llip s o id a l d i s tr ib u tio n fu n c tio n [13, 22] w ill g iv e th e same e x p re ssio n in (2 .2 7 ). Upon s u b s titu tin g th e above e x p ressio n s (2.27) in to th e t h i r d - o rd e r moment e q u atio n s (2 .1 9 .5 ) and ( 2 .1 9 .6 ), a c lo se d system o f th ir d - o r d e r moment eq u a tio n i s o b ta in e d , n u r2 = c o n s t. (2 .2 8 .1 ) < W r = 3 n CR T r ) 2 Q r m = n(RTr )(RT± ) (2.27) Q 1 ± = n (RT± ) 2 W " 3nCRTx )2 (2 .2 8 .2 ) = j An(px - pr ) (2 .2 8 .3 ) - | a»(Pj . - Pr ) (2 .2 8 .4 ) 23 ^ 4 < A f r ) ♦ s r t . ) § * I ^ r u u + [ = 3 F ( r H ) n - ^ 1 (2 .2 8 .5 ) u ^ < $ f o ^ ) * C r V g - ♦ = c| f ^ r - ] + 2 ( ^ 4 . ) = A n / 2 (2 .2 8 .6 ) u E quations ( 2 .2 8 .1 )- (2 .2 8 .6 ) to g e th e r w ith s u ita b le boundary c o n d itio n s a re s u f f i c i e n t f o r so lv in g n , u , pr , Px » r an<* <^± • 2 .5 . D if f e r e n tia l E quations and Boundary C o n d itio n s o f S p h e ric a l Source Expansion i n Non-Dim ensional Forms The p re s e n t stu d y on th e gas expansion in to a vacuum w ill be based on Boltzmann moment e q u a tio n s. A tte n tio n w ill be p a id on th o s e up to th ir d - o r d e r moment. The flow i s assumed to be s p h e r ic a lly sym m etric, so t h a t th e moment e q u a tio n s ( 2 .2 8 .1 )- (2 .2 8 .6 ) may s e rv e as governing e q u a tio n s. F u rth e r assum ptions w ill be t h a t , a t an i n i t i a l ra d iu s r = r t , (a) The flow i s h y p e rso n ic , i . e . , u , 2 V - - 5 5 7 ~ 1 where U j, Tj a re r e s p e c tiv e ly th e mean v e lo c ity and tem p eratu re a t r = r . (b) The flow i s in e q u ilib riu m , namely T B t t = T, 24 (c) The flow can be c o n sid e red a s continuum n e a r r ■ r j , so th a t th e r a t i o o f flow tim e to c o l l i s i o n tim e a t th a t p la c e i s v ery la r g e , nam ely, Flow tim e ^ n . r 2 » 1 (2 .2 9 ) c o l l i s i o n tim e u 1 where n j i s th e number d e n s ity a t r » r ^ (d) The flow i s allow ed to expansion f r e e ly so t h a t as r approaches i n f i n i t y , n w ill approach z e ro . T h e re fo re , th e boundary c o n d itio n s giv en a t r = r w ill be s u f f i c i e n t to d eterm in e th e flow f i e l d . I f n , u , pr and px a re giv en a t r = r ^ th e n one has boundary c o n d itio n s : At r = T j, n = n j (2 .3 0 .1 ) u = Uj (2 .3 0 .2 ) Pr - (Pr ) i (2 .3 0 .3 ) Px = (.V ± ) 1 (2 .3 0 .4 ) From th e m o lecu lar view p o in t th e component o f th erm al flu x in th e r d ir e c tio n qr , i n th e n e a r continuum re g io n may be ex p ressed a s , % = m(J T * 2 ^ _ ) « nakA ~ o r J r + 2 j ± « naRA U- where 'a ' re p re se n ts th e speed o f sound a t th a t p o in t, k i s th e Boltzmann co n stan t (equals to mR) and i s th e lo c a l mean fre e p a th . From th e above e x p ressio n , one sees t h a t , in th e re g io n near r = r j , in which th e flow is continuum and near eq u ilib riu m , th e l a s t two boundary c o n d itio n s may be w ritte n a s , At r = r 1, J x ' ° r n > * l Rlli 3F*r = r (2 .3 0 .5 ) J x - °x V l M l C- § ) r . r j (2 .3 0 .6 ) where or and a re c o n sta n ts o f o rd er o f u n ity and w ill be determ ined l a t e r . Subsequently, i t w ill show th a t th ey depend on th e c o llis io n model. The moment equations (2 .2 8 .1 )-(2 .2 8 .6 ) and th e boundary c o n d itio n s (2 .3 0 .1 )-(2 .3 0 .6 ) form a com plete system and a re ready to so lv e. I f one in tro d u c e s th e follow ing non-dim ensional v a ria b le s : th e non-dim ensional p a ra m eters, RTi e = — — « l (acco rd in g to assum ption (a )) u i 2 (2.32) A nirj a 1 =----------- >> 1 (according to assum ption (b )) U1 and th e non-dim ensional independent v a r ia b le , * - 7T V - <2 -33> 1 1 th e n , in term s o f th o se d im en sio n less q u a n t i t i e s , one may w rite th e th ir d - o r d e r moment eq u atio n s a s: w here, 27 « 2 { - l 2 L - [p - p 1} ( 2 . 3 4 .4 ) 1 d 2U d ,2 / ^ ^ / 2” n dU , „ r3 d , S Pr., 6s Pr5 . n H j (* - ^ r ) + 3 (s Pr ) + E t d7 C -= r-)“ as r r as u as n U n + (*2^ J " a i 2 t— ^ - i> (2 .3 4 .5 ) U ds r 1 u 1 «♦- - S-2 d , u 7 \ , k- , dU . rl d ,s pr p x , 4 S P J-1 di- C * t4 ) ♦ (s Px) d ? + e [ - d J (“ = — ) “ = ~ ] ♦ - XT (■**<£) * ®[2 {“ - * 0 (2 .3 4 .6 ) U as 2U ^ “ x / r M axwellian M olecules - 7 B-G-K r (2.35) j („ / - 7>j^_) Maxwellian M olecules - 2 * /j _ B-G-K The boundary c o n d itio n s (2 .3 0 .1 )- (2 .3 0 .6 ) , in term s o f dim ensionless 28 q u a n titie s , can be w ritte n a s: At s ■ si ■ — a l nCsO - 1 (2 .3 6 .1 ) U(S!) - 1 (2 .3 6 .2 ) p ^ S ! ) * 1 (2 .3 6 .3 ) P |_(si) - 1 (2 .3 6 .4 ) J r ( * i ) - kr s i e ( 2 3 6 5 ) ^ ( S j ) « k S ie (2 .3 6 .6 ) where kr and k ^ a re c o n sta n ts o f o rd e r o f u n ity , and w ill be determ ined su b seq u en tly (they depend on th e c o llis io n m o d el). In d e riv in g th e boundary c o n d itio n s (2 .3 6 .5 ) and (2 .3 6 .6 ), th e r e la tio n ^ 1 * 1 a i r i r i •oc oc (2.37) U1 u l A i M i £ i o r r i M 17 * M i “ i has been u sed . There a re two sm all p a ra m e te rs, e and — , in t h i s problem , a i hence th e s o lu tio n s w ill be fu n c tio n s o f e , aj as w ell as th e independent v a r ia b le s . The sm alln ess o f e allow s one to expand th e s o lu tio n s a sy m p to tic a lly in to a power s e r ie s o f e , which w ill be 29 tr e a te d in th e nex t c h a p te r. Corresponding to th e sm all (o r s ^ , l two d i s t i n c t reg io n s e x h ib it in t h i s problem , nam ely, th e re g io n where s i s o f o rd e r o f s i (th e in n e r re g io n ), and th e one where s is o f o rd e r o f u n ity (th e o u te r r e g io n ) . In th e n e a r-e q u ilib riu m , in n e r re g io n , th e d e riv a tiv e s in eq u atio n s corresponding to th e second and h ig h er moments appears as h ig h er o rd er term s. Such degeneracy cannot hold in th e re g io n s ■ 0 ( s ) . M ath em atically , one could t r e a t the problem by applying th e method o f asym ptotic expansions in an in n er and o u te r re g io n s (in a 1 ) . In th e fo llo w in g , however, th e expansion w ith re s p e c t to a 1 w ill n o t be pursued, although th e assum ption o f a 1 » 1 w ill s t i l l be made. Thus th e development w ill concern p rim a rily w ith asym ptotic expansions in e a lo n e . The r e s u l t s and i t s degree o f accuracy a tta in e d t h i s way is expected to be eq u iv a le n t to th e one based on double expansion (in e and a 1 ) , a t le a s t fo r th e p re se n t problem . In Van D yke's term inology [2 3 ], th e p re se n t approach may be re fe r r e d to as th a t based on th e "com posite eq u atio n s" (as f a r as a 1 i s co n cern ed ). The use o f com posite eq u atio n s has th e advantage o f b y -p assin g th e procedure o f m atching s o lu tio n s in neighboring re g io n s. Since th e dependent v a ria b le s (2.31) a re norm alized by th e q u a n titie s in th e in n er re g io n , th e s c a lin g f a c to rs should be i n tr o duced to th e dependent v a ria b le s which p re se n t in th e o u te r re g io n . The choice o f p ro p er s c a lin g f a c to rs can be in fe rre d from th e m atching req u irem en ts in th e o v erlap re g io n , where both th e in n er and o u te r expansions a re v a lid . In th e in n er re g io n , where s » 0 (s ) , in view o f th e dim ensionless moment eq u atio n s ( 2 .3 4 .1 )- (2 .3 4 .6 ) , one expects th a t th e flow is n e a r-e q u ilib riu m . Under h y p erso n ic approxim ation, one may w rite th e fo llo w in g e x p re ssio n s: s i 2 n ^ (— ) (2 .3 8 .1 ) U » 1 (2 .3 8 .2 ) “ “ T~ ^ C|— (2.38.3) — i i Si n 1 1 p_ j _ j — ■ (2 .3 8 .4 ) n 1 1 J T ** kr e Sl ( ~ ) ( | ^ ) 2 (2 .3 8 .5 ) T S1 T 2 k e s x (— ) ( ^ ) (2 .3 8 .6 ) In th e o v erlap re g io n , where s << s « 1, th e n e a r-e q u ilib riu m s o lu tio n s a re s t i l l v a lid , alth o u g h th e degree o f approxim ation becomes m arg in al. From th e above e x p re ss io n s, one may d e fin e th e d im en sio n less v a ria b le s o f th e o u te r re g io n , where s = 0 (1 ), a s : 31 P_L P-L P. - -----— r— - ------- t ------------ (2 .3 9 .4 ) a S l 10/3 S ilo /3 n 1KTl S = ■ . - £ - L ------------- (2 .3 9 .5 ) r e m /3 c 1 b/3 — _ DT Sj e £ u ^ j RTj S « — ^ ---------------- (2 .3 9 .6 ) ± _ 1^/3 1 U / 3 - RT Sj e s1 e U jiijK ij Thus th e o u te r v a r i a b l e s , N, U, P , and S ^, a re a l l o f o rd e r o f u n ity in th e o u te r re g io n . Some o f th e same s c a lin g f a c to r s were a ls o found by Freeman [2 4 ]. In term s o f th e d im e n sio n le ss v a r ia b le s d e fin e d by ( 2 .3 9 .1 ) - ( 2 .3 9 .6 ) , th e moment e q u a tio n s become, NUs2 = 1 (2 .4 0 .1 ) U I ? a £ Sl { 2 [2sP-l " a ? C * Pr )]} (2 .4 0 .2 ) Ns 1 - f s 2 P 1 * [ 3 ( s 2 |> r ) dU 4 / 3 I d . . 2 S , ds Cs V 1 .. . s l U ds Cs V U ds - e s l ‘‘/3 r - 5.1 - pr ) (2 .4 0 .3 ) ± - r ^ p 1 + r C 5 du , E S "/3 l i - ( V s . i ] d s [ s p J-J + 1 u ds 1 U ds ^ 2 Ns1 * - i r p i - pr ] C 2 -4 0 -4 ) 32 ^ f-2e • ) + * , 3 fs 2P ) — + — — ( r ) ds t s V ^ 7 T V ds U ds 1 N J e Sl P P . 6s r x 4 dU f 2 < 5 • » s Ns r IT T T " + U 3T (S V 1 T L1 s *tp p !» _ ,dU . 1 d , r 1. ds (s S ± ) + ^ 7 T Pj^ d s + u ds C N ^ e S j 4S^ 2. dU - M S ' 1 U N U ds C 2U 2 w here, S - S M axwellian J . r 1*1 = - S B-G-K l2 = 3 (S - 7s ) Maxwellian r i - 2S± B-G-K The boundary co n d itio n s o f th e o u te r v a ria b le s can be As s ■ s j , N (sx) » - L . s l 2 U(S l) = 1 (2 .4 0 .5 ) (2 .4 0 .6 ) M olecules M olecules w ritte n a s , (2.41.1) (2.41.2) Pr ^ S l ‘* " 1 0 / 3 s l (2.41.3) Since one i s in te r e s te d p rim a rily on th e t r a n s i t i o n flow which appears in th e o u te r re g io n , in th e follow ing c h a p te r, th e s o lu tio n s d isc u sse d w ill be based on th e o u te r v a ria b le s d efin ed by eq uations (2.39) and th e governing eq u atio n s (2.40) su b je c te d to th e boundary c o n d itio n s (2 .4 1 ). CHAPTER III SOLUTIONS In t h i s c h a p te r, s o lu tio n s to th e d im en sio n less moment eq u a tio n s ( 2 .4 0 .1 )- (2 .4 0 .6 ) s u b je c te d to th e boundary c o n d itio n s ( 2 .4 1 .1 )- (2 .4 1 .6 ) f o r e << 1 and >> 1 a re going to be looked f o r . The stu d y in C hapter I I su g g e sts th a t one may expand N, U, P^, ^ , S^ and Sj_ in powers o f e, N = N0 + e Nj +............. U = U0 + e Ux + ......... P ■ P + e P + ......... r r 0 r i (3 .1 ) P -L = P±0 + e P±1 + ........... s = s + ......... r ro s ± - SX0 + ......... ( I t i s found su b seq u en tly th a t th e f i r s t - o r d e r s o lu tio n s a re indeed o f same o rd e r o f m agnitude as th o se o f z e ro th o rd e r in th e re g io n , S > _ Sj .) To c o n s is t w ith th e i n i t i a l c o n d itio n s ( 2 .4 1 .1 ) - ( 2 .4 1 .6 ) , th e i n i t i a l c o n d itio n s to th e above expansions can be w r itte n a s: At s = S i, 34 N0( s i) = 1 / s i 2, N .(S l) = 0; i = 1 ,2 ,3 , U0(S!) = 1, U .(s i) = 0; i = 1 ,2 ,3 , Pr 0 ^ = 1 /S l1 0 /3 , Pr i ( s i) = 0 ; i = 1 ,2 ,3 , P ±0<S1) = 1 /S l 10/3, PL i( s 1) * 0; i = 1 ,2 ,3 , Sr o ( i l> ■ v s!1/3> s r i ( , i 5 ■ 0; i = 1 ,2 ,3 , S, 0 ( s i> = k / s j 173, Su ( s ,) = 0; i = 1 ,2 ,3 , On s u b s titu tin g th e expansions (3 .1 ) in to eq u atio n s (2 .4 0 .1 )- (2 .4 0 .6 ), and eq u atin g th e lik e power o f e , one fin d s th a t to th e zero th o rd er o f e, N0U0s 2 = 1 (3 .2 .1 ) 36 c c 4(S „ - 7S, ) M axwellian S J.O - ro T 6 r0 ^ M olecules L10 = h20 ~ - S n - S .„ B-G-K ro J-0 i , s 2P 2 n dU, n _ i _ p p __— _ * L _ ( 1 - ( ) 3 fs 2P )___ b l Un Nn ro i o U n ds 1 N. J 1 k /3} 1 rOJds O 0 ( J 0 Sj ' dU’ 0 " 0 u o u a N 0 S, G - 4 s 3 p 2 _ 1 _ fL_ f r ° -LQ') _ ( 1 1 fs^ p • )____! 2 UnNn ± 0 Un ds 1 N J 1 4 7 3 m s -L0J ds '1 w ith boundary c o n d itio n s , At s = S j, Nq(s i) = ~~t (3 .3 .1 ) S1 U 0 (S l) = 1 (3 .3 .2 ) ProCsi) = - n 7 T C 3-3'3) S1 plo(s i ) = T H T T (3' 3' 4’ S1 sroCsi’ ’ 7TT7T (3-3‘5) S1 S io^si' “ T T 7 T (3 - 3 ' 6) s i and th a t to th e f i r s t o rd e r o f £ , (up to seco n d -o rd er moment eq u atio n s) where F. = s , k/3 [4sS - 4- ( s 2S J ] - 3 (s 2P J l l 1 j_o ds v r ( r J ^ r o ' ds - J Ui C P l o - pr o 3 p2 = i s2 u i ( P i 0 - Pr0 ) - 2(s-Pi 0 ) - s , * ' 3 . 2 j j j ( s - S ^ ) w ith boundary c o n d itio n s, At s = s , Nj CSj ) = 0 (3 .5 .1 ) U ^ S j) = 0 (3 .5 .2 ) P ^ S j ) = 0 (3 .5 .3 ) P ± 1 (S l) = 0 (3 .5 .4 ) In th is c h a p te r, s o lu tio n s to th e above z e ro th -o rd e r and 38 f i r s t - o r d e r eq u atio n s w ill be p re se n te d . 3 .1 . Z e ro th -o rd e r S o lu tio n s 3 .1 .1 . UQ and Nfl E quation ( 3 .2 .2 ) , to g e th e r w ith th e boundary c o n d itio n ( 3 .3 .2 ) , g iv es a c o n s ta n t z e ro th -o rd e r r a d ia l mean v e lo c ity . UQ = 1 (3 .6 ) S u b s titu tin g th e s o lu tio n (3 .6 ) in to th e z e ro th -o rd e r c o n tin u ity e q u a tio n ( 3 .2 .1 ) , th e s o lu tio n o f NQ i s o b ta in e d , Nq = 1 / s 2 (3 .7 ) which s a t i s f i e s th e c o n d itio n ( 3 .3 .1 ) . 3 .1 .2 . P „ and P, ro -Lo Upon s u b s titu tin g UQ by u n ity and NQ by 1 / s 2 , e q u a tio n s (3 .2 .3 ) and (3 .2 .4 ) reduce to th e fo llo w in g form , I f C**«W ■ T (P X0 - P r„J t3-8-15 * I s2[pl o * pr „ l <3 ' 8 -2> For convenience, ta k in g ( s 2Pr ) and as dependent v a r ia b le s and w r itin g , Yj = s 2Pr() (3 .9 .1 ) Y2 = s 2Pl0 (3 .9 .2 ) 3Y = Y t + 2Y2 (3 .9 .3 ) 39 a sin g le second o rd e r d i f f e r e n t i a l eq u atio n is o b tain ed by combining eq u atio n s ( 3 .8 .1 ) , ( 3 .8 .2 ) , ( 3 .9 .1 ) , (3 .9 .2 ) and ( 3 .9 .3 ) , s 2 — + (3s+ l) ^ + i - Y = 0 (3.10) ds2 ds 3 S Yj and Yg are r e la te d to Y by th e r e la tio n s : Yx = 3Y + | s Y (3 .1 1 .1 ) V2 = C 3 .ll. 2) The i n i t i a l c o n d itio n s are given by, At s = s x, Y = Y} = Y2 = 1 / s ^ 3 (3.12) In view o f th e s o lu tio n ( 3 .7 ) , one w ill id e n tif y th a t Y j, Y2 and Y a re th e sc aled z e ro th -o rd e r non-dim ensional te m p eratu res, Yj = (S j- ^ 3) ^ (3 .1 3 .1 ) T Y2 = ( s ^ 3) ^ (3 .1 3 .2 ) - . 4 / 3 C T ) 0 Y = ( s , 4 /3 ) - = A (3 .1 3 .3 ) 1 Thus, eq u atio n (3.10) is e x a c tly th e same one ( f o r o » = 1) o b tain ed by Edwards and Cheng [12]. As shown by Edwards and Cheng, eq u atio n (3.10) can be solved in term s o f co n flu en t hypergeom etric fu n c tio n s. Y(s) = Cji|)(- i , -1 , I ) + C2s " 2<j»(|, 3, j ) (3.14) 40 These fu n c tio n s are d efin ed in R eference [25]. They have th e fo llo w in g p r o p e r tie s . For s >> 1, T> - 1 ’ ¥ = t r (2 )/ r ( | ) ] [ i + y J + .......... ] - 2 .,2 - 1. -2 2 1 5 1 ^ S ♦Cs* 3 * s 5 " S ^ 9 s 108 2 s (3 .1 5 .1 ) For s << 1, . , 4 . 1. -4 /3 r, 8 , t ( - j , -1 , - ) = s [1 + ^ s + ...........] 1 _ 1 _ s "2* ( |. 3, i ) = es s 3 [ r (2 )/ r ( | ) ] [ i + .......... ] (3 .1 5 .2 ) The i j > fu n c tio n has th e same c h a ra c te r as th e in v is c id hy p erso n ic -*♦/ 3 s o lu tio n n e a r s = 0 , nam ely, T s ' , and approaches a f i n i t e lim it as s -+ » . The < J> /s2 fu n c tio n has th e c h a ra c te r s^ es n e a r s = 0 and approaches zero as s •* °°. T h e re fo re , in th e case o f expansion from th e continuum in v is c id lim it, i . e . , w ith an in v is c id i n i t i a l c o n d itio n (3 .1 2 ), C2 must be zero and Cj eq u als to one. ( I f an i n i t i a l s tr e s s ap p ears, th en C2 w ill n o t be zero . T his case w ill be d isc u sse d in th e fo llo w in g c h a p te r.) By e q u atio n s (3 .1 1 .1 ) and ( 3 .1 1 .2 ), Y j, Y2 can be g en erated from Y. They can be w ritte n a s: Y = U - j , - 1 , j ) (3 .1 6 .1 ) Yj = 3ip(- -1 , | ) - | U - y , 0, i ) (3 .1 6 .2 ) 41 Y2 - j U - j> 0, I ) (3 .1 6 .3 ) The same r e s u lts have been o b tain ed in Reference [1 2 ]. In b r i e f , a s h o rt d isc u ssio n on th e above o b tain ed s o lu tio n w ill be given below: (1) In th e h y p erso n ic li m i t , th e r a d ia l flow v e lo c ity is co n sta n t and th e number d e n s ity i s in v e rse p ro p o rtio n a l to th e square o f th e r a d ia l d is ta n c e . (2) The z e ro th -o rd e r av erag e, p a r a l l e l and tra n s v e rs e tem p eratu res can be re p re se n te d o r g en erated by th e co n flu en t hypergeom etric 4 1 fu n c tio n ^ (- -1 , j ) . Near th e so u rce, re v e a ls th e in v is c id s o lu tio n , namely, Tro ' tj.o * CT)o " r ‘ ‘,/3 w hile as s - * ■ « , th e asym ptotic b eh av io r o f < J > fo r sm all 1 /s g iv e s, (T )0(«) = ^ s " /3 T. (3 .1 7 .1 ) r(§) (T ) (") = 3 ^ 1 1 s , ^ 3 T. (3 .1 7 .2 ) F (|) r(.3J I Thus an u ltim a te "fro zen " p a r a lle l tem p eratu re and hence a "fro zen " average tem p eratu re w ill be f in a lly reached as th e flow expands f a r away from th e so u rce; w hile th e tra n s v e rs e tem p eratu re d ecreases to zero lik e 1 /r . The b eh av io r o f T^Q as r • + • 0 0 i s unexpected, sin c e 42 one would expect th a t a t la rg e d is ta n c e as c o llis io n becomes le s s fre q u e n t, w ill f in a lly reach th e fre e m olecular s o lu tio n , which i s , T. „ * 1 / r 2 , as r - * > 0 0 . ±0 . B ut, in view o f eq u atio n (2 .1 9 .4 ), w ith n - — 1 /r 2 , u = co n stan t one fin d s , ^ ( r 2T ± ) « An r 2(T., - T ± ) As r_+ “ , Tr approaches to a c o n sta n t value w hile Tj_ d ecreases to zero , hence th e c o llis io n term approxim ately eq u als to Anr2Tr , which w ill be a c o n sta n t and can not be n eg lected even though th e c o llis io n frequency d ecreases to zero lik e 1 / r 2, as r - * ■ < » . There f o re , due to a fro zen p a r a lle l te m p e ra tu re, th e tra n s v e rs e tem perature w ill be a ffe c te d by th e c o llis io n even i f th e c o llis io n frequency approaches zero as r - * ■ « . C onsequently, th e fre e m o lecu lar lim it o f Tj_ cannot be reached. (3) Since th e r a d ia l mean v e lo c ity is c o n sta n t, and th e tem perature d ecreases as flow expands outw ard, th e lo c a l Mach number M in c re ase s as r in c re a s e s . The lim itin g Mach number M ro can be o b tain e d by eq u atio n ( 3 .1 7 .1 ), m « 2 ■ r <!) s r ‘,/3 mi 2 I t can a ls o be c o r re la te d w ith th e r e s e r v o ir c o n d itio n 2 where Knudsen number Kn i s based on th e mean fre e p ath 1Q a t r e s e r v io r c o n d itio n and th e so n ic ra d iu s r* as th e c h a r a c te r is tic le n g th . The so n ic ra d iu s r* i s r e la te d to r j by, and th e mean fre e p ath 1Q a t r e s e r v o ir c o n d itio n i s d efin ed by, where a i s th e e f f e c tiv e c o llis io n cro ss s e c tio n d ia m eter, and th e s u b s c rip t "o" d en o tes th e q u a n titie s a t r e s e r v o ir c o n d itio n . (4) Y, Y and Y2 a re shown in F igure 1. 3 .1 .3 . S and S. ro________ lo_ (A) B-G-K Model For convenience, ta k in g s 2sr0 an(* st+Sj_o as t * ie dependent v a r ia b le s , one may in tro d u c e , 7 3 *1 - r . ( i ) 2 [M;]2 [1 ♦ - 2 - ] z M,2 1 1 16 M o /2 n n 0 o 2 5 p 0 z 1 s 2S ro (3.18) z 2 s^S 1 0 The co rresp o n d in g i n i t i a l c o n d itio n s , th e n , are 44 (T ) t o o / 5 10 / o-/ F ig u re 1 . —The Z e ro th -o rd e r T em peratures, Y j^ Y2 . 45 At s = s i , ■5/3 (3.19) Z i(S i) = kr Si Z2( s i) = kj. ,s.i1/3 For th e B-G-K model, equations (3 .4 .5 ) and (3 .4 .6 ) become, dZi Zj ■ ■ ■ + " = Gj (3 .2 0 .1 ) ds s2 dZ2 Z2 + — = G2 (3 .2 0 .2 ) ds s2 The s o lu tio n s to equations (3 .2 0 .1 ) and (3 .2 0 .2 ), su b jec te d to boundary co n d itio n s (3.19) are sim ply, I s _ 1 _ 5 _ 1 Zi = {es j s Gj e s? d s ' + k ^ 3 e S l) (3 .2 1 .1 ) I s - I I - I Z2 = es { / G e S* d s ' + k ± s 3 e S l} (3 .2 1 .2 ) s i 2 For s « 1, Zj and Z2 can be in te g ra te d and have th e follow ing form, 1 i _ i _ £ _ L_ _ £ 1 3 s 3 s , . ,c 3 r “ Zl = es (4s 3 e s - 4 s 1 3 e sl + k s 1 3 e S l} (3 .2 2 .1 ) I I _I I _I_ I _I_ Z2 = es { I s3 e s - I sx3 e S l+ k^Sj3 e s l} (3 .2 2 .2 ) In th e o v erlap re g io n , where s s 1, _S_ _ 1_ 5 ^ _ 1 _ j_ 1_ 1 ^ 1_ ” 3 ” si ~ 3 ~ s , 3 si 3 si Sj e 1 « s e 1 ; Sj e 1 << s e * Zj and Z2 , r e s p e c tiv e ly , a re , Zj = 4 s"5/3 (3 .2 3 .1 ) Z2 = (4 /3 ) s l/3 (3 .2 3 .2 ) E quations (3 .2 3 .1 ) and (3 .2 3 .2 ) show th a t in th e continuum n e a r e q u ilib riu m re g io n , where s << 1 , th e energy flu x term s do depend on lo c a l s ta t e s . The s o lu tio n s in th e o v erlap reg io n should match th o se in th e in n e r re g io n , where s = 0 ( S j) . T h e refo re , k^ and should b e, k = 4 , k = 4 /3 (3.24) r » j_ In th e o u te r re g io n , Zl and Z2 can be o b ta in e d by q u a d ra tu re , sin c e Gj and G2 a re known fu n c tio n s o f NQ , UQ, PrQ and P . The r e s u l t s o f s 2sr0 and s2sj.0» which re p re s e n t th e energy flu x e s p e r u n it s o lid an g le a re shown in F igure 2. (B) M axwellian M olecules The coupled z e ro th -o rd e r energy flu x e q u atio n s (3 .2 .5 ) and (3 .2 .6 ) o f a M axwellian gas w ith f u l l Boltzmann c o llis io n i n t e g r a l, may be w ritte n in term s o f Zj and Z2, which a re d efin ed by (3 .1 8 ), 47 6-S-K £ .'* 4 g u , r > , R T i j(u-aff < * C J ( u-u)(v+ v*) f d F igure 2 .—Z e ro th -o rd er P a r a lle l and T ran sv erse Energy Flux Per U nit S o lid A ngle. s 2S and s 2S . ro ±o 48 dZ. . 7Z2 1 = 1 [Z ---------- ] + G2(s) (3 .2 5 .2 ) ds 6 s2 Equations (3 .2 5 .1 ) and (3 .2 5 .2 ) can be combined to y ie ld a sin g le d i f f e r e n t i a l eq u atio n o f Z2 , — - + — — * [ — - T — ) Z2 * A t 3 ' 26) d s2 6 s? ds s* 3 s 3 2 where, dG A = [Gj + 6 — - + — G2] (3.27) A c e 2 ds s Using th e tra n sfo rm a tio n , 2 1 X = | i y = e 3 s Z2 (3.28) th e homogeneous p a rt o f eq u atio n (3.26) becomes, x ^ + (2-x) ^ 1 y = 0 ( 3 - 29) dx2 dx 5 which i s o f th e form o f th e co n flu en t hypergeom etric eq u atio n , . ^ ♦ ( e - a g - v ■ 0 dxz w ith p aram eters, c = 2 and a = 6/5 The g en eral s o lu tio n to eq u atio n (3.29) i s , y = C ^ C j, 2, x) + C2K j , 2 , x) 49 T h erefo re th e complementary s o lu tio n to th e eq u atio n (3 .2 6 ) i s , 21 2 1 * _ s - 3 s */»6 . 5 1^ p 3 s , ,6 a 5 1^ (Z2^c = 1 e ^TT* ’ 6s^ 2 » ( > T ^ The p a r t ic u la r s o lu tio n to th e eq u atio n (3.26) can be deduced from (Z2) c . I t re a d s , w here, (Z2)p = + * *2 ^Z2^2 (3.31) 2 1 t y > - * ’ s 5 ♦ ( ! • 2- H > 2 1 (Z2)2 = e 3 s *c|, 2 , 1 1 ) u s (Z2) 2 A 1 = - / d s ' S1 W s ( Z ,) . A u = I ------------ d s ' S1 W W = (Z2)i (Z2)l d_ ds (Z2) (Z2) 2 2 >2 I - $ 13 6 1 s The com plete s o lu tio n to eq u atio n (3 .2 8 ), th e n , i s , .30) z 2 - ( Z 2) c ♦ ( Z 2 ) p (3.32) By eq u atio n ( 3 .2 5 .2 ), Z, can be o b tain ed by, dZ2 Z2 Z, - 6 -r— + 7 --------6G2 (3.33) 1 ds s2 In o rd e r to s a t i s f y th e boundary c o n d itio n s (3 .1 9 ), and C2 in th e com plem entary s o lu tio n (3 .3 0 ) have th e fo llo w in g v a lu e s , I _ 3 1_ __7_ Ci * - i [8 + kr - 3k ](|)5 r ( |) e 7 Sl SX 15 (3 .3 4 .1 ) I _ 13_ C 2 = i- [8 + kr + 2k ](|)5 e 2 Sl Sx 15 (3 .3 4 .2 ) In th e o v e rla p re g io n , where s i « s « 1, Zi and Z2 can be in te g ra te d and have th e sim ple form , _ 5 Z1 * 6 S 3 (3 .3 5 .1 ) Z2 ■ 2 S 3 (3 .3 5 .2 ) The m atching o f th e s o lu tio n s in th e in n e r re g io n to th o se in th e o u te r re g io n d eterm in es th e c o n s ta n ts kr and kj_. For t h i s c o ll i s i o n m odel, th e y should b e, kr » 6 , k± = 2 (3.36) Upon s u b s titu tin g th e v a lu e s o f kr and k^ in to th e ex p re ssio n s (3 .3 4 .1 ) and ( 3 .3 4 .2 ), one f in d s , In th e o u te r re g io n , ag ain Z\ and Z2 can be o b ta in ed by q u a d ra tu re . The r e s u l t s o f s2SrQ and s2S^q a re shown in F ig u re 2. The above stu d y shows th a t th e energy flu x e s o b ta in ed by th e s e two c o llis io n models have th e same q u a lita tiv e r e s u l t s . Near s * 0, -5 /3 both s2& ro and s2S^0 a re p ro p o rtio n a l to s" , and th e r a t i o o f th e s e two d if f e r e n t p ro p o rtio n a l c o n s ta n ts i s kf /k = 3. Q u a n tita t i v e l y , th e r a t i o o f p ro p o rtio n a l c o n s ta n ts o f d if f e r e n t c o ll i s i o n models i s : (kr ) M axwellian m olecules (k ) M axwellian m olecules 3 (kr ) B-G-K (k ) B-G-K 2 S ince kr and kj_ a re r e la te d to th e th erm al c o n d u c tiv ity , and th e v is c o s ity c o e f f ic ie n ts a re th e same fo r b o th c o l l i s i o n m odels, as t h e i r seco n d -o rd er moment e q u atio n s a re th e same, one w ill fin d t h a t th e r a t i o o f th e P ra n tl number o f th e se two c o ll i s i o n m odels i s , (P ra n d tl No.) M axwellian m olecules 2 (P ra n d tl No.) B-G-K 3 which i s c o n s is te n t w ith th e fin d in g s in R eference [26]. As s ■ + • » , b o th c o llis io n m odels show th a t s 2S reach es a ro lim itin g v a lu e , w h ile d ecrease s to z e r o -lik e 1 /s . The v a lu e o f th e lim itin g v a lu e s o f s 2S f o r th e two d if f e r e n t c o llis io n ro models i s , 52 ( s 2S ) M axw ellian m olecules v ro'' 3 ( s 2S ) B-G-K v r o ' 2 A " fro z e n " s2SrQ a t i n f i n i t y im p lie s th a t a c o n s ta n t th erm a l flu x flow s tow ards i n f i n i t y . 3 .2 . F irs t-O rd e r S o lu tio n s 3 .2 .1 . Ui and Nj With th e knowledge o f th e z e ro th -o rd e r s o lu tio n s , eq u a tio n (3 .4 .2 ) can be w r itte n a s , In th e in n e r r e g io n , where s = 0 ( s 1) , U1 can be in te g r a te d and ex p ressed a s , which approaches to 5 /2 , a s s approaches tow ards to th e o u te r re g io n . In th e o u te r r e g io n , where s i s o f o rd e r o f u n ity , (3 .3 9 ) “ z d [ — - 3 ? y i J i s o f o rd e r o f u n ity o r l e s s , hence dUx ds 0 (Sl^ 3) « 1 53 T h e re fo re , Ui i s o f c o n sta n t v a lu e (su b je c te d to an e r ro r o f o rd e r o f s i - / 3) . Uj must match th e lim itin g v a lu e o f th a t in th e in n e r re g io n , and th u s , U! ss* | (3.40) th roughout th e e n tir e o u te r re g io n . P h y s ic a lly , i t means th a t i f th e Mach number a t th e i n i t i a l s ta tio n i s f i n i t e , th e flow v e lo c ity w ill reach i t s lim itin g v alu e w ith a d is ta n c e o f o rd e r o f r j . Nj i s r e la te d to Ui by eq u atio n ( 3 .4 .1 ), Ni = - M i/s2 (3.41) which i s shown in F ig u re 3. 3 .2 .2 . P and P _ r i________ j_i_ In term s o f th e v a r ia b le s : W j = s2Pr l (3 .4 2 .1 ) W 2 = s2P x l (3 .4 2 .2 ) 3W = W j + 2W 2 - (3 .4 2 .3 ) e q u atio n s (3 .4 .3 ) and (3 .4 .4 ) can be combined to y ie ld a s in g le e q u a tio n , d?W + I S s i i l t 4 W. . p ds2 s2 a ? 3 s3 where 54 4 ! U , ' _ M l No 3 2 / O Z 4 6 2 to 5 /2 s = ex',/l, A ni^> z, L ^ m Ot, - U - N — - f - | S, U 4 3 2 / o I 2 4 * 2 to o — I— ---------------1 ----------------1 --------------- ■ --------------- 4 F ig u re 3 . —The F ir s t- o r d e r V e lo c ity and Number D en sity co 55 i rs+n ^2 d ^2 F . 1 [ISLll (Fl + _ ) * 4_ (p, * + _ _ ] W i and W 2 a re r e la te d to W by th e r e la tio n s : 3 d s ^2 «» * 3W t l 5 3 ? w - f IF‘ * ~ 3 (3 .4 4 .1 ) w 2 = f [(f, * ! i ) - * Is-m s (3 .4 4 .2 ) The homogeneous p a rt o f eq u atio n (3.43) i s th e same as equation (3 .1 0 ), and th e g en eral s o lu tio n to eq u atio n (3.43) i s : W = W + W c p (3.45) W c - C i* (- -1, i) ♦ C2 i - ♦(§., 3 , I) S W - ♦ (- -1 , i ) Ix - i - ♦ (§•, 3 , | ) I2 * s - 1 (|> F 81 sc W d s' [2 . j I E . d s. Si w W -T * (4- ) ’ S2 r es S“3 In o rd er to s a tis f y th e boundary co n d itio n s (3 .5 .3 ) and ( 3 .5 .4 ), 56' i . e . , W i (SO * 0 , w2 (S l) = 0 Ci and C2 in equation (3.45) a re found to be, 7 p 4 T s 2 Cl = T Si [t (Fx + — )1 „ i 3 i 1 4 v i 2 s = s l 1 L - r h p 4 c 3 " Si rr V , r s . r * , T 1 ' W ] [ 4 (F‘ 7 ) ] s=s, As Si « 1, [f C F , * - ) ] - - *£ S, s 1 v 4 2M 25 ~ 3 T h erefo re, one h as, C i = - S O Si (3.46.1) r (— ) - — - — C2 = - 5 - Sj 3 e Sl (3.46.2) r(3) 2 i_ 3 s In th e in n er reg io n , Ii = 0 (1 ), I2 ® 0(S e ) , and hence Ci C2 - 0 ( s i) , = 0 (1) and 4 _ 1 _ * I l - 0(S 3) , - 4 - I2 - 0(S 3) s2 Thus, W i s o f h ig h er o rd er o f ( s i) in comparison w ith W , which c p i s m ainly c o n trib u te d by a c c e le ra tio n term s, 57 dUj C s2pLo> d T - (S2P ) -7 — v r o ' ds and in ^ ;Ii. W can be ap p ro x im ately w ritte n a s , (3.47) As s approaches tow ards th e o u te r re g io n , W approaches to th e v a lu e , In th e o u te r re g io n , 1^ = 0 (1 ), I2 = 0 (1 ); th e r e fo r e W i s w in th e o u te r re g io n . The v isco u s s tr e s s h e a tin g w ill cau se W's to r i s e . In b o th re g io n s , th e energy flu x term s a re term s o f h ig h er o rd e r, and hence th e y a re n e g le c te d in th e num erical com putation. In th e in n e r re g io n , th e y a re o f o rd e r o f s j in com parison w ith th e a c c e le r a tio n term s; w hile in th e o u te r re g io n , th ey a re o f o rd e r o f If/ 3 Si in com parison w ith th e v isc o u s s tr e s s term . The r e s u l t s o f (3.48) a ls o o f h ig h e r o rd e r in com parison w ith Wp. B esides th e term a p a n g-iji(- j , -1 , j ) , which i s in h e r ite d from th e in n e r re g io n , Ii and I2 a re c o n trib u te d by th e v isco u s s t r e s s term , W , Y Y x W i W 2 xr~ and rr~ 58 a re shown in F ig u re 4 . The b eh av io r o f W ’s a re s im ila r to th o se o f th e z e ro th -o rd e r te m p e ra tu re s, nam ely, W and W j re a c h lim itin g v a lu e s , w h ile W 2 to ze ro lik e 1/ s . The num erical com putation shows t h a t th e te rm in a l average p re s s u re to th e f i r s t - o r d e r approxim ation i s , p . ■ [p0 J . i 1 ♦ — * i - ■ [p„ ] - t 1 * 2 -5 e] p o and th e number d e n s ity a s w ell as mean v e lo c ity to th e f i r s t - o r d e r ap p ro x im atio n , from s e c tio n ( 3 .2 .1 ) , a r e , At s °o, U = {U [1 + 2 .5 el} 00 O J 00 N = {N [1 - 2 .5 e]} 00 O J 00 By d e f i n it i o n , th e te rm in a l av erag e tem p eratu re to th e f i r s t - o r d e r app ro x im atio n i s T o . Pi Nx T ^ - ref) S,*''3 [1 ♦ « (pr - 5-)l - [ff]0 [1 ♦ 5 «] I n te r e s tin g ly , th e te rm in a l Mach number M w, su b je c te d to an e r r o r o f o rd e r o f [s^ + 1/Mj1 *], can be o b tain e d by, to th e num erical p r e c is io n , i s th e same as t h a t o f th e z e ro th -a p p ro x i- m ation. 59 W Y -2 to IV r J4L Y . W i r* 2 0 -2 - 4 C O to W1 W2 w F igure 4 . --The F ir s t- o r d e r P re ssu re C o rre ctio n s — , rr CHAPTER IV „ v OTHER ASPECTS O N THE ZEROTH-ORDER SOLUTIONS In C hapter I I I , b o th z ero th and f i r s t - o r d e r s o lu tio n s to th e h y p erso n ic source flow have been sought and d isc u s s e d , under th e assum ptions th a t th e gas i s o f M axwellian m olecules and having an in v is c id i n i t i a l c o n d itio n . In th e p re se n t c h a p te r, th e same problem w ill be examined ex c ep t: (1) The i n i t i a l c o n d itio n i s n o n -e q u ilib riu m , (2) The gases a re o f non-M axwellian m o lecu les. 4 .1 . Z e ro th -o rd e r S o lu tio n s w ith N on-equilibrium I n i t i a l C o n d itio n s Suppose, a t th e i n i t i a l ra d iu s r = r( , th e flow i s n o n -e q u ilib riu m , nam ely, (Pr ) l * ( P j h (or ( T J i + ( T j h ) th e non-dim ensional boundary c o n d itio n s (3 .3 .3 ) and (3 .3 .4 ) w ill be re p la c e d by, At s = s i , *1 - ( s2Pr 0 ) - S i*/3 [1 * | A,] (4 .1 .1 ) v2 = (s2P l 0 ) - s;1 * 73 [1 - I i l l (4 .1 .2 ) w here, 60 61 [ T - T , r«*r. (4 .2 ) T 1 which c o n tro ls th e degree o f i n i t i a l n o n -eq u ilib riu m in T. The r e s t o f th e z e ro th -o rd e r eq u atio n s and boundary c o n d itio n s w ill be th e same as th o se in C hapter I I I . Thus, th e z e ro th -o rd e r s o lu tio n s o f U Q and N q w ill be th e same as th o se given by eq u atio n s (3 .6 ) and (3 .7 ), namely, N0 - l / s * The z e ro th -o rd e r tem p eratu re Y i s found to b e, Y Cj. *(- 1 -1, j ) + C2 S"2 3, | ) (4 .3 ) Yj and Y2 a re r e la te d to Y by, Y, ■1 - 3Y * ! S 3 ? Y Y « — S — __Y 2 4 S ds Y (4 .4 .1 ) (4 .4 .2 ) In o rd er to s a t i s f y th e boundary c o n d itio n s (4 .1 .1 ) and ( 4 .1 .2 ) , Cj and C2 must b e, SZ 1 ,-4/3 °2 = ,-4/3 *1 T V i 7 s s 2 1 3 3 ^ 1 - f v ; - t s^ \ The s u b s c rip t *1' den o tes th a t th e v alu e o f th e fu n c tio n i s ev alu ated a t s ■ s j . 62 and I f th e assum ption th a t Sj « 1 rem ains tr u e , th e n , 1^ 1^ , c -4/3 _1 . r(3) si - 3 'l> , ~ s , — — A e s i 1 s 2 1 r(|) Cj = 1 ♦ 4- SjAi + 0 ( S ^ i , ) (4 .6 .1 ) r (— ) - — _ C2 = - I - 5- (SiA j) e Sl Sj 3 [1 + 0 (S l)] (4 .6 .2 ) y r ( 3 ) hence, 4 4 i '1 * 1 Y = ^ + J (SlAl)[* - —i - e Sx ] (4.7.1) r(3 ) s2 By eq u atio n s (4 .4 .1 ) and ( 4 .4 .2 ) , Yj and Y2 , th e n , a re : *1 [1 + | (SiA1)][3 t|-(- - 1 , i ) - | iK - j , 0 , i ) ] r ( l ) - I - I - £ (SiAO - 5 - e S1 Sx 3 . - i - < 4 4 , i ) (4 .7 .2 ) y r ( 3 ) 3 s3 * 5 Y2 U * J (S iA j)] J» °* 7> V (— ) - — - — 4 a 3 S i „ 3 r 1 , , 5 . l w 3 1 ■ o’ (s l Al ) e s i L — ~ ♦Cj* 4 » 7 ) + y — y r ( 3 ) 6 s 3 s * s 2 ♦ c f 3 . i ) i I t w ill be in te r e s tin g to examine th e developm ent o f th e i n i t i a l s tr e s s in th e re g io n where s •«< 1, in w hich, by eq u atio n s (4 .7 .2 ) and ( 4 .7 .3 ) , one may w rite appro x im ately , 63 Yi - Y 2 c 2 1 1 . 1 3 s ” Si 2S + Aj (— ) 5 e Sl (4.8) The f i r s t term on th e r ig h t hand s id e o f eq u atio n (4 .8 ) can be i d e n ti f ie d as th e one due to th e s o lu tio n w ith in v is c id i n i t i a l c o n d itio n . The second term re p re s e n ts th e c o n trib u tio n by th e n o n -eq u ilib riu m i n i t i a l co n d itio n . I t decays very f a s t as th e flow leaves s li g h t l y from th e i n i t i a l s ta tio n , because o f th e ex p o n en tial p a r t, exp (1 /s - 1 / s i ) . T h erefo re, th e flow always ten d s to re tu rn to th e n e a r-e q u ilib riu m s o lu tio n , even i f th e re e x is ts a n o n -eq u ilib riu m c o n d itio n a t th e i n i t i a l r a d iu s . In o th e r w ords, one may say th a t th e upstream eq u ilib riu m s o lu tio n i s a s ta b le one. Since Ai e n te rs in to Ci as w ell as C2 in e q u atio n (4.6) th e i n i t i a l v isco u s s tr e s s has a lso a s lig h t e f f e c t on th e tem pera tu r e s . As s • + ■ « , a l l term s r e la te d to < f > w ill d ecrease to zero lik e 1/ s 2 o r 1/ s 3, w hile ij>(- 4 , -1 » 4 ) reach es i t s te rm in a l w 9 v a lu e . T h erefo re, th e fro zen tem p eratu res (TQ) 0 B and (T 5^ having 4 v alu es which a re [1 + SjAi] tim es th o se o b tain ed in eq u atio n s (3 .1 7 .1 ) and (3 .1 7 .2 ) w ill f i n a l l y be reach ed . C onsequently, th e lim itin g Mach number w ill be sm aller th an th a t w ith i n i t i a l l y in v isc id c o n d itio n by a f a c to r 64 4 .2 . Z e ro th -o rd e r E quations o f Non-M axwellian Gases and I t s A sym ptotic B ehaviors The c o n tin u ity eq u a tio n and eq u a tio n o f m otion a re n o t i n f l u enced by m o lecu lar fo rc e law . Thus, th e z e ro th -o rd e r s o lu tio n s o f N q and U Q w ill be th e same as b e fo re , i . e . , For g ases o f non-M axw ellian m o le c u le s, th e second moment e q u a tio n s in which th e c o l l i s i o n i n te g r a ls a re n o n -zero w ill be d i f f e r e n t from th o se o b ta in e d in C hapter I I . The Boltzmann c o l l i s i o n in t e g r a ls o f a non-M axw ellian gas can n o t be e v a lu a te d w ith o u t th e in fo rm a tio n o f th e form o f th e d i s t r i b u t i o n fu n c tio n . However, f o r th e B-G-K c o l l i s i o n m odel, one may w rite An = p /y and y « Tw t o re p re s e n t non- M axw ellian m o lecu lar fo rc e law as done by Edwards and Cheng [12] . In t h i s s e c tio n , moment e q u a tio n s f o r a non-M axw ellian m olecule based on th e B-G-K c o l l i s i o n model and i t s asy m p to tic b eh a v io r w ill be d i s c u sse d . Assuming th a t where a i s a c o n s ta n t, one may d e fin e th e n o n -d im en sio n al p aram eter in a more g e n e ra l form , N o 1/ s 2 a 1 amRT where y j i s th e lo c a l v is c o s ity a t r » r . Using th e same dim ensionless q u a n titie s d efin e d in s e c tio n (2 .5 ) and expanding them in th e same way as done in equations ( 3 .1 ) , th e second moment eq u atio n s o f z e ro th -o rd e r approxim ation can be w ritte n a s: The same procedure as done in s e c tio n (3 .1 ) w ill y ie ld a s in g le n o n -lin e a r d i f f e r e n t i a l eq u atio n : Yi and Y2 a re r e la te d to Y by eq u atio n s (3 .1 1 .1 ) and (3 .1 1 .2 ). E quation (4.10) is o f th e same form as th e one o b tain ed by Edwards and Cheng [12]. The u > = 1 c a se , which corresponds to M axwellian m ole c u le s , has been stu d ie d in th e p rev io u s c h a p te rs. In th e p re se n t c h a p te r, a tte n tio n w ill be c a s t on th ? cases when u > ± 1 . (4 .9 .1 ) (4 .9 .2 ) where 3P o (4.10) 66 E qu atio n (4 .1 0 ) can be tra n sfo rm ed to a f i r s t o rd e r d i f f e r e n t i a l e q u a tio n , I j - . (2u-i) e2 + m ! * (i+3p)[i ♦ p e] — ^ + — t1 r P . -id p - 9].8 . . o (4 . i 2) The s o lu tio n cu rv es o f e q u a tio n (4 .1 2 ) and th e asy m p to tic b e h a v io rs o f e q u a tio n (4 .1 0 ) fo r b o th p - * ■ 0 and p - * • » w ill be s tu d ie d below . In p a s s in g , one w ill b rin g up th e term in o lo g y " S t a b i l i t y , " which can be d e s c rib e d in th e fo llo w in g m anner: I f , as tim e in c r e a s e s , th e s o lu tio n cu rv es co n v erg es, th e s o lu tio n i s s a id to be s ta b le . On th e o th e r hand, i f th e s o lu t i o n cu rv es d iv e rg e s w ith in c re a s in g tim e , th e s o lu tio n i s s a id to be u n s ta b le . In t h i s flow problem , as tim e in c r e a s e s , th e flow le a v e s th e o r ig in and expands tow ards i n f i n i t y . Thus, a s a d d le p o in t s o lu tio n cu rv es a t s = 0 and a node p o in t s o lu tio n cu rv es a t s - * ■ < » w ill g iv e s ta b le s o lu tio n s ; w h ile a node p o in t s o lu tio n cu rv es a t s = 0 and a sa d d le p o in t s o lu tio n cu rv es a t s - * ■ » w ill b o th p ro v id e u n s ta b le s o lu tio n s . 4 .2 .1 . Small End p - * • 0 In view o f th e s o lu tio n (4 .1 2 ), as p 0, 0 i s im p o ssib le to have s o lu tio n s o f o rd e r o f u n ity o r l e s s . Thus, th e r e a re th r e e p o s s ib le s o lu tio n s to eq u a tio n (4 .1 2 ) as p - » ■ 0, nam ely, (a) |0| » i (b) | e | « i 67 (c) |e | = 0 ( i) In case ( a ), when to / 7 /4 , 1, eq u atio n (4.12) reduces t o , 5^-+ [(1-u2) + j (l-o»)3 + 0 (p)] 03 = 0 which g iv e s, 6 « ( I)15 V Since ( i- ) * 2 << ^ , th e re fo re th e assum ption 10 1 » i s c o n tra d ic te d . For th e case u > = 7 /4 , eq u atio n (4.12) reduces to , do 21 , 3 e2 . dp" " 16 p + 4 ~ - 0 which g iv e s , _ 8_ 1_ 0 = 1 1 + o ( l) + C e 7 p2 7 P2 The above s o lu tio n curves has a node p o in t a t p = 0 . Using eq u atio n s (4 .1 1 .1 ) and (4 .1 1 .2 ), one fin d s , 3 i S « e7 p , Y ■ S-“/3 which shows th a t p 0 maps S - » ■ ». T h e refo re , fo r t h i s s p e c ia l u > , Y has a s ta b le e q u ilib riu m - u / 3 s o lu tio n , Y « s , as s - * ■ ®. In th e case ( b ) , eq u atio n (4.12) reduces t o , ajr ♦ i - 0 P3 which g iv es 0 = o i ) P 68 and v io la te s th e assum ption, I 0 I « - I I p In th e l a s t p o s s ib le case (c ), one may p u t, f> 0 = A + z(p ) (4.13) where A i s a c o n sta n t o f o rd e r o f u n ity and 1*1 << 1 Upon s u b s titu tin g th e e x p re ssio n (4.13) in to e q u a tio n (4 .1 2 ), and eq u atin g th e same o rd er te rm s, one fin d s , [1 + (l-io)A]2 [1 + j (l-w)A + j ] = 0 I t fo llo w s t h a t , 4/3 x A ----------- > - T—r w ^ 7 /4 ,1 i . f (I-..) When p i s tra n s f e rr e d back to th e p h y sic a l c o o rd in a te through eq u atio n s (4 .1 1 .1 ) and ( 4 .1 1 .2 ), th e s o lu tio n A = - w ill cau se p . = o dp which im p lie s t h a t , as p 0, a l l p mapps to one p o in t on s , hence i t i s n o t o f i n t e r e s t . 4/3 The h ig h er o rd er eq u a tio n in z f o r A j ---------- i s : 1 ♦ f (l-« ) dz n m _ + — z = - (4.14) dp p2 P m = ■ [4u> 2 - 14w + 11] (7 - 4w)3 69 3 a ) / 7/4 n 7-4o) The s o lu tio n to e q u atio n (4.14) i s : n C e p By eq u atio n (4 .1 3 ), th e s o lu tio n to e q u atio n (4.12) i s : + — + n _1 1 C I e7_4w p (4.15) P The b eh av io r o f th e s o lu tio n cu rv es o f 0 in th e v i c i n i t y o f p = 0 depends on th e v a lu e o f w, For a ) > 7 /4 , p = 0 i s a node p o in t, (F ig u re 5 ). one o b ta in s , A ■ T which w ill c au se. a f " = 0 and hence th e mapping does not make se n se . E quations (4 .1 1 .1 ) and (4 .1 1 .2 ) w ill b rin g s o lu tio n (4 .1 5 ) back to th e p h y sic a l p la n e : For ta < 7 /4 , p = 0 i s a sad d le p o in t, (F ig u re 6) . 7 On s u b s titu tin g eq u atio n (4.13) in to eq u a tio n (4 .1 2 ), f o r a > = (4.16) 3 S « p 7-4 (4.17) 70 ing to s ta b le / c O•/ O 0-2 F ig u re 5 .— (w = 3 /4 ) . S o lu tio n Curves n e a r p = 0 (c o rre sp o n d - s = 0 ). A sa d d le p o in t shows th e e q u ilib riu m s o lu tio n is ra* / • « 1-31 O cog F ig u re 6. — (w = 5/2). S o lu tio n Curves n e a r p = 0 (c o rre sp o n d in g t o s - * ■ °°). The node p o in t in d ic a te s an s ta b le e q u ilib riu m s o lu tio n a t s - ► « > . 7 For to < th e o r ig in o f p mapps to th e o r ig in o f s ; w hile fo r 7 u > th e o r ig in o f p mapps to i n f i n i t y on th e s . 7 S ince a sad d le p o in t s o lu tio n curve n ear s = 0 fo r to < j w ill g iv e a s ta b le s o lu tio n , and th e node p o in t s o lu tio n curves fo r 7 to > w ill a ls o g iv e a s ta b le s o lu tio n a t s ■ + • ». -*f/ 3 One may conclude th a t th e eq u ilib riu m s o lu tio n , Y * s , e x is ts n e a r th e so u rce, as long as to < 7 /4 . For a gas w ith to £ .7 /4 th e s ta b le eq u ilib riu m s o lu tio n occurs a t s - * ■ ®. 4.2.2. Large End p -» « For convenience, l e t t = 1 /p , th en eq u atio n (4.12) can be w ritte n a s: - (2to-l) -— + to(1 -to) ~ + (3+t) [ t + (l-to )0 ]2 — d t t 2 t 3 t 3 + | [ t + (1 -to)0] 3 \ 2 (4.18) A gain, th re e p o s s ib le ty p es o f s o lu tio n , as t - * • 0, w ill be examined (a) 10 | » t (b) | 0 | « t (c) 10 1 = 0 ( t) In case ( a ), w ith to f 3 /2 , 1, eq u atio n (4.18) reduces t o , d8 v ,0.3 A -.2 ,0 .3 = to (l-to )(-) + 3 (1 -to) (-) which g iv e s, 72 and c o n tr a d ic ts th e assum ption, |0 | » t . For w = 3 /2 , eq u atio n (4*18) red u ces to d6 82 1 83 d t ~ " t 2 6 t 2 which a ls o can be shown th a t i t i s im p o ssib le to have s o lu tio n o f ty p e (a ). In case ( b ) , eq u atio n (4.18) reduces t o , 3 ® . i t d t t 3 which y ie ld s th e s o lu tio n , 6 = - y t 2 + c t 3 This s o lu tio n has a node p o in t a t t = 0, fo r a l l u > . In th e p h y sic a l p la n e , i t g iv e s , Y = c o n s t. (1 + i — + ....) (4.19) 5 p S « p (4.20) which shows t h a t , a t s -> » , a "Frozen T em perature" s o lu tio n e x is ts f o r a l l i d , and i t i s s ta b le (F ig u re 7 ). In case ( c ) , one may w r ite , | - A + z ( t ) (4.21) where A is a c o n sta n t o f o rd e r o f u n ity , and |z | « 1 On s u b s titu tin g eq u a tio n (4.21) in to eq u a tio n (4.18), one f in d s , 73 Jncretilnj of tir»e ^ \ \ V i Figure 7 .--S o lu tio n Curves n ear p - ► 00 (corresponding to s ■ + <»). The node p o in t shows th e e x iste n c e o f a fro zen s o lu tio n a t in f in ity fo r a l l w. (is i (For o i - 3 /2 , one w ill fin d th a t A = 2, and consequently ^ = 0*) At t = 0, th e s o lu tio n curves have a node p o in t f o r to > 3 /2 , a sad d le p o in t fo r w < 3/2. Upon tr a n s f e r r in g back to th e p h y sica l p lan e, one fin d s , as p •+ » , Y « s"2 (4 .2 2 .1 ) 1 _ S < * p3' 2w (4 .2 2 .2 ) E quation (4 .2 2 .2 ) in d ic a te s t h a t , fo r u > 3 /2 , th e p o in t a t p -> « . mapps th e p o in t s - * • 0, and fo r w < 3 /2 , i t mapps th e p o in t s ■ * < = ° . T h erefo re, i t appears th a t th e s o lu tio n (4 .2 2 .1 ) which is p o s sib le a t s - » ■ » fo r o i < 3 /2 , and a t source end fo r u i > 3 /2 , is u n sta b le . In summary, one a r riv e s a t th e follow ing conclusions on th e asym ptotic b eh av io rs o f s o lu tio n s fo r th e non-M axwellian g ases: (a) The s ta b le "E q u ilib riu m s o lu tio n " e x is ts a t s - * ■ 0 fo r u ) < 7 /4 ; a t s ■ * 00 fo r w > _ 7/4. (b) A s ta b le "Frozen so lu tio n " e x is ts a t s < * > fo r a l l a). (c) The " S e lf- lim itin g so lu tio n " Y * s-2 is p o s sib le a t s - * ■ 0 , fo r u > 3 /2 ; a t s - * ■ < * > , fo r (ii < 3 /2 , but is u n sta b le . CHAPTER V drawn: CONCLUSIONS A N D R E M A R K S From th e p re se n t a n a ly s is , th e follow ing co n clu sio n s have been (1) For th e s p h e ric a l source flow , under h y personic tru n c a tio n , th e f i r s t- o r d e r as w ell as any ev en -o rd er (except zero th - o rd er) moment eq u atio n s form a s e lf - c o n s is te n t c lo se d system . As a r e s u l t , Mach number e n te rs in to th e asymp t o t i c expansion only as in te g ra l power o f 1/M2 . (2) The f ir s t- o r d e r c o rre c tio n to th e le a d in g term in th e asym ptotic expansion, d isp la y s an in n e r (n ea r-e q u ilib riu m ) reg io n and an o u te r (no n -eq u ilib riu m ) reg io n . The mean a c c e le ra tio n o f flow ta k e s p la c e only in th e form er re g io n , w hile th e e f f e c t o f th e v isco u s s tr e s s appears m ainly in th e l a t t e r one. The r e s u lts o b tain ed show th a t th e f i r s t - o r d e r c o rre c tio n to th e v e lo c ity is 5 u l2 ■j e (where e = ) R Tj throughout th e o u te r re g io n ; and th a t th e f ir s t- o r d e r p re ssu re c o rre c tio n s , due to a c c e le ra tio n , drop in th e in n e r reg io n u n til they reach a value 25 " T 0 75 76 ! | j then th ey are r a is e d up by v isc o u s s tr e s s in th e o u te r re g io n . In t h i s co n n ectio n , one may n o te th a t th e Mach number a t th e in v is c id i n i t i a l s ta tio n may n o t be very h ig h , although a v ery high te rm in a l Mach number may be reach ed f a r downstream. T h e refo re , th e f i r s t - o r d e r c o rre c tio n could be a p p re c ia b le in th e p r a c tic a l s itu a tio n . For in s ta n c e , i f th e i n i t i a l Mach number is 3, th e f i r s t - o r d e r c o rre c tio n to th e v e lo c ity co u ld be as high as 17%, and 28% to th e p re s s u re s . (3) The s o lu tio n o f h e a t flu x e s o b ta in ed from th e M axwellian gas i s q u a lita tiv e ly th e same as th a t o b tain e d from B-G-K model. At i n f i n i t y , s2Sr reaches a lim itin g v a lu e , w hile s2S _ l v an ish es lik e 1 /s . Q u a n tita tiv e ly , th e B-G-K model y ie ld s sm a lle r h e a t flu x e s , about 2/3 as much as th o se o b tain e d from th e M axwellian g ases. (4) N on-equilibrium i n i t i a l c o n d itio n w ill r a is e th e fro zen p a r a l l e l and average te m p e ra tu re s. (5) Based on m odified B-G-K m odel, gases o f non-M axwellian m olecules w ith m < 7 /4 , have s ta b le e q u ilib riu m s o lu tio n n e a r th e source and fro zen s o lu tio n a t i n f i n i t y . Using th e same model e q u atio n and stu d y in g only m ath em atically th e w > _ 7/4 c a se , which could have s ig n ific a n c e in th e th eo ry o f s o la r wind, one fin d s th a t th e s ta b le e q u ilib riu m s o lu tio n appears a t i n f i n i t y . Of co u rse, one should be aware o f th e f a c t th a t fo r such a s o f t m olecule (w _ > 7/4 corresponds to a fo rc e index v £ 13/5) th a t th e v a lid ity 77 o f th e Boltzmann eq u atio n is q u e stio n a b le . In c lo s in g , se v e ra l fu tu re works f o r follow -on o r p a r a l l e l to th e p re se n t study a re su g g ested : (1) The p re se n t stu d y can be extended to gases w ith a g en eral c e n tr a l r e p e llin g in te rm o le c u la r fo rc e law. An assumed form o f d is tr ib u tio n fu n c tio n th en has to be in tro d u ce d to e v a lu a te th e c o llis io n moments. The second o rd e r c o llis io n moment i s computed in Appendix A. The th ir d o rd e r c o l l i s i o n moments can be e v a lu a te d in th e same way i f th e d is tr ib u tio n o f th e form (2.26) is used. (2) When th e m olecules are f a r a p a r t, a weak a ttr a c tin g in te rm o le c u la r fo rce e x is ts . I t e f f e c ts th e c o llis io n in te g r a ls a t low tem p eratu re. In Appendix B, t h i s weak f a r f i e l d e f f e c t has been s tu d ie d o n ly in th e lim it - 2 / 3 e3 Sj - * ■ 0 , th e more g en eral problem i s y e t to be analyzed. (3) The stu d y o f th e source expansion in to the vacuum i s an id e a l one. In th e r e a l i s t i c a l s itu a tio n , even in th e sp ace, no m a tte r how sm a ll, th e re always e x is ts an ambient g as. The problem co n sid e rin g th e flow expansion in to an am bient atm osphere w ill be o f i n t e r e s t from b o th t h e o r e t i c a l and p r a c tic a l view p o in ts. (4) P a r a lle l a n a ly s is can be done to th e f u r th e r study o f Edwards and R oger's work [15]. (5) An a n a ly s is in c lu d in g th e e x te rn a l fo rc e , such as th e g r a v ita tio n a l fo rc e , may be o f i n t e r e s t in a s tro p h y s ic s. APPENDIX A EVALUATION OF COLLISION INTEGRALS The c o llis io n in te g r a l (2 .1 1 ) can be w r itte n a s , A Q = / f £ x [J] d c ^ c _ (A. 1) where J is d e fin e d a s , 00 2™ J = f Q f Q [Q] gbdbde (A .2) [Q] = Q’ - Q g: r e l a t i v e speed o f two m olecules b : im pact p aram eter £ : p o la r an g le. The geom etry o f th e en co u n te r betw een m olecules i s i l l u s t r a t e d by F ig u re 8 . B efore c o llis io n th e r e l a t i v e v e lo c ity o f th e two m olecules is g. Let m olecule P move r e l a t i v e l y to m olecule 0, which i s a t th e o r ig in . P lane R O Q i s p e rp e n d ic u la r to g. I f th e m o lecu lar model has a p o in t- c e n te r fo rc e , th e n th e fo rc e betw een th e two m olecules i s d ir e c te d along OP, and th e r e l a t i v e m otion w ill be co n fin ed to th e p la n e c o n ta in in g g and OP. The r e l a t i v e v e lo c ity g ' a f t e r en co u n ter has th e same m agnitude as g ' by th e co n se rv a tio n 78 79 N' S' Figure 8 . --M olecular Encounter. 80 law. The v e lo c ity components a f t e r b in a ry c o llis io n can be re p re se n te d I 'b y th e i n i t i a l v e lo c ity components, th e c o llis io n d e f le c tin g angle and ; th e p o la r an g le. In th e v e lo c ity diagram F igure 9, by th e s p h e ric a l ; trigonom etry and co n serv atio n o f momentum, one fin d s (R eference 16, p. 216) u ' = u + (u j-u ) cos2 h 6 + h[g2 - (u i~ u )2] s in 6 cos 6 (A .3.1) 1 / v ' = v + (v i-v ) co s2 h 6 + *s[g2 - ( v i- v ) 2] s in 6 cos (€2 - 6 ) (A. 3.2) wf = w + (wj-w) cos2 *5 6 + %[g2 - (w j-w l2]^ s in 5 cos (€3 - € ) (A .3.3) (u x-u )(v x-v) cos € = - •2 [g2 - ( U j - u ) 2 ] ^ [g2 - (v j-v ) f t (U j-u )(W !-W ) cos £ 3 = - 1 -- r ~ [ g 2 - ( u i- u ) 2]^ [ g 2 - (W x - w )2 ] S u b s titu tin g th e ex p ressio n s (A .3 .1 ) -(A .3.3) in to eq u atio n (A.2 ), a f t e r re a rra n g in g , one h a s, For [Q]j = u'2 - u2 00 r ; J j = 2 T r ( u x 2 - u 2 ) g J cos2 -j bdb y 0 ^ 0 0 + j [ -2 (ux-u)2 + (v j-v)2 + (wj-w)2 ] g J s in2 5 bdb For [Q]2 = [ ( v '2+w'2) - (v2+w 2) ] 81 Z Figure 9 . --V ec to r Diagram f o r an E ncounter. 82 J 2 = 2ir[(v i2-v 2) + (w j2-w2) ] g / co s2 y bdb 0 2 00 + ^ [2 (ui~ u)2 - (v i-v)2 - (w^-w)2]g / s in2 6 bdb. For [Q]3 = uf3 - u3 0 0 J 3 = 2ir(u i3- u 3)g / co s2 4 bdb. 0 2 00 + J [3 (v j-v)2 + 3(wj-w)2 - 2 ( u i- u ) 2] [u i-u ]g / c o s2 -| s in2 < 5 bdb. + j { 3 u [-2 (u i-u)2 + Cvi-v)2 + Cwi-w)2] - 2 ( u ! - u ) 3}g / s i n 2 < 5 bdb. For [Q]t* = [ u 'vl2 - uv2] + [u 'w'2 - uw2] 00 J4 = 2tt[(u iV i2-u v 2) + (u1w12-uw2) ] g / cos2 7 bdb 0 ^ O + J - [2(ux-u)2 - 3 (v !_v)2 - 3(wx-w)2] (ux-u)g / co s2 y s in 2 5 bdb. + j {-6v (u i- u ) (v i- v ) - 6w (ui-u)(w j-w ) + u [2 (u x -u )z - (v j-v ) 00 - (wx-w)2] - 2 ( u j- u ) [( v x -v)2 + (wx-w)2]}g / s in 2 6 bdb. In th e above e x p re ssio n s, th e term s w ith c o e f f ic ie n ts o f 0 0 0 0 j co s2 4 bdb and / c o s2 y s i n 2 6 bdb * n 2 J n 2 83 a re in th e sym m etrical form o f v e lo c ity components o f th e two m o lecu les. They w ill n o t c o n trib u te to th e in te g r a l ( A .l) . Thus th e in te g r a l (A .l) i s s o le ly r e l ie d on th e in te g r a l 00 / s in2 6 bdb ' 0 which depends on th e in te rm o le c u la r fo rc e law. The k in em atics o f b in a ry c o l l i s i o n o f m olecules w ith a p o in t- c e n te r o f fo rc e r e p e llin g acco rd in g to th e law F = — (A .4) r v has been tr e a te d in most t e x t s [15, 16]. S in ce th e d e f le c tio n an g le 6 depends on F, b and g, th e in te g r a l 0 0 / s in2 6 bdb J 0 w ill be a fu n c tio n o f g and F, and can be ex p ressed a s , 2 v-5 n / s in 2 6 bdb = C 2m K)v_1 gv_1 A,Cv) (A .5) " o 2 where A 2 (v) i s a p u re number depending o n ly on v. The c o n s ta n ts A2 (v) a re e v a lu a te d by q u a d ra tu re and th e y a re l i s t e d in R eference [1 6 ], p . 172, fo r c e r ta in v. Maxwell f i r s t p e rc e iv e d th a t when v = 5 , 00 g / s in2 5 bdb = C 2m K D 5 s A2 , which i s independent o f g. Thus A Q can be e v a lu a te d w ith o u t knowing th e v e lo c ity d i s t r i b u t i o n fu n c tio n f . Some c o ll i s i o n in te g r a ls f o r th e M axw ellian m olecules are given in C hapter I I , ( 2 .1 2 .1 ) - ( 2 .1 2 .4 ) . 84 (A -l) C o llis io n in te g ra ls o f non-M axwellian m olecules For th e non-M axwellian m olecules, an e llip s o id d is tr ib u tio n fu n ctio n v2+w2 u'2 _ r 1 n "2RT, r 1 A _~2RT_ ^ ” n ^2irR T j.^ ^2irR T ^ r r is in tro d u ced to e v a lu a te th e c o llis io n in te g ra l o f second o rd e r moment eq u atio n . The assumed e llip s o id d is tr ib u tio n fu n ctio n may be q u ite d if f e r e n t from th e a c tu a l d is tr ib u tio n , [1 4 ], b u t i t does c o r re c tly re p re se n t th e f i r s t fo u r low est moments, namely n, u , T and T. . X J- I f one in tro d u c e s, Xj = u* - u1 , Yx = vj - v ' , Z = w ^ - w1 X2 = uj + u ' , Y2 = vj + v ' , Z2 = w| + w' g2 = Xx2 + Yj2 + Zx2 (A. 8) h2 = X22 + Y j2 + Z22 The s u p e rs c rip tio n denotes th e v e lo c ity i s a p e c u lia r v e lo c ity ; w ith th e assumed d is tr ib u tio n fu n ctio n (A .7 ), tran sfo rm in g in to th e (g,h) c o o rd in a te s, th e c o llis io n in te g ra l A(u2) = / dCjdc ~ becomes, 85 3 X i 2 Y ^ 2 2irRT 1 •)* * n2 / e 4RTr 4 RT l r The in te g r a ls w ith re s p e c t to X, Y, Z may be e v a lu a te d in term s o f s p h e ric a l co o rd in a te s as s p e c ifie d by th e fo llo w in g e q u a tio n s, Xj = g cos < j ) Y = g s in < j ) s in 0 Z1 = g s in < t > cos 0 (0 _ < _ < |) _ < _ -n, 0 < _ 0 _ < 2tt, 0 < g < » ) Then dXx dYx dZ1 = g2 s in + d$ d0 dg and 2 g [ 1 - 3 co s2 4 > ] s in < { > d d > dg (A. 9) X T r l e t t i n g , and in te g r a tin g (A .9) o v er ( c , < |> ) p la n e , one o b ta in s , 2 T v-5 A(u2) = (2mK)V_1 [ ttA - ( v ) ] n2 (2RT ) (4RTL) 2Cv“ 1) r v-5 1 H S ° vT T + 4 - t2 A 1 - rv2 [(I)*5 / e * ] • [/ ------- ~ C X v- . 5— « -1 (1 - Xx2) 2 ( v - l) (A .10) When v = 5 , e q u a tio n (A .10) red u ces to e q u a tio n ( 2 .1 2 .1 ) . (A-2) P e rtu rb e d M axw ellian M olecules C o n sid er th e g ases o f Lennard Jones model w ith index 5, [27] m 2K m2K’ . 5 r 3 (A .11) At sm all d is ta n c e , th e second term i s sm all compared w ith th e f i r s t o n e, so th e g ases behave lik e th e M axw ellian m o le c u le s; a t m oderate d is ta n c e , t h i s model in d ic a te s a change in th e index o f th e r e p u ls iv e f i e l d . I f K1 > 0, t h i s model a ls o ta k e s in to acco u n t th e m utual a t t r a c t i n g fo rc e betw een m o lecu les a t la rg e d is ta n c e . I f K1 i s so sm all t h a t th e e f f e c t o f th e second term on th e moment e q u a tio n i s I sm all as compared w ith t h a t o f th e f i r s t one, one may th in k F in (A .5) as a p e rtu rb e d M axw ellian M olecular fo rc e law. I f one ad o p ts th e fo llo w in g d im en sio n le ss q u a n t i t i e s , 87 m K1 kTjO2 * - £ 72= J L g" 2kTi w here, a i s th e c h a r a c t e r i s t i c le n g th , f o r exam ple, th e c o l l i s i o n c ro ss s e c tio n d ia m e te r, Tj i s a re fe re n c e te m p e ra tu re , and k is th e Boltzmann c o n s ta n t; fo llo w in g Chapmann's d e r iv a tio n [15, p . 18 0 ], th e d e f le c tio n an g le 6 can be w r itte n a s , 6 = [1 - -1 — ]~H (A-12) w here, vo 12 ° 6 S 2 / ? ° ° [1 - - !j ( f - ) '*]■** dC o ^0 v ; = b [ xl. ] K' n 1 ih b V [ 1 ------------P , V = - V'2 5 = v [1 - — ]% , v = b [ ^ ]** *0 o v , 2J O 1 jr J o and ^ i s th e p o s itiv e ro o t (th e l e a s t such , i f th e r e a re mere th a n one) o f th e e q u a tio n , 88 1 _ £2 . h f h l ) k * 0 oo ^ '-e J In tro d u cin g th e dim ensionless param eter, K' mK' th e n , 6 may be w ritte n a s , 6 = [1 + e 2 ~ T = ] ^ 0 ( A- 1 3 ) The term o f £Q > th e in te g ra l / s in2 6 bdb o can be w ritte n as, oo 00 / s in2 5 bdb = [2mK]ls — / s in2 6 £ dE (A-14) J o g Jo 0 0 C onsidering th e case o f weak f a r f ie ld c o rre c tio n to th e Maxwellian m olecules, namely, | | << 1, one -may expand s in2 6 by T a y lo r's Theorem, 6 , s in2 6 = s in2 < 5 n + •* — s in (26n) e0 - H N e„2 (A-15) u & U — _ o 2 2 N = - ig d -2 -(.c ° s 2 _ H 2 * e o 2 2 , 0 < e ' < e . e 2 2 - 2 - 2 Thus, th e second o rd e r c o llis io n in te g ra l fo r th is m olecular model is : A(u2) = / / f j f j [ -2(u j-u)2 + (v j-v)2 + (Wj-w)2] e„ o o (2mK)!i • [ j n s in2 6Q £ dZQ + ~Z J (2V sin (2V 4g • 0 89 - Is e22 / o N C0 « „ ] d ^ d J . :N i s bounded by i ---------- [(26q) + (26fl) ] as " and is o f o rd e r o f ! i ® i as 5q 0; and SQ < _ i t . T herefore th e in te g ra l | g2 /„ * ' ®2 n g" e0 d50 ‘ is bounded. Using th e tra n sfo rm a tio n (A-8) and th e d is tr ib u tio n fu n c tio n form (A-7 ), one f in a lly reach es, u B A(u2) = (2mK) (T rA2)[n(Px - Py) + - £ - ± * + OCe.,2)] (A-16) T 4 > = 2 /2 ? n R(T - T ) fl(X) (A-17) r n / v^ 1 si . 1-X l+/x 3 ri 1-X „ l+v^n f a = - Y {1 + — An ------ - Y [ 1 ---------A n ------ - ] } (A-18) 2/x l - / x K 2/x 1-/X A2 = A2Cv*5) B2 - /„ T C2«0) s in C2«0) T - T X = - £ ----- i . T r The co n sta n t B2 is determ ined by q u ad ratu re. According to R eference [27], B2 = - 0.4829 (A-19) 90 In th e range 0 < X < 1, ft v a rie s between y |- and 2. In view o f e q u atio n s (A-16) and (A -17), one fin d s th a t th e in flu e n c e o f th e f a r f i e l d c o llis io n in c re a se s as th e tem p eratu re d e c re a se s. APPENDIX B HYPERSONIC SOURCE EXPANSION OF GASES W fTH PERTURBED M A X W ELLIA N M OLECULES In th e h y p e rso n ic l i m i t , u n d er th e same a ssu m p tio n s, s u b je c te d to th e same boundary c o n d itio n s and u sin g th e same d im en sio n le ss v a r ia b le s as was done in C h ap ter I I , w ith th e z e ro th o rd e r s o lu tio n s N = - , U = 1 s2 th e second moment e q u a tio n s o f th e gases w ith p e rtu rb e d M axw ellian m olecules d e s c rib e d in (A .2 ), a re , | r Cs2Pr ) = f t p , - Pr ) [ l * e 3 SKX)] CB-1) 2 h = - ! C px - V I I * e 3 a(X)] (B-2) ! i JL e3 " e2 A 2 2 s u b je c te d to th e boundary c o n d itio n s , As s = Sj S j2 Pr = 1 (B-3) Sl2 P ± = 1 (B-4) e3 re p re s e n ts th e d eg ree o f th e f a r f i e l d p e r tu r b a tio n to th e M axw ellian m o lecu les. 92 Assuming, . 1 1 Ieif I = 3| « 1 Cor | e31 « Sj3 ) th e follow ing expansions can be made, < 0> < 1> Yj = Yj + Yj + .... (B-5) < 0> < 1> Y2 = Y2 + S Y2 + • * '’ C B "6) w here, Y. = s 2P , Y„ = s2P , , 3Y = Y. + 2Y l r ’ 2 j. * l 2 On s u b s titu tin g th e se expansions in to eq u atio n s (B-3) and (B -4), and eq u atin g th e lik e power o f e^, one fin d s th a t to th e low est o rd e r o f V J <0> o 1 <0> <0> Y, = * ! [», ] Sz (B-7) < 0 > o <Q> <0> S >- Y 1 - ' T 2 3 - [s2 Y ] = - 4 [ Y2 - Yj ] w ith boundary c o n d itio n s, As s = s 1 <0 > Y x = 1 (B-8) < 0> y2 = 1 E quations (B-7) and (B-8) are e x a c tly th e same d i f f e r e n t i a l eq u atio n s and boundary c o n d itio n s as those appearing in S ectio n ( 3 .1 ) , C hapter 93 <o> <0> <0> I II . Thus, Y, Yj and Y2 a re e q u iv a le n t to Y, Yj and Y2 re s p e c tiv e ly in e q u atio n s ( 3 .1 6 .1 )-C 3 .1 6 .2 ). The eq u atio n s o f n ext h ig h e r o rd e r o f a re , d C l) , , (1) (1) Yl ■ f 1- [ Y2 - Y!] ♦ F3 (B-9) Sz (1) 2 (D ^ 2 ! r(l) (1) s5 2 h y 2 + 7 Y 2 ) = - y — [ y2 - Yl ] - f 3 (B-10) 2 C O ) (0) 2 C°) CO 1 k 0 4 0 - Y2 f 3 * - 2 - [ Y2 - Yi j [ ^ P fiCX) [S 3] , X = 3 s2 Y Y1 w ith boundary c o n d itio n s , As s = Sj CD Y! = 0 CD y2 = o W (B -ll) CD Combining eq u atio n s (B-9) and (B -10), a s in g le d i f f e r e n t i a l o f Y is o b ta in e d CD CD m d U L + .C3S+D d V + 4 Y = Fit CB-12) d s2 S2 ds 3 2 1 CD CD CD Fn = 4 4 p 3 3 Y = YX + 2 Y2 The g en eral s o lu tio n o f eq u atio n CB-12) i s , w here, ♦ - ♦ ( f . 3 . §0 ♦ - M - f . -1. i> „ = IC|1 e* s'3 r e f) (1) (1) Ci) Y j, and Y2 can be derived from Y by, 0 ) C O 3 Y i ■ 3 Y * f s l r C1) 3 d C Y} v2 ■ 7 s a r In o rd e r to s a t i s f y th e boundary c o n d itio n s ( B - ll) , and C be zero and, Cl) S s *F Y » i/i f -----1 d s ' - f —i d s ' s l s ' 2W s2 i W ( 1 ) Cl) Cl) Y Y Y _ _1_ and _2^ a re shown in F ig u re 10. ( 0) ( o ) Co) Y Y, Y2 must 2 The r e s u l t in d ic a te s th a t a f a r d is ta n c e a t t r a c t in g f i e l d , which < • > r < •> r < •> * « i> Y J < i> < # > < # > <•> o> <»> o.S IO « o t/l Figure 1 0 .—Temperature c o rre c tio n s Due to th e Weak Far D istance A ttra c tiv e F ie ld . 96 co rresp o n d s to a n e g a tiv e e^, ten d s to in c re a s e th e p a r a l l e l and av erag e tem p eratu res and to d e c re a se th e tra n s v e rs e te m p e ra tu re . On th e c o n tra ry , a s o f te r r e p e llin g f a r f i e l d w ill cause low er p a r a l l e l and av erag e te m p eratu re s and h ig h e r tra n s v e rs e te m p e ra tu re . LIST OF REFERENCES 1. A shkenas, H. and Sherman, F. S ., "The S tru c tu r e and U t i l i z a t i o n o f S u p erso n ic F ree J e t s in Low D en sity Wind T u n n e ls ," R a re fie d Gas Dynamics, Academic P re s s , Inc., New York, V ol. 2 , 1965, E d ite d by J . H. De Leeuw. 2. Fenn, J . B. and D eckers, J . , "M olecular Beams from N ozzle S o u rc e s," R a re fie d Gas Dynamics, Academic P r e s s , Inc., New York, V ol. 1, 1963, E d ited by J . Laurmann. 3. P a tte rs o n , G. N ., "A S y n th e tic View o f th e M echanics o f R a re fie d G a se s," AIAA J o u rn a l, V ol. 3 , 1965, pp . 577-590. 4 . P a rk e r, E. N ., "The Hydrodynamics Theory o f S o la r C o rp u scu lar R a d ia tio n and S t e l l a r W inds," A stro p h y s. J o u r n a l, V ol. 132, 1960, pp. 821-866. 5. Whang, Y. C ., L in , C. K. and Chang, C. C ., "A V iscous Model o f th e S o la r W ind," A stro p h y s. J o u r n a l, V ol. 145, 1966, pp. 255-269. 6 . Molmud, P ., "E xpansion o f a R a re fie d Gas Cloud in to a Vacuum," Phys. F lu id s , V ol. 3 , 1960, p p . 362-366. 7. N arashim ha, R ., " C o llis io n le s s Expansion o f Gases in to Vacuum," J o u rn a l F lu id M ech., V ol. 12, 1962, pp. 294-308. 8 . Bienkow ski, G ., " C o llis io n le s s Expansion o f Gas Clouds in th e P resen ce o f an Ambient G as," Phys. F lu id s , V ol. 7 , 1964, pp. 382-390. 9. Owen, P. L. and T h o r n h ill, C. K ., "The Flow in an A x ia lly - Symmetric J e t from a N early Sonic O r if ic e in to a Vacuum," R ept. and Memo. 2616, G reat B r ita in A e ro n a u tic a l R esearch C o u n cil, 1948. 10. Brook, J . W . and Oman, R. A ., "S tead y Expansions a t High Speed R atio U sing th e B-G-K K in e tic M odel," R a re fie d Gas Dynamics, Academic P re s s , Inc., New Y ork, V ol. 1 , 1965, E d ite d by J . H. De Leeuw. 11. Fenn, J . B. and A nderson, J . B ., " V e lo c ity D is tr ib u tio n s in M olecular Beam from N ozzle S o u rc e s," P rin c e to n U n iv e rs ity R ep o rt, A ugust 1964. 97 98 12. Edwards, R. H. and Cheng, H. K., "Steady Expansion o f a Gas in to a Vacuum," AIM Jo u rn a l, Vol. 4 , 1966, pp. 558-561. 13. Hamel, B. and W illis , D. R ., "K in etic Theory o f Source Flow Expansion w ith A p p licatio n s to th e Free J e t , " Phys. F lu id s, Vol. 9, 1966, pp. 829-841. 14. Edwards, R. H. and Rogers, A. W., "Steady N o n -Isen tro p ic J e t Expansion in to a Vacuum," Fourth Aerospace Science M eeting, AIM Paper No. 66-490, June 1966. 15. Edwards, R. H. and Cheng, H. K ., " D is trib u tio n Function and Tem peratures in a Monatomic Gas under Steady Expansion in to a Vacuum," U n iv e rsity o f Southern C a lifo rn ia Report No. USCAE100, September 1966. 16. Chapman, S. and Cowling, T. G ., The M athem atical Theory o f Non- Uniform G ases, Cambridge U n iv e rsity P re ss, second e d itio n , 1960. 17. Je a n s, J . H ., The Dynamical Theory o f G ases, Dover P u b lic a tio n s , I n c ., fo u rth e d itio n . 18. Gard, H ., "On th e K in etic Theory o f R arefied G ases," Communica tio n s on Pure and A pplied M athem atics, Vol. 2, 1949, pp. 331-407. 19. Maxwell, J . C ., "On th e Dynamical Theory o f G ases," The S c ie n tif ic P apers, V ol. I I , pp. 26-78, E dited by W . D. N iver, Dover P u b lic a tio n s, I n c ., New York, New York. 20. Bhatnager, P. L ., G ross, E. P. and Krook, M., "A Model fo r C o llis io n P rocesses in Gases. I . Small Amplitude P ro cesses in Charged and N eutral One-Component System s," Phys. Rev., Vol 94, 1954, pp. 511-525. 21. Kogan, M . N ., "On th e Equations o f Motion o f a R arefied G as," PM M , Vol. 22, 1958, pp. 425-432. 22. Holway, L. H ., "K in etic Theory o f Shock S tru c tu re u sin g an E llip s o id a l D is trib u tio n F u n c tio n ," R arefied Gas Dynamics, Academic P re ss, Inc., New York, Vol. 1, 1965, Edited by J . H. De Leeuw. 23. Van Dyke, M., P e rtu rb a tio n Methods in F lu id M echanics, Academic P re ss, New York and London, 1964. 99 24. Freeman, N. C ., "A Note on th e S o lu tio n to th e Boltzmann E quation f o r a S p h e ric a lly Symmetric Expanding F low ," Im p e rial C o lle g e , London, 1966. 25. E rd e ly i, A ., Magnus, W., O b e rh e ttin g e r, F. and T rico m i, F. G. H igher T ran scen d en tal F u n c tio n s, McGraw-Hill Book Company, Inc., New York, V ol. 1, 1953. 26. Liepmann, H. W., N arashim ha, R. and C hahine, M. T ., " S tru c tu re o f a P lan e Shock L a y e r," Phys. F lu id s , V ol. 5, 1962, pp. 1313-1324. 27. J o n e s, J . E ., "On th e D eterm in atio n o f M olecular F ie ld s I . From th e V a ria tio n o f th e V is c o s ity o f a Gas w ith T e m p e ratu re., I I . From th e E quation o f S ta te o f a G as," P roc. Roy. Soc. A, V ol. 106, 1924, pp. 441-477.

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Creator Chen, Tsu-Yee (author)

Core Title A Study Of The Kinetic Theory Of The Steady Spherical Source Expansion Into A Vacuum

Contributor Digitized by ProQuest (provenance)

Degree Doctor of Philosophy

Degree Program Aerospace Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, aerospace,OAI-PMH Harvest

Language English

Advisor Cheng, Hsien K. (committee chair), Laufer, John (committee member), Pitcher, Tom S. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-594770

Unique identifier UC11360109

Identifier 6801675.pdf (filename),usctheses-c18-594770 (legacy record id)

Legacy Identifier 6801675.pdf

Dmrecord 594770

Document Type Dissertation

Rights Chen, Tsu-Yee

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