by

Peter f an Bnooker B¡ Sc. (ttons )

A thesis submitted. in accord.anee with the reguine-

nents of the Degree of Doctor of Phllosophy,

Department of Mathematical PhysiesThe University of Adelaicle,

South Australia

Felrruar y 1970

COÌTIEÌ\TIS

ABSTRACT

STATEMENÎ

ACKNO\¡LEDGEMn{TS

CHATTER 1 IIVIR,ODUCTTON AI{D SIJMMARY

1 .1 The Boltzmann Equat ion

1.2 Extension to Dense Gages

1.3 SummarY of Present I¡fork

Par¡e No.æ

1

2

CHAPTER 2 FEDUCN ]ON OF BOLTZI/IA}TNI S Eq'AT IONTÐ D TIIFEREI\]IITAL TORM FCR ASIMPLE GAS OF RTGTD SPHFRES

th

24

31

322.1 Boltzmannf e Equation for a Gas of

Rieid Sphenee

2.2 Nornal So1uti ons of Boltzmannt sEguation for a Rigid Sphere Gas 38

2.3 Reduction of the Collision Inte-gnal

2.1+ Differential Eguation for General66n

2.5 Diffenential Eguation when n = 0 81

2.6 Differential Equation when n = 1 87

48

2,7 Dlfferentlal Eguation when n = 2 95

cotulpNîs (com, )

PaEe No.€

2,8 Dlffe¡,ential Equation when n = 3 1O1

2.9 Dlfferentlal Eguation when n = h 1O8

0HAPTER f

'3.1

OHAP:TER 4

4.1

l+.2

4.3

4.4

ÎTTE SrcOND APPROXIMAÎION TOBOÍ,TZMANNI S EQUATION

lhe Second Approximation to tJreDistr lbut ion tr\¡nc ti on

3,2 Transport Coefflcients

118

118

134

140

141

146

154

161

251

T}TE 'IHTRD APFROXIM,MION TOBO T"[I ZMANN I S EQU¿I lON

The Form of  for the ThlrdAppnoxination

The Thircl Appnoximation to theHeat Flux Vecton

the thircl Approximation to thePnessure Tensor

the Third. Approxina tlon to theDistribution Funcbion

l+.5 Comparieon of Results

cnA?TER 5 CONCLUSIoN

APPENDIX 1 TI{E TENSORS Gn

2 SONINE EXPANSTON

3 r,IArr{EMAT ICAI RESULTS

255

ool{IENTS (COI{!.?-)

APTENDTX 4 'hN nX¿Ctr SOLUTION OF ¡Ol,TZ¡nAW¡tf sneu¿TroN FoR A RrcrD sPHERE GASIAust. J. PhYs. 21, 5L+3, 1968

5 FCRTRAN PROGRAI/IS

BIBLIOCRAPHY

E]GI,ANATORY NOTE

Equations are numbened. consecutlvely in each

section. They are referned to within that section by

the equation number on1y, and in other sections of the

same chapter, lcoth the sectlon number ancl eEration

number are given; liker¡¡ise in othen chapters they are

nderred. to by glving tJle chapter, sectlon and eguation

num'l¡en. The chapter and seetion number of a particular

page 1s denoted. in the upper right lrand corner below the

page nu¡nber. References in the te xt are numbered in

ord.en of oecurrenee, and- these numbers are supen-

scnipted. to distinguish then from eguation numbers¡

ABSTRAClI

Using the successive approximation schemer normal

sol-utions of Boltzmannts equation are obtainecl for a

d.il-ute simple gas of rig1c1 spheres. The integral

equation of cach approximation is reduced to a set of

ordinary differential equations by the use of the

c oll-isi on d-ynamics of tr,vo rigid spheresr âff,1 cer"tain

auxiliary f\rnctions which are d.efined; This rcd-uct ion

is quite general, being applicarbIe to all orclers in the

d.evelopment of the distrÍbutlon function in EpherÍcal-

harmonics; The secon,L approximation to the ri-istri-t¡ution

function involves terms of orcler rt = 1 and. rt = 2, ancl

the thircL approximation, terms of ord.er rI = O11 t213t ancl

4. Accord-ingly the general theory is specialized- f or

n - Orl ,ZeJ and \, thereby giving thre differential

equations necessary to solve the first two non trívia1

approximations to Boltzmannl s equation.

The clifferential- equations of the second- approxima-

tion are Solved- numerically¡ and the seconcl apllroximation

to the cListrÍbution fr-r¡ction is obtained. from the

rel-ationship with the auxiliary f1rnctlons in which the

d ifferential equations are vur itten. Integrat ion of

this to calculate the pressure tensor and læat f lux

vætor gives tLre coefficÌents of shear viscosity and

thermal conduction exactlyo This procedure obviates

the need. to expand the distribution f\rnction in terns

of an infinite series of Sonine polynomials which is

usrlalIy cLone 1n calculating transport coef fici ents.

Using the second- ap1aloxiration to the d.istribution

f\¡nction the third allproximation to the pressure tensor

ancl heat flr¡r vector are caf crrlated" by trrerf orming cer-

tain integrations, The val-ues obtained are checked- by

a seeond- calcuLation corresponùing to that macle in the

second. approximat ion, Thls is marle by solving numerically

the differential- equat ions f or the third- approximation

to Boltzmannrs equation. The solutions give the thircl

approxima tion to the clistnlbution furrc tj-on through the

relationsLiip with the ar:x11iary f\-rnctions, arr} this is

integrated. d-irectly to obtain exactly the pressure tensor

and. heat flulc vecton d.ependent on terms non linear in the

gradients of nr:mi¡er density¡ temperature and. rÞan veloc-

ity. In ttris approximatlon the necesslty of enf orcing

the srrbsldlary conditions Ís pointed out.

STATE}/IE}TT

I hereby d.eclare that this thesis contains no

material which fe.s been acceptecl for the awand. of any

other d-egree or di ploma in any University, and. that to

the best of ny knouledge and. belief, the thesis contains

no matæial previously puìt1ished. or urritten by any other

personr exeept uihene due reference is mad.e in the text.

Peter I. Brooker

ACKNOÏITLEDGEMENIS

I must finst express my gratltud.e to my supervisort

Professon l{. S. Green, fæ suggestlng the topic arxl method-

of apBroach of this thesis, ancl f on his valuab le advice

ancl encouragement <luring thre course of this vrork. Also

to Professor C. A, Hurst arcl Dr' P. 1¡1. Seymour, and-

incleed. to maqy menrbers of the ilTathemat ical Physics

Department, Univenslty of Ad.elaid.e¡ I am ind-ebted f or

some helpful di scussions.

I should. l1ke to thank c.s.I.R.o. for the fi nancial

assj-stanee provirìed through a C.S.IoR.O. Senion Post-

graduate Stuclentship from 1966 to 197Ot and the University

of Ad,elaicle for a University Research Grant for February

I 970.

Finally I wish to thank Mrs. B. J. UlcDonafd fon her

lnvaluab le assis tance in Wping this tJr esis.

ì !iI tj

I

'') '. -.-.'' - ítf: 4¿rUt

1.

CIII\T'IER 1

Tlil'RODurj'.il IOIT AND S UImiAlìY

Dlscussion of 'che transport phenomena in gases is

generalll/ macle in terms of a kine bic cquation l'¡Ilich.

c-lescrlbes the irnevæsible apiJroach of cr gas to equil1-

'briurn throu¡4h a velocity, or one psr ticle clistril¡u'cion

function. The first such equation vrras the Ro1'bzmann

eqr-lation vÌrieh is useful vuhen the i;rs is clil-u--Ûe, In the

first ¡eotion of this chaltter TVe sìral1 cliscuss bnieflSr

tln historieal development of this equation ancl solution

of it, 0riginally the equation l'¡as cLerj-ved[ in'cuativelyt

ancl in Sec'úion 2 rue give a summary of the rri¡ork v¡hich has

been clone to put -btris in'i:ua.tive 't'heor5r on a- soun'dl llas-Ls,

ancl- also to cle bermine corrcctions to i'c t'lhen the c-lensi-t y

of the BAs increases. This involves the Ìrse of Liouvillers

egunt ion for 'che lil r.lar'cicle Ê)rstem¡ anc-[ in 'bhe cliscussion

which ¡¡'¡e ¡;ive lt shoul cl- l¡e borne in ninc.1 tirat th.c iF.s is

cousiderec-L to h.rve sllort ran¡1ecì. repr,rlsive in'l;ermolecula.r

forces so that the iclea of a collision as L r¡¡el1 clcfinecl

2.'l,.,1'

event is neeu:.in¡;fLr1, Other sÍtuations '.rr¡here the 1on¡5

t an¿ec} rnture of 'bhe forces make such a concept cliffieult

to clef ine oecl"Lr in pl-asrnas, but v¡e r;Lrall no i: consicler

those hene. In ,Secti on 3 vì/e inclicate ihre proìrlem vre

slrall consicler in this -i;hesise nrme15r, the exac'b nuaerical

solution of lJoltzmarur?s equation (using the orlglinal

form approi)riate to a <lifute ßas) for a simple gas r¡f

rigicl spheres, The solution Ís carnied out to scconcf

orcler in 'bhe groclie nts of num'ber clensÍtV, 'i:emperatu,re

and. rnean velocitSr, and. thus gjves ân extension of tne

linear tnarrsport eguations to incluc-le non linear -btri,rs.

I . 1 tþe P19_1_! z4gr¡¡_.B^qggþ j-,.o¿

The kinetic theorSr of iF.ses in its moclenn for¡l crn

l¡e saicl to hnve started- with the '¡,¡ork of l{axr¡¡ell and.

Tloltzmann in tJre last half of the nineteen'ch eentur¡r.

In 1 859 T,.faxrve 1l d.i sc ov er ecl t he 1ow of c'l-i s'br i'bu-t i on of

moleculan veloci'úie s for. a gas in equllibniurn, nncl in

1866 ne gave the first proper mathei¡a'cica] f orilul¡.'i:ion

of 'bhe tireory of a non unif orm gas(1). Prwious

3.1..1

c11scu*:isions f or a non uniforn gas had been ¡iacle mos-b1y

r.'rith a Tnol-ecl-llar moclel of rigicl ela,stic ílphcres usin.P,

approxima.be methocls based- on the menn free path -technique.

(see for example, Jenns(').) The equ.rtlonsof trans¡fer

,¡,¡hich l'laxr'¡e11 cìer ivecl tevcj the rate of chan¡,;o of îJr¡r

mean rnolecular propert¡l assocj-atecl t,lii;h the i-tas; ',,11e

nate of char¡1e beiryi separa-becl in'uo thc ltari;s clue to

nolecular encounters, thc motion of ihe particlcs¡ êtLì.rl the

ac'bion of thc extcrnal f oFCo¡ IIe usecl these equa.iÍons

f or a [Jas r¡¡hosc rnoleculcs, considerec-l as polnt particles,

r,vere sr.p¡rosecl bo exert f orccs on each othcr proporbional

to thc f ifth power of the d-istance ltetr¡ue en litenr nnc-l

ob tainec-l the f irst neeurate, theoneticel values for 'clte

coefficlents of viscositSr, i;irermal conc-luetion ancl ':Lif-

fr-rsion, fon any rnolecular mocle1. Calculation of i;he

inte3rals involveil in the mole culan eneounter con'bril¡u-

tions in tJ-e tnrnsfen equa'bions wa¡ ilnrpossilrle withoutt

n lcnot',r1i:dge of thc vef ocity clistnibutiou f\rnc i,ion excellt

in this case of 1rl,{axr,vel-linntt molecules.

4r1 ..1

In 1B-12 Bo1'r,zrnanr.(f ), lly an in'¿Lrativc ar¡4urnent noi

l:ased" in a rigorous way on thre Jaws of neclunicst

ob'bainerl the inte¡Jro-cLiffenential equati.on r¡¡hich the

single particle c1i stribution f\rnction vi¡as reguinecl to

satisfy reganc-[],e ss of the il'bate of thre i,as or the forces

ncting on it. Ile vlas able to solve the ec.{11-&-bion 'co

ol:'uain the cfis'¿ril¡lrtion fi-rnc-i;ion for aItj''iaxl¡'¡e11inn-" 3ì[lsv

ancl ol¡taine cI ilaxlrcllr S reS-rltS f or 'cire 't ransport Co-

efficient,s for this mocì-el by clirect intcgretion.

I¡or altou'r, seventy yenrñ the Roltzmarul equa-bion l¡¿''l';

regnrc-lecl- as the þar;ic egua'bion with v,lhich a gas not in

eguililtriuir should 'be stucÌieci.. I'¡e thocls of ,so lt't'i;iou of the

(tL) .eqr.:aÈ1on wæe flnst iìiven lry Hif-ber¡\LI-l (l9lZ), Cho'pman

/r\ IE\(lglz11916¡ \:'t ancl llnskog\') (tc-ll1¡1917) t who l-nid- the

l¡asis of t]re so callecl Challnan--iÌnslco¡1 methocl in i-i;s

mode:rn fortn. This ie the most eommonll/ gsecl metþoc] of

olti;ai nin¡; -r,he transport coeff icicnts of a Jas fron c.

Ìcno,illecÌge of 11rc intermol-ecular inter ac'r'i ons '

Ã

i.\Jn ob,ceinin¡1 -bhe ::olut]on ìty this mc'bhoc-l 'r,hc s'ì:¡.te

of the ,s),stem in ar;sumerl to l¡e no L fan fro¡r local

egr;i1iìtri uln, r¡,ftr ich is e s-bate tr"dlere cquilillr ium is es'bc'll-

lishcd in volumes smnll com¡]ared. to the 'cota.I voluinet

i:ut of suf'ficient ç:,j,ze i;o contain a largc numller of

molecules. It is of the form of ì,{axlvel]rs eqr-riliirrium

cli stributi on exc ept that bhe numll er ilens i ty clt -bempera-

ture 1ll, anrl- the mcan vel-ocitv.9o in this caise are functions

of Slliì.ce arr.L timee $ince in neighì:ourin3 vo]umes cl-e'I'ancl

c need. not þ¡rve the Sane value. 'Ihe equation '"'i'hich iS-onon linenr in the clis'cri'br-rtion f\rnction cau thcn lle solrrecÌ

l:y a ne tirocl of successive a¡lpro;citn¿ltion in r''¡irich a- Élys't'en

of line¡.r intcnral ecìuatioirs is sol-vecl in succes;sioÌt'

,I.he stneaniing and- collision terms in Ure intc¡1ro-clifferen-

tial eguation are ana.l-ysecl to mirhe such a- schclle ilostlible,

ancl the orems on inte ¡¡ra] equat i ons a re uged ''c'r sh olr 'cha L

solutionc of tlr-e integ¡al eqgat j-ons are po,',rsillle if ancl

onLy if certain subs tclirry conclit ions as soci¡'"t'er-l- ltii;lr 'uhe

c o11i si on invar iants ar e enf orced. Tn e ach ajlilr oxima--i: i on

6,1.1

the ctistribution f\rnction is su¡lposecl to clepe nd on tj.mc

only through tl.e c'ìc¡lenclencc on time of c1, T :urc-L gor ancl

tþe eqr,lation is erìal)/nect in s,rch a V,ia3r that in aqy eplrroxi-

mAtion the tirire derivnbives of clr 'I and qo are rrniqucl-y

c1etermínecì. in 'cerns r¡f cl¡ !l arr] go ancl" their flpfl.cc

,-Lerivative s. flhcse S olutions ilre '1,he norrnal solu'bion's

of Rol-tzrnannt¡ sctruation ancl ctescri]le tkre ,so-c¡-l]-ecl

hyctr oclynnrnicnl ap pr oach to eqni lillri um.

The f irst nlìJro)rirnat ion is il]e local ;ie>¡¡¡e11ian

clistribu bi on. In cacþ successive nÞlroximr.'bicn tltc

c orr €c ti on to thc ctis i;riilu.bi on fbnc'ci on is '¡'r 1.i;terr as

the proclr-rct of e local- î'inxwellian state multi¡:ljecl lly a

lrr\ rnctio' o(N) is expressecl inperiurba'¿ion 0'-'. T.ri¡ fi

terms of senies of Sonine polynoinials in ihe 1¡eculie-r

speec-l (ttrc; I,tecuJiar velcei'cy J{ irs thc velocity of a

rrtolecule nel-¡,'tive 'Lo tire re 3-n vclocity), multipliec-L l:y

tensorr: it-t l¡- i¡dricÌr tranllform r.ccorc-lin3 to irrcducible

r epr cS entat i- ons of the th r ee cl irnensi onnl r o-b ¿r-b i on lir oup.

Tiru.C cS,Sen'úielly vre lrave a clevelo'rment of the ¡olution in

7.1.1

sphcr:ica1 harmonic,s. l¡l/e I'ril1 call threse bensors Ìn Y-

inrecluciltle tensors in 'Lhe f o]lowing. Scalar proc-lucts

nre f irrmecL betryeen the irrecluCible te nsors nncl tenSors

clepenclent on no parameters 'ltu-b cl, Te and- co enr1 thein

spacc cl-enivntives, thr-rs giving a ,scalar expression f or

- (trt )a' 'o

The second ap,--'roxim¿rtion f or a simlrle 3,ir.s involve e

only the gredients o¡ go and'I; and- the values of 'c1Ìe

plleSsr-lre tenson pr ancl- hea'c f 1ux vector Qrcnlculatecl ìl¡r

integr¿Ltin5 tþe first correction 'bo 'rfru lo""t equi-1i1:rium

s-ba-be, give -bhe cocff ic ient of shear visc osit5r ancl 'che rmal

conclucti-on, In the sar¿e Tr/ay the coeff icie n1s of cliffusion

can l¡e olttainecl f or iì gaS mixtltF€o '.ì-'hu,S lvc have a tÌreo-

retical- basis of the maeroscol-ric linear transpor'u equa-

tions l',4rich are useci. to describe the fincl stä.qes of a

,s)rstens api:roacl1 to equiliþriu-nr, ol3 the beh¿'.viour of a

n1'¡tem itr an extæneIly maint¿rined steacly ,st¿rte jn t^Ârich

thrcne is a smnll- consi;a"n'c ff-otn¡ of íiomo qunnt:'-ty.

8.1.1

Clearly l;y going to hi¡þer approximations it in

llossÍlrle to olttain Íencralizations of the macroscollic

laws, r')e cruse these hiilre r eplnoxima ti ons; 'co 'chc c-lj-s -i;n j.-

l¡ution fìrrc tion clepencl on hii{her orcier grarlienbs nncl

proclucts of grdients of d, T and co. This non linear

d.epenclcnce in the clistributiou fLrnction ¡ives ihe flame

non lincnr clepenclence on -ihc grnclients of clr '-l and cln'-o

'bhe moments, In the thircl .rllpro>cimation the expressions

f or p ancl- q contain procLucts ancl Í;ltlr.lares of grnclien''ccr

anc-[ sec oncl r:r r]-er graclien'¡s, a-ncì, are conmonl;yz call-ecì -tìre

Burnc i,i, equat i ons. ()ne of -bhe ch i cf conc cr nß of this

'che sis is tire cval-un'u ion of -thc thirc-[ appro:cimatioirbo

Lire d-isiribr:tion f\rnction and. bhc coltesponcling non

lincrrbrans¡rort equetions for Ð. sin-n1e ijas of ri¡;icl-

sphetr' es.

The apllr6,.rriatcness of the exìtansion of 'cho pertur-

ìtation in Sonine polyno:'tials raiher -blrnn in povters of V

wl:rÍch $¡as aclollte c1 1rr,z Ql1''lm't¡ ancl 1tìnskog, Ir"/as no-ticer'ì- ìly(z\

llurne¿-¡\ t r, v'iho also l-)rover-l the convcrg'ence of the series

o

1.+

uêecl f on certain mofeculan moclels including flre r.igicl

sphcre noclelo Ii:l this methocl the final resLrl'b,s fon 'cl,e

transr¡ort eoeff lc icnts ane ol¡tai necl as rat ios of infi nite

cletermin¿rnts. 'rhe ratios eonvenge rairicìly, so th.r.t in

genæa1 the fqqr'ch appnoxlmation to them is su-fficient.

lr complete c-Lescription of flris method_ of oì:ta1nlry¡

nonnnl solutions of Bo1-bZrirâr'r.rrr s eguat ion j.;:l to l¡e fou-ncl

in Chaprnan ¡nc'L Colrlingt s 'r'l-,hc Ì,'iathematical llheor5r of lTon

ilnif orm Gnl.,"o" (B),

,Ln equiv-r,l-ent methocl of o1l.tainin¿j 'rhe

normal solutions ræing a vâl:ia bional- technique has bcr:n

3iven by curtiss ancl lÏirschfelc-ler, ancl is c'r-ee;cribec]. intri"iol.ecular fr'heory of cï¿rse,s anc-L Liquiclstr rrSr 1li¡-"ìrfclcrer,

cuntiss an¿ gir¿(9).(t o'Kumart' "/ ( 1 967) tias reformulatecl tLrc Cha¡man-Jlnskog

oohr.tion of 'che 3oltzroann e qua_tion, r'rriting the proltlem

in matrix forrn, anc-l empl.oJ¡ing the methocÌ of cìealing v,¡ith

irrecLuoibre l,ensor.s r;rhich has been usec'l extensivel]r by

r1¡ii-aner ancJ Racah in atornic ancl nuelear lrh¡rsics fon

cleal-ing r,'ri'bh prol¡lerns in an.-Íular momenbum coupling;. Tn

1Q.1..1

this f crm 'chc suecessive approximation proceclure anil the

slrl:sicì.iary corcl itions as sociatecl r,rith the collision in-

varlalr'cs can þe accoun'i,ec1 for ,¡¡ithout 'the u;¡r-Lrl recourÍte

-i;o 1;he theory of inieCrnl equations, 'L'he 1ntr" oduction

of r.æL t¡.lccs at bhe st;art gives a simplif ica'¡ ion of presen-

tation ovæ the Cirapnan--F,insko¡1 formulation r;¡hich arrives

at resu"lts in ternÊ of infinl'be clctermin¡nts from a.

ìreg;innini; in terms of the integnal equation. -À1sor 'the

col-"1-ision inteflral and'bhe .\ri$ociatecl bracleet guantlties

of th<; lcinetic -ùreory are clerivecl. in I{umarrs theory in

ictrms of the 'Ialmi coefficients l-Àrose proper'r,iec ar¡e

knolvn from e xteu,sive stuclie¡ of the harmonic oscil]-ai;ot'

sheIl moclel of nuclear the or¡r. The integrals in 'chis

f cnm arcJ convenient for nume ical calcuf¡Lt ion. I(umnrts

u,se of ircclucil:le -r,errsoT.S in spherical polar coorclina'ces

i,s nlso aclvanteigeouls vr¡iren calcu]-ation for 'cl:le momentg of

the clistribu-tion functlon are consiclered- involving higirer

orc'ler tensor eqr,r,ations. 'I'he question of the sJ¡mmet:rics

involvec-l as the ord-er isl incren"';ecì is no't encoun'cerecl in

iri s f onmul-a ti on.

11.1..1

Ano'thcr me'LJrocl of rsol-vinß 'bhe Bol'czrnann- equl'.'i;icn

d.iff cren'i fron the Cltapntan-]lnskog me thocl of nornal so lu-

tions has been d-evelopecl b], GracL(tt). He e;cpanclecl'Ûhe

clisi,ribu-bion fÌ;.nctiou in a comitlcte set of orthogonal

func ti ons in ve locity (usl n¡r three climensi onal- ile rml.'ce

fï.nc tions since th. ese ilr'e âpìtropria'cc 1',¡]r-en ihc veloCity

is expressecL in rectangi-rlar coonclinates li¡hich he chose,Thls

T:fermite expansion in rec-bangr,rlar coorcU-nâtesllecOmes a

pnocÌuCt of spherical harmonic;'; anc.[ Sonine polJ¡nomi'r]--

';¡/lren expresse<l- in sirheical- coonclin¡..tes), .L'he f irst term

in i;he ex'pans:ion is e local ['iaxr'¡c11ian clis'brilltL'¡ion

ftrnc l,ion jus'c .r"s in 1he normal solu'bion of the .¡rethocl

Just dcoCrlbed; . anc-t tire coefficients am(gr-r) of 'bire

I.Ierrnite func'¡ions vrhich inul-tlply this l-ocal j,'iax¡¡e Il-lan-

in the succeecling terrrls are fLrnc-tions of $.pace ancl tit:tet

ancl arc just sirn¡21e f\rnc-bions of the momcnts of 'cl¡e

cl-istril¡uti on fì;nction. To 'chird. orclcr in .bhc cxllansion

there are trventy coef f ic ien'cs :rncl Corrcsp<¡nÔin¡¡ to -tjl ese

are -br,.fent:¡ noments. illhis set is reclucccl to -Lilir'¿een if

12..1,1

thc moments of V.VrVr. nre contrnctccl over a pair of

indicies, the moments'bhenltein¡1 d.r gor ll'r !r ancl q.

The expalrslon is sul;stitutecl into the lloltzn?.rrrr

cquation ¿ncl momen'bs of tjrc e qr.r¿¡tion are f orme c1. ll'he

infinite set of ,l-iff eren'cia1 equations forbh.e orn thus

f crmed. is equivalent to Boltzmannrs cor'la-bion, anc-L it

1s nore useful +"o '¡rr ite tlrese equzrtions in -bernl,s of the

moments of 'che cllstribu-bion f\rnetion tt'ran thc &tn. ITolr

it hapnens that thc equr-'r,ion for a given moment lnvolves

higher momentg ancl rio to solve the system of equations

thc expnnsion has to 'be iruncatecl bc5ro¡fl ¿1 cerbain ord-er'.

It is usunl to use 'bhe 'chirteen rnoment al?pro.xílnnf ion,

fn this f ormulrtlon -bhe Llressurc tensor and. heat

flux vecton are treatcd. on the iiame leveJ as-bhe nunbcr

denslty, mcan vclocity ancì. tenperaturce 'Ihey satisfy

c'liff erential equa i:1ons in tlr.eir ol'un ni¡1ht jusi; as -t]re

others sniisfy tire conserva'cion cquations, l-ncl'i,he¡r nn6

no lon5¡er nelatecL just to the gnaclients of a1l- orclcrs of

c1, T ancl c^ ns ûre y were in tlre normal solution¡s. llhe' r-o

13.1r1

cquLl'i,ions fon tþe pressure tensor ancl heat flux Vector

can lle ,Solvecr. v¡ith given inltial cond-iti ons ancl are

a.rl.pl-icaltle to re,:qions \Ã/herc 'bþe Chapnan-lJnskog normal

solu-tion,S cannot be useclr becausc -bhe norlnal Solutions

arc restricte¿. to near cq.'-lili¡riuln situ-ationr;r Gracl-r s:

theor"¡' d-oes not re15¡ on the funtional clepenclence of the

clis'bniltution furction on ci, T ancl gor so tha.'c hj's eqlr-a-

tions are no-b restrictecl- to smaIl cleviatlons from eql-li-

libnium anc-L are ef:illccial]-y Useful f or shoclc \ilÐ.ve 'ureat-

t:ùCn'rl, lthis a.lJ1)roach is no'c clesigned aÉl a meihocl -uo

enaltle bhe transDort coefficients to be câlcu1r--'ccd-, f or

-i;lre heat flur,x ¿-'.ncl the pressLlre tensor are in gencral ini;er-

connectecl in a cornllllcatec--l !i/û}zo illhe prime purpose of tire

tne titoaÌ d.evclo1:ecl ìry Grad. i s bo Sive the clif'f e::cntii-t1

egr-rl'bions satl,Ef iecl lly a f l-ot'¡ rirhe n 'bhe normal sol-utlons

aI" e inacl-e qLlA'r,e ,

I¡or certain ,slo\¡ILy varyinÍj flovrs the solutions of

Grad-ts equations Show that û gu.15i equil-ijlriurn is íì'e'tl up

af .ber the orcÌer of a few collision times, airc-l 'bne

1l+'1.1-2

pressurc tensor ancl hea'c f]ux r¡ac tor can bc elqlresücrl in

tæns of thc grad-ientri of cl, T r.ncl co, Thc decn5r of 'ti-re

s5/stem to equiliìrriurn can then be expresneð lty i;hc

Chapnan-iìnsr,iog normal so lr'rtion$.

Solutions of Boltzmannrs equations have l:een oJrtalnecL

ìry cx1:ancling the di,stribui;ion function about aì;sohr'ùe

equilibniurn n¿rthe r tha.n dr out l-oeal e quili]trium Drs in

'clte Gracl exiran.sion. 'Ihis expansion, which h¡.s ,rreen usecl

lry lr¡1¡.r Chang ancl Uhler.lr"ct(12 ) in a d,iscr:-ssion of 'r,he

propaga'bj-on of sou-ncl in ¡r monatomic Basr i;: d-escribecl 'l:y

ç"-u(t5) in "llneyclopcdla of Physics, Vol. 12t' along wi ih

-bire normal- solu'cions, ancl Grarlts oi,"/r'l solutÍon of

Boltzrnannr s cc-pat ion.

1 . 2 Hqqfo¡1_i:p__pç;n_s,c*_G*ae-e¡i

Boltzilannr s equn'cion v,¡ns inj-tÍalt¡r d.cvclo',-recl on

rrhysi eaI gnouncls, a r:sr,nnin¡1 f ir st t hat the $ys'ce rn coulcl-

be c-lcscribecl bJ¡ an cquation for -thre one partlele cÌisbrÌ-

bution f\rnc tion, ancl second 'bha'b the coll-ision 'i:æ¡r vas

glve n 'by -bhe "Stosszahl-nnsatztt. Tlr.e la't,'cer ûssr-tnl,ltion

essentiall5t ç¡"anr-r that oul;r ltinnry collisions are

15.1.2

cOnsiclerecle anrl- '¡irat tÌrei'e is cornplc'ce CllSence of Corre-

lation l)e -bïrcell the Bositions ancl velocit j.e s of collicì1ng

i)crticleri. Also inrptiecl in the "Stos¡zahJans a"Çzt' iS 'r,he

assrunption'bhat thc cr.ir:tril-'ru'bion ftLnction clLocs rro'c Vrry

¡ruch over the region of 'frhe interporticle potentialr or

ove¡l the clura-i;ion of a collision, So thût rnllicl- f]u.c.Lua-

'¿ions in tho systctjt arc not c-le'scribecl. C1ear1¡' the

restriction to ltina-ry collision¡:; ic only goocL if tl're Ílas

is clilute e ârrcl in this renl-m tire transport c ocf f ici e nts

ob'taineci from the normnl solu-cions of Bol'¿zm¿rnnt'1

eguc.tion are iä c¡lj-'ue Soocl e-'.rcclûent ",¡ith cxperiltcnt.

I¡or a moderetcl-y clcnse gaß ,,;'hich hovcVcr is not nenr

to conclensin¡¡, thc cf fec-bs of multipf e collisi ous ancl

par tic le-pnr'c i cle Co rre l.at i onS ÌleC ome nir.tch morc imir''ir -bnnt

nnc-[ nrust lte taken intO ¿LecoL].nt.. Thc firs'c ai,ternlt l;o

solvc 'che pllobleiil of a moclerntel5r clcnse 'g¿Ìrl r¡''¡-as mecle 'by

trnsrtco¡-ç(1ll'), f or a [.irr.s of rigicl spheres. IIe usccl an

intrra-tivc approach 'basecl on the original Bof i,zrnann

clerivation, and- toolc into ¡-ccount the finltc voluine of thc

16,1.2

nolecules ancl the trcollisionel transfer" of nrome nturn ¿rncf

eîef[|Vrbutstilfassulneclìlinar5lcol]-isionsandmolecular

chaos Of bhc ttStOsSzahlansatZ'r. HiS resuf'{,S Vriere lnber

verified. b¡r ç*'tiss ancl Í:inicloo(t5) on the l¡asis of ttrc

mocÌif iecl Boltzmann equatic.n of Green ancl Bogoliullovr not

taicing into account morc 'bhan tvro bocly collisioils"

Ilo clevelop a gener¿r1iza'blon of ihe lloltzntalrn egua-

tion vrhich ac counts f or the corrcctiorrsarisin3 rì"s ''t,he

cle nsitSr of 'bhc Sru inc¡e asese re searche.'rs in the i !l¡'Or;':

turnecl to the l,ionviIIe equation vÀrich g)verns'blre 'cilne

cvoluti on of the N partic1e clisiribu-i, ion func-i:ion" Llllis

equation is l:asecl on tl-te c\ynemics of the 1{ par ticle

c¡¡stenn so t ìrat eny genc;ralizecl Boltzmann ecìua-b ion

cLerivecl front it has a souncl lllSis. First ¿rL-i;elnl¡ts/¡f\l lt),t '-' , Rorn and. (i'ree n

(tt)in thi-s

aiæ1a]1 ea 1¡'/C f e

Bo6¡oliuì.: ov

mncle by Tiirk\¡¡ood(18)

Bogoliu'l)ovt r-¡ me'choc}s ï''el:e based- on a f¡nc-i;ional

ttA¡satzrrsiililar to ürat u$ecÌ in tLre Chalrnian-ì,ìnstrro¡| -bhe ory

in ',vh ich -bhe tj rne c-lelrcniì-ence of th e d-is tr i'bu'ci on func 'ti on

17,1.2

1s ijiven through the time t--lel.renclence of tre local equili-

'brium varial¡lcs, cl¡ T anc-[ co. In Bogoliu'bovrs the ory the

s partlcle c-Listribution function f(") fon t'i'¡o or more

particl-es cloes not depend- exitlicitly on t ime; r'ather the

time vaniation of thesc f\-rnctions is i-ivcn throu¡1h thc

onc ¡nrticle clistrilrution functinn f (1 ). This means the

system is cleseribecl- by a kinctic equation of the form

¿#.=A(r,slr(r), (r)

u¡here A(g, v I r(t ), is a functional or r(1). This

equation is of course valiC only f æ times much []:rcr-ter

than the collision tì-me ; smallcr fluctuatlons harr in¿

been averagecl out by the fì-rnctional- ttAnsatztt.

By making an exr?ansi on or r(") ".o # in powers of

the cle nsity, Bo¡¡oliuìrov ol¡tai necl a systern of eguations uftrich

coulcl in principle be solvecl to a]l orclens of the clensity.

The flrst oncler eguation c-Lescriltinlt only two lrarticle lnter-

actiOns is jr.rst the Bolt zmarLYI eguation. 'Ihe first cornec-

tion obtained. l:y i;oin[,; an orrr.er hi¡:her in c-Lensity has

18.1,2

l¡een cli scussed extensively lty Ct-roh ancl Uhlevrllecl<(t9,zo)

"

illhis conrecti on talces into .lc count interac-b ions of 'chree

partiel-es in the [a,Sr lìach higher orcler in the cìeusi-b5r

talccs into account interactiono of successively more

par ti cleri in the gas.

'Jlhe func-biorurl rhrrsa'czrtof Bogoliubovr s AÐ ilroach iras

lreen qr-lcstioned. ity Dorfman ancl Coþe n(Zl ) *i'to sholvecl tha-b

i-b helr-l only for the finst tlvo -r,elrllìs iri 'chc cìensi-'c5r

exllansi on of f (z Recent work of Ll. ''j. Green anc-l

(zi)

) (zz)a

Cohen l'¡ho use cluster expansrions comllon in cquili-

ltrium tJre or¡r f or a c]err¡e ijns, has provicled. iln al-bcrn¡r'c ive

clerivntlon of the generâl¡i.z,eÒ. Boltznann cqr-r-a't ion Without

uslnLl tl-rc functional ans¿rtz. O'¡hcr alterna'bive rncthocls

ìrnvc been ¡ irren l:¡r .'.1o1tinger ancì. cur bi-u(zll) , 'I{ritz ancl

(zt)íianclr i (zs)

,(25)

Ono ancl Stccki ancl Taylor to name a

feït/,

Solution of the generaLizc:ð' Boltz:"nann eqltation

f o1f cws a similar beehniclue to tþa'c usciÌ in 'l,l¡e CÌrapnian-

Lìnsk o¡1 th eo r:7 . 'llhe cli s br illu.1,i on funct i on is e'.:pa ndecl

19,.'1.2

alrout locnl- equjlibriu-n thus line¿¡.r,izi-ng the eQLr-ntion an.c1

then nornal soluti or,-.i ar c ob bainecl. Choh and llll-]-cn]te clc

chowecl tìi¡lt Euch ¡- llnocecÌure leacls -bo an ex]ransion of 1,1-re

transport coeffi.cicrris in por''iers of ',,he clenr:i'i¡z of -i;iie

f orm

€: = to + a4tr + d't';' -t- (2)

The coefficicnt fo is just tirat ob-bainec-l from bhe

.rloltznann eqrla'¡ionranci. tlt e f irst densi-rl5r corr ection E"l

is clcpenclent on threc y:article in.berr-ctions¿ '1lhey

(Cfrofr arul Uhfenbectrc) obtainecl an expression for iL trlhich

involverl- 'che 4y nnnrics of i.,hrcc bod,¡r Ìntcractions ailc-L

r'rhich thcy rircre un¿iblc Uc cxplicÍtly eveluate. Ïn prln-

ciple the higþcr -berins are lcncwn as inte¡1ra1s c-Lepencì.ing

on th.e cþnamics of four arrl more particleso

'ilhere is nn altennativc method. of c-leriving transi.rc.,rt

coefflej ents basecl on th<l -bincì correction o.rl'.lroach of

g"""r,,(28) ancl t(u]¡o(zl) ,*t"n ¿rvoicls using a iienerelizecl

Boltznlann egua-t,ion. lllhese methocÌs have ]¡ecn shor-rn 1t5z

Kar,vastci anc-[ otrr'rcnheim(lo), .Ernst, Dorfman ancl Cohe n3l¡

20.1,2'

and. o'chers to give the s¿'.tne clensity cxllansi on (Z) f or !ì^ ^tJ.t ì. ti

transpont coefficients.., À rliffcrent approílch to thc'tirne

3z)correlation pro'blen is; iì.ue -ûo Zvtt:nzig ancl b o'c]:..

ncthocls of olt1;aining trans;1tont cocfficien-bs fron tire 'i;ir.ne

c orrclat ion lìrnc tlon are i[ j.scr;.s secl by I')rnst, Tlaines anc]

Do'fr,ran (ll)

.

It vra¡: expectecl et first ttra'b the inte¿rals in Ûhc

clensit5r 61lra¡sion (Z) involvin5ç the cþnamics of three,

fonr nnd- more par bicle¡i 1'Iould- ltc found- to llc conver¡¡crLt

when rreolrlc Jearnt -i;o cvalu.ai,e trerL. Llolvever i'c r/es

founcl that sL-Lch was not 't-,hre c.rrte, ancl in fact the in'¿e-

¡5rall: clive rije, !--c¡ estaltli,sh the cx1¡i bence of the -berrrt

onc tries to slroi, '¡hat f'or any rcasonabl.e ,(t ) ,¡he llhrase

volume corresponrlin¡5 to -bhc cll,'n"-nltool- events -lhat contri-

ltute to thc inbeijral is finite. This has ìlecn verifiecl

f or the "brittl-e o¡ llision terri in 'chree climensions by

Dorfman ancL cohen(i1'21), ono ancl shj-zume ßS), Green and'

.pi-ccirel nQ') on¿ lleinstock(¡6). on the other h¡'ncL it

1/.is founcl 'úha't -i;he -i)hase volume assoeieied. rvith -bhe

21 ,112

relevtnt cLlznalliCal even'cs involVing four ¡rni-1 more

particles is no b finibe (Dorfman ¿rncl Cohe n r ':¡cinst oclc

3l), Goltrrnan anil rrricn',"^(,Bll sor,g"*o(39) has clonc

calculations f or e two c.Lincnsional gc-s r,vhich shoür bhe

cLiveri;ence ¿lr ises in thc triltle co llisi on ten m (ttre Choh-

Uhle nl¡eck tenm). It i,,ras pointecì" out pnevioi-rs}5r þ¡r

Dorfman ¿r-ncl Col-lerl(ltl) that such a d.iveri,ence ,,¿-oulil occr-lr

in t,,i/o d_irrension,._: slnce its origÍn is irhysi cally bhe

,sî.me ¿L s thcr rìiv er ?;enc e all irearin¡1 in thr ee cline ns ioni: vrii th

'bire fo ur ¡:¡,rr ti c1e co 11it: i ons. In the tvo c.li nens ir:na-]-

cnse the phase s,oitce ar¡soci¡.,'bcd, r,yi'¡l: cer'úain iiequenceÍl

of ll inar ¡' co 11i,s i ons, an o¡¡l tlt c bhr ce ;n rt j c1e s gr o!',/fl

logari'cirmically in thc'ciinc ltctvi¡een the first and last

collision. The limj-t as ti-ie time ijoes to inf ini'c¡¡ has to

hc e valuatecl so thc elq)r ession cliver íles 1o3'ar.ithmically.

Since these cliver getlcc clif ficult,ies exis'c it i s 'ìry

no means eviclent that any of 'l;he previous resuJts in the

-thcor.y of tranrsl-lort coef ficicn bs f or a cli Iui,i: or rnoc'l,e r-

ai;ely cÌense gas have a sounc-L basis ancl tÌlc v¡hole 'bheorSt

is uncler rr:vierrv.

(zt )

22.1.2

3t)ïn thein paper llrn,st, i'lniner ancl Dorfman sih oll

'uhat both nethocls of obtaining the coefficients of sirear

vleooef[rardsc]f cT-ffusion from time corre la'c j-on f\rnctions

give the seme exr)ressions for thc first ttvo tenns in the

c-lensit,' cxl¡ansion - bire Bof tznann -bern anÖ thc Choh-

llhlenl¡eck tcrm, I-Ior'¡cver ''che5r point out that the cliver-

gent tænr.¡ beyoncl thc f irst ttre, t¡¡hen tr)roperly resummed,

might moc-lify thcse f irst tr¡'¡o terms, Thcy stress t.lrnt

uirtil it is knov'¡n that such ¡. moclif ication cloes no'ù

occur, the clerivation of Boltzrrrannf ¡ equ:l't ion cannot be

tru15' ri¡1orot1,s¿ 1tt the ¡rre sent time it i r¡ usual- to asffLïrìe

-birat fon gases with short, range reÐulcive ini;ermolecu.lar

i:otentiaÌs, 'bhe valuer: of thc first '6'¡,10 termÉi are un-

moclificù ìry the cliver¡;ent terms of higher orclers,

.ll.btenp',,s h¡-ve 'been macle bo clotcr"nûre tænel;eyoncl -bhe

f irst tlo in the d-ensit¡z ex-?rnsion by Kalvaso-lci nnd

(tro) (11 )0p'penheim arrcl lntcr by other authcrs fn thi,s

appronch thc frce notion beti;,¡e en binary collisions of 'bìle

four parLlcles ir.; consiclererl to irc clampccl by llinarSr

23"1,.2

col-li sions ,;vith all othen rnolcculcsi of ihe [F-r'i¡ Tllis

leacls'Lo a resurnmatiou of cCrtain ctiVergent ol]erators

anil 'bhc result fon tlre i;trincl term in ,'he densi'c¡¡ expan-

sion is then Lrroportionaf i:o d 1og c1 f on self cliffusiont

ancl d.2 1og c1 f or shear viscosity.

!'/e ¡.re only concernecl- iir this thesis '¡uj. bh a cliltL'cc

gas of ri¡1ir1 spheres ancl shal1 malte ihe usua] assrlnil'ci on

that ihe f irst tæm in t-,he Censity expan,sj-on of the

transirort coefficients cnn 'ltc obtained- from tlie Bol-bzinann

equation, -T{igheryberms in the clensity expansion are not

consiclered; ra-bh.er we generaJize the linear trnns;ort

eqr_r.ations lt5' $ohriniî the BoltzmaYffr cqr-ration r-rp 'r,o tite

tþirc-l al?llro;cimation in'bire Chantna.n-llnslç-q)3 mc-bhod of

successive api)roximl-bi on.

2hr113

1 .3 Summ,any of Present Work

It r¡¡as poinied. out ea::lier that the lloltzmann

eguation yielcls exact results f or the tnans',;ont coef-

fieients ancl. the corresponcling distriltuti on function,

(fn orn rryhich they can be clirectly calculated), when the

gas consists of Î'fax¡lel-l-ian nolecules. tr'or other molecu-

lar moclels 1t lyas long consid.ered neeessary to use i;tre

expans ion in Sonine polynomials as clevelopecl by Burnett

(and d.escr lbed. in Chapman and. Co*t ir.g(B) , Chapters 7 - 1o),

to d.etæmine the transport coef ficients, Althou11h the

results of thi,s tLreory give scries f on the transport

coef ficients li¡hich ane rapiclly ,summabl-e, itr is neverthc-

less of interest to cliscoven an exact solution of

Boltzmannf s egua+"ion in the successive ap-,lroxirnat ion

schemer for an lntemolecr.rlar potentj-aL other than the

fifth poïr/er L.a.w. Tcleall¡r this *roul-d- be a potential of

the Lennarcl-Jones type, but as this 1s too com:tlicated-

f on an exact treatment a rigid- spherleal moclel has been

ad.opted. in thls thesis. In faet the rigld si_rherical

25,1.3

mod.el does offen a good. approxination to the s'r,ron5;ly

repulsive interaction bctli¡cen molecules ¿lt short d.is-

tances, ancl the attr"aetivc f orces are certainll¡ much less

inportant if not negli,gible, Iìven at liquicl clensíties

ri5¡id spheres have been found to simulatc '¡ve11 the

behaviour of 1,hre rad.ial d"istribution fUnction of equil1-

briu:n of ætual theorlr.('h'z).

Once the d-istribution functlon is known exactly

simple integration gives th.e pressure tensor ancl heat

fl-rlx vector so -bhat the trnnsport coefficients are also

o'bta ineÔ exac'cly . 'Ihe f irst cxact calculat i on of 'bJr e

transport eoefficients of a gas of r igirl s'phe res (a1;arrt

from the intricrte ard special solution of f i.l¿rr"¡(¿13)

f or self-cl1ffusion, r'vh.ich vuas cLeveloped in 1915 prion

to the complete d-evelopment of the Chapman-Iìnsl<og methocl),

r,.¡as made lry Cott.r(4t+) u¡ho obtained the coefficient of

ther-ma] conduction aften a 1on6¡ and teclious calculation.

FTe cl.1d- this by rec-lr-rcing the Boltzmann integral equation

in the case of heat concluctlon to an ondinary clifferential

equat j-on whieh he solvecl numerically,

26.1,3

In Chapters two, thnee ancl four of this thesis

cotterrs spcciallzecl- tneatment is simplifieil and. ¡,;cnera-

lizecl to make possi'ole üre c-leterrnination of -t he clistribu-

tion functlon and- 1ts moments for eveny approximation to

the Bol-tzmann equation, thus extenclin6; the transport

cquations to incluc-Le terms non linear in the grac-Licnts of

.l-rTanc-Lc¡--:o

Flrstly in Chapter tlvo the Roltzmann eEtatíon for

riticl sphe res i s set up f on each appnoxirnation¡ anc-L th is

is reclucecl to a sct of ord.inary crifferential equatlons

r¡r¡hose orcÌen c-lepencì-s on t]ie orclet n of the irreclucibrc

tensor in the clevelolrment of the c-ris'Lribu-bion function

in tens of these tensors. This rec.luction is achievecl ]:y

makinÍl use of centain auxiliary f\rnctlons, ancl the

co11ision clynanics of two rigicl spheres. ,Ihe ¡¡eneral

tneatment io specia l_izeci f or orcl,ers rL = Or j ,Zr3 ancl 11,

the values which arlse when clealin¡j with -the thj-rc1 approxi-

mation¿

271.

In Chaptæ thrree the second approximatlon to the

a

3

Boltznann equat ion, (tfre fi rst nc)n trivial ap proximation) r

is consiôered. This requires the rlifferential cguations

of Cha¡rten tvuo for n = 1 anc-[ n = 2 since irreclucible

tensors of first ancL seconc-1 orclen apilear in tkre solution

in this apilroximation, The clifferential equations cre

solveC nurûæica1Iy to give the rlistribution functiont

ancl the values of thre r¡e l-ocity c.lepenclenee of this f inst

correction to the local cguili"brium state are ':resentecl

¡;raphically. From the solution the exact values of the

c oeff ici ents of thermal concluct ivity ancì, shear viscosity

ar c ealculated by numerical integ;rat ion. These exact

nesults are comlrarecl r¡vith other cal-culations l:oth approxi-

mate ancl exact.

The -uhircl altpnoximation to the c'[istribu'üi on functlon

ls cllscussed in Chapter four for all tqms except those

rri¡hlch involve pro:lucts or ,squares of the seconcl apl?roxi-

nation to the äistnibution function. Once a¡;ain the

corr ecti ons to the cli stnibuti on furc tion are pre E:entecl

28,1.3

Sraphicatfy, the resul-ts lteing obtai ned from the numtr ical

so luti on of thre âppropn iate diff enential eqilat i ons ' Tt

is pointecl out 1àat the zubsid.iany conclitlons nelateiì. to

the existence of colllsion invariants must be satisf iccl

1n this ca,se, Divergent solutlons at the Õr'i81n which

reSult vvhen theSe are not enforcecl ane removecl l¡hen the

conr'ì-itions are propenly accountecl for. Ari aItænative

methoal of ca-lculation of tlre pressure tenson and heat flIrix

veetor is a1-so ¡-iiven in aclctition to the usual methocl of

cl-irect integnat ion of the r1i stribution f\:rc t ion. Tloth

exact ealculations are seen to agree exce1Ient1y, anil

they are comparecl ',vith ttæ altproxinate calculations in

this apl?r¡ oxi ma ti on.

After our exact calculation of the exact solutions

for the seconcl appnoximations to the Boltzmann eguation

it was poin-bed- out by Cor¡åiry: (privatewas eonple tecl,

communicati on) that Pekerl.(l+5) ,,rr¿ Pekeris anr-L ;\lterman

(ue ¡ hacl caleulated. the coefflcients of self cliff\rsiont

Fhear viscosity ancl thermal concluctíon exactly' l.lhey

29.1t3

appear to have been unaware of Cotterts earlien workt

and. their method is nathen nore particarlar than the

genenal mebhod. cleveloperl- hene in Chapter two and. applied-

in Chapter thnee i although it ]ead.s to the same results

fon the coefficients of thermal conduction and shear

viscosity. The theory here goes beyoncl that of Cotter¡

and- Pekeris ard Atterman 1n developlng d-ifferential

equations to d.eal with terms in the d-istributi on function

clepend.ent ôTr zero) ürird and. founth ond.er inreclucible

tensors in add.i.tion to the tensons of f irst and second.

ord.er, Tn faet the general method- of obtaining the

differential equatlon for arbitrary orcler n is g'iven'

tr'r:rther, the ttrird. âpproximation to the Boltzmann equa-

tion is solved. exaetly, anc-L thc third. approximation to

the Bressure tensor ancl heat flLux vector are also calcu-

lated exactly.

The ealculations in tris thesis ane mad.e f or a

sirnple gas and we clo not consid.er mixtures' After being

mad.e aware of Pekenist work on the coef fieient of sel-f

50.1.3

d.lffusion we checked. his wonk in our scheme ancl obtained-

the same value for the coeffiei-ent. However we wll-1

not present this here as the theory of eeLf difftrs lon

comes as a specializatlon of a Bart of the theory of

ni xtu:r es.

31 a

2.

CIÌAIì'I ER 2

ticTlOIT oF rr TO

DIf,T¡ll,REI\.IT T, TTOR1rrri I¡OR /i ST¡{,?L.E GAS OF R IGID gPlÏr-ìiì -ri,s

In this chaptcr the inteilral equation l,'¡hich occLlns

in each approxirnation to Boltzmannrs equationi in the

Successive apllroximation Scheme of normaf solutions' iS

re(Ìuced to a Set of cliff crential eqrtìt ions wlr.en tire

moleculår moclel- uncler consid-epatic¡n is th.a-b of a sirnple

gas of ri gicl spheres. Each d.ifferential- equat ion corres-

poncls 'bo a ptrticular orcler in the cLc¡¡elopnent in spheri-

cal harmonics of the c[istribution ftnctionr and- the

nec-luction is qur- ite generale a rìethocl bein¡; tjiven whereby

the cli ff erential equ at ion correslroncli n[f t o arl¡i brary

orcler can l¡e obtai ned. Certain auxiliary f\rnctions, end

the collision dynatnics of tr,^ro rigiil spheres a.re used in

obtaining the òifferential equationsr ancl since the ci-istri-

ltr-i.tion func-bion is cl-efinccL throu¡¡h thc auxil-i¡.r5r fi-rnctj-ons

so lut ion of the cli ff crent inl- e qr.rat i ons en¿tr le s i .i, to 'b e

cLet erntineC.. ItÍrs-r, ri[e begin by Summari zing brlef 1¡r the

col1is ion clynamics alii?r oi:riate to th i s i¡rob1em, ancl ''ch en

3z2.1

from the theony of normal solutions the integral equation

rrhieh is to be reducecl- to cliff erential forrL is ur i'cten

clor¡m. tr'ronr tÀis point we proceed- to perform the inte-

gra" ti orrs invo lvec1 in th e col-Ji si on integ rale arrl ihe

tTcneral preecription l-d:rich lve prod-uce for olttainin,g the

cl-j-ff ercì1tlal- equa tion corre sponcli np, to th c original

inte gra 1 equat ion 1e speciatized so f ¿rr as is neecìccl to

facilitate the l,,¡ork of Chapters J anc-L 4¡ The red-uction

is of ne cessity rrra the ma-b ical, ltut 'l.¡e shall at various

points, attemp't to inrlicate vi¡hat ce rtain cal-cul-ations

are intenrlecl to clo, arrcl also u¡e shall sumlnarize some of

the impor tant re sur lts A s rì/c pr ocluce th etn.

2.1 Roltzmannr s lrlqu¡.tion for â Gas of Iìip:id. Spheres

The Boltznann eqr-rat ion, v'¡hich clescribes ihe non

eguili'brium. 'behaviour of tþe si n¡1]e panticle clis'¡ribution

function of n clilute grsr can be clerivecl on th.e llasis of

physical arguments aíi r¡les originally clonc by Rol-tzïlí1T\tr

hirnself ; or it is possible to olttai n it uncler cert¿r.in

a

simplifying assul¡Iltions from Liouvillef s equation r,vhieh

33.2.1

d.escribes the N particle system. Thesc of necessity

restrict the gas Cescribecl by the eguat ion to lle <l-ilute

as inctic¡rtecL in 1,2. iil/e shall not tr)ncern ourselves

with a clerivation of this equation as it is r,o lle found

in arry nurirl¡er of plnces, ins't,eacl we just r¡¡r j-te r1own -Ûhe

forrn of the cquation, which we sh¿tll- use:

FOT

r- ãs'#*='S* du cllr de pb [r(go) r(yo)

- r(s) r(s) Ì

tI

,l

a (r )

Hcre m is the mass of the mol-ecules of thc gasr anc-[ ì;he

velocity d.istril¡uti on f unct ion, f , is a fl-r nction of time

tt posì.tion L ancl mol-ccular vc l-ocity y o-il a molccule,

and. -Il is the external force actln¿ on the mol-ecules of

tire gâs (ancl is consi clere cì inile;rcnclent of g). T]re inte-

grTl l- term of (t ) invol-ves the d¡r namics of -bi nary collis ion

l¡e'tvleen mol-ecules of the IFI S: p iS th e re lative slreed. of

two molecules which collide '¡ith an impact :parameter b¡

the precollision velocj-ties 'being g aTxl v anc-''ì- '¡he post

eoll-isi on vel-ocities u^ arr] v^..*o -O

34.^)éo I

I'r, is corruron to consicl-er collisions in terrns of the

rel at ive ve locity of the partic les̡ef orc arrcl- af ter

collision, d.efinecl r¡est,lective ly by

.g=.!-Lr Q)

arrd

€o=9o-Yo t 3)

ancl the angle s in (t ) 1s that lvhích the plane in r,.,¡hich

the scattering occurs rnakes with a fixed. itlane through

e,. (fne fact that p ancl ¿o lie in the same plane is a

consequence of the assumccl spherical syrnmetry of the

molecules r¡,¡hich nequires the force iretween them to be

cÌirectec-l lr1r.lng thc l1ne joining -i:heir centree.)

\Ve r¡'¡ilI nov¡ use the result¡s of t,;¡¡o particle en-

corrnters for a gencral central f orce, and. then for r1gid.

spheree, to,.'ririte clov',¡n the form of Boltzmannts cqulLtion

(t ) f or a tjas of rigid. s!:rhç¡s",

ral Res l-ts of T,,l o tic c

The ¿rniil-e ¡ tLrroug-h r,'¡hich the relatj-ve veloci-by 1s

clcfleeted in an encounte n cì-epencls in gerere"l upon both Pc

the magnitucle of the initial relative vel-ocityr and. bt

the iml-rnct iÐ.ramctcr.

35.2.1

In general vrre can v'n ite

p'bclbd.e = C(Xrp)dg (¡r )

where C is a function v¿hose form clepenÖs on the larv of

interactlon betvreen the moleeules, anrl _q is a unit vector

in ltre clirection of the line Joining thc centres of the

molecules at their closest point cÌuring the encounter,

thc apse 1ine.o- Dseo

lp - pol

From the conÈq,ervaiion of energy ancl momentum in a

collision lyc have

p-po ancl go*Io=u+Ir (¡)

and. thc rela.tionshipsl¡ef orc ancl ¡rf ter collision arc [jiven

'l-r r t

a , (g. * €o) = o e

Bo - ! = (p'g)o = (co'q)q, :'

v

Eo = E Z(.P,'q)q

a

-1'I

and-.Þ = ,8.^ + 2(p^'a)+ .& €o .-o (6)

62.1

ticle Encounter For Ri c1 Srrhere s-Á- --._.=..-

Diergramnatically .1 collision be t¡i¡een tv,ro rigicl

spheres is sho'irrr in l¡ig" 1.

fn a gas of rigicl sirheres thc line joinÍ.ng the

eentre s of t'¡r¡o collid.1ng nolecules at i-mpec'b is the apse

l-ine.

Thus tl-e iinpact irararneter is given by

b = o- sin t/ t Q)

,r,¡here o- is the ctianeter of 'r,he molcules; ând- X is ind-e-

pend-ent of p the initial relative velocity.

Then

3 a

pìrdlrcle = 12 p sin / cos /clpde I (B)

Since a make$ an an¡i;le þ vrith !r ancl the planc through

q ancl p makes an angle e wiih a fixed. plane throi-rgh p,

p and. e âre the polar angles specifying bhe d.inection of

år and so

clg = sin þdrlde . (e)

Thus

pì:clbde = Êp.qdg

the special form of (4) for rigid. sphere gases.o

(ro)

372.1

a

t"

e

f

Ffû' I

38.2.1-2

The f orce be bvueen th c rlgicl sphe res is repulsivc so

that for a givcn clirection of pt the limits on q, are

given by the relationship

€'3)0.

Thus the Boltzmânn equntion fon a gas of ri¡;icl

spheres þcglecting the external force) is

(r)2.2 ltzmannr s Equabion f or a

Rigicl Sphere Gas

the methocL of successive appnoximation which L,ìnskog

introc-luced to o1¡tajna solu-ui on of Bol-tzmannr s cql-retion is

¡;iven in Chapten 7 of Cirapman and. Co¡irfl:-n,'(Q) ,o" a simple

gas. T'here it ls shovr¡n that the equation is solvecì. by

,;'m itin.gI=I +I +I (1)(r ) (z(o)

= t(o) [' +O +O

)*(2)(r ) + .... l , (2)

3)

where the f irst apirroximation f is given by( o )

,(o) = rr (#f" exp t- crv'] ,

39.2.2

(tt)with cl - d(g, t), the nurnben l1ensity at (I, t),

T = T (g, t) , the ten,rera ture at (I, t ) ,

c^ = _c_^(rrt), the mcen velocity at (Irt) ,-O *o'-'

mq=ffi,

k = Boltzmannfs constant

andI = I - g_o the neculiar velocity.

This is the loca1 llïaxvrrellian d-istnibution.

(5)

(6)

(7)

(B)

(g)

r.¡ie shall not iliecuss the vra¡r 1ìnsle og subclivid-ed. the

Boltzmann equat ion ancl usecl the conservat ion equations to

make bhe approxination sclreme r¡rorkr as this is cÌone

fully in Chairman and, Co,,,¡f ing" Using 'Ghe theory given

there, ancL the f orm of the Boltzmann equation (t, (tt )),

lve fincl that the integral equation lvhich hos 'r;o be

solved. in the N + 1th opproxinnation for a gas of nlílÍc-l

spher es i s,

(r'r)d2r(o 12

+Õ (u)

) [[n.ni'å t(o) (')ro(s) to(tt) 1u)

r(w) (yo) lai au( )

f\ a

(w)uro

(10)

40.2.2

rn (to) 4. is a known scalnr, clcpend.ent on thc previousry

cleterminecl N appnoximatlons to the clistrilrutÍon function

given bv f1o¡, t(t), ... r(N-t ). rtryond,che seconcì-

ar-rproximation f contains tens involving the inte¡irals(N).of prnclucts of the lower or<ler O These terms are

trutsicle tire scope of ttrls thesls arrl they shall be con-

s id.e red in f uture wo rl<.

Not¡¡ -4. can l¡e r¡¡ritten as the produet of three types

of t ems. Firs'b1y th ere ar'.e yl (v) , knor¡¡n scalar f unctions

of q\12. Thcse multiply tensors of v¿rrious orrr.ers in [r

the orcl-er being d.etæminecl T:y the apllroximation consic-Lerecl.

rn this tkresis u/e consic-[er the seconcl a¡:lrroxima.bion (N = 1),

which involves tems in scalan \I, y, and. V V, and. als<¡

the third- allltroximation (ttt = 2), which involves tæms in

scalar S, Y, I Y, I V- I, ancl y y, U g. Finatly there are

tenms r¡vhich are of thc oncler lI in the spatíar clcrivatives

of d., T ancl ,qo, Thus the seeoncl apnroximetion invc_rlves

only terms linear" in thre ¡lrad icnts, the thirc-l al¡?rox1-

mation has terns vrith irrod,ucts of the gracL ients or

41 .2t2

seeoncl d-erivatives, and so on. These terms Lìre tensors

Zn(Ò-t 1rgo) of apÐroprliate orclen n; ancl thc scalar

proclucts which are formed. l¡etv,/een these tensors ancÌ the

tensors in 1l give a scalar quantity for ^.

The ten.sons in \I are wnitten in term¡; of the ,rth

orcler tensors cn(Y), wirere

G,, = (--' )"v;Ë#+ (t)for n ).o, In (tt ) the opcrator ._i",

òvnis o- tensor d-ifferen-

thtial olrernt or of tlre n orcler be ing simpl¡¡ òãî aIrP I lcd.

n times, so that for example ,G, = Nor¡¿

the tensors G-r, transform accorcling to irreclucibfe repre-

sentat ions of the three clime nr:i onal rotation groulle so that

 ,,,'ritten in tenms of these G' is essentially cleveloitecl in

spher ical- harmorrics.

Thus the expression f or Ä f or a particulan aprtroxi-

mation can be i¡lritten

(tz)

yeÒròlæ2 òV i¿r¡

), Ivi(v)Gn(u) î z,',(d,r,eo) ,¡,

1n

l+2,?,2

u/here i is summecl over all the independ-ent terns of a

particular n, (1.e. term,s involving .Jifferent gracLients

of d-, T, ancl go)r and- n is su¡nmed. ,over the val-ues v'¡hicli

oeeura in L of the irarticular apL-ìroximation considened..

The sSznrbof l inclieates n scalar proclucts have been

formed- between G.r. ancl Zn lthe expressions f or .4. are

given in Chapman and Cowlin¡1 for i\T - 1 ancl ll = 2, ancÌ at

the appropriate point v'¡e wil-l- use their expressic)rrso For

N - 1t Gr ancl G2 are involvecl; and f or i\T = 2 lve requ.ire

G'o, Gl, Gn Gr ancl Go. These f irst five G. are dis=

eussecl in Appenclix 1 ,

To olrtain a solution o:f (tO) the d.ir:trij¡ution

function is similarly c-levelopecl in spherieal harnonics

thus,

o(N) (y) = t' t D. d. (n)

1v¡err(u) î zn (et,r,9o) . U3)L, L, 1'r

where the lÍmits to the sulls are the same a,s in (lZ).

D is a multiple of ot, d., Í, anc-[ T requirecl to m¡r].ce O1

lnz .l = 1.- n1 '

1n

cl j-mens ionless,

(n)

43 c

2.2

U¡,ron su'bstitu-bing (13) into (to) ancl separating out

thc inclepend.ent sphæical harmonicsr we arrive at the

foll-owin¡; equation for each of the i incle¡renc-lent terms of

a particular orcler n:

)( å/'u r(u)nF

T ø2exp(- c,úV" )

+ ;i"1.ï,li':'n]':';,1ï,,j]"",n" ü,

ør(") {vo)r,.(to' !) Jda ¿u , (11+)

Ngwhere Y = m r ancl P,' is the T.,egenclre polynomial of order

IÌ¡

In clcriving thls equation u¡e have usecl the lrroperties

(a,tr(tt) (16)). rt is the íntegral ec¡:ation whlch now

has to j:e solved fon each i ancl n to obtain ør("){o),

the cÌimensi onless scalar f\-rnctlon u¡hich

in the expansÍon (1Ð for ,(*).

is th e unkncnffn

Bef one cJi scussing the solution of the equation we

point out that there are certaj.n auxlliary conclitions

placecì- on the O(u)

whie.h ensune conrect clef inition of

)lll,2.2

c1t arrl T in each al:rproximation for f o The y f ol-lovu frolnc:o

the icl-entificat ion of the loca1 lfaxlvellian state as the

first approximat ion f (o).

In the f inst approximat ion the id.entif ication

d-

is m,rde. But the def inition of cl requires

I

ï

t (o) .s

so that

d-= fdr,

(r-r( o )¿.y = onI )

C1earIy, then, the co nrlition

Ï t ro)*(t) (l¿)ay- = o , (1 5)

for al-l- I'Tr ensures correct clefini-tion of cL at eecl-r sta¡¡e

of the approximation to f.

In a similar manner by identifying -9o ancl T with thc

f ir st âppr oximat i- cn, rv e have

./ ,t - t(o))v- e;'- = o,

ancl

[ ,t - t(o))v'a.s = o,

452.2

Thus the conclitiors

It..r(n)(y)yds= o (16)J

-(o)-

ancl

"[ t,o¡o(u)(y)v'ds=0, (17)

for al-l Nr ensure correct d-efinition o1'go ancl'lI in each

appr oximat i on.

Now the rnlution of the integral equation 1tt-r) has

previously r:een made by expressing ø.(") as an infinite

series of Sonine pol-ynomialse f irst usecl by Rurnett Q)

and- described. in Chaprnan and Cnrti^g(B). Thene it is

shown that the results of this method give expressions

for the trans rort coefflcients r¡rhich are obtai.ned. a-s

rapi,ìly conveÍ'gent series. In this thesis ho'¡levcr the

inte¡1ra1 equation (14) is reduced- to d lfferential f orm

by the introcluct ion of centain auxiliary furnc tions. The

d.ifferential- equation ii'r solved- numerically ancl so an

exact numerical solution f on ø. (") is obtainecl through

the relationshi,¡ of ør(n) tc.r the auxiliar¡r fu¡stions.

a

Dineet numæical integration of the clistrihution function

then gives an exact expression fon the pressure tensor

ancl- the læ¿it flux vector up to thrc al:)Lrroximation con-

siclered.. The integra tions are

¿+6.

ZCé

(18)

(1 e)

(zt )

D-

¡11 fY!c1vIand.

o = 4 [r v" v c]-v .-¿t-I

From the expression (¡) f or. the f irst apl?noximation

t(o)r the f irst approximation to the pressr-Ìre tensor is

$iven by

r¡rnd.o(o) = +AÁ )¡o13/2 ¡ *(#) .1.

exu (- av2 )v4av {

= d.kT î. (zo)

Ilere d lu the unit tens:or. The f irst apllroximation to

the he at f 1ux vec tor is si milarly iiven by

1, .V2 V clv(o/)_e t

2.1(o

:

=Oo

In closing this section concerning the normal-

solution of Ensko¿¡, we must point out that the equation

(tO) has a sol-ution if ancl only if certain subsicliary

47,2.'2

cond itions associatecl with the col-lision inv¿riants are

enforcecl. These are talcen account of in the v,tay thc

diff enential tøm of Boltzrnannt s equat ion is sul¡cLivicled.

for the successive apj?roxir';lations to the clistribution

f\-rnction. Á,s \¡ie said- ltefore \¡re ere not conccrned. wlth

Ciscussion of üre subclivision, but since tJrese con-

clltions are invokeÖ latcr in Chapter 4r u,re give them

hene for completenesso

tr'or ,þ(i.

m V anc-L * v""¿

in tur.n, the suJrsidlary)1 ,

cord- iti ons ar e(i)

a .¿1.d-v - 0 . (zz)

Any integral tenms in .4. satisfy (ZZ) (see Chapnan

anrì. Cowlins (1 .5\.r()-v), (¡))) so that ignoring such terme

',vhen v¡e deal ,,vith the thir.d. aBl?roximation rn¡i11 not affect

the cond.it ions (ZZ) .

2 . 3 Bs{.tc*!¿-gp-_-oå _b-þ9_ C.-o-}J-*lçjo¡¡ ¿I1!-e¿-qg}

In this section v¡e r,vi11 sirnl:lify tJre collision

integral

l¡8.2.3

(z)

tt ^.p.a exu (- cxu")tø(")(u)pn(û'i)

i 'p'+>o

ø(^) (uo¡r:rr(ûo . i) iae .ru

sÞeeds Uo In 1:an'r,icu1ar,

,(n) 1v) = .p.-a exp (- .ou')t(n) 1u)r

which occurs in the equation (2, (tt-¡.) ) f or each aplroxi-

mation to the .iloltzaann eqr:ation, The eqr-ntion (2, (t1,,))

cÍ..,i'r. ll e r,¡ritten(") (") (,,t) (n) (")

T;II' (v J (v) + K J (v) - I( (v) (t )o o

(n) (") (")vrhcre J a nc1 1( involve þ as a fl.r.nc ti on of the

(")prec o1lision speecls tl ¿rncì V respc ctively; an.c-Ì Jo and

(") (")K involve ó as a fìrnction of the post collision

o

)

and. V .o

nII, ,-p 'a >O

I((") (v) = tt ^p.s exp (- cü')t(") 1v

I .tp. a >u)ae d-u t 3)

Jo

and.

(n)

(")

(v) = tt p's cxLr (- au")ø(")(uo¡e,r(ûo ü)¿q ¿,, e

'.1-g'g>o u r'L u

(lr )

tlo'/z

\9.2.3

(6)

K (v) = tt p'a "*p (- øu")ø(") (rr^)p,,(g^ ' ü)on .ru ,l./p'aîo- ' o/ Ïr'-"o(¡)

o

illhe superscript n in (t ) (5) clenotes 'Lhe orcLer of

the irreclucible tensor G in the clir:tribution ft)netionn

(Zt(13)). ilie v,¡ilI notï recleice each of these ;:;ix dimen-

sional Ín-begrals in turn to ârr intqral over one variable

b¡r using'chc cl¡¡nanics of thc collision ltetlreen rigicl

sphcnes given in Section 1 of -bhis chapter.

I¡irs.b,;re notice

2sin ty' cos {tdtþd,eï"

.9'3 clg = Pa>O

= *r(=t [" o¿-O

Also¡ fron (1 ,(z)) ¡.urct (zr(g))

.Ê=9--Y

=U--l ?

so thntp' = U12 + V2 2lIVx, (7)

50.2.3

l,'./her e

Then consj. cler ing ü(^) , and usiini:{ (6) (B), rïe have

VU

Iu"n (- .,u")t(n)qu)rr,(û . û)''p¿u

o(

(B)

P, (*) )

(g)

"(

n)

c0

= l+rr2 U2 exp (- oN')ó (u )å (x) (u" + v2t,lo

(n)ï',

Pn

1

2Wx)zclx d.U.

Nor¡¡fonU<V

I u-"v1ì

(#)' rro (x)

tri(r,(*)

nL-

P, (*) I

P" (x) ) +

(")

(") (#)'r"r

+ G)'(p,(*)1

-fFì

+1

õ

+ ....oo

1\-t_,

Ipm+1Pn_1 (") l

D=O

whene r.rc clcfinc P_t

2m+1

o,

Thus

a15213

(")= tvÍ2 fuu" ex' (- cru,,)r(") qul $ Ll""

[ Ë) .t

(r.,*. (,,) - P,,-1 (*)Ìo* au

D=o ,"*

(n) tu) E / 1r"t-)

[n

(x) [ro

(x)J

+ Llrr2 U2 exp (- øTl2) ç[,

oo

(")o

r/_,H=O

1

2m+1

But fnom ttre onthogonality of the Legenclne polynonlals,

"1 26/n-(*)P-(x)ax=i-BI}, (to).l-1 2n+1

oo nr¡4

), (#)"-'(n**. (*) Pm-1 {-))l o"

ïr1=O ;r^.,

=;Îu [r," .;h(#)*' h($) ,-.

r/e get 1f ' (-

/-., o"

Lno -

(r r )dto I)

) a

Sul¡stituting this in the expression for t(n) xïc

fi naIly

,(n) = htf |tu" exp (- cxu")r(t) 1u¡ =V- [-*L=- lÐ*'.ro 2:n+1 LZn+3 \' /

get

*,öl*

+ )l.rr2

J

to

[ *u,

exp (- o,rr,)r(") 1u¡ -u- [*L-Æ\"*2,/v 2n+l Lr"*¡\u/

# e")'l du

5?.2.3

(t z)

afrl 1n ¡nrtleulan f or rr = O , which we laten speeialize

(o) - rnr, fuu" exp (- cru")r(n) {ulv ! + + (#)'] *

¡ LylT2 U2 ex1l (- cru" ) Ø(") (u)u f1

["+ nG)"] ou

The expression (z) f or "(tt)

has novr been nectucecL to

(13)

Ig norrT re cltrcean integnal over the singl-e varia'lt1e lI,

tí(^) , (3), to a sj-mil-ar i.n,cegï.a]-.

I.ls ing ( 6 ) o.ncL ( 7 ) once a.51 ni n, Ì/i'e fi ncl-

)( n

t/o

[ ""n (- o.rJ") þ

[ fl"" exr)

(")

I1(")

I(

(")

(v ) øp¿u

oo 'l?:.= Ll-r2 þ (v) U:? cx¡l (- aU') å (u"+v2-2wx) clx clU,

1

Then ernployiryl (g), ancl (iI ) v,¡ith n - O, v,¡e recluce I((")

= Lrqt ó (v) (- ølI2 )v

i'+*G)'l uu]'

['+*Ë)"1 *( )

t.oo

Kn

+ U2 exLr (- øu2 )U (rLr)

53..2.3

This is the form v/e rr-,/ere seeking, ancl so rve pas$ to "

(o)o

ancL necluee this to an inte,gral over tl.

trlcking use of -bhe fact tÌrat bhe relative speed. io

u¡.rltereil in a eolli si on, (t , (¡) ); ancl the relations

L.nfån'rs=f þa,(15)

(re)

(17)

(1e)

wirere d in (16) is the cle1ta fìrnction; \4/e can v¡ni-be (ll)

and.

(")Jo

From (1,(5))

so that

Po' p2 = 4[g' (y, - uo) - 9o' (y - uo)J. (18)

Tl:ren using (ll), (tg) ancl the relation

lald(ax) = d(x)"

iive fi n<l-

54.

.ro(') = [[ üy. (v - uo)

go' (y - u-o)Ìø(") (uo)r",r{ûo.i) (- cri]2 )c1U d.ll ,--oexp

2.3

(zo)

(zt )

lde nolrl cLefine

(")atrcl- on sulr st itu-'cing 'cl-lis in ttre expression for Jo the

result is

iii -U-U .ú-O

i- ø[Uo + yl" ]d.,.,T dgo

Jo

d[g . (y - g.o) lø(") (uo)r,r(üo' i)t

The intq'ration ovtr I{{ can ì:e lrerformecL b}t consiclering

exp

qt

a

toV-U-. Pu.btlng:o

L=3e

d(zly - gol)

axes paralleI arrl perpenclicular

(y, - ri^ )2 -W'***--*Y-- ancl

lv - u I-"o'

vhere _!lr ie Lhe comtrronent of !T perpendicul¡.r "o I - lJ^r

lve c¿ì.n ¡:¿r ite

- (tt)?)=

o

(") (u )r, (d' o' n'*ot

r/extr) l- a(z + lJ '-L \ l

x exÐ i - ø[u, + uo"l'7az d[ dgo,

55'.Z',r3

llhe in'bep;ra1 lr¡ith r€spect 'bo g is

('r g) , ro (n)

bec omes

,(üo. f)

Ed. , ancl usi ng

(")Jo

(r¡) = 5t lv - u I=-o'

exp l d.u-o

(zt)

(zt)

a

rn f'unther evalr-rating the intc¿-ra1 in (zl) the srrbscript

oon U- is clropi:ccl r,,¡i i;trout loss of gener ali ty, ancl lre

d.ef ine(21+)

(zs)

so thatG2 = V2 + IJ2 2Wc (26)

ancl

G_V-U-

"=ü'üe

llhe variable -Q shoulcl not lte confusecl vrith p r'.ihích

is an expl-1cit ch'namical varia'ìrle whereas G me rel]r has the

same fonrn as - p now ihat rvc have d.ropped- the su'bseniBt

on l¡^. Q is intnoclucecÌ simply to all-ow the exr¡ression-o(ZS) to ire inte¡1ratecl, ancl tt-rls integratj.on is done

nathematicall5r lrrith no collision cì.ynamics entering in.

'Ihen clefining

and-

**_- -* e *:.*_ Pn

56.2.3

(c)acfor.u<V(28 )

(¡o)

(c)ae ,(Zt)

(n)\.I/itlr tire definltions (z¿r) (zl), Jo

l¡e come s

ro('){v) =+ I.*[t,ruSc)exp t- o(U . ë), Jr,.(c)ac du .

Ln(uev) = e+ulq-u-?) |

t exp[- cx(g õ), ]

r,rr(vru) = aë+uj-) t

"*rr[- o(_U . d)"------_G -'*-*.

nrr(c)dc for v ( u'

(2e)

I

1

1

\,ve fi ncl

lTorr fron (zB), (27) anct (26)

ro(t)1v¡ =ry fuu,,(t)çu¡ exp t- øu, lr,n(u,v)clu

. ry- [ *

urr(n) qu) exr) t- øu2 ]Lrr(vru)au ,ot Jv

exp (- øu" )r,rr(urv) = # [:;tf c{ n2 +v?-G2ñvñt rrJ-.¡.*"^t' l- G2 \, 2

_u2 P)'l n

wher,e'che varialtle of intepation is nov,r G, ancl Prr(e) is to

l¡e consiclerecl as a function of G, through (26).

57,2.3

Thus

r,rr(urv) =e-*¿:tr

2W exp

U+V

-V

Ft_

exp

qIr (.- G

v2 - 1J2\2-ì .* u'ã-"-)_l Pr,(c)oc. 3z)

S imi la¡, ly

But

so that

exp (- du" )Ln(v,u) = # [ Ëf;(-.v2 -u2

L.f+ \ I Pr.(c )ae , (sl)

t - . *=-u=)' = f - * ui--.- L'*)' + Z(yz- - u,),,

exp (- cru2 )Ln(v,u) = s-ru"-l%ffi-J:ll tî""n L- f; (-

* * Le'Y-1)'] n*,c)ac ,

orl

r,rr(vru) = lB*-ff Å:;" ""n f- f; t o * É-a*u=)"]r,,(")ae

Gtr)

T'hus intercL:,ange of the vaniables U and V transforms

t rr(urv) into r,rr(vru) so that on1¡r onc integralr sâv

Lrr(Ur\r), need.s bo l¡ e cal-cula-tecL .n/hen vte evaluate .fo(t) (t¡) '

Zi;(3o). ProceeiÌing no¡,.: to c-letermine (Sz), J.,r(Ur\r), ri,/e sec

tlrat sincc err(c) is a f\rric'bion of G throu¡ih (26)s tlre

integral will contain terms of the Wpe

, (m) = f

u*u exp [- g (* - vÎ :-^-rE\"1 e2*ac (ts)r - ./v-u L + \ G ) -J-

\/

for integen m such that O -( ilt.( r¡ Accordin¡¡I52 u¡e first

evaluatc ,(rn)r ad on changlng the variable of integra-

tion to X = ft-É *" ilct

(') fq"lLrI exll (.

V2 U2

)"12

t.2m

+(v, - u2)

G2m+2

"f1

)"v2-u2

Y2 clx.X

3e¡

Coml¡ination of (ZS) and 3e¡ in egual amounts gives

,(n)-+ ¡v+ue] r a/^ Il"-u2\21.. Jv_u :p l_ [ (G - *T-*"

) _]

2m+1

l c1G , 3t)

But

GG2 l+

2n

G2m+

2m+1

G2v2 2m+1

=G2m - u*=-rlllt * %+Y: * (%-o=)[1

a

59'.2.3

(:s )

(lro)

+ r.. +G

2-m

tv2-u2+ -?-"-*"=]I

j=o

Then on substitution of (lA) in (l'l) ancl changing thc

variabJe of integration fronr G to T<t ,,vhere

r=+lc.\v2 - u2\

LÏ/, 3g)

ìj're cled.rrc e

, (rn)2mI

J=0

*--?I}.¡--*(z^-i) I ¡l

-ßf=*ffìjn2i-2nr

iu exp (- oK")l-u

1<2m - þe^ l: +

Dm1 \- /2n\=þ l,\¡ ) '

dK

j=o

In (llo) G is to læ exl?ressecl in ternõ of K -r,hrou-gh (lg).

lTor¡r 2m

(;il-,)

(lrj )

(t+z)where

a

60.2.3

Cornpan in¡4 (¿r.o), (Lr1 ) and (LtZ), vvc see ttrat it is

cl-ear that 1ùre sum in (LlO) will be ne,Jlacecì'b:r a sl-tm

)',22' o* , x2L1u' - v2)*-¿ , ancì 'bhe coefficients **¿L=0

can be calculatecl for each €j-ven value of m ]¡Jr comp¿rrison

of the series, e.gr when m = 1

G2 + (u, -v:ì) *G.Î-t-J-,:.)-1 -t¡f{z (u" -v2) (tJ)

[see laten (6, (Z) ) J.

The relationship def ininii a* in general is

m

2m\-',)

L-t G2 j-Zn

J=o

I (')=2 tro

(u" - V")i

U

(u" - v2 )*- ¿

a

^2 j-2 LL'

(rltr-)

ïn

\-'/_,L=o

22L V2j

)

2La

m¿

j=o

Thus at thj-s stage r,'re have an exllression for I m( )t

m

exp (- oK" ) I 22Lm¿

*2 t (J, - V2 )nt- üdK,

¿=o (! 5)

= (u, - v2)nho(u) + (u, - \r:r)t*1n, çu¡ + ... + nr(u) ,

(ue ¡

where the R. are knor¡.,rr functions of u, for a [:;iven valueL

of m7 - a ncl Ïr.r ve -r,he fo nr.t

61 .2.3

n, (u) = z2L*1 o^, f u

exÞ (- o¿aa ¡N2 tar, (¿:7)

L' mL lo

lrTor+ we have in princillle a known expreÐsion for ,(rn)

for all rn, ancl so rrre reverb to the expres,sion for l,rr(Urv),

(lz), to cLetenmine lrow the ,(tn) contril¡ute to Ln.

Firstly 116 remark that lrr(c) is of the form

- n-2+llc'nnnln(c ) + .eo. + Z [",t], (48)

n

vuh ere

[v'1] = v is n is ocld,

1 if n is even. (lr-9)

I

nen(c )

TÌrcn exllressin¿i Prr(c) o" a f\rnction of G as re,çrired,

Y,/e ol¡ ta in.n

)u2

zn 2W

oï'e (n*t )znu\nrrr(e) = (- 1)nan

[n'" - (i)

z. *yz +bn

+ .... +

(u, + v2 )o

+ .... l(- t)n(u" + r.r,)n

62,2.3

( (u" + vt )Gz(n4)

z Ic, u2 V,,

[ãz tn-z )

(";')

+ .... (- t ¡t1u" + v2 r"-t]

,[n-1 ,nJ In-1 ,n] [n-1 ,n]U V n

In (¡o)

+ v')r ("-1 ) t )n (u"

( ro)

V" ),tI , ", ]

U [n-1 ,n] u^-1 fon n odd

ut fo' n even,

Thus the equation (SZ) fon r,rr(Urv)

r'r. (u'v )ext) ( ø= )

Un+1

1S

(rL

nr) l-it_

n )(

1

an1n+ ut*2

+ ... ( +

(il" + V")I (n-3 )( 'r )nr¡s"y"r," Lì

(n-z ) - ("-21

\

+ ... r ) n(u' n-2v") ï (o)l

,[n-1 ,nJ V [n-1 ,n] (1)n Ir (u, + v2)t ,

In-1 ,n]

(

z(o)

U

(o)T l (rt ¡

Ij

a

(')Nor¡ir the form of f is given in (Ue¡ for all n,

ancl s o r-,rr(urv) is elearly of th.e form

Ln(u,v) =ç¡p ( dt?JUn+1Un-r1

In + V2I)orr

(u) + .óoø v2^¡

,6j.2.3

(u)l ,1n nn

$z)v¡here 'bhe fìrnctions D are in princiille kno!!n,

¿n

ijumma r.i zi n¿f i;11e n,

n, !.i e fi ns t fi rrcl the coef f ici enis

Legenclre polynomial of that order.

to calcul-a-be t,rr(urv) for a given

a 'l¡ oc. z front'r,ltennnAlI the values of

(u) + v2D

mI for in-beger m slrch'bha'L O -< rn -< n are c,..Jcu1etccl

1A sLrov¡n in (lS) (l-r7). Thcn zubstitution in (lt ¡ $ives

T,n(UrV) wlriclr will'l¡e of thc form (52).

It har:pens that for a ?iven n > 0 .i-,hr.e ftrnc'Lions D¿n

nre zero for L > Lt vuhene Lt is some integer less -r,han

n. This w111 be illustr¿rtecl l¡_ten v,¡hen ure calcula'¡e

Ln(urv) for n - 4 r2r3r11.. 1116 r¡¡rifs Lrr,

( )

r,n(tIuv) eæ*-LqtJ-'-) rnUn+1Vn+1

Lrr + .c..a1n(u) l ton

$t)lvhere the sLm covcrs the nequired numlter of terms for a

par'cj-q-rlar rt. i\l:sorltin¡t the exponentÍaI factor" in-i;o the

Dr* bÍ "''rritlng

64.z.j

we get

1

þ*1-vf;i'Lr.(urv) = [arr(u) +v"nrr(u) +."",] , $S)

This ernl¡1es (;O¡ to be rrirritten

exp (- cxu2 ) ø(t) (u)

ut-1farr(u )

+ V2B (u) +...]du ae]rjp--G- c¿U2 ) d

(") (u)

"o(^){u) ="j;ff., t Ën [* ut-1

[.trr(v) + u2Brr(v) + .".]uull (re¡

$t)

rivhere the Arrr Bar, . c. nre in principle kno,¡,,rr ft't"nctions.

1ìq-ration (SS) has ttre form of Íìn integral over U

',vhich we have been seekin¡;,

we have nol¡r only to reclucc ro(t) , $); and. it is

easil.y shown either directly or f rom symne'br.y consid.er¡r-

ti ons that

(n) (")K J

oo

Thus ,ri€ can rirr ite (t ) as

"(n) 1v) = "(") qv) * rr(n) (v) 2J

o(v), (58)(n)

r,n¡here nor// \¡/e have reclucecl the Iì.ï1.Íj. to an intqnal over

tl.

65.213

At tliis point 1t is convenient to introcluce centain

futrctions '¡¡hÍeh facilitate op(rations udth some of these

exl?r es s i on-,!i o

Define

tr (v) = ti exp (- cü" )It'o

/*u cxp (- cru")rv

dU,(¡s)

(6o)

(61 )

(62)

\or/

(6rr )

(65 )

(66)

clu

+ )

"V,/(v) = IrOexp (- qI(z)d.I( ,

ancL

so 't hrì-ij

ancl

e(V) = exp (- crv") a

Then uilon integrntini, t<(V) by parts rne have

r.(v) = # f(zøv'+ i)./ + ve L

t<'(v) = fzorv,lt + e ] ,

tr" (v) = *

1re

rC'ol

,

k,-'(v a

The equation satisfiecl 'by k is

k"+zqk'V-l+cdr-O

66.2.3-l+

anrL the clef inition of k enables equation (f Lr) fon le (n)

be ,,,r itten

to

n(n) 1v) = r,.,,." ø(t) (v) qP (67)

2 .t,- n:_!Íe_1ie¡-þ;e_l _I]_g,,1cfrj_qn"Jlog*_S9¡1g1l-¡1

Tn this section rJre r¡rill outline the nethocl lvhich

eriables us bo f\-lrther recf,uce the inte¡;ral eqru:rtion

(¡r (¡g) ) , by removing the finnl- remaini.ng integra'c ion..

ilhis enables us t,o r,nn Íte clol,¡n a diff erential egua'cion

r¡rrhieh nusb'be sa'cisfiecL in the N +'1 th

"p1r*oxima'bion to

Boltzmannts equation, in place of 'r,he integral equation

(¡, (¡S)). As in -section 3 ,¡re vrill attempt to c1o bhe

eval-uation of the finnf integration fon ge.nenal n, ti'rc

order of the irredu-ciltle tensor Gn(I) involvecl in the

clevel. opmcnt of 'Ll:re clis-brilru-ti on f unct i on (Z- , (13) ) . TÌre

equation r,l¡herr Ír = O is horn¿even slightl;r specla.l- l:ecause

of the form of J(o), ancl rn¡e will clerive the equation for

n -. O separabely in Section 5. In Sectioru; 61 7, B, ancl

9 vr¡e will speciaLize the general me thod d.eve1ol:ed. in ihis

section for anbitrar5' n j. 1, for the particular values of

67.z.11_

n which we need. later in solvin¡¡ tlr.e seconcl (lt = 1), aricl

'i;hircl (lt = 2) ap¡)roxim¿rtion.s to Boltzmannts equa'uion.

The egr,r-a.tion rrrhich mus'r, l¡e solvecl is (¡r(¡S))

(r )9

u¡here J(tt), ,,(n) ancl .ro(t), redu.ced- in ç3r(z), (67),

(fe ¡¡ 'bo at most iniegrals over a siry,-l'le vaniable, are

collccted. bclo\¡rr¡

n)n+1U2

3

<":Ál¡':ut-'l

U2n+1 (v2 rJ2 ì)--:-- - -="--.\cl-ULzrr*l 2n-1 \2n+1

yiac!'l-l = k(v)v'ø(n) qv)

(z)

G)l4tr'

avn+1 Jo(") (")

e(u) øn-1

CD [nrr(u) + v"err(u) + ... ]c'LU\,rf U

+ f-* e{fu-!ÙO- [n,r(v) + u'arr(rr) + ;. . ; ]du/v IJ" ¡

(l+)

Now the id.ea is to clef ine certain ftrnctions tÈrich

ena'ble the integral eguation (t ) to be reclucecl to a

68.2.Lr

clifferential- e qu,at ion. llhis is achievec-L b;' ini;egnating

(z), ß) nnd (tr) b5r pnrts, anct differentiating v,¡ith

respect to v as we .'¡,1i11 nov'l c-lemonstn¿rte. T]re solution

of the differential egLration in one of bhe net/ vano.ibles

glves t(n) from its relationship with ttrat nev,lr varia'l¡le.

The firs'c auxiliary function vu-e cìef ine Ís*

e(rJ) ø(") rulí3 (") (v)

[, du $)un-1

In his expansion of the cListribution functíon ih

terms of l3onine polynornials¡ tsurne tt provecl 'í;hc co11-

vergence of the series fon certain molecular mod.els

incluc-LinFj thre one \,ve hnve usecl here; the ri¿¡1cl sphcre

(")mod el. I'Ie ex1'lancled in our notat ion, scì th¿rt ó

is a polynomial which near zeyo is pnoportional to vm

fon m )- h., Thus þ',-) ctef inect in (¡) 1s a r.¡et_I c.lof ined_

function over the rang.e O ..( V ,( * .

Then on integrating by parts, equation (Z) red.uceÊ

to2n+2 u2\¡" \fu

JO

p (") (u)n+1 2n-1

clU

69,2.4

+ ,f*ø1,¡

{ul fL

2W2n+1

(zn+1 ) (zn-t ¡c-tU

[uoq,,¡ rul

2n+1

2n+1 n-1

clU

(u ) cru.

n+1 U2n-1yzn+1 2n-1 )

tco

u13 ( n )(6)

0)

(g)

trquat ion (3) , wr^ i tt en in terms of fl 1ê'LÐ( n )

u2n-1r<(v)0¡r.;'(v)e(v) a

Again on inte5¡rating i:y ¡tartse \,i/e cân exnress (¿l) in

-berrns of fr6¡;

ovn+1 .ro(^) VD

(

l,ïu,"

, (u) # [n,.(u) + v"Br,(u) + ,.. ]du

¡ (u) [zuerr(v) + 4u"crr(v) + ,,.. ]clu ,

(B)

îr')t n

+

pr ovi c)- ecl Arr ( o ) = O, = O '.. etct

If oh15r f\,n y' O ancl, B'

ancl (B) have thc form

/ o in (B), tkien (6), (7)

ß (n) (u)wr (u) riuun+1 "(n)

en(o)

Ër,", (u)mr (u)au + v2

tlv

rt'ro

oo

( )

)t'tr'

up n(u)¿u, (i o)

70.2rU

n+1K(tr) u2n-1 , (11 )

4îr' e

and

ovn+1(")

Lvtr'

with the l'T, some knov¡n fl-rnctions.

If lve 1et

FV= /. t{*)(u)N'(u)au + v2

+ zRn(\r) [*uorrr¡(u)au ,rr .lv

Dol?,rr.(V) = #

J(u)rir, (u)¿u(")

[+ ^tvr -l

(tz)

(1 3)

D

d-ef ine the opera'bor Dotr, ancì write Donm f on the operator

appl-ied. to a f\rnction n times;

Fj-n " ruå.-1^Ñll= -oP \ l¿'nz )

then calclllatin¡4op

(r ¿r)

ror u(n) = "(t), ,,(.),

"o(^) in turn; $/e rje,ú ter¡rs

Ínvolving É ancl 1ts cle riva -uives uit to thind. c-Leriva-(")

i;ives, fon all but the.i;hird. tcrms in (10) fon n ) 0r

(")(wirich is the case vrhen clealir.rg with J terms) anc-[ the

thircL te¡m in (12); botÌr of ,ivh ich conta.in

t,@

rJ 13 (n) (u ) au.

71 -2.1+

To ernble such a tern to be hand-lecl, a fu::thcr

vnr labletiy

oo

7(rr) =1J

13 (") (u)au

is clefinecl,

Then (6), vrrri'uten in termo of Tçrr¡ r i s

)t

(1 5)

(16)

""-T#9=Frç,.¡(u)e#

Ër,", (u) (u2' - v'rrzn-')ou

U2n-1 Ue

2n-1

V

o

+

,UZn+l- Tãib-C2n-if v1r'¡ (v)

["]u,', (tr¡u2n-2 qïr" - v2 )du '

trc1u,e.tion (7) reduce s 'rlo

un+1.,,(n)æ,-

,"¡ (v)41.' € V

ÂLI

clvv2n-

)

[vy(") r6,{v) J, ?7)

ancl egr:ation (B) on lntegrating'b¡r ¡_rerts, becomes

LV

l) n(v) + ,..(u)

II

l- a.a I+ du

+ rrçrr¡(v) [znrr(v) + Lrv2crr(rr) + 6vanrr(v) + ... ]

ffi1")u[ecrr(v) + zhu'n,r(v) + .o' ]c1rJ,+

72.2,ll-

(18)

(1e)

1rovicl-ed.

f5l'h = oL u -Ju=o

Én'(u)j--*u* = O ... e-bc.U-O

ITovrr if only Ãn y' o arrl un / o in (s)' then (16)'

(tZ) and (ta) have -bì:Le forrn,

Ltçr'

un+t "(n) f n,", ru)rq"(u)o.u + v2.ilr,", (u):iru(u)au,

'Yr*l

(20)

(22)

Lt-n"

and-

oun+1 "

(n) "v-#t -- ="ê,lv)n'(v) + .kä,t")Nu(u)au

+ v2 kälrtltn(u)ou,

ruith tìte N. knov'¡n f\rnctions.I

Then calculating; 11 of (tlr) gives terms involving

?(rr) ancl it¡s clerivatives up to the f ourth d-enivative.

Thus the lntegral e gua'cion (t ) can l¡e transf ormecl to a

73.2.4

founth orcler d.ifferential equation in 7(rr), when only

f1 n

anrl

ard. B are not zeYo in the elcpression for JLJn

The eqlation is

(")

('') l_ f-vn+2tr,(r)roorr, L*u-t-J = oop [or+f (¡(n) * r<(')2 2J

o ) a

(zt)

If in acld.ition C / o then there ane two tærns ofn

(f g) which will st111 cont¿in an integral of y(rr) when Oon'

is calculatecì, They are

1j,,", ,u,# Ë3#12) uu

oo

?(r,) (u)ucrr(v)au.

Tþe first of these can be red.uceil to terrns involving

on15z ?(rr) ancl its derivatives by simply calculating

2 (")J

o (In the same \¡/aY the existence ofop l+æ

D3 l(")

furthen similan tønsr Dn, li .... in Jo , lrrould justn

necessitate ¿ù:L acld.it ional operat ion Don to be applied f'or

each f\rrther, term. 'Ihe orcler of the equation would- thus

J¡e raisecl by one for eaeh term beyoncl Brr.)

7\,2 ¿Lt

The second- term cannob be taken care of in the name

It is

ne cess ary to f ur ther d-e fine

\¡úay, l¡ecause Cn(\r) i* not sim¡r1y a pov/er of V.

L* ,r",r,¡ (u)au.

o(rr)(v)[ecrr(v) + 24v"Dn(v) + 'o']

,L* o,rr, (u)u [tr anrr(v) + I 92u2nn(v)

(.t)'4n2opD

cx, n

Then the l¿st terrn of (tA) 1s

+

rvould give terms in y

n+2 nJ IJ

3

)(

+ ' ' . .] clU

(zs)

1zr+ )

v n

ancl

ancl its

(26)

In this case only Cn y' O, Bo tir.'rt calculating(")2

JoY( n)

(")

)t (

Tn"' , and- o(rr), o(r, c[1rr¡ "r dçn¡" "

Prut Y

its d"erivat ives can be expressed, in terms of o(

derivatlves through the nelation

øqrr¡'(v) = - vv(r.)(v).

),

n

)(

Thus D op 4rr'will involve only o(.r) ancl.o

its clerivatives Lql to ø(t )

(rr) , where the sullerscnipt (4)

inclicat es the f ounth cler ivat ive, Cal culat ing

75,2" tt

op T2l ro" ¡r(t) = "(n)

rrnt r(^) gives tæmc 1n

-J

7(n), y(.r) ' ' ' ' "1n¡

(5), and through (26) these ar e

erçnesseo in term,s o, o(.r), o(rr)' .r,.. o(rr)(6).

Therefore when t\ny' o, Rn/ o,,n/ oe but Dnr En, "

all are zeyo in the expression for Jro(*), the integrnl

equation (t ) can be transforrred. to a sixth orcler d'iff eren-

tial equation in o(r.)t

r\ "rf,fo!9¡ =n "[IY.(,r(")*N('),op I*TF:J = "on Ltr.rr-

The existence of further terms Drrr Er, """ in

J(") not zerot is taken caT'e of in the same vlay as Crar

(")o

D

2Jo

o

since the i¡noblem is no14/ equivalent to th.'ìt '¡uhen C' uras

supposecl non zero.

eogr Suppose nn I O; then the terms

(") l)

f"1"¡cur #unr.,' (u)

U\ du

ancl

of (zS) and. (18) c1o not

when Oon is aÞl?liec1 to

o(r,) (u)utrenrr(v)¿u

reduce tou"*t lo_!_:l

terms involving o(rr)

. A firther Do-, musthrf

76.2.4

be applied- to reduce the first term to differcntial formt

and definition of

t"lv

oo

^(.r) = o(r,) (u)u¿u

is necessary to red-uce bhe Sec onc1. The equatiou procluced

then 1s

(")

D l-o " Æ'zil1l\l = o l-o " Æ: [;(n)"op ljop \ )+n' ,: )- "op i-."oP \4tr ( -

* N(t) ,ro(") l)l ,/iancl is of eigth order tr n(rr).

:ile have now proclucecf a general pnescrip-i;ion '¡'¡herelly

the integral equation (1) can l¡c reduced to a clifferential

equation for arbitrary n ). 1. tr1rst v,¡e mUst calcula-'¡e

lnlerqpl_icitly the form of

"ot"' to d.etermine holv nany of the

^D/.ìr{nr -n, -n, r o. are non zero' If /"rrr or Art ancl Brt are

non zero¡ tit/e Can obtain a four'ch ord-er d.ifferenti¿11

egLration (Zl)t t. "(rr).

However if there are m tenms

non zero l:eyoncl An ancl Brrr v,/e ean obtain a d-ifferential

eguâtion of orcLer 2(n + 2) in the variable n(.r). Tlre

var ia]t l-e A ig the mth ar¡xil-iary function def inecl af ter

the Pqrr¡ and. y(rr) of (¡) ancL (lS), ancl has blrc forn

77.2,)1

f¿( tr) (v) = UA

Dm

op

Slnce the expressions D op

(28 )

'llire eqtLa.tion

for I,{(")

(n) (u)du,

vuhere A' is the m - 1th au*iliary f\rnction"

¡È

and- so

2 (n)2 I

"(t), -la(t), anct lo(t) are required- in each eqr:ation

rqJard.lcss of the value of n, they are calculated- nov/.

From (1 6)

DËt,r,)

(II )52n-z du, (lo¡

$änt9¡ = rr(n)(v)v2n-2,

?lrop

D2 ßt¡op

Similarly, from (17)

Dop

71rr1 ') + V2kllrr ¡" ' I( )+ 2s,Y2j](vv n o 3z¡

78.,2.4

Straightforu'¡arcl algeì: ra then gives

g1(_ll\ _ l-t'-6 (rr)

\ t+rr, ) - iFr [v"kv(r')oB

+ v"y(rr) "'12[(ztt - 3) + 2d{2 ]tn * Zvlr-'I

+ (vv1n) v(rr)') [vzx" + vk'[(lr't' - Ð + 4øv2 J

+ k[(zr' - l)Qn - 5) + 4crv'(zn - 3) + l-r'cr"vaJJ]'

3t)

From (18)

av^+2,r,-, (t) '(v)

D 2

V+ VB '(v) + ooo.n

'ìIoon

-¡æ

+ r1,,;(vl [aï ê#l + v' d,_

dV,nn'(v)(.-.i-: + ... I

(

+ l_r, ¡j",,,)(u) # (4f1) ""

+ hv" .|]r,,,)(u) # t+l d,u + '""1

+ ?rr'(v)[znrr(v) + Irv'crr(v) + 6v+¡rr(v) + "']

+ rzçrr¡ (ir) [znrr'(v) + evcrr(v) + l+v2cn'(v) + " n ]

. a+ ,[r*r,^, (u)u[acr,(v) + zt+u"n,r(v) + '"'l1];,.,

79.2.1+

Cancelling among the second, thirct and sixth tenms we get2 (")

J+ vBr.'(v) + v3c;(v)

Gs)

Do

Then

/3*:(Ð\U

d, f- t.. # /u v(r,)(u)u[acrr(v) + zhu"orr(v) + ..o.]du

)

+ Y(rr) (v) favcrr(v) + 24v3Dn(v) l+ ..o.

,.,vn+Z" (n)Dop'(---#j) = - v(r,) " (v) t+ + e,,' (v) + v2ci(v)

)

+ err'(v) + v2crr'(v) + .oo 3# - 4vcrr(v) - ...]

acrr(v) zhvrnrr(v) - .. ..'l * y(rr) çv¡þcrr'(v)

+ zhv'orr'(v) + ..oe * 4Airorr(v) + ...1* 2rlrr¡(v) #

f*å . # [uu'/-"1,,¡ {ur # (+) au * ...]

. # É # [ Å- "(.,)(u)u¡scrr(v) + 24u2Dr,(v) + .." ]uu] ]

ßø¡

Bo.2.Lr

The relationship rretween t(tt) arrl the varia¡les of

the cli ff erenti a] e quat i on f ollolvs irnmecliat e l-l¡ frorn tþc

definition of P(") t T(n )' o"(") aoao fì ( n )'

rì'rom ( ¡)\rn 1 V )(

( )13

þn (v) , 3t)

€

but

so that

so that

!3 n( )(¡e )

3e)

(Lr-o)

(ø

n) (v) =t:

€

If r,ve neecl to go beyoncl an equation in y(rr) and its

clerivat ives \¡'te use

rln¡ (v) = ,

(ø

n (v)) vl-? çv2 oçn) 3v'-rrr) " + 3ø1rr¡ ') (trt ¡€

and. so ofrr until ,(")qv) is r¡vrj-tten in tærns of n(rr) ancl

its cLerivetivcs.

81 .2.5

2,5 _Ð_=_!

The general thcory of section l¡ d-ocs not ho1cl f or

n - O becau.se the ternn "(o)

reguines special treatmort.

In th-is csse \ffe are conceneil with only the tq¡ms in-

volvlng the scalar Go in tJre clevelopment of thc clistri-

lrution fhnct|on (Zr(15))- The integnal equation v'¡hich

has t o b e si-r lvecl to .g ivc t ( o) is

u(o) = "(o) * r(o) ,"o(o). (r )

r\¡e shal-I 1n this section obtai n the cli ffere ntia1 equat icn

eguivalent to (1), the methoC" usecl re sembJini, closely

the general mebhocl clevelopecl in Section l¡ f or n ) 1'

From (SrW))'

(ç. w'\ clU

cr.u (z)+ ue(u) p nI3LT

ru.Jo

t

ue(u) ó (u)(o)

(o) (u)

ue (u) 6(o) (u)

+ VU2Ioo

oo

t €.w"\¿u

f *

u"(,,) ø( o) (u)å.(u - v) 3 ci.u

.ly

82'-2.5

nOOlpJv

(u) (u - v)2au + A(v)'

(o)clU

ß)

(l+)

(5)

(6)

Q)

(g)

(10)

(o)

where

ln (l)

and"

Thus a(v) = o,

ancl

tr'r om 3, $7) )

p (v) = ue(u) 4(o)

n(v) = ue(u)d lE\5

+ [IV2 clU

But using the cldinition or o(N) QrUÐ), arrL the

arrxiliary cond.i ti ons (2, (15) ' (17)) we Îi nd-

i *

.(u)ø(o) (u)u,au = o,Jo

L*(o)

)(u)

(u)u4¿u = o

a

e(u) d(o)

VJ!.1l+rf

oo

(3( o )

(u) (u - v)'au. (B)

(o)ø (v)t<(v)

þi n\ (v)t<(v)= - ---ve(v)

(o)Iüe now must evaluntc J., explieÍt1Y fnom the

rrrescripti on given ln Se cti on 3. TJsing (¡t (¡O)) riue cnn

wri te

83,2t5

"%(:.)- = [u u,,(o) (u)e(u)Lo(u,v)cru*-'n":- =

J"

+ I *

u't( o) (u) e(u)Lo(v,u)¿u/v

(t t )

so that ïvc have to calcul-ate Lo(UrV), which from (lt(æ))

1s given by

This is just

Lo(u,v) = %#Ð t:Ï "*n [_- f; (-

* . o: tr]¡j.)',-l o*.

r,o(urv) = ,(o)

(12)

(13)

from the clcf initi on of I (*) in (ltß5)). The exPres'sic¡n

(1Ð agrees vrrith tkrat oþiainecl from ttre general rclaticn

(¡r(¡t)) with zo = 1 as it must.

lTow from (¡r(Llo))

I o2 e(r)ar,

Thus

"-]ûÐr,n(u'v) = ÜV

( )

a

e (r)cue

(14)

BL..2.5

To obtain (f U) we hrve usec-l th.e fr:ncbion t/ clefinecl in

(:, (eo¡ ¡. on su¡sti-tuti ng (14) into (11) r\e get

ov,lo( o)

--'ljrr" .Ër,(u¡p(o) HH+ . .lu- u,(u¡p(o) H+l

= Ë u.,,u) # (*{S) *

pr"ful %,e)

d.u

1r r)

(1 6)

d_

clV

d.u

No,¡¡ rlifferentiaticn of (B), (1O) ar.r-[ (16) rn¡ith

respect to V gives

,v"(o)-ll="ç1L A-tr- ì

=2 î)l) (u)(u - v)au(

d. (o) (v)t<(v)

)o

clV Ve(v)

hdk' (V ) v)(17 )

e

CIearly f\rrther clif ferentiation of (17) with nespcct

to v twicee woü1cl give a fourth orclen cquation in,61o).

It is hor¡¡even pcssibl-c to rùtsi n a thircL orcler eguetion

in a neui var iaìrle W, clef inecl bY

oo

85,2.5

(u)ou, (18 )w(v) = v p (u)au + rJ 13 (o)(o)

Then,Lifferentirt ing (18) rirc ¿4et

vir'(v) = IJ t, (u ) cru, (t g))rì

\ru" (v) = lg (v), (zo)(o)

w",(v) - - ve(v)ø(o)(u), (= pio¡(v)). (zt)

On substitutlng (te) Ql) into (17) ancl mul-tlplying

by €' rre have

ancl

- 4a]r-'W," + Ze\x'l

(zz)

To obtain the last tøm in (ZZ) u/e have again invokccl

the nuxillary reLrticn (¡),

Equat icn (zz) can be vvr itten

, Dop eeï = - # (nq# . 2o(krv'' ¡ k'titt' r."ru))

(zt)then if

¡(v) =

oo

on integrat ing (23) t TIIe ol¡tâin

Hç:: . za(t<w'¿ .¡ k'rirl' - k"vu) = ,r(v). (25)

In inriting (25), the cond-ition thiat VII ancl all its

d.enivatives tcncL to zero as V - oo has bcen u'sec-L. This is

casily clemonstnatecL as fo 11i¡øs.

The f orm of IAJ"'y (Zl), ancL t/\/" e (ZO) anc-[ (l{-), show

th¡rt

86:2.5

(26)

ßo)

AS v - @r \N"' .. 0 exPonentieJl;¡2,

w'(o) = o,

W" - o exponen tia l-Iy . (27)

IrTow using (¡) ancl (zl) r\E hn¡e

Ii"r(u)ø(o)(u)clu = [-urn,"'(u)au = o (ze)lo o

But on integrnting by parts"V/ u*"'(u)¿u = vlrl''(v) - ui'(v) a iv'(o). (29)lo

Also fnom (19)

s o t ha-t (zg) implies

V - Ø, nr'(v) - vw' '(v)AS

- o expon(rl tial1y, (lt 7

Finall-y from thc ddinition of \rI/¡ (18), r¡/e have

V + oos W(V) * O exPonentiallY. 3z)

,r(1)

87.2"5-6

(r )

The equ:.tion (1 ) hns nor¡rl bcen reclucerl to a thircl ordcr

d iffcne ntial equat i on in ì:/" ìve can obtain ,(o) frr¡m the

solution of (2Ð through the relation (Zl), vr¡hich is

,(o)1v¡ = 5#F 3t)

as

2r6

a

This cgwltion, (zù, r''¡i11 be usec-l in Clr'lpter 4 to

oi:tain ssl¿tf ons of thc thirc-l approximat ion tc Bc¡]-tmannrs

equ.rt ion. We wi 11 apply the L.H.Sì. Ôircc tl¡i ¿infl cal-cula te

I'¡(v) frcm thc expressir:r¡ p\o/ lvhich in turn is re ls'tcd to

the,t\. given in Chapman arrl Ccwling.

The intqral equat ion to ]ce solvecl f or that part of

thc d.1striþution f\¡nction involving the irreclucible

tensor Gl is

Dif fer cnt ia I Efirat ion 14J'hen n = 1

(r ) (1)J +K

"(r ) a

o

In this section we lvi1I fo]Iow 1hc general prescription

]a1c1 c1,¡,;,/n in Seeti,-rn l-¡ tO obtain the c-[if feren-bial equntion

in thls câsc¡

BB.2.6

The first thintJ we nruú clo is establ-ish thc form of

_(r )Jì ' , which is to say we must cvnluate A¡,8¡ o.o explieitly'-o

Thls Ís done by calcul¡tin¡' ¡, (urv) tt" inùi catecl 1n

,Scction J.

rn thi s ca€j e err(c) of (¡' (LrB) ) is

P¡ (c) = zte where z7 = 1, (Z)

Then on substituting this in (¡r(¡t)) u¡c ol¡tain

-1r,,(urv) = ## t-r (r ) + (v" + u")r l

e (r) ar,

(o)a 3)

To eva]uate this we mdcc usc of Sectlon J ' ßS) -

(¿+Z) to otrtoi., r(o) on.-t l(1).

tr'rcm (3. (40))

(o ) fu"l-U

(4)I

anrl

tU

,(r) exp (- od{')U

lc, + (u" _ V") +It_

(5)

\2

dK-l

aG2

In (i) G must be expressetl in tems of K through (lrjg))'

Frcm (1, (lrt ) )

l+K, = G2 + 2(1J" - V") + , (6)

89.2."6

On zubstitutÍr¡g (7) 1n (S) v,/e get

-(r )U

s o that

u¡hene

rclat ion

(7)

t U

and then ombinlng (4) anc-[ (B), the form of L,(Urv) :-o

Ur,, (urv)

i¡riri-bten in ttrc f,-,frÌr (lr$n)) thls is

Ar (u)f,, (UrV) = ffiiz** t

'lt -W2q"

-1R,(u) = € (u) e (r) (u' 2rt2)c1K (t o)

(g)

(tz)

Equation (tt ) follovvs from (to) 'by making use of the

e(tt)x'dK =

which is obtained by lntegrati.ng thc L'-TÌ.S" by parts.

a

690

a

Also from (tt)

op

A¡'(u) = ryi#Ð * u'

¿

(1 3)

(14)

Notv 1n qlplyinq the Seneral- mcthocl of Scctlon lL we

notice that l, (Urv), (g), cc¡ntains no terms v4cr (u),

VuDr(U).,, êtco¡ so thr,at it is llossiblc to cíbtain a

forrrth orc-ler c]ifferentinl cquation. in y?), nnnely

2 ¡¡"¡,(1)(Eã- /v3\ñã)=D (,r (1 ) (t ) 1)+K 2r( )D t

op

(1 5)

(artrrou¿1h thcre is no Br 1n ".!t

), thc form of ,(t ) .r-oes

not al-Icv¡ a thircl .¡rc1er equation in FU) to be obtainecÌ

by opeftrtin¡1 on (1) lvitir o,rn . Thc a<1clj.tional- variable

be clef i nec1. )

a rurlys is of S e ct i on I¡ vue as sumed c er'cai n

(4r (9) ) arrl (:l' (t 9) ) rwere obcvecl by l!r, and

clevelopment of the eqrnt ione (t 5). These

obeycd in this cûse since from (to) ancl' (1lt)

Ar (o) = oe r"d l-A+FLl = o. (16)' u Ju=o

v(t) has to

In the

c oncli ti ons t

Br., in the

are in fæ t

6

)

1

a

)

92

n2Now we rruill takc ov(T the form of D op

for ,r(n) = ,r(t), ,,(n), ancl t5") from scction l¡, for this

particulan ease of n = 1D ar¡-l so writc the cxplicit form

"'12(- 1 + 2ot\12 )r + z{:r.'l

"Ci)[v2k" + vk'(- 3 + 4øv")

or (15),

ancl

DoB

tr'r om (4, (11 ) ) ancl (tt r Gl))

(17)

(18)

(1e)

¡¡" ¡-(1 )

(æ-:

-lrfti lu"o")=¡¡"6(1)\ l+tä-

vúi

Dotr)

(4)2I( )

1 )+ Y'y (

al+ (v

+ k(3 - 4øV2 + )-rol2\f a) I

Similar ].yr from (¿l' (¡6 ) )

D2op

which on srbstltuting frnm (11+) 1s

92.2.6

D..5. ry (k., + zavrc,)]

a,

k?(r )

op (r )

(zo)

The equation ltf) for n = 1 can then be written(t+ )

eD + lttlv V2 Lz(- 1 + 2aY2 )to * Nk' I

(zt )

(1) to

op

+ [vzk" + vk'(- 3 + Bofr/2) + i<(3 - l+o¿Y2 + tla'va) j

KLya [(v, - L¡o¿ya)k" + vk' (- 3 + 4av2 Bs"2va) + t<(3

- 4aV2 + l+azva) |

'r (t )"Solution of this equation for y() endr Ics f,

be f ound. tLrou¡1h thc re l-at ion (4, (¡g) )

(zz)

There is an altænat ive nr:thod. of arnivine: n'b nn

equation for n - 1 which v¡e wi 11 r-.¡utline"

From (4r (¡o) ) ancl (¿r, (¡ ) )

""(1 )

) (r )D op hrf- =2V

/""v (u)ctu, (zl)

ancl

95,2.6

þ (r)(26)

(28 )

oon #1; = # (vri(v)r(r)(v)). (zrr)

D

Sim11,:uly, frqn (¿lr(B)) \^/e crn obtain an exirression for

, Þil1) *r.:-"h is t4ven explicltlv bv using (rrr) as

f o1lows,

)

" (r )

D op 4n2n_P (r )

(v)nr'(v)o

)

-lr (v ) . (zs)(t )

Thc eguation r¡thcn n = 1 cnn the n l¡e tnritten

eD = 2e\IlJ ,,r ¡

(u)au

+€ (vt 6(1) I +ç-. a

äqy:K:€

op

c1

dv

(zt)

In olr tai ning (Zl) vüe hm¡ e usecl th e re sult of (1, (66) ) ,

k"+2øVk'-4crlc-0.

l4/e ncru rLefine a nevl variable X, by

= # (: /,,"",., ) (u)¿u - I:P zút2oþ(,, + vke/tt')

x(v) = ilr,r; (u)au,

so trat

ancl

u/e ol:tai n

x'(r¡) = v11¡ (v) ,

x"(\r) = -'[p(1 )(v) ,

x'"(v) = - l3(.,)(u) * v.P(1) (v) a

9ll.216

(2e)

(ro¡

ßt)On r:ulrstituting (Zg) 3l¡ into (27) ancl integratin¡1

.. úJ u. e- å Å' + å Xqd

(r )clU ßz¡

Gt)

The f orm of X ancl lts clerivatives, definerl ttrrrough

a^u "(rr)

o.U É(rr) of Section 4, is m.fflcient to cnsure

thc constant cf integration on the left is zcro¡

The solution of (SZ), thc iliffercntial cguation rrhen

(1 )Ir = 1, thcn gt'-ves ø thnough thc relation*rip

(r )þ (v) = frilvT

1

L"'" (v) - x::fy)tvl a

In the er-suin¿J v'¡ork in rnirich,¡ve use the n = 1 equa-

ti on in obtai ning soluti t)ns to thc sec orrl ancl tr ird.

approximation to Boltzmamrts equation we,¡ri1l use the

thircl crcle r egur.ticn (SZ¡ rather than -r,tre four-bh onrfer

equat ion (zl ) .

95.2.7

2 . 7 D_r€_fg_ç_qtai a L-$q¡_-+tj"o_q l&ç_n*;1 :, _ ?_

In this section \¡ve will fol-]ov¡ the mebhocl ou-blinecl

in Scction 4 to obtai n thc Oif fcrential cqua.tion ,ivhen the

clistribution function involves the irrccluciblc tensor of

orcler 2e Ga. The intcg rnl eqr-rc.t ion in this casc; 1s

"(z) = J +K(z) (z) ^ -(2 )¿t)

o(1)

As in Se ction 6 thr: finst cons icleration is to cleter'-

mine the exl¡l-icit form of f on the pnr'r,icular value

of n concenccl¡ hcre rr = 2. This me anrl r,vc have to calcu-

late l, (urv) .

Norvforn=2

Pr(c) = ãze2 + 22

whcre a^ -z rnc-L 22 = +. (z)12 - 2

The form of T,2(UrV) whicfi we gct on su̡s-bi'cuting

this into (lr$t)) is

,( n)o

(r )z(rJ' + v2)l (u" + v2)2r lT,z(u,v) = #+ E Ir (z) (o)

(o)I_ + 22\J2V2I

+

3)

96.2.7

(6' (h) ) encl (6, (B) ) nespec tive ly' so to complcte the

fltrìecification of L2(urV) we have to ealculate IQ).

lVe alread.y have elqlres,sions for -(o) ^ -(t)I. ' aYTCL I. ]-N

+ G2(U2 _ V")

From (¡, (Lro) )

,(z) exp (- *'r[nï

U

U

4

+ (U" - Vr)2 +

+

+ c-LK

.U LI

= [ " ex-û (- ,,K') t fs+îIí t1 ;l-u L, e'j=o

anct fr om (1, (41 ) ) ,

16,!K4 = G4 + L(U, _ yz)çz + 6(U" _ Vr),

# (u, - v''j)3 * G=-*,v:)"'

U,2

(¡)

(6)a

Then using (6) ancl (6, (6) ), the expressi om f or Ka

ancl K2e lvc cat'r ¡¡rite the sum in (5) as

16 I<4 1Z(rJ2 _ yz )62 + (U, _ V, ) = I4 at,

G

s o that

j=o Q)

(z)

2

s(r)å lsl$' 2N= ¡z

97¿217

(B)

ancl

(g)

(t o)

U2K2 JclK,(11 )

(12)

I exi? (- ci¡<')lt6 t<4 tz(uz - yz )62

- v2 )" ]dK.

Sui¡stituting in (Ð the values for t (o), r (1 ),

I gives after a 11ttlc algdtrae(z)

where

t,(uI)U

+(v u2 )n2 J - u"v, ]ar

¡'Jï= [n"(u) + v2B2(u) ],

U-1 â [(u" zl<, ),A, (u) =€ (u) e(t<)

and.

B"(u)t

€ tJo

U(u) e(x)LGt<' u2 ) dK.

Usl ng (6, (t Z) ) anrl the fact

f1

Ir(r)n4aN=#3|'(u)we get

? o,$) æ-2q, ,

ue(u ) (¡ + 2o[J2)f, (13)

nJ3

Lã#tl t"4 åut

o'\ (1h )

gB.2t7

and.

s o thât

B" (u)

Br'(u) =

? o,'(u) = #Ì F"us - ua . å] .- ["' - 5] ,

=.4{s:ñ-+1-ffi,IJ22

# (.%9. B,'(u, - 'l+u') = *{S fzou" + t)zou"

2, r,r.(urv) rf (ZrGS)) r,os only A, / o

#B aU3 I ,

(15)

(1 6)

(17)

+ Bz'(u) ry = #H+ zaus + Ír2, (tB)

anci.

+ zu(øu2 + 1), (19)

Nowfor n=

ancl 82 y' Q. The tpneral [hrcory of Scction h. ind-icates that

(1) can therefone lce rcclucecl to a f ourth orcl-er cliffenential-

d.- \--úþ)clu ) - e(u)qu(1 + 2onJ2) - o[J2. (zo)

cgr-ration in Ve)r namelY

D __2 re,ÍÐl - D ._2 s# (¡(z) * *(z) ,r\r))l . et)-op i-4n' I op ,41T' o 'l

OO

2.7

The concliti ons (tr, (g) ) ar¡i (l+' (t g) ) which A2 arr.-[ B" must

satisf y f or the c-levelopncnt of the gcnenal equa-i;ion as

¡1ivce in ,section l¡ tc¡ bc approprinter are in fact oi:eyed

herc since from (14) ancl (16),

Ar(o) = o, B"(o) = o , (zz)

ancl from (lS) and (17)'

rk_ (ul-r

l--T---lr=o=o (23)

U=O

We nov'¡ sr)eciali.ze f or rt = 2 tLre val-ues

D2

,Section L¡ nncl thus ,.rrn itc an e xplicit f orm of (Zl).

From (4, (¡t ) ) en.J. (ttr3l))

ffi *(n)_] ,"' r,,(n) = ,r('), K(')¡ ad J(") of

op

ancl-

o

D 2 (24)op

D 2 ¡yo*Q)Lu"

o',, ) ('' )\=

)v-2

.T\ryop

+ Y'y '"l2l1 + 2d{2l:r- + vt:x'I(2)

+ (vvp) y(z)')fvzk" + vk'(1

(25)+ k(- 1 + /+øv2 + hc"va)J .

+ hcxV')

'loo. '

2.7

Similarly from (lr (J6))

,q1aJ e) -

o,,n"(=+-) =- v(z)"(v)

) (26)

rvhich, when v,¡e su]¡st itute the values (1S ) r (t g) and'

(zo), neduces to

(z)4Jo

D op l+xr'+v",1

'(v)fr*,lz.,,v2

+ ,f *{BY (z)

- Yk-y

+ 2V(orv2 + t ) II

+ ,r(r¡ (v) t- øvf"(r * 2q\t2) #B .* vJ a

The equation, (Zl)t for n = 2 ean then be 'l¡ritten

(27)

(h) ' ' ' Í.211 + 2qY2 ]x * zr1:r-' I+ vQ)(2)CD op2

+k(-1+4eV2+4ot"va)

IAD h,L

)

u

+ 4V4t +2V3e I

Icr)

fy'y" + vk' (1 + l+crl/') + rc(- 1 ç l¡øv2 a t1c,2qa)L

4val3 + 2çttl2l1 -'*v3 (øv2 + ,t )rl

01 .2.7-B

(28 )+ YQ) (v) [¡y(t + 2o¿Y2 ),/(v) + 2v2e!

= A(uqr¡ (4), Y72¡"' , Tq2¡" , Tç2¡' , '(r))çzo)

comparison of (ze) ancÌ (zg) clefines rr(v(r)(h),

Tç25"'r Y727" ' Tç21" v(z)) rrvhich we shal-l- l¡'rter use

n,s a short-hanc,1 form of thc R.II'íì. of (28) t¡¡hcn we use

this equation in oþtaining arlluti¡ns of the seconcl and

third. appr cximations to BoltEmannt s equaticn.

S oluti cn of thi s e quat i on f or v e) gives fQ)

thrc'ush (!t t ßg)) ,

t(z)çv)=#lvvç4 v12¡'7. (3o)

2.8 iffer ent rj[Lren n =

IlVhen the ci.istnibuticn f\¡netion invoÌves the irre-

clucible tensol G¡, the intestnel eqUation r¡¡e must solve

to olrtain thc unknown f\rnction ,3), is

G) 3) ß)

102.2.8(1 )

Forn=3

(¡)J +K

a

nu = B

F 2Jo a

In thls section v/e use thc methocl of Section h to recluce

thls equation to cllffenential form¡ anc-l show how / 3) is

rel"atecì to the d.ependent varirible of thls differen.tinl-

eqrr,etion. The methocl is the same ns we have used. in the

l?r ev i ous two s ec t ions "

Firstly we must evaluatc Lr(UrV) so thr¡.t lr¡e carr ir,et

an explici t form for Jo ß)

Ps(c) = âgcs + zsc (z)

ß)with

ancl zs = 22

The n on sub st 1. tuti n5¡ th i o int o (1, $l ) rv e ob tai n

r,.(u,v ) = # l_ â Ir(¡) - i(r, + v2)r(z)

(r )

3)

(o)

(,r) (r )

+ 3(ü" + v2 )27 (u" + v2)l l

- 7 x 22TJ2v2[r(t ) (u" + y2)r(o) f] t,*l

the only unknovrn in (4) is I r sinee I

urr.r l(2) have loee n founcl in (6r(4), (B)) and (2, (B))¡ arrl

,I

vr¡e have an expnessirn for this frcm (¡r (ho) ) t

Ub

i---, (Uz _ V" ) Jcxp (- od(') ) ls-ry-Ër¿- ¿lcLt -æfr

(1)

103..2.8

(¡)

(6)

0)

(a)

(z)

T(r)

, (t)

6\-L,j=o

a

Uj=o

fn- (¡) G has to be expressed. in tæms of K thrcugh

(t, 3g) ) . Ilnom (¡r (Lit ) )

6

64K6 = T 9

j=o

ancl using this in conjunction wlth the exl?res,sions for

Ka ancl K2 ¡1iven in (Zrle¡¡ ancl (6r(6)) lve can write the

sum in (¡) oô

6l+t<6 Bo (u2 - v')K"a zLt(). - va )62 (U" - V")"+

(u, - v2 )-1

^2 j-6

LT

exr) (- o[(') [64Ku ao (u" - \r')K4

ThusU

tU

+ Zl1.(U2 - V2)'N' (U' - tt2 )3 ]clK

'substltution in (l+) of the val-ues for r(o), T

I G) gives af ter some al¿ebrae

I , I

r,, (urv ) / u'(r),

I r[tu' zN3)"

1.Ol¡.2.8

(e)

3u" (rJ" - v2 )r<r+h(u, - yz)sa

3uzvz (u2 2K, ) d_K

Tlr [4"(u) +v"B"(u)], (10 )

IJ(u' 2t<')" 3rJ4K2 + 4tJ?rr( clK

_l

j

ì(

r,vhene

Äu(u)

B "(u)

(u) e(r)

U

-1e 5

2 tL(tt )

-1=€ (u) e(r).,lt5u'*'zL

usins (6n (t 2)), Q r?S)) arrl the f act

+ 2OI<4 3U, (rJ, 2r<, ) dK

(tz)

+ 1ootYz + l¡a2va ) )

0t)

tl9

.Ë'(x)N"ar< = 5þ Us,tt(v) - ve(v) (r I

vre ge t

Ê o"(u )ogr * 12v:_ _ 41 .|s - 42q. ' d.2 cx," I

- lla d2

? n"'(u) = H# ?"u' 3tJs +6u: _

d"

+

(ru)

(r¡)zu4+ t"

6_q +

61J21-ããJ I

6u-l"-ls.')

1 o5t^o1¡ ()

2Bs(u) = HË+ [- 3ua +# F] th H'(re)

Qt\

(¿+a'uu * 1oøua)

+ (4u" + 2cru5) , (lg)

t- 6q2rJs - 3otrJ"J - 3arJa . (zo)

ard

and.

zn u' (u)

Thus

tug - 49+ 2.,rrs + rra, (18)+ B"'(u) - *U

e (u)

# + 8".(u) #r) = *€+

d.

d.u

3)Now f on ïI = 3 thcre are only As y' O, B, / o in Jo

so that by the [leneral theory c¡f Section l-¡, (t ) can be

re,luced to a fourth orcler eguation in yß),

oon |{#l = oon' t# GG) * ^ß) ,ro(3',-l .

(zt)

The conditions which must be satisf iecÌ 'by A5 ancl Bs ,

(4, (9)) ancl (!119))rane in fact obeved since from (14)

ancl ( 16)

au(o)=oe e"(o)=o e

106,2.8(zz)

and" from (lS) ¡rrcl (17)rÃ='(u)-l:;'-- I = o ,L u Ju=o =o. (T)

U.-O

To wrlte an e¡-)Iicit form of (Zl) we ßIreciat-izc th.e

values D

and.

tr#9l] r* ,'"(n) = ,r('), K(")

9

(")J

oof

(ztr)

(zs)

op

fìection l¡- for tLre c.nse r4/here rÌ = 3. From (ll, (¡t )) ancl

(t: , (ss) )

D2 g#l =-zvav7r, (v) 'op

+ k(3 + 12qY2 + L+q2Ya) I .

oo' W2l - ,-1 ¡v"oy(l) (¿*

+ V2 y G)' ' ' Í.2(3 + 2otT2) tc + zvu' !

+ (ur(r) v(Ð')[vzlc" + vk'(5 + lrcrv2)

107.ZrB

When we substitute the exì)ressions (tB)r (19) encl (zo),

into (26), it recluees toß)s¡

Do

op 4r' ) = vß)" (v) þ"v" H++ * v'J

[4o"vu * loqla) g{B . (4v" + 2av5)

[{eo"uu + 3..,rs) #B + s.,va),

213 ¡ ' (v)

zvçr¡(v)

II

Then

eD op

+

ß)

+

2 2c¿r,f 2 )k.

(27)

(28 )

+

+ zVk' +vy ?,o" + vk'(5 * hcxv')l)

t(l + 17qy2 + \otr[al+ 4yatþ +

yG) ?'u" + vk'(5

2V3 e-l

--J+ k(l+ 4øv2 ) + 12aY2 + \.ø,2Y4)

/W\ø

+þte "u' 3v")þ + lovae

z(Lro,v6 + 1 ov4 ),! - 2 + zVs €) l

YG)

is the eguation f or rr = 3.

ß)Soluti on fo r YG) then sives Ó throui,.rh (t-r,3l)) ,

3)'(s)'lþ (v) =6

I (v ) [vvß) a (zg)

2,9 lffcrentia I tì-

108.2Lg

(5)

Thc thircl appnoxination tc, the Boltzmann equation

involves irreclucible tensors Crr(Y) for n = Ot1 e2e3 and. 4.

liì/e have alread,y o't¡taincd the clifferential equaticns vrrhich

cleterrninc the un]cno\{rn fbnctions t(n) whic}r are involved.

vrlth the corresponcì-ing G. in the clevelopmcnt of the d.Ís-

trilrution function (Zr(ll)) for rr = Or1 ?2 ancl 3. In this

section we makc thc finnl spccinlir,aLion of the gcne ra1

rncthod- of Scction L¡ to cbtain the cliffcrential cquatÍon

r¡vhcn ri = l¡, from the inteSral eguation

F(h) = "(lr) + r((4) zro(l+). (r )

In this casc Pr.(c) of (¡, (t+B) ) is

P',(c) = ^¿c4 + bac2 + 24 Q)L1.

t¡¡i th +, b¿=-+.1 4 - and- ,o = 28.

Srfrst itution of thls into (1, (54)) ûives

r,o(u,v) = #.,# rc tr(4) - h(u, + v2¡r(3) * 6(u, + v2 ¡,7Q)

- ti(u" + v2¡"r(1 ) * (u" + v21r1(o) 3 - # zzü2vz[r(z)

z(rJ, +v2)r(t) + (u" +v2¡r1(o)3 *âzasayay,",]

(Lr)

lìquati.¡ns (6r(4)), (6r(B)), (Zr(B)) ancl (Br(B))

give expressions for" t(o), r(1 ), ,(,2), ancr t3). To

e omplet el-y

, (tr.) r ,¡ihich

r (4)

109,2.9

spec ify La(UrV) lve have theref¡,.re to cLetermine

fnom (¡, (¿+o) ) is ¡1iven byBU tJ2 j2

exlr (- sI<" ) I^2

j-B\.Tj=0

To rvrite thc series in (¡) in terms

f c1I(. (5 )U

expressions (8, (6)), (2, (6)) anct (6, (6)) for r(6, K4 ancl

I<2 ard also thc r.clationB

j=o

which rn¡e otrtain from (3, (Ltl)).

2D6r<8 - b!_B(u2 - v2 )ru *

- 4o(U" - V2)3K2 + (u' - V2

Thusr (h)

of K wc use the

fis2 - v2)ic2 i-e

The result ie

2t+o(u2 - v2),r,o

256r<8 I(:) (6)

(t )

(B)

A

\4l= t/l na J-óLTj=o

Iu"/-V

exrr (- oK') lz56xe 448 (u, - yz )6o

z4o(u2 - v2)rt<o - llo(u2 - v2)"t " + (u, - vr)4+

( o )aa)

11O¿2.9

r (4)Substitutlon in (4) of the rralues of I

L4 (u,v) = f+= /" .(r)å þsf<u' 2r<2)4 - 6u4(u' - vz)62

gives e,f ter scme len¡ithy al¡¡ebra,

U

+ (t7u, - v2 ) (u" - yz)6a 1zFJ2 - v:ì )r<u]

l- fou2v2[(u2 ?-K")' + (v" - rJz)r"J+ 3tJ¿v¿ d.K

(ar(u ) +v2Br(u) +vacr(u))

(g)

(to)

whereU

l"A¿(u) = € (u)

-1nr(u) = € (u)

+ 36otJ¿t<z - 3ou6]crr"

e(rc) # [(u' 2Y'")4 - 6uôKz

I

+ 17'¿aNa 12IJ2K6 ]AN

"U

[,'tc^l + [u" 1tLu6K2 + l+1 uaKa - l¡'l¡u2K6 + r 6x8]r.

(tr )

a (u)

ancl

[" 4Ð [uzoro - T*ot*r<4

0z)

l35t<o 3ouzr<2 + JU4 ]ar.

ht)cr(u) = r-1 (u)

Now 'by lntqratin¿'l:y pants we can show

ru.loe(r)r8¿r =

-V V1

111 .2.9

2ïa2v4 B 6

(1 5)

(18)

(1/r )

Then usins (tl+), ((', (12)), Q, (3)) ancl (S, (1 3)) , the

exBressions f or I+aeB4 ancl C 4 arê evalurted with lengthy

but stncightforwarcl eff rtrt to ¡,_1ve,

3za4ao(v)35= trd

*{Ë+

[4o'uu 28cx3u6 + 123o¿2u4 33oqü2 + 42o]

+ lzofrJ' - 15q2rJs + 5ocru3 - ll2ou]

f8øsgs 2¡c¡jtJ7 + 7Bø3Us - 16Ba2U3

+ 1BOøU2 ]

+ ll+oouu 1t¡ø3u6 + h8ø2ua lBoaua] (te)

1 6a3B¿ (u) = t- 6oø3u3 + 36ooÊtJ4 -11zDo'TJ, + 15751

+ t- 3oqztJs + 75ot]3 - 157DIJ)

#s

16a384'(u) = HS

(tt)

t- 12oq'trJ7 + 36oo,3u5 - B1ocx2u3 + goocru]

+ t- 6oq3u6 + ZloaztJa gooau2 ]

l1za,rJ4 6oauz + 1 05 ] + t- l ocus 1o5u ]

(tg)32c,2co(u) = g{B

32a2Cr'(U) = fr# [z¿+a"us - 72a2tJ3 + 9oøU]

+ (1za'rl4 !oøu2 )

Thus

11212.9

(zo)

(21)

zno (u)A"'(U)Jfu=ts + Bo'(tl) + 1J2C 4'U

(u ) - t¡uC, (tr )

= HooÌ 2orr7 + u 6

= Z('J) ,

# (+ + tsa'(u) +"1 + ulc o'(tr) - r+uc.(u))

= 98+ [L¡ø2uB + 14øu6] + lzauT a 6uul (23)

= z'(u),

u-lã_J

þz)

(27)

(zì+)

2ct. f B¿'(u)rffi [ u:-] = *f,,] f 3oa2tJz - +

,lr 6;jz-2

a2]gf:|2 a-J+t

15qU6 + 3J u4 (25)

(26)

2

= v(u),ancl

h # ffil = .fiS l-?o,uu

- 6øv3 . å ul . ltov" - Z u")

( za)= w(u).

1'13.2.9

Here Z, Y arrcl \,\l ane functions d.efÍned. by comparison with

the expressions immediately pneced-ing them¡

Now for n = l+ thene are A¿ / ot Bo / o, C.t / o in(4)

Jo so that followin¡j the general theory of Section hr

another varlable beyond y(U) has to be d.dined. Thls is cx

d.efined- byoo

(+)

"(¿r )(v) = Uy (u)au ,(4) (2e)

so thata,u¡ '(v)

(¡o )V

Then using (zg) anct (¡o), the egr.ratlon (1 ) ean be ned.uced.

to a slxth ordæ eguation in a(¿+)'

= oon, L# ( ¡(4)3D op +K (4) (4) l2Jo )

ßt¡

the oud.iticns r¡¡hich have to be satisflecl by A¿r B4

and Ca¡ (¿t "

(9) ) ancl 14, (t 9) ) are ln f ae t ob eyed si nce

frorn (lS), hl) anct (t g)

A¿(o) = o' no(o) = o' ca(o) =o (lz)

arrct from (te ), (tA¡ ard (zo)

c¿' (u)l aoB¿'(u)-

o ' -T-Ju=o oU =o 3Ð

114.2.9

\l/e now proceecl to calculate the e4?licit f rx'm of

(ll7 j.n terms of o( arrl 1ts d.erivatives. Frc;m (1r., (¡t ) )(4)

D2op

so that

D3op

(:t+)

3t)

3g)

(4)

6 (4)

Fnom (t-r' 3S))

D2

with

and

+(,+)

(4)

a(v) = zl(zw" + 5)k + vk']

(J0)

opæn!9t - -v:( fiF:) = ail [v"i'Y + a (tr )v'y(ir )' ' '

4r" ) = zvaøçU¡' ' (v) + Bu" o(r') ' (v) . (35 )

(h)

!lÐ. l" -3q"€

e-GD-6(v) [- v"o (4)

+ [- v"o (4) "Q + jya (4)

1

a

1

2at5

9

w'o(Lr) $) vaa Yt+v--(4)

€+

l+ la(v) - ¡(v) +

expre ss ion,

Thcn

[t+x - .(v)] *6

(¿ (v) * "' (v) )

(h) r 1zLt€

Vø(l) [zt*. - 6a(v) + ]b(v)l€

(5)

)

. :OÈ t- Jok + 6a(v) - Jb(v) l

= "{v)vo,ur(5) + a(v)uo(r,-)( + r(v)uo(r,)

+ s(v)vø1¡,.¡" + t(v)va,Ur'

with c, cle rt s ancì t iclentifÍed from t]re prcvior-rs

(t+o ¡

(Lrt ¡

(l+2 )

(trl)

+ "(u)

+ o(Lr) (4)

1¡ çrr) + d' (v) ) + o(1,) ' ' ' (" (v)

+ "'(v)) + "(Lr)"(t(v) + "'(v)) + "(¿+)'t'(v)

gc-LV

_c1

d_v 4) II

_l(+Õ r1

LVd. co(v) i

116DOLa )

Z,-(v l, Lt-_u)=CX (4)

+ "(u)

+

+ o(t'-)

v2

l+V V

T4_N)Lvs

u_N) lLel*.--v:-s(tÙ)V

] + o(¿r)zw(v) + 2v lurrr,(u)rnr(u)au

D

Thls can bc put in tÏre form

u, (4)o

4æ ) = o(t,) "'Vm(v) + "(¿, )

"vn(v) + "(U)'vp(v)

+ cx¿Vq(V) + 2\,1 Iu/e

v¿(u )'v(u)¿u

with m, n, p ancl q clefinccl by comparison witii (l-r4)'

îhus

2

(h4 )

(4r )

op

+ o(¿,.)" [p(v) + tt'(v)] + "(¿*)'[q(v) + p'(v) ] * "(1,)q'(v)

"J+L (ue )Then

",u- (4)

onn (-#--) - o(1,) (! )'(i,) + o(¿*) "'[r,(v) + m'(v) ]

a

V

I'r(vlV J + Bv"l - zq'(tr)a2

117,2.9

zIo,(v) + p'(v)

(¿+z )(¿+ )

is 'che equation for h = l¡,

In thls eguation thc f\rncbions c(V), .1(V), r(V) aaa

'¡¡(V) etc, arc all lencrn¡n fUnctions of V clcf inecl as \¡ie

dld the analysis leaclin¿l to (42). To prevcnt thc equa-

tÍon J¡ecoming unureilcl5r, th e expl_ic1t form c¡f these

functions hns not l¡een usecl. The e quation requircs the

cler, ii¡atíve s of th c f unct i ons also ancl it is a si m rll_c

matten to obtain these frorn the Ìrnown eq)ressions for

thc f\rnctions, thoupìh we sha.ll not bc¡thcr to (Ìr-. 'chis.

(!)the solution of (47) for d (Lr )Sives ø through

(rr' (41 ) )

ø(4) =-#[v'o (¿r) 3\ra (4) '¿ + 3a (l+) I (4e)

118.3.1

C].LA3IELJ

THE SECOND APPROXIIfiTT] TON TO BOLTZ}IAI\I\TI S EOIJA"]]TO}'T

In this chapter we shal1 obtain the first corrcetion

to itre d.istributlon f\-Lnction fnom the local e quilibr.ium

stiltc. It invo lves: si?herical harmonics of orcLcr tì = 1

and. n = 2 and r¡¡c makc usc; of the egruetions of 2.6 and-

2 .7 t o calc ula te nu¡ner ical-Iy th is s ec ond lp¡troximati on

to thc distribution f\rnction, Thc values of the heat

flux vecton and. pnessure tcnsor we obtain b5r integrating

thj-s, enrible us to give exact expnessiclns for the

coefficicnts of -i;hermal condrlction ancl shear viscoslty

for a gas of rigicl spheres. 'llhese coefficients are

compnrecÌ urith the apl?roxlma,te results ob'beincd ,','hen thc

cl-istribution function io e xpand.ed. in Sonine polynornials.

3.1 The Sccond A'oltrox

The s cc oncl aj)proxima tion to the clistributi,¡n func-b ion

r = t"f., + ,(r ), (r )

is obtained by solving (2,2, (to) ) r,uith )T - 1, nnmety

119.3.1

(r ) (r ) (r )

Io(r ) (s)

(v_o) laq au Q)+Õ

=À.

Fron Chapman ancl Ccv¡linij (7,3, (Z) ) thrc exLrression

(uo) o

for  is

in v'I-r ich

rI=f (v)[t"u' - |lu' # * 2o sou, rt nol,(o)

uou=y-v-Eg

=lu, Gz (u) .

',I/nitten in thc form of (2,3r(lZ)), 3) 1s

¡, - r(o) ff, [(*"Ã\- á)v cr(Y) '

ß)

(1+)

(5)

(6)k crv2 c, (I) t Ê n" (

)

+3

(1)The clistnibuti-on functic¡n O ùs si milarf;r exB¡. nclcd.

tirus e

r(') (y) = - # t(t ) (v) cr (y) "-#.l

- # t(z) çv) s,([) , * go. Q)

Then follorlring the ttreory of 2.2, subs-bitution of

(Z) into (Ð gives equations f or each of thc f?) arrl

fQ) 1n Q) . The se eqra tÍ ons ere

120.3.1

(B)

(g)

(ov' - å\\ '/

1

otzV rt (t ) ,

ancl-""'* "{r) * K(2) _ ,r (r) _ ¿"çr*gv-: = "(r).o5

Nor,v eguatic¡ns (B) nnci. (9) are integrel cquations of

the forn r,vhich rtye recLuced. fon anbitrary n(t ) urr'¿ ¡,(2) to

diff erential equations in ?.6 and 2.f resllectivcl¡r. Ii'rom

(2,6, (32)), equr.tion (B) in ùifferential form, is

('kx"' + øV- KX, , W1V)

ê+-ÅL¿q

ï,

co6(r).#ffi1 -' (r o)

5V3 e (r t )=-*._-rB (øoF )z

The non homogeneous terrn in (tt ) is derivecl by substi-

tuting the expression (B) for n(t ) in (1o), ancl penformÍrg

the indicated integra tion.

o

It is corrr¡cnient to '¡,r,ite this; in d.ÍmenÊionless

1213 r'1

(tta)

c3 6 , (lz)

f orm ancl rve dc this ity substitu.ting

e2 = ot72 anc o2X(t') = i(").

Then (1o) can bc expressed.

N / ¿ \ - d ru tr,kX"' + (2c - r\ kx" ()N' + 6X = - {ri,\ cl

In 'bhis equation k nrut i'bs clerivative s nne no\¡/ io

be considered as functionr: of thc clilncnsionless vanialrle

c clcf inoct by (2.3, (Sg) (66)) rvith d. = 1: orgr

e(c) = exp (- c") t (13)

ú(c) = exp (- c12)clc¡c

(t lr.)

and-

x(c) = *lQ"'+ t)i,(") + ce(c)1. (15)

Diffcrenti¡rbion in (12) is witli respect to c.

trqua.tion (lZ) can be solvccl nuncricall¡r by a Rungc-

I{utta techniÇuer l:ut before vr/e clerscril¡e this lve must

I

fÍnd tIre 'bou-nd.nny conclitions satisfiecl by X and. its

clen ivat ives,

To integrate re.6,(Zl)) v,/c macle use of bhe d-ef,inition

of X, arrd. its <l-erivatives (8.(rr(zg) (tl¡¡, U (1)

(2.)t,(¡)), ancl y(r) (2,tt,(15)) r¡rhich show,

as V ¡ "" X' - O exponentially like exp (- crv") (t6)

122.-,3.1

(17)

(18)

(zo)

(21)

0

xt" o

rim x(v)

lr It il

ll 1t il

In additj-on on integratlng lry partsr w€ can r,,ir ite

(1)x(v) - vx' -V x"(v) . + Å\"6(u)ø

=*.[* uoe(u)ø(')(u)au

(u)au. (19)

Thus

a

V-co

But the auxiliary conc-lition (2.2, (16) ) w]rich enrlures

correct def initi on o¡ go in a1l approxirnetionsr givesl

rvith the d.ef,init1 on (7) or o(1),

I"(r ) (u)u"c1u - o

oo

e(u) p a

Thus

V ¡ oo r X - O exponentia]-lyr (zz),v

so that X ancl all lts ci-erivatives tend to zeyo exponen-

tially as c ¡ oo r

Thc form of X, þ() ard. y?) aIs o ind-1ca te s that

123.3.1

(4)

( z¿+)

(zs)

(26)

x(o) = o

ancl

X'(o) = a¡

i"(o) = o

i"'(o) = à2 e

vtrhere n¡ and- a z are constants to be dotermined-,

ITow the homogeneous equation obtained- fron (lZ) by

neglecting the R.Il,,S.e lras solutions which near zero are

proportJ-onal to co, ca ancl c3; and. only the solutions

starting lilce c and- cs 'can contribute to thc required

solution of (12), the solution starting like a constant

violating (Zl) and. (25). Vic ctetenninc the bounclary con-

d.i-t'ions rcquirecì. for these two homc)gcneous solu-tions by

consi cler in¡4 a llorfi/cr series soluticn of the homoge neous

equa.t ion, anl then lvith these bounrlarSz conclitions we

clcvelop numtrical solu'cions by ltre Rungc-Ku_tta step by

step integration techniquc from c = O to e = 10. .An

arbitrary soluti on of -bl-re non homogeneou-"q cqurt ic-rn (lZ)

satisf ying (Z:r) ancl (25) (actr.i.n1-1y we start it 1llce c" )

121+r,

3.1

is also olttainecl numerical.llf . Thc homogeneous sol-utic¡ns

ere proportlonal to co sì.n.d- c for large c ancl the non

hcmogeneous is like cr

Now t hc co rnple te so luti on of (l Z) suì: je c t to al l t he

requirecl" ltounclar¡r condi tions t'r¡il-1 bc the line ar" combi-

nation cf these 'bhree wlrich clecre ases exponentially, ancl

it is unique bæause neither of the homogencous sofutions

clecay exponentially, In cletermining this conbinat ion we

are effectlvely fixinfl ar ûrr1 ã2. l¡/e d.o it 'by requiring

that i an¿ i'l¡e zero for sone lnrge value of c = ct.

Lari,,:e in tJ- is sense means that c¡'shouldl be zufficiently

grect so that the solutlon r¡¡hich clecays exponentiall¡t

like exll (- c') shoulcl in fact l¡e zero to the ,lceuracy

'ye work. the vrlue c1 = 10 which vrc use is certainlSr

sr-rfficie nt, I{aving fixecl a¡ arrd- Ð.2 to ensurc an expone n-

tially clecaying solution, vue clevelop the reguirccl

nunerical solution of (lz) anc-L obtain t(t ) froi¡. it 'b5r

r-rsin¡1 (2.6, (lÐ); vrhich in terms or i is

e(e)r(r)1c) = * [î"'(") - i':ts:-1 . Qt)c I \-/ e _J

125,3.1

(r )The function eþ is like c f or c ncar zeyo, and.

it clecreases elcponentially o_s c * o" ¡ We have graphed_

thi's f irst correction to the d-istributi cn furrc-bion in

Fig. 2e ancl it is interes'bing to eonpare this gnaph vr'ith

that of lrig, A1 . which shol¡rs the form of .p(t ) f or

i\iiax'r¡ellian moleclrles. The latter graph is of ,.r3¡r?)

€ trvhich is pl:oportional to thc f irst term

in the expansion of the distribution functlon in ,sonine

polynominls f or an anl:itrany molecular modeJ a.s usecl by

Burnett' l,lle see there is 1i'btre allparent d-ifference

between the form of the clistriburticn furrction for a gsË

of rigid. s_phreres ¿rnd that for I'änxwellian molecules in

thir..; in,s tance ,

The solution of 'thre homogene ous cqua'b j-on v,¡hich is

l-ike c at the ori 6qin is in f act the analytic solu'ci on

i = þ(")" 'Ihu^s (lz) can be rectuccc-L to a sccond- orcer

cquaticn in À'lty su'bstitu'cion of i = ty'^, On d-oira3: ihis

llt^ æa *(f !. tr)v u

1263.1

0.r

- 0.1

0

I¿

c

fr)€0

- 0.2

.0'3

.0'4

'0'3 frG.t

127.3.1

k,lt:^"'+ rr(56 * (2. - fl(/)^"

,!" + 2ek c+ -Ç c"e . (zg)Bæ

There is no second analytic solution of the homogeneous

eqr:ation so that (Zg) cannot be red-uceil to ¿r f írst orde r

cquation rvhos;c sol-uti.¡n wrruld be t::ivia1" ']ljris recluction

of the L.i,i.'i. applieo to .rts rr ='1 equation but in this

thesis we always numericall-y sol-ve the thirct orclen

. s /¡ \equntion in X ard- obt¡.iir ø\'/ bhrough (2il; rnther than

sol-ve the second- orcLer cgr-ration in À'r.,vhose solution

must be integnatccl to oÌ¡tain þ(r )

a

As a check we nlso solve the n - I eqr,rltion in(1)v(l), (2,6, (2t ))' for the partÍeuran F in (B). In

this eE-lation, writ'ben in clinensionless fonm, the

lromogeneous solutj ons ncar zero []re like co, e, c2, and

c3, ancl bhe bounclary cond-iti ons at zero reqr-rire that of

these only tlre solutions sbarting like ¡. constan.t or c2

can contribute to the solul,ion of 'che non honoaleneous

cqurt ion. Accordingl¡¡ we cl-evelop numer ical- so lu'r,ions of

128.3.1

these two homo€Ieneous solutionsr ancl an arbitrary solu-

tion of -bhe non homogeneous eguation (v'ihich we star b

like ea). The homogeneous solution which Ís constant at

the origin is just

it.> = € e

correspond-inll to i - þ, ancl it is pos,sible

(zg)

to o'bt¿r in 'chc

(lo ¡

requirccl exponentinll5r decaying solution of the non

homogeneous eguation by taking Ð. combin.ntlon of the

horrroS-eneous sol-ution lilie c2 at zeroe ancl the ar'bitrary

non homogeneoÌ],s solution, It then apllearÊ th¡.t we can

obtain nny nuniber of soluti ons lthrich are ex¡rLnentiall¡r

cì-ecnying by addinil ¡u}tiplcs of the homogenecn-Ls solurtion

(29). l'Tor¡¿cver in this easc there is a concli tion '¡¡hich

v/e have so f ¿rr nqlcc bec1" It ¿tr ises from the auxiliar¡r

relat i on (z .z, (16)) , ancl is

ï"

oo

v(t)(u)¿u = o ( )a= rim X(V)

V-oo

On enforcing this øn,'lition vre f ix the nlultiple

of e which can be aclclecl 'c,o thc exllonentially clccn.¡ring

eioluti c.n, encl so obtain n unigue solution i¡¡hich satlsf ies

129.3'1

ar-r the requirements. The resul-t obtaj-necl for t(t ) l¡y

solving this n - 1 eqrl,-,bion is just that olÏb.ri necl in

solving eqr-ration (lZ) to the ¿lccurtcy u/e vrork ¡-s of

coÌtrse it shoul-cì be. In Chaptcr 4 vrhen ,ve d.eal v'¡i'bh

othen ït = 1 equations in the third" al?proxim¡-rtion to

Roltzmann's equatione xre always u$e tJrei equailon i^ather

'bhan the y(f ) equation, a,s 'bhe la'b-ber requires thc

adcLitir-rn integra.tiOn (lO¡ to be perf ormecl in obtaj-ning

the requirec-[ so]-ution.

Ilaving solvec-L (B) ancl o¡tainecl ø(1) vüe norï proceecL

to solve (9) to obtai n çQ) irncl so complete 'c¡e slpeci-

f ication of *(t ). The cîifferen-tial form of (g), ruhich

is fcuncl 1ry using (2.7 , (ZB) , (Zg)) anc'L 'bhe e4r lÍ-c it f orm

(c\of F\'-' , is

' (L) '-¿(v(r)t'*', Yçz¡"', Tç27"' Yç2¡" u(r))

= .t_-, d 114 / ,{+ n(z)Ì, = y;S . (3t)= "rv) dT vL.lv \t"-trtr /\ *ão

Theind.icia.l.oquatlonofA=osh.¡wstlra-r,the

homogeneous solutions near the origin are rilie $ , Vo,

130.3.1

Ve ancl V2 o Since tit,: f irst of these i s not anenalcl-e to

numerical- solr-ltion from a starting value V = Oe the

equâtion soti,f ied by

,e = vve) Gz)

is solvecl numer icalJ-y. This equation is

. ll,L )ks.*,* e.,,ftv" *Æ.*-#)r. (ç.#)'l+ s"f(z'vo + Dv, - *,. fr)ø . (t" . T - 4fu)'l

+ u'[(r"u" 5v + * - #""')t . (v' - â. #.,o')'l

+ u[(e"u' + s - # * tå"o,.=)ø . (t . *, - r#-u=).]

5eYz=R:. 3l)tfa

'vl¡e put this in cÏ mensionless form by substiturting

c2 = oLr,rr, ancl ã(") = o3/r*(u), (ltt)

anc-L this ha. s the ef f ect of putting oL - 1 in (Zi) .

Differentiation i,s nov,¡ v'ri'LÌr respect to crancl k(c) , 'l'G) t

¿rnd- e (c) are clefined by (2.3r$g)' (6o) r(çr) ) wi'ch a = 1'

as 1n (tÐ, (rh) ¿rn<l (t5). rf wc let Y(;(tl) , å"', Ë",

i', ãl be the l,.ITn[J. of (lÐ in climensionless f orm,

the eguation to l¡e

-/1, \ NY(gt t', g"'

so lved is

g t1 I tJ5ec2

Éa

131.3.1

(sn)

By making use of thc form of 8g 3Z), and- Ve) ancl

^lJ of (2.3r?5), (¡)), the bounctary condltions for õ

)9,

(g

(2)

and. its clerivatives are Íìeen to lte

I

g'(o) = âsr

I (j6)

zero exponentiallyGt)

o)o

o,ã,- ( o)

at.t \?: (o/ = ã4t

(¿r) (o) =o,

and. g ancl al-l its deriva'cives tencl toaSC+cX)r

In ßA¡ â.s chd- aa are const¿rnts to be cleterminecl.

From the lrtlici.al equa't ion of Y = O we ,see thflt the

homogenecrns solutions s'bart likc co, ct e?' ancl c3 at bhe

origin, anc-[ the concli'cion 3e7 allows only the so}-rtions

like c ancl c3 neâr the ori¡iin 'r,o contril¡u-bc to thc re-

o,uirecl solutir:n of (lS). i¡/e obtain these tt¡¡o Ìromog;eneous

solutic.rns numer jca115i b5r Rurnge-T(utta technique , nncL

132.3.1

sirnilar.ly olttai n an arbi'cnary soluti on of (ln), urhicìr we

start at zarc¡ like c3. Tt-ie comltinat ion of the se 'chree

numcrical solutions ivhich d_ecays exponentially as e + oo e

Ís cleterminect by Îlxinp, as ancl a4 in the sarne manner ac

v'/e obtainecl a1 ancl a, of (Zt-r) anct (25) in the n = 1

equrtion. Then intqrntin¡i from c = Oe wi'ch tire boun-

clar y concLiti ons (16 ) knovi/ny v/e obto.in the re qLl irecl

numtricaf solution; anct f rom it øQ) from the rela-tion

(2.7, (lo) ).

In terms of ã, through (Sz) and (3t), this is

.(") ø(') (") = å fã.. {") H#). *"!*]. (lB)

(z)The f\rnct ion e þ v,rhich is like c2 f or c near zero

airc-l cÌecrerses exponentially as c * oo , is plot'i,ec1 in

Fitl. 3. In tr'i¡1. A',2 ure il¡rvc plottecJ. the graph of

ec2S 5/rþ) = €cQ r¡¡hich l'-s Proportional to the first ternr

in tlre Strnine polynomial ex;rnnsion of ,6Q), ancl which is

the only tenm r¡rhich contnlbutes to ,øQ) ,or liirxr¡,¡ellian

mole cu1es. On compar ison \Me []ee there is little

c-tiff er ence in f orm Tre t'¡r¡een the clistril¡ution funct ions

for both. tttese exaetly l(norifn solutior].s.

I a t

0

0.0

t 'gtj

@¡

0

0

'11.0

I

l'ççÇ,

9t.0

131+.3.1-2

Since tÌ-re homogeneous nolution of A = O which ic:

Lvpropontional to cloes not contribute Lr: the regLlirecl"

(2)solution we cculct have solvecl (ll¡ to ol¡tain þ insteacl

of ¡ioing to (lE). Tn fact we clid. this also ancl both

ansvrers for fQ) are founc to -t¡e in excell-ent agreement.

I.,ater in Chapter h, where we eml-tloy n - 2 equ-a'bions to

get soluti c..,ns of the thirc,l- apllroximation to llol-tzmarutt s

e qurtion we always r:rie equations in g ancì. its d.erivativ€s¡

t.2 !gtrr.sÌq-q_t- 9qçåqipj,q$5i

(1) Esc-qnê Aptrr o;cim aiion to bhe }Ieat ]Ilux V rt,

Thc s ec ond. a!ìr)r ox irnati on t o the irea t f hrx

vec tr:r is

n(t)=+ [r,^,0(1)v,yo_v, (r): 2./ -(o)- vrò\¿r

/¡ \Af ter subst ituting from (r, (Z) ) f or o\ ' /, ancl

ne¡ilccting vanishinll integrals of oclcL f\-i.nctions of 'bhc

corni.''onen-bs of Yr llre ca"n write th in

n(t) -- # [ t,.1ø(t ) y' # v' I c1v,,J

(t ) (c) csclc = À òT

(z)

rò,#Gf" #[o*u*n (- c') ó

ct3

wlrere ^

, tlrc coefficient of ther.mal concluction in

Founierrs lar¡¡ f or heat concluction vlþich we hrve Just

pn ocluc ed-, i s ¡15- ve n b Y

155.3r2

(tr-)

(5)

exp (- c')þ (t ) (c)csac " (=)

U si ng t hc numer Í c¡. I so luti on (zl ) f or ,þ(r )

ol:tn j-ned- in Section 1, \Me can perforyn thc lntegration in

(l) numer ically using Simpsonfs rule and find-

:

"=yÊ(#) ,[*

^ J5, iEir.\å [rÀ¿=6æ\ffi,,

^=#6=f x0.637153

- J5-= Ælå\å x j.oz5z2 .= 6l+o-z \rrn /ll.lhen the coefficient of 'bhermal cond.uction is

(1)obtainecf as a series, af'r;er þ has been exllressecl as a

series of Sonine ÞoI5r¡e¡nials, 'che expression f or Àa t the

value of X up to ttre fourbh tenms in

+ '02273 + 'oO2o9 + 'oOO31 ] (6)

0)

Chapman ancl CowI ing, Chapter 10, 'Secti ort 2.1 t

1

\2\ 1. 02513 .JL /K"rx.{oU.Í' \øm

remark that 1X-ris series is napicÌly convcr¡1en'ü anc-[ t]re value

given Ín (7) can lle supposecl correct to wi thin. '1',,,,.

From our exac'6 calcula-bion (¡) we can see thnt this

estimate of ihe error in the alrproximate calcula-bion is

suff ic ient.

(2 ) S e-ç q-ncl -.Allpq o¡i{La ti _o-q.-La_ :!.ltç, -PJ.çSSgq q-jl-e-qs-9=q

The second ¿ìllproximation to the pressure

tensor is(r ) o(1)VVdv e

136.3.2

(B)p = lll (o)f

(1 )which r¡y(hen we sul¡stitute from (t, (Z) ) f or <Þ ancl

ncglect vanishing intqrals of oclcl f\rnctions of the

components of V ise

(r )1

m4?-.

ð,o-2f þ(o)

(z) (v) G. (v)Àåc VVd-vdr*o---p

= -EP(#)' L-"*o (- e") *rÞ ead'e; , 0)

or,vhere g ¡ thc shean viscosity tensor, is a symìnetric non

cLivcn[{ent tenson formecÌ frc-,n tlre components or $ go.

l\Tow in the Navier-Sto1ces equation of macroscopic tran.s-

(r )o

= 2ng, (r o)p=

p or t lXre or'5r vve vvr i t e

137.3.2

\dhere 4 is bhe coeff icien-b of shear visc os i-by. il'hus in

(g) u/e Leve d.erivecL o value of -bh.e cocfficient of shear

v is cos i ty,

2 /znltrlå î,=îæ- (*" I /"c\o (2)enp (- c')þ (c)"4ô-c, (tt )

The integral is calculatecl numerically using

Simpsont,s nule fron the solution / (¡A) of Scctiori

1, the result being

(2),

1

2 /2mL'T\2, = Tþ (-f) x 0.561285

'l

5 ncTm\ 2- +r: i Èt-Ëi\ x 1 .01603 .16f \ ø t\'- /

5 /t¿rm\zTtd = T# ( " ) [t + 'ollr85 + 'oo'loJ + 'ooo12]

1

= å (#)" 1 '01 600 '

The series expresßion for tì-ie coefficient of

(z)shear viscosÍty obtainecl r¡¡hen ø is exp¿:ncled- in

Sonine polynom.lals is, r.rp to tile fourth term,

(12)

(13)

(r ¡)

(t ¿r)

This is in good. agneement with ot-lr exact

ans\¡rer t (13) t certainl¡z at least within the accuracy

Chapman ancl Covrling gtate qf .1'/o.

138.3r2

The resultfl of Co'¡ter for the coef ficient of thermal

concluction, and Pekeris and Alternan for lloth 'che trans-

port eoeff ic ients ealcu.l_ated_ in this chaptcr a[!ree

exac'cly !t/ith our ansyIers. Thcir method of alpproach VúaS

,tihe samc ari thnt which ¡¡.¡c h¡.vc usecl in tira'a 'Çyre apl)rolriate

integ ral e Ef,a'ui on was re clucecl to Ír- cliff ercn'c j-41 cquat i on

lt¡r 'bhe introcluction of auxiliary fbnctions. Thcn '¡he

coefficien'ús wcrc oþ.Ûainccl cxactly by mlmeric¡-ll-y inte-

grcting -bhe exact $oluti on for thre clistrilluti-on f\-r-nction,

r¡rh ich v¡as ob-bai nec-[ b5r solv ing nume rica11¡r the i[ iff eretrtial

cqr-ra'c i on. Cotier t s recluc'ú i on of the integ ral eqr-trt 1on

f or heat concltr-cbion, ïl = 1e WÍr.S Very conplex arrcl resirltecl

in a Seconc] orc-ler c]-iff ere.n'cÍa.1 cq111r.ti(.)n essen'ciL1I5r

e quiva t-ent to (f , (Za) ). Ä.s â reËLr.It of integrn-bing tlris

equation he obtainec-L the coef ficient of thernal- coniluc.bi-

vity whj-ch is exac i;1y (5). Pekeris ¡incÌ .1l'bcrman 1n

,¡lreir nec-luction to cliffer.entiûl form of -bhe equations f or

he¿t q-¡ncl-uction ancl Shear viscOsity, uscc-l anxiliary

function,s which \ryerc rather l0ore partio-r1an than the Very

139,3rZ

genenal ones l're intnftluccd i¡ 2.1. The¡r ol:'r^:rined- fourth

onder equations in both lnstances, ancl integration of

these enerbled- tkrem to calculate the transport coefficientse

the results being exactly those of (¡) and (13),

Our necluction of tire integral egu.ation applies f on

arbitrary n and" so can be used. to ob'bain the cListrilluiion

frrnction lvh"rr O(lT) involves spherical- harmonics of ۓny

orcler. ,li/e sha]l in the next Chairten capitalize on this

arì.Vantag-e wlten we calcula'¡e the clls'crilli-rtion function in

the 'ohincl apirroximat ion to Boltzmannr s equaiion.

1 l+0.4.

CHAFîER ¿}

TH ILP,TRO TON oTIt zl'l lA ATÏ

The thlrd app roximat i on to th e di s t ni]cuti on funct ion

vrhich is quad-ratic ln finst ord-er spatial derivatives of

c1¡ T and. por or contains second. d.erivatives ,¡f these

macrosccpic V3l1 iableS, involves spher ical harmcrnics of

orcler ri = o 11 ,2¡J ancl l-f, In this chapter vüe shall use

thg c-lifferential equations of Chapter 2, Sectj ons 5 9t

f or these values of n to olltaln soluti'¡ns of the third

apl,Toximat ion to Boltzma.nnt s eguatlon' Inte¡5ra'cion of

tþis seconcl copreetion to the local equilibrium s'cate to

d-etcrmlne the heat f1u-x vector ancl pnessure tensor era.bles

us to o'btain corrections -i;o the linear trans't¡ort egue;b ionst

the equations non linean in ttle grad-ients being knovrn as

the Burnett eqriaticlns' There is an alternative method- of

calculating the Second correetions to the preSSU-re tensor

and heat flux vector '¡drich 1s given in Chapmori and

Cor,vlingr s book and 1ul1ich cloes not invol-ve calculating tf€

scc orrl correc t ion to th e c-Li s tribu-ti- on ftrnct iono Ii/e sha1l

141 .l+''1

f ollou¡ this,initially and tñen la-ber solve the cliffenen-

tial eg¡ations ancl calcul-ate the momerts by in'bqgration

of these soluti-orlSo Finally, the resLllts of both these

exact method.s are eomparecl V'¡ith the approxirnnf e val-ues

obtained. by usirg the ,sonine polynomÍal extrnnsion of the

first cornec tion to the loca1 equilibrium state.

4¡1 The Form of A. fon the Third. Altpr

The third aptrrroxima.tlon to the clistnibuticn fi-rnctiont

r - r(o)(1 + r(t) + Õ(2)) (1 )

is o'l¡t¿únecl by sclving (Z.ZllO)) uiith N = 2, nnmel5r

^¡t a

"4. is d-ivicled into thrce groups of terms, 'Ilhose involving

only scalar V are

ß)

1^n

/2(")[3 c,,Yz 1) l---(t I .è ,- - .-è . "(t )-115 òr åo - òn 'ìa Il._ .)

Òr_Òr

A' Òp+:--C|m dr t (tr )

Those of oclcl ólegnee 1n V are

Òå òr.

(y, y, ' g)ðnòg ?s'v+-:d.m - or

X'

oe

:t*) ) - z o ('-\

ò4,ãv?

142..:1.1.1

òTòg

+ A'g /Dn\Dr

+V2

òîòg ar t{l Òf,

òg

p IJ

))

2òA,'ov'

òA'_òT

(r

/ò\¿r

(v

/òT\(Ë) c òT

òn

) +2 ùB:

drn òV2+

;).

o

+ A'vv' È G). ##uo' #Ë. SFoo' Ë Ë

o

=) +

/òB'(ãr \ty y , lllv

/\

+J J

ancl flnal-ly the tenms of even clegree in V- are

a'ì o

$ er'Ytrr,ol"'yy,,ëoiv, .#),i -- =\*/

å ^ (, #*u' #)tuu, !l + B'ss' èËo

2e

(¡)

)2

ò8,ãîã (y g ' s)(g v : e) + r

+ ,r (B'Y- Y , g Br'IlSr ).oe=

Chapman ancl Covulin6i (15. 3, (tt), (lz), (tt't¡¡-

o(6)

These expressions (ll), (¡) nrrl- (6) f or -¿\" come from

11+3.

h.. 1

rn (tr), (i) arrl (6),

òãE

p=clkT e

e'(v) = - â

A 0)

(s)

(g)

(tz)

Do

ffi =(*.n" òòr )

¡(rçl ) = c2 t t ^€'s (e(u)ç(g) '¡'(!o)e(go)þg '1'¿ ,./ J p. e>o- (r o)

(t )ancì A' ancL IJ' irnrolve þ ard. ø nespective Iyr trLrich

are the solutions fcuncl iu Cha-,:ter 3 for thc second

ar¡proximation to 'ü,he clistr iltuti on ftrnction;

(i t )

(z)

I

,

ancl )2

ð"æv2 a

Iir (t t ) ancl (lZ) arrl in subseqr-re nt parts ',i'heire \¡¡e Llse

A' anc'L B' the superscript iloeS not clenote iliffcrentintion

but merely a neìJr r¡arinl¡l-e.

Thc terms of the f orm (to)r in (¡) anc-L (6), involve

(1) (z;oroclucts er¡-1 sguares of þ ancL þ 'rn/c clec i0ed- 1n

Chapter 2 to i¡;nore such qurclratic terms¡ arn-[ rften this

point, vlr- ere tl-rcy are gj-vcn fc, r comDletcncss salccr Iilre

)a

she11 neg]-ect them.

Nonr A' can 1lc r,rcittcn

x'(c) óÍ]-rc:T3e

14j+.4.1

(13)

1r u)

(1 5)

(16)

(17)

Nt'.F

:.?l."i7c¡'er

I I

where

where

"=Ffu (å)"'

and Î ancl its d-erivatives are the values founcl vvhen v'¡c

nuner ica11y gefrrecl 3t I , (12)) .

Similar 1Y,

¡ -ecdsfl-c¿1e c3

bF +

r'; =#;(*I ,

and. I an¿ its clerivatives yúere o¡t:rined numcrically '¡lhen

v,rc s olved (3. I e 3¡)) .

Fnom (1L) arrl (16) uie can expressthe tæms c¡f ^

invol-vin¡¡ d-eniva'i:ives of A'¿lnd B'rrrrlth rcspcct to'-ll and

V, in tenms of the kno¡n s¡lution i an¿ ã. By carryini5

out the cLlffenentiatir-rns, remembering c = c(T)r !'üe finçl

and.

dÒr

ÒA',æ

ò8,òT

4Òv2

Also fnom (18) ancl (lg),

T +V2

ancl fnon (zo) ancl- (zl),

(v,r) =#(+ .*;r-#-),

=-#(#-%.#),\z'

(v,r) = #4 (V W. #¡,

=-z# (v +.+*-ry).

145.ll. 1

(rs)

(r g)

(zo)

(zt )

àÀ1ÒT

èB_:òT

(zz)

T +V2 òB' -^.ãffã=- )b , çzt)

Thesc relaticnships which are true for a gas of ri¡1id-

sphcres are not tnue in ¡.Je neral f on other nolecuhr

modlels.

It ls not necesssny at this point to t¡vrite À in

tenms of the tensors G' of (2. Zr(ll )) ancl À'pcnd'ix 1,

AS Vte inte nct first to calculate the thircÌ ap.j-)roximations

to thc heat flux vecton ancl irres$ure tensor by e method.

v¡hich c-lcres not involve lntegration of tlrc thircl appnoxi-

1l+6.Lç.1-2

mation tO the cli s tributi on f\rnction. Lat er vrhen we sÔIve

(l) to obtair. o(2), we rn¡ill of course rrv:rite '¿\ and- o(z)

c-Lv (t )

ln tøms of G fol-1ov,d-ng. the general method- of Chapter 2tn

Se cti ort 2.

4.2 The thir d- Airpr oximat i t e tFl cc tor

chapman ancl- cornrlin¡1 shor¡i¡ thnt the thircl approxi-

mation to the heat flux vcctor is

n(z) = s I t,.rv'y dgé. I \-l

1

= pd"' t(z)

)A I(O ,

a result prove c1 'b5r mlking ¿se of thc auxilinry rel-ation-

ship (2. Zr(16))r rvhich cnforces correct c-Lefinition of

c^ In each ap¡,tnoximatic,n. AIsc,-o

certain'bheorems on

anclintqrals are used'

l+.1r, (Z), (B) ).

In (t ),V (z)

(3)

Now  - - dzr(o(')), so using (1 ) ancl (z) '

1Lt7.4.2

(4)ê_ ^t(o)

since A' is proportional to 4(1) arrl in Secticn 1

(z )-kr2 f3V clv

The intçg;ral (h) requires knorvled3,e of only the

seconcl aptrnoximat icn to the clistributl on function, O(r )

,

TVC SAVU

 involves only *(t )v and ø Thus it is possible to(z)

a

perform the integnation in (4) numctrically, using the

exact solutions of (3. 1r(12)r (¡¡))r ard so ol:tr jn an

exact ansïver f or n(Z). Chapman ancl CorvlinES havc usecl

only the first *""* in 'che Sonine ptllynomial exìransion

of f?) ard. ,Q) to calculn'ce the integral (tl), ancl v¡c

shall compare the resul'cg of their calcul-ation with our

exact answers in Scction 5.

Because $ is of ocld. d.egree Ín Y¡ there is no contri-

bution from (t , (4), (6) ) to q Follorri ng ChaPman ancl

Cowling¡ ïife will consicLe r the inclividual terms nj of

2( )a

(1r (5)) in turn, ancl calcu.late thc

eac h.

contributi "" 1t3' from

( t ) tr'ins tIy we ha"ve

L -(t )

148.l+.2

(5)

(6)1

òv òg't(o)p

Then on suJ¡stituticn of (¡) into (4), and using (l , (t t ) )

\¡ue gct

:,(') =¿n 1",",ø(o)üs*,!(t) ou

Iòg

clv.[",o¡ø(t)u

(t )P

The integnal vanishes, a consequence of -b]re auxiliary

ccrnd.ition (2.2, (le ¡ ¡ t so that

qr(z) = o. (7)

Thus the first tøm cf (t ' (¡))' r¡rhich is the onl-y

one not involving the scconc-[ epproximation to the d'is'cri-

bution fìlncticn, gives rlo contribution to the hcat flux

vec tr¡r.

(2) The seconcl term of (f t (¡) ) invo]-ves a proc-Luct

of spatial clerivat ives of T ancl gor ancl is

I\z = - f, x lt #*- ' v2 ÈA'-\ 1r òr t<

5 \ or'-* ' òv')t 'Ë' (B)

I,4aking use of (l ,(ZZ)), this 1s

Lz=2a.'au #' (9)

1l+9.4,2

Then, using the expre ssions (l+) anrl (g)r lve c¿rn vrrite/¡\

thc contribution to q\t ) in this cÍÌse as

q r(2) = zw2 L tÀm

VV'*.'-dv.dr

(1,1 , (f z) ) by using (t, (th) ) ' and perf orm ttre inte¡5ra'tions

We ex¡rress this in terns of Xe the sc¡lution of

Do

õE'

.rven the an¡11es to oJ:taint

(z\ g / t< \ ^ òT r* c't-- /i."t lot\2z"'' =* (-þ) ^#./" ;f"T("--F¡u"'

(10)

since the integrand. is knorrn numff ica11y there

is no ctiffiorlty in perf ormln,g the integratlon in (tO)'

Using Simpsont S Ï"Lle f or numerical irrtegr¡f ion lre ob'cein

n_,(2) =1.17).82 (-r#)^#. (tt)

ß) IText u¡e c1cnl rvith the tæm

.As = A.v, /þ fèr\ - (4 "\ . +) . (z)= rr v - \ot \¿g/ \òg =o/ Ò!-)

The expression (#) can l¡c written

D\.)

Dt/òr\ 2(ãã') = - 3

òròe

èÒg )

c a 4òg

(13)

1 50';4,2 "

(see Chapman and Cowl-ing (l5,zr(B)). Thus the contri-

(2''¡ution to q' / f"nm tIlS tcrm ínvolves pnoc-lucts of

spatial clenivatives of T ancl gor anl also sec oncl cleriva'

tive s of c^. The tcnms (9) and' (lz) havc bhe same f orrn

so far as yclocity clepe nclcnce is concerncc-[' so thet

= ,587\11 d'T Lk (BË)/ì-^\..4.1\òx =o/ òs.__J

a

(tr) the founth term of (t, (f )) is thc first tcrrn

invclving B''

(2)Q¡

(14)

(t ¡)2 òB' t- o /r\¿ = á ãtr (Y, g ' g) (g

g::òg

on sLr.]rstltuting (15 ) Ín (tr) ' ¡rc tict

:(r)u = q;3 /ft # (y s, !l u u # .'".

The values of A' ^"a ffi ^"" subst itutcd fron

(f , (f h), ( 21)), and' tlrc integration over an¡;les calcu-

l-¡.ted. to gi ve

(z) òuòn

x"e"

¿_T5qr'

(# a

)

/#..t

\c-O ¡@el

./o

(16)

Q¿

#"n1. q# - #¡ "" .

151 .'\..2.

Thls time the lntqral involves the knoin¡n

soluti ons of b oth (3 .1 , (lz) , (SS)) r rìlrd once a¡iain we

calculate it us1n.¡¡ Simpson?s ruler obtaining, f cr this

contrlbution to q

(2)Qa

(z),

,252078 a (17)

(z)In (17) ïve havc n contnibutionr 9a , depending

on prod-ucts of spetlal clenivat ives of p ancl $or so thiìt

since Þ = dkT, a grnclient in number clensityr ancl mean

velocity will pnocluce a contribution to the heat flux

vec tor regarcl-less of rnÀretircr T is constant or not.

(5) As in (tr-)r the f ifth term of (t ' (¡)) clelrcnds on

spatlal d.erlvat ives of p arrl co rnr.l invo lves B'i it is

20As = =- ß'Y 3f : c. (18)

Calcu]-rting the cclntriblrtion to thc heat flt]x

vector ln the ,saae mar!î.cr, using (L-), (f B), (l ,(lll)r (16))t

anc-t evaluating thc integrnl in (19) numerically, $/e finct

A:u-t(o)I vv @,3\oo

-- \or =l\- /(z\ zr{t2

^ \ I _

----

.95 ' clm

2

='rf H*) #' 9Å-(ä;:- q'-) (Ë %

152.4.2

qæ)

(zo)

(zt )

+Ãn

= < .o6ooz68 (#o) #oe

oe=)

(z)Then com'bining qa ancl Q s lrj,e get(z)

Qd +9s = '31?-105OJòn

(z) (z) @ a

(6) The sixth term of (t, (¡) ) is

¡oÀ6=B'V-'*(UUt:) )' (22)

so that from (l+), (zz), 1t , (tll ) ' (16) ) \¡¡e have

åþ:t (o)

ò t.¡t¡ o

vl/. * (v v t 9) clg+ +- Of,

l /.ET-\ è o r* /i"'; - #l # #= # (õr) E' 9 /o (iä- - c3 / \""

. 4\ ='& dc . 1zz)' e") €\e

Agai n ],ve eva].uate the integra} in (23) by

slmpsonts rule since the integrnncl is known nunericallyt

qu(z) - r.r" t

and. obtain

(z)Ço = '214.3l,+BB 9

oe

Òj[.

Òg

O/

e) (s(y, s

c-Lvar_òr) yu '

/kr(ffi=

153 ¿

4,2

(zt+)

(zz)

(.2)a contributi on to q .'ivhich d-epencls on seecncl deriva-

tives of c-.

Q) rhe last term of (t ' (¡)) is

lrz =2òA'îrã /\ ?

so th¡.t golng thnough the samc procec.lure as with the

previous tenms \rue ean writc the seventh con'r,ribution

(z\to q' ',

y,Q) = kr2 I* (# - 3#) (v s' ;

= #(#) Ë 3 [-[u(%- #xu.Ël # å-)fil

/kFdF

X(r )

c"

(26)

= 2"s3877 (zt)

another contril:ution lvhich c-lepcnds on llrcttucts of spntial

clerivativcs of T tncì- cn.

Itlsinterestlngtonotethatthesecorrec-

tlons to the lreat flux vcetor which Íl're non line¿lr 1n the

)òTÒn ,

oea

15\.Lt-i2-3

gradients involve in each case spatial d'enivtrives of $or

the nE an ve l-ocity; s o 'chat if the g3' s i s at re st or in

unifonm nrotion, n(') = o, Thus up to 'c,ho 'chird olll.'roxi-(t ) - LÒJ,

mntion -r,hc hea'r, flux vcc'r'or is Just i{ivcn n¡t !. = - -ÒI

in this cnse, on the oLher hancl if there is no temlJerature

grnd.ient, F,!o that n(t ) = oe trere can st ilr -e a ührermalo

fJux if c1-r,hcr d. (l*r.1 so Þ the presourer P = dkÎ), A gr 2

is non unifcrÍr¡ as shown b¡¡ (zl)' (rll) and- (1f)'a'na (zt+)'

t+,"3 &s.:rèic"g-AlrtJ.o¿!"!Ì9'!jp-n- *t*a- the e ss Llre Te n$ or

(2)Tnasimil¡rmarjnertobhe!v3'}/inrn¡hic]rq

i,,,ritte n in terms of ,(t ) and fQ) in Íìccti on 2, it cnn be

sho,¡¡n (oee Chapman ancl Cowring, 15'3r(2)) thr'¿ thc thirtl

approximation to the pnessurc tensor 1s

g(t) - m Itrr¡ y. Y dr.t

= p f e r(o(z)¡av-,-l=Thc auxilinry relnticrn they use in this cnse is

(zrz" (17) ) vr¡hich ensLÌres corrcct clef iniiion of T at eacli

stage of thc epproxlmetion to f '

rn (t )

155.l+13

d_B' --q,Tì-LJFs.:.-= It \\o/

(a )

(l)_22

a

KT (¿+ )

The lntegral (4) delrcncl-s only on thc solutions of (3'1 '

(z)

using .i\. = - ctrr,,(*(z) ) an. (i ) ancl (z), l¡rie can ,f"mitc

f-P' n vcv av ,JÍ'( o) - -

(tz) r(lù)r iust as the inte3ra1 (2, (tr)) for q cli c1 ;

sinee B'involves ø(z) ancl À involves onlv t(t ) anCr /

2( )!

(z )a

Thus the cnlqrlntion of P fo11or¡¡s a sj-mi1af Pa'l;'r,ern(z)

(2)tr thnt of q in Sectiort 2,

on15r those terms of '/\t (1, (6))t rr¡hich nre of even

(c\cì.egree in I contribute t.l p,t-'. Thc tøms '.r¡hich ¡.re of

odd clegree vanish as thc integrand in (lr) is thcn of oclcl

cLcgrc;e ln y, ancl tlre scalar tæms (t, (Lr )) givc no con-tri-

l:utic)n since for orbitrer''/ scalar n(fr)

I n (v)vt'v clv = o.J'The incllviclual terms, A* of (t

' (6)) nre novr tnkcn

J(z\

inturn,and.thecor.responclingct¡ntributionsto!.,from. (z)

these terms we clenote bV P j ' '

156.4,3

(r ) The flrst term of (1, (6) ) is

A., = _å^e èB_1òT

o)3ãva+V" (u

)

oeV ) (¡)

o

= 2 Æ'-'{-U : e , (6)

where ln ,¡r¡riting (6) we have m¿rclc use of (l ,(ry))'

Then in subst ituting (6) into (LI) 1ì'Iê fie t

(2)pt

ruhich, when u¡e pcnform tho integratiÇ-n ovæ angles ancl

(1.1 , (sn)), is jusù

= 2kra fff u"u (u u : c-tl,r)

u$e (1r(16)) to expne€is B'in terms of the solution g of

oe

s,(" =# (#) ^; [-.Ì:r (F # +\2\a"7r¡.4

c" a

The lntegrarrlis known numerically, so simpsonts

rule is usecl to ev¡.Iunte the integral, the result being

(z) /m("rF

(z)

(B)!t = 'lt.O9O1 B

.o\¡u,/=

whlch is a contril¡ution to the preFsure te nsor clepencling

on procLucts of spatial clerivatives of eo'

157.lr-.3

(z) Next ve co nsicler'

^-rr..Ðo!\2 = ,'g y , (ut t 2

o .ôê.'=-C= dr*o )

a (e)

(r o)

o

Thistæmcanlleexpresseclintermsofspatial

d-erivntives of È- å S\.ovhere I is the erberrral-foree acting\n cLm òL/"

on the gi4grand procluc'cS of spatial- clerivatives o¡ 9o (see

Chapmm ancl Cor,uling ?5.2, (g))r Jrut we kecp t'he prcsent

f orm f cn convcniGncc.

The f crm of (g) is the same as (6) so far as

vel-oc i t5r d-e penclc ncc is concerned, so that

(z)pz '2Ol+5O9 3r

=/o

¿c=

a

llen e ihe notatiot * clenotes â symmetric non

rlive r¡ient tensor. in wr itlng (1 O) u/e have also useóL

o.oooer*O=C.e

=or-o==a symmetric non clivergent tcnsort

n result which fol-lo'¡vs because I lsI

0

3) The thlrcl- term of (1' (6)) gives rise to n

eontriltution p"(2) involviry:* second- clerivatives of the

temperature. Tt is

ðrÀg = - /-r'gg : (rt )

158.l+.3

Then following the same pattern as fon previous

.,(2) = kr 1*; uoY v s ' È (Ë) "t ,

(z\ h /r<\fÆ"-L\ f-Þs" = n tvp ) ã.ã \ãr./ /.- 5n2 \l'" / 'z \-Ë",/

(t:: - 8;VË-::I c, c3 /\c3\ /\

which on substitu'cing fron (1, (14)) arrl (t ' (t6)) ancl

performint the inte,gretion over an¡1Ies iso

terms vre ca. n v'¡r ite

(13) numerical.ly is

(12)

(13)+ a

The rcsult of perfonmin¡1 the integration in

o

pu(2) = '2Lr3,,BB (*+) È (Ë) . (rh)

(L ) 'rhe fourth tcrm of (t '

(6) ) is

^. =4"4u,r, è!+, (1b)rL¿ = ¿m ¿V" å r- ÒI òå

so thnt usiru (tr), (r l), (r ' (t 6) ) ancl (t '

(.i9) ) ' ancl

calculatin¡i the inte¿ral ln (16) numtrica113r, we fincl

(z\ 2:lr f B' g$i vov v v . --æ Q! u,,P4' = -m / ff"l òV2 å .L å å = bI òg **

159.4',.3

o

(# -&;:.Ë)Ëc6ãGT

*:s-*-

= .o686i63 (*r=) Ð# .

It is. òA' -. .. òT òT¡1 _-.- V V :

----.¡¿15 - òT r- -L òI òf

so t 1n t sub st itut ing (t e ¡ in'c o (l+) 1.'r/c ile t

EuQ) = o, I+ # Y,"g g, u ' # #'t-.'-'=- "l t'(

o ) o'r

Then sulrstitution from (rr(1S)) anc-l (t'(t6)) fon

4;'-þ+ d.c (16)

(tt)

(18)

ot\òT

This oontribUtiçln to the pressure tensor cìeì:rencls

on cLerlvatiVes of p ¡.ncl T¡ ancl since p - cllcT this 'uerm

contnlbutcs 1f T is nct eonstant or if bcth cl anrl T are

no'ú constcnt.

(¡) on thre other ìr¡rncl the fifth tæm Â5 of (t, (6))'

/o\ancl so p u\' ' , involves just th e prccluc t of fÍ rst or c'[er

cl.er'lr¡at ive¡ of 'll with eac h other.

ancl

B'gives

(z\ z / lc\rrs" =-:T t¿iFf')t'tf' \ '/

-c-6-- /eg¿ n Ê'::'ãfð \-- " *;t-

#.') (Ë '# .#)

ãr*€

i 60.Itr,3

1r l)

(zo)

(zt )

c1C ,

whlch on numtricel- ewtluaticn is

(z)o

!s = .0397221

+

a/k\(*ñ4') òr òr

(6) Finnlly

ancl usin¡¡ (zl) and. (lr.)r we ha.ve

pu(')--2kr fji__-t(o)

ÀDt o

i3=' v"v (v v : eov' =

(r,(zr)) rcr R'r"a l$ to giveuv o

),

o

=)(ss ) cl-I .

This is evaluatecl after using (tr(t6)) ancl

EuP)-- #e(#);';.#Xe.6 -øi;-

=.76s1ttl (#) îÏ

(zz)

o

(zs)

a contriþuti on cleperrling on products of clerivrtlves of

t

C.-o

161 .4.3-l+

The seconcl approxirnntion tc the pressure tensor ancl

the third- approximation to .bhe Ìreat flux vector vanish

l,¡hen the g3S is at nest or in uniform motion a's lre have

Se en. I:IoWeVer ',]te resUlts of -Lh is Sect iolr sh oVr¡ ttrllt

there is a coniribution to thc prcssure tensor pro-

portional_ to (") proclucts of flrst derivntives of

temperature with each othcr ancl with the first clèrlvn-

tive of pressurc p, (¡) scconcl c1erivntives of the

t emperature nncl pre s surc a ancl ( " ) f irst clen ivatives of

E; in ad,clit ion to con-i;ributi ors cì.cinnclin,.¡ on prcc|-rcts

of cleriv¡Jives o¡ go, Thus even if thc gnEl is in uniform

motion or at rest the pressure tensor cnn sti1l l¡e mr¡eLi-

fiecl- from p * dkTli v'¡hich it i's given by up to secc¡ncl-

cr.l?proximation in such n câsG¡

4. l+ Lhq Tbi r"!L-þpgg-r.i-m.+:E-q,4--!g -!¡s- :lå's-!:-i!-qt! -o¡r---F-ugc-t*¡-o¡

Tn t his s e cti on ïrle hri 11 r¡olve the th ircl appr oxinetl on

to Bol-tzmannrs equation (l , (z), (l)), to ol¡t¿rin Õ(2) ' 'Ì'he

methocl tiye use ulas clescnillecl in Chailter 2 ¡ Sectj-on 2,

r:vhcre ï[c c-liscussec]- ihe Sener¡.1 N+1 t']rapi:r'oxim¡Jion, It

invcl-ves expressing -A' in terms of the

Grrl ancl here ¡"nre neccl the f irst five-

Then on substituting

162.l+. l+

i rr eclucible 'ce ns ors

Got G1 aaa G¿cGrrt

( ¿Jl+

In=O

f ordr(n) qir)er,(Y)? ,n (¿,t,ço)

L

0

in (1r(2)) ancL separrting out the incì-epcnc-Lent spherical

harmonics, v'/e obtain egl-ta'cions f or cach of the i ind-epen-

cle nt t erms of a lrartiCular orclcn rl-¡ These in'ue31t al

equations have been red.uccrl to clifferentirl- f orm for thc

pnrticulnr Val-ues of n conCcrnecl in tkris approrcima'ciont

rr = orl r ZeJ tnÒ. )t-, in (2,5 9); ancl solution of the

diffencntial- eçrations al-lows us to o'btain ør(") f or a'11

i tqrns of anY of the giricn ri.

r,I,re sec at oncc the re is confusion in notetion. Tn

tfp s ee cnd. ap i¡¡oxim^t i on (tt = 1 ) vÍc l'lr ve ol:'t¡-lnecl solu-

tions t(t ) an¿ øQ), anci now in this case tye are locking

f or solutions ør(") , ør(t) , þr(') , ör(t) "t¿ ør(L ) ' rhe

t(t ) arrl rQ) of the seconcl approximaticn enter i-ntc t'he

equatione rrhich cleterrnine the t(n) of the thircl approxi-

163,)+.4

mation since thcSr ¡ùlpcnr in À, So tLn'c there ís i'r. possi-

bility of conf using the *(t ) and- fQ) of thc seconcl

ap3¡roximation, ancl &ny of the t(t ) ancl fQ) of this

the third ¿pproximat ion, This ar ise¡: llecause in t''rn iting

*(w) as (z.zr(1Ð) wc ctic-l nct specifSr thc orctr:r of the

approxim¡.tion wlth r¡ftrich \,/e are conccrned. Ä means of

ovøc:'ninij this cllfficulty r¡,roulcl bc to l:Jlr:l all i;hc

var i¡il¡ les of n pft. ticulm appi3oxim¡rtion wi th nn aclcÌ iti onal

subscriitt c-Lcn¡tin¡3 '.nihich npl2r<-rximet i on is cons ic-Lcrecl: Crfl.

l/jIC COUI-<1 i,i¡f i'r; C

þ (v )err(v) (clrTrgn),I

and_ then in clealin-4 r,vith thc reclueticn of this to cl-iffcrcn-

t1a1 f orm vre v¡oulcl u,se ihr: functionsOrÉ1rr¡r Wy(rr)r Nct(n)t

Nx , Ng ete. However slnce thc integrnl terni of the

IT + 1 th approximetion is of the same form fon nl1 N we

clicl not bother to c¡.rny ,suc:h a l-abel through the gcnenal

the orl¡ of ch¡.ptcr 2¡ vr{re rc thesc functions carrying the

onclen n, ihc angrment v¡ ar¡l thc super="¡ir.lt telling the

order of tlre clerivetive lvere alneady encumbæed' enough'

*(w)(n) n,z

NniN

The me ans \¡/e ad-opt here to avoid- confusi,:n 1s to

d.efine netu vania'bles precisely equivalent to those of

Chapter 2 rryhrenever an embiglity could- arise' Thus for

this the thircL appnoximati:'in we v'rnite

Diúi (") 1v¡crr(y) ?rn (¿,t,9...,), (r )

the t/r(") replaeing the dr(") of the EpnerirÌ thcony,

ancl thcn we re labef the varia'lcl-es of the diff enential

eguatlons as fol-l,:¡ws. For the n = 1 equation we 1rr ite

*(z)

so that

164.4.4

(z)

r.l_

exp (- cru"),/i(1 ) (u)ou, sr(v) / *

u", (u),

1sa(u)au ancL Qr(c) = ø"ar(v)'

-.t. (r ) ar' -evi = ca

The recluetion to thre dlffcrentia] form goes through

exactly as in 2.6 exeept that nolv r replaces Fç1¡'

s replaces ïq1¡r a replaees X, and finally Q is t¡ritten

instead of i , Similar.ly for the n - 2 equa'cir,n, lve

vr'¡. ite z instead. of 0 e) i ' e.

165,.4,lr-'

r oo t., (z)

1u)z, (V) = l_-- o*n (- ,rUr) j-r--_----- d-Uy m insteacl of ye)''.lvu

t instead of gt and finally R(c) is thc relal¡el]ed form

or ã(e), sc that

,þr(') = å þr"iR.' ':-J-- lRi\

c'/+ a ß)c

Although we havc re label-led. ,(n) witii t(n) in this

npproximation for alL n, v,rc will not bcthen to nelebel

the varicù:les of ttre Íl = Or Il = 3t ar¡1 n = 4 clifferential

cquations cxccpt in so rar as É(rr) = / - rsÆ-t"gi3t

,/r(t){u) du is now \,ïrlitten in terms of ,/r(tt) not ør(t) .

This is pæmissible since rve clicl not 1n Chapter J hlrve

cau6e to use these eguationsr so that nc am.bi¡1uity can

¿rn is e coneer ning which approximat 1on V ß)r sftVr occurs

in: it must be this the third.. Thus for example we get

where \G1 is the solution of the 1th n - 3 equation

in this the thlrd- approxinrat ion. SimilarlYr \18 get

wi " '!u-lV

(o) (¡)eþi

where ![i is ti.e solution of the i th n = O equation in the

thind approximat ion.

166.l+.1+

The first thing we have to d.o nou¡ is to r¡4" j-te  in

tæms of Go, Gt, Gu G5 ancl Gai

Finstly the terms involving Go are ol¡tainecl' These

âre the term,s r¡rhich are Just scalars in \fr ad there are

in add-ition to the terms of (1, (4) ), contributions from

(r, (6) ), the terms of even clegree in I. The tems in-

volving I I ean be'rrritten in terms of G.--, arrL G4 since

fnom (¡..1r( ¡))G

d\=/

I3II=v'

Yz òi\' òT+3ãfl8

Ç*'+a (6)

(z)

Thus the ct..ntnibuti,rn from the finst five tcrns of (1 t (6))

1S

23

v2B',oe

O tr2V ^te+:.l-r ò

òr/òT.\ 2V2 òA', .èg . 4\ ar / - J,1m òv2 Òr Òq

ÒT

õE'

The first term of (Z) comes from the seconcl- pnrt of

the seconcl tg'm cf (t, (¡)), ancl the remalning three terms

come from the thircl, fi;urth nrrl f if th tæm's of (t, (6))'

There is no contribution fnom-bhe first term or the first

part of the seeoncl term of (t, (6)), This is because they

167.ll.4

both involve a double scalar product of y I with a

syrrmetnic non d.ivcrge nt tensor, so that on su.llstituting

(6) f or \r V, the contribr-rtion involving G^ involves the

tnace of the non clive^gent tensor u¡hich is zaTQo

There 1s just one te rm lnvolvlng u y, g y in (1 r (6) ).

It is-?oE:vvvv::òv2 .L.Lå-

oo(B)

Frorn (4.¿t,(g))' this is Just

-## ["'*¿ii I I + V2 (rrtv¡t¿oo 5o(J

+ dt juoo, * dixv .vJL

ooy4e. . g. - =-lJ i(L o

t'

d.1

o\ ya o

el-îe=/ 4 = #L-16

35ç/o+*v2vv: le¿ - - \=\

oe

oG^q+

t. v.v. )IL J K'

o o -l 16 òB'xB..o. l=-BælJ KüJ

* ? u,

+

* vivt di¿ * viv¿djL

d.. d.:tK JL.6. +J KL

(

ooeo

+ 0di jk )

ooeeaGaz

z

L

()

ti

lo

_9

oe v4

J-12 ].

(g)o

In obtaining (g) u¡e havc a,gain usecl' d: 9 = Ot and (6)'

Thetenrnsof./lwhichinvolveG..arenovrfcollected

from (r , (l+) ), (Z) ancl (g); theY are

)(

1 68.l+,4

oo9:e+

yz- $ v'n'A-' òp- clm òr

V2 ÒA' òT

-æ-3 òr ògòoòg

o

:Ë u"n, : 3Ë ; . Ë (uid'r * vjd:.r. * v

o

9+Glèpòn

(r ) l:

J^Jdm

V2

òTÒg

OJl'

ãFÒT

òg

A ":àòn

ògòg3

+ +

VVV:

òTòr.

, r-, O O

. !,va g=e: e. (to)15 " òV2; =

VThe tensor G¡ is Just ? uo thnt the terms of

^which irn¡olve G¡ coile from the terms of od-cl degree in !r

(t, (l)). In acìdition to the terms invclving U in (t ' (¡))"

the terms in I ! \I also contribute, slnce U y, Y- can be

written in tæms cf G.5 ad Gl. The fi rst i)f these terms

1S

which on us irg (A.. t , (Z) ) 1s

5chn t-

r.di ¡ ) Ð

ò*i

. (#' ;)l'

2 òB'- r*r1crm ov -

ò8,òv2

2clm ro]

5òB'æ

ye_4:

(t t )

169.l+.h

Similarly, the o.ther terms j-nvo1v1ng g g V recluce to

v3?5

- fr r(o)v G,

!¡,

f,v" (St--trF) L-.:#! *n, (Ë

9+Gl

Then colJecting all the terms of -A' involving G1

B,Þ

:" òÒg

o

)

/ò\¿r (tz)

(13)

and

+ VA' Gt

from (rr(¡))' (tt)' (lz) and (t3)' we have

!)-l

/ðT|.¿r

:\=/

oe2

oe

i)?vsdm

I (r ) + 2VA'A G tò!òg

tkGl¿r\ònJ

_+_5dm

v3#n, (* G¡ (w\òr

o\el+-t

=¡ . f,v'Gþ/ò

li¿r

1621

Ò.A,òT*?v')

+

+2 V3B, G¡5

zò,\'ã.r) G, !¡.1t tr)

The terms of .(\ involving G2, are obtained- by substl.

tutlnc; (6) in (r, (6)) ancl nlso by usine (g)" They ûre

* r""'a G z2.3

+ V2B, G2 :

_4_Jdm* zrv't' G' Ò_

ògyz æ

ÒròTãs

òT òTòg ôg

ò8,òv2

ooG2 t (g ' ").

(t r)/1v2 ya

170..¿+.4

'vllrcn we ]il/ere ealculating the contr.ibutlon of the tenms

in y u u of (r, (¡)) to the terms invc,lving G¡, v'/e vfere at

the same time flrrling the tenms cf .¿\ invoJvini'i Gs.. tr'rom

(11), (tz) nnc-L (ll), these ane

5$;v" #n" i#! *f,v"n'Gs;*;D

. E \rv_l--v/òB'(tr

-ßvo35

o

9.2òA'ãrr

ooee4

ò8,ãtr

\ o, : Sl-dr/-

(z)

(16)

(18)

T,ikewlse tfE tsm involving Ga wa$ calculated' in (g)'

It isG (17)

Now ,4. has been wnitten in the form of (ZrZr (12) ),

i;e.' in terms of the irrecluclbl-e tensors G-¿ Thus then

equation of this apirrr:ximntion¡

c

(s)Io

4

t )- u, 1v)cr,(r)lrn (¿,t,s,.,),/-: Lt rh-^ JIf-v l-

(z)can nor/v 'be solvecl by sul:stituting f cr O from (t)r anc

usin¿ the methods of Chepter 2 we obtain clifferential-

171.lr., h

eguations for each of the i terms of a given n. Solution

of these eguations then all-ows ,/r(n) to be calculated- and.

so t(Z) to be completely silecified. r.iile will nírïrr consicler

the r/nnious val-ues of n in turn, ancl solve the i incli-

vid.ual clifferential equeti,rns f on tlre give n n¡ tr'ron the

so.l-ution o(2) thus obtained-¡ we câr cal-culatethe third

approximation to the heat fl-ux vector ancl prcssure

tensor by inte65rat ion, ancl bef r,r€ v/e start to solve the

(2)equationsr r/ve nm.ke a fer¡r remarks about g and E(') .(z)

aThe integrals for q rr1(z)

vsf O Gr dy ?(

are r€ ,spective 1y tD

( 2) =l "[

u"(o)o(')u uo3

ml=î l (r g))o

(2)and p = lll ft o( d-vVg.(z ))

= m ]u,r(o).(') $ n,,- þ 4) u* . (zo)

On sulcstitution of the e¡rression

r(z) = )- t Dtúi(t)er,{u)?r,',(¿,so,r)

11' ,

into (19) and (Zo) and considenation of the integrals

(z)

172.4.¿r

(") d.c o

(zt )

which occurr we fincl that only those terms fn O

involving G¡ contribute to q(z),, and. only those involving

Gz.t contribute to O(z), In the latter case lve need- to

use the auxilinry relatic¡n (Z.zr(17)), v,rhictr ensures

eorreet clefinitlon of T in each approximation to shot¡¡

thnt n(z) has no contributi cn fnom scalar terms i-r, o(2).

The contrÍbution t o n(') ,"o* each of the terns of

(z)o lnvolving G¡ is

9i(z)

Iu"t(o)úi(1 ) er 'ziGr d.v=-äot

2rn3

oo (r )D1]-

t,u v5f ,þ dVi

(r )1

c5e(c)r/rz.-1

(o)

t.lo

oo

D

(2)Similarlyr the contribution to ! from each of

(z)the tæns of Õ lnvol-ving Gz is

(2)!i. =-mD u'f (o) úi_

t_ I(2) zi3Gzz! G2 clg

W5

D

o

z.ÉI Ï o*

uot( o) øi

(2 ) dv

o

T[" ,(c)Ør(z){c) ¿o . (zz)

1

D I

173,,4nh '

Thus ,¡/hen we calculrte the lndividual ./i(1) and.

pr(') , we shal1 also d.etermine the corresponcling contri-

butio.ru nr(') orra pr(2) by nurner ical1y evalu.:rting the

int egra Is 1n (zl ) and. (zz) re spe ctive l-y.

rr = 2 Equations

tr'irst we de a1 with the i incllvidual terms of Âe

(15)¡ and- a(z) rvhich involve Gr, wheren=2, This

(z)enables us to ceil.cul-ate the contnj-butions ]]t to the

pnessure tenson,

The first tæm of (15) ¡1ives ri se to the e graticn

(2) l u'"'L G, :

v ÀGe:

(zt)

(zt+)

=.F(2)(26)

oed.21 ( 0,

ancl s ub st ituti on of

,,(2) - -q. 2)

zq4d"q4

) ,

( oã=

-2Jo I

(25)

rn¡hichr on srbstituti,.)n from (l ,(lZ)) f or B' , recl-uees to

\v,ø' = r(") (#f" #Y (rQ) * r (z) (z))

a2Jo,(z) *

^(z),åt . #)

1

lll'In (25) and. (26) we h-rve sunpressed- the sulrscript

on Je) , *(z), "o(') "rra ¡,(2) wh.ich inclicates we1

are clealing with the fi¡st of the terms of -4" for this:

value of n = 2. ì.Uhen wc cleal- rvlth other te¡ms of -4. ure

shall- lilcewisc suppress the rrc-lue of i in thosc; ce ses nt

the similar stafle of the reduction of the integral

cqua;bion, althou¡þ vre skral-I incluc-le the subscript i in

the final- clifferential equation we procluce.

irTow (26) is an intcgr.al equation of the form rvhich

vre red-uced- to a cliffe::cntial equâtion in 2,7. From

(2.7 , (zB) , (zg)) the d.if f or ential fo rm of (26) is

Ä(ml(4)rr t""tfrt" rtnt" ml) = # (zt)

Here m¡ is just the relabellecl equival-ent of yl(Z)'

Following Chapter I rre make the substitutions tr (V) =

mr(v)v ana then R¡(c) = ;/'t,(V), (these l¡eins; the

relabellecl equivalent of el (V) = \ e¡ (V)V anr

ã, (.) = o3/'r, (v) ) . This recluccs (27) t o a c'Lime nsi on-

less eqirat i.,'n in R ¡ ancl sul¡ st, i'buti ng the ilart icul-nn f orm

nr ¡'(2) from (26) we ,¡btain

175.4.Lt

Y(R, (tr)

rn t " "Rr'''rïl "Rl

)

Wry,#(# %.#)JÀLI

ãã

4a[.;(t+)L-

g (1+"" - l+c )ã' ' 1Zc2å, + 12egl+ +

(28)

Tn (zO) the R.H.S. . iB a known expression which was

obtainecl rn¡hen we sol-vect (3 .1 ,(lED j-n the sec oncl al?proxi-

mation to Boltzmarutrs equation. This equ.rtion ill-us-

trates the necesslty for relabelling varia'bles 1n thc

var"ious approximations, fon hacl this not been clone the

L.H,s. wcul-cl have been v(ãl (lr) r6, "' ,ãr" ,Ér' ,Ér), withru ltrl N d

Y ôef inecl by (5 ,1 , (lÐ) containing g, t*' rfi,," "

" ' 8l r 1

ancl clearly it woulcÌ be necessary to d-istin¡1uish between

N,-

the knol¡{¡n g of the R.H.Íi. anrl the unknown g of the L'ÏT'sj'

as we h¿ve done.

Nor¡¡ the only difference l:etwe en this equntion and

the h = 2 eguation vle solved- in the seconcl ap¡:roximation¡ i

the shear viscosity eq.".tlon (3'1 '(ls)) '

is 1n the non

homogeneous term (al'ch,tugh in this case it is proportional

176.4.1+

to c2 for c near zer.), aniL tenrfs to zero exlroncntially

just as in (3,1 ,(lf))" The non homogcneous tæm of (ZB)

is not known analytically as it was in (3.1 ,(Sn))¡ but

we know it numrr:ica]l-y over a rnnge of c from 0-bo 10 so

vüc qrn cbtain a sorution of (28) by the seme proeeclure

as \,rie usecl in solving (3.1 ,(S>)). Thi,s is tLre Runge-

Kutta step by step numtrical- intcgnation techniquer and

the methocl of combining the homo6let'reous solutions ancl

an arbitrary non homogcnec)us solutic.,n to form the

reguirecl exponentially cÌeca¡,ri¡g so ]-ution t¡'rhich also

<rbeys n,'(o) = Or Rr "'(o) = o, (tne rel-nbcll-ecl- equiva-

lcnt or f,'(o) = O, ãr"'(o) = o) is just as we usec-L in

Chapter ).

Tn solving thc equ¡.tion r¡le once rtgain use a step

sizc h = '01 as r/ve ilid. Ìn Chnpter 1, ancl application of

the Runge-K.utta method- take,s us from c¡ to c1 + hr c¡ + h

to c¡ + 2l¡, e1 + 2h to c7 + 3h nrrl so on from the initial

value of c = Q7 = O' Thc cal-culations in any one ñtep,

require knowledge of thc non homr.iieneous term f or the

177.¿r-.4

mid. point of the interval beslcles both encls* Thus v¡e need

to know the R.I{.S.., at intervals of e = 'O05 from c = o

to 10. 1rt/e achieve this by solving (3'1 ,(¡¡)) a-gain with

a step size h = 'oo5, and punch the R'IT"í]'' of (28) on

eards fon the requirecl vcÙues of cr These ane fed- into

the computer as the necessary data when we solve the non

homogeneous equat i on (z g) .

The valuc of €qrte) is oirtainecl from thre s'¡luti'rn

R¡ of (28) by using (l)'

,,1,,(2) = å (n," - 3Ri,' - ++) '

ancl we have presentecl- thls gnaphically 1n tr'ig. [¡.

Wc now rs e equation (ZZ) to cvaluate the ccrritri-

butin., n, (t) to the pressure tensor.

p,(2) = - q (+-\ ^ ! [* r(c)ú,(') (e)cadc (zg)

=' 5d,r'z \zt4 d'rn I = ./ o

,Subst itution of t he numæ lcal value of ..¡' r(Z)

in (29) and- cal-culation of 'r'he intqtral by simpson's

rul-e gives

p,(2) ='roeol. (#)^;. (jo)

178J+.4

3

0

€W,

. 0.1

t2'

- 0.2

' 0't

FrG., ¿

179'4.h

The remainin¡5 n = 2 eguntions are treatecl in exactly

the same fashion and. ule vri11 discuss them very bniefly'

(z) The next egrmtion ure have to solve is

which, on m.bstitution of

'Dovzri' c2 t (# t

v2!\.G2,Ê(3i)

(z) z3

;),o

=2

o2e

).2

(¡t7- d21(O,

(z)J-

\¡z flz qa

ÐoGz' (u? g 3lc[

,l)(2)

2@2 3z¡

3t+)

into (lt), gives

This is Just (ZS), so that immediately/^\ (Z\

þr\t) = þr\'"

(#)(k;(z)ancl (ss)Pa = .2O45O9

3) The next term of Â. involves second. spatial cleriva-

tlves of Trand the eguatíon We must soJve 1n this case is

(z))

z3

3e¡

3t)

- d2r(Os a

On using (tr(lt)) for A'; ancl sul:stituting

1

6tFaz út ú"(z) Gz,Ê /òT\

(ã" )

1 80,4.h

in (J6) ' we oi¡tai n

(z) (2)J +K 2J

le'zi( 5)+

lr.# ¡fi"'?\-F(z)o

x"ïã "(z

)

)c2

;GT a

(¡e )

In d-ifferential form this is

Y(Rs(tl) rR"" "R5"

rR""R") = 6,(C)d.

d,cILã

"" orp(2 )Â

e 41r?_

ìt(

(4c" + he)i(4)d

(taea + g.z)Y" I+

d

8c )i'' J,(1r"" + Øg)

vuhere we h:,ve substitutecL from (38) f or "(z)

and' per-

fcrmed the indicated. differentlatlons, As before, this

eguation is solved by the Runge-Kutta technique, using

a step size h - .o1 , ancL this time the ncn hornogeneous

term invo l-ves the knov,¡n numer ical- s'¡1ut ion of (3.1 , (t Z) ) '

i¡l/e feecL this intr¡ the computer as clata which is 1:unchcd'

from a solution of (3.1 ,(12)) f or virhich h = 'OO5 is usecl'

The d.istn¡rution function ,,1, r(2) ," ot¡tainec-L from

the numtrical- solution by using (l) t ancÌ is lllotte¿ in

Fiij. 5..

181l+.1+'g.r

0

0.t

.A

0.3

€

0.2

0.1

I

0 !I

-0'l

trG. I

182.4.1+

By making use cf (ZZ) ancl the nutnerical solution

for €þ s ancl pe rformin¡,t the intep;ration in (¿r.O) by

S irnps onr s rule r vË fi ncl

(z)

= ,21+31+BB

(4)

of p and. T:

(z) OL

6tF cf o-alm

a

(z)þo G2 -.

(c)caac

(t+o¡

(trt ¡

(1"'.3)

(44 )

o

/k('iF

o

The fourth equation involves a prclcluct of graclients

sg.òv2

V2

òg

_L*3dm- d2ï(04(2)) 2G

anJ subst ituting

òT9æ,òg

(42 )dl"

usins (1,(19)) fcx.#

a4

in (41), \¡/c get

C,u è+òq òË

+ I)

)(z-

The equivalent clifferential equation is

y(Ro(4),R 4"',R))',Ro',Ro) = ,(") # [å # ettr#2)l

183.4,4

(tr5 )

Thi s e qr-rat i r:n is so Iv ed numer ieally in the same

manner as the previous ¡ = 2 equations¡ tle clata for'¡he

R..ll.S, bein¡J punchecl 1n a tr)rogram in t¡rhich (3"1 ,(lz)) ls

solved for f;. The numerical- value of ,úo(2), olrtainecl

fnom the solution R,t of (45) by using (l)ri= grapirecì- in

Fig. 6; ancl the contribution to the pressure tensor in

this clse is*q-

(z\ lrclm / d \ãp*-¿î fÐ'4 5arà \.6æa" o-¿Tnj dË då lo

oo (z)e(e)td (c)cacle.

(ue¡

(2)Using thc numerical value of €þ¿ the intqralI

in (Ue ¡ is evaluated numerically anc.L the result 1s

o

PA = ,0686163 òr òr (t17)

(i) I:rext we solve an equation involving the prrrcì-uct of

grarf i ents of T t

(z)

- crz r (ou

(-"_*) 9! òll

(z) òT òTòr òr=Zv"t

òA' ^ãT: ", (lrB)

t '9tJ

¡.1.

0.2.

3.1 .

0't -

I¡.0 .

1.0

,^t

¿ I0

trl tjg l

tÄlr iti ng

185,.4.h

(t19)rñ Q) -L

* ,l,uQ) o, , #Y5 = - 1zoid'oÁt'

in (48), arrl using (rr(le) to" ff r we find

"(z) +r((2) zro(z) =#(+

+;) ;?4 = n(z). (5o)

ô1,

rò t

+vx"'

c2

y(nu(4)rn u"'rRs"rRs'iRs) = e(o) * [å * (q#¿) I

= ""g(6) + (uco + 1oc2)i(5) * (r¡es + 2eco + zoc)1(t+)

+ (12c4 + 2\-c2)i.''' (tze" + ZL+c)i'' (51)

The numtr ical value of ,,1,u(2) obtainecl l¡y nulnerically

solving this equation for R5 ancl- using (l) is itraphecl in

Fig. 7i and. by integrating this solution ure obtain vi-a

(zz), o

In ùifferential f orm thls is

o

(c)caac

$z)

= .o3s7zzo (#-Ð HE , $t)

186 'Li.h

1.0

0.5

- 0't

t2t

- 1.0

-t.5

- t.0

' 2.1

€VIs

0

32

G

I

FIG. '

187.4'l+

(6) The final term of .4, whieh inv6lves G2 gives rise to

the eguation

- dzrqou(2), = - #v, ffi e", (; . !), (¡Lr)

This is solve,1 by substituting

^(z)!/5' =

in (¡U) to obtaln

,(z) * n(2)

)

oe=

oA

=

of-{rÆ_q2

Y(Ro(4)rn6""R6"rRu"Ru) = e(c)*rå# )-l

ct þ,(z) G2 : ( $n)

$6)

Trrza'ø4

2J (2)o

Il(z)

+ þÉc6

ïn cLeriving (le ; we have used (t, ( 21)) ro" ffi,a.nd now writin."; this equation in diffcrential fo rme we

get

+ (lzc 2Lrc3 )i' ' + 6oã' 6oc[.

$t)

The soluti on of this equation by Runge-Kuttn tech-

(z)nique allows us to get €þ o fnom

\

/ -

'(c).¿u(') (c)cadco^(r)=--e+ \--YÞ _ 6 I - =twr' /

l.Åa- +

1 BB.4'lt.(¡e)eÚo

'76911+8

(z) f3oe2c= å (*u" ,

and- this 1s gnaphecl in ri'i,î' B' From (zz) u/e calculate

the contributi on to ! from this term bY inte¡¿ratin2;

the numer ical sol-uti on the result 'l:eing

2( )

(¡e)9

o

a $g)

The soluti.r*..¿r(2) for i = 1 ,.o. 6, which t''¡e have

prcscntecl graphical-ly in Fif,' 4-8, n1l- start like c2

near zero aS \¡,re expe ct, 4rc1 d.ecrea,se exponentially as

the first term of the Sonine PofY-(o\

€ú,\'', which is ProPontionrl to'l_

', has the same bchaviour near c c O

nomial expansion of

(o)

c*oor SimilarlYr

eez$ 5/Z(c') = €c

ancl c = oo , Hot'uever by comparing these exact solutions

forcf rQ) vrith €c2t Fig' I\2' r¡re see that the form of the

first term of the approximation is not veny similar to

the exaet d-istribnticn f\rnction f or i = 3t I¡- ancl 5. -Llven

fon those cases where the clistribution function has ¿l

simil-ar shape to €c2t nnmely ,rl.,t(2) "^a ,{u(2) , the

.l89l+.1+

0

0.2

(2l

r6

D.¿

¡.6

0.f

1.0

FIG. '

I

1ga.l+.h

function cannot 'be representerl 'b¡r the f irst term cf '¡he

expansi<)n as r¡,rel1 as the solution ,þQ) nr the secontl

api)rroximntion rr¡hich we founcl in Chapter J.

Tn Sectj.on J ,¡¡herc wc calculatcd- itre contril¡u'bion

to the thircl a'irproximation to 'ühe pressurc te nsor ¡.

ïvas not r¡mitten in terms of the tensors Gn(-V-). 'I'hcrc

we showecl- that only the tæms of even cl-egree in I I ,

(t, (6) ), contril¡ute to the inte gral (¡, (ll)). Actuaì.1y,

onl¡¡ the lrarts of these terms which involvc G2 eontribute

to the integral, in the saräe manner as onl¡r tl1s pæts of G2

(z )in the c-Levcl-o,rment of 0 (1), contribr-rte to thc,

(z) (z)valuc of p calculatecl 'Ìry d.ircc b1y integrat ing Õ o

l^ tThus sinc" o. \t/ comcs f ron thc samc sourcco the

^.-41 -'r

involvj-ng Ger wc expect thc values calculatccl- lrere -br-r ¡"

iclentical with th.¡sc of Secticn J since both calclrl-ations

are exact numtrical- evaluationei. 0n comr¡arison lïe see

that the enswers âre in f ¡ct iclentical, thus inclica-t,ing

our nunerical calcuJetions hnve been accurate.

191 .4.1+

e--=J*-EÆ-q!ågl1s

The rext grou-p of equations which we d.eal with are

-l^tthcse invc¡lving terms in Õ\t/ and- ^r

(14)r vrþich contain

Gr, Sol-utj- on of these equations enaJ:l-es us to calcul-ate

(r ) (z)vi ancl sc Ai via (zl). Tn Section 2 v'¡e have already

(z)calculated. the val-ue of q by looking at the ind-ivi-

(o)(6o)

cLual- tenms of  rivhich a¡e of od,d. cle.Sree in Ir (t r (¡) ).

The terms inyllL eive a contribution to n(') ooa it is

just the contribution from -r,he part invot.ri* n, in 'ure

clevelopment of y I Y in terms of G5 arrl Gr. Thus when

$re so,]-ve the equ¿]tions in this section for eacþ of the

indiviclual terms of ^,

involving G¡ ¡ (14) r rve might

expect to or¡tain the samcì results nr-(') , as lve obtained-

in section 2; just as the rcsutts for !r(') calcul-ntecl

by integration of the torm"".i þrQ) *"*" seen to 'be in

e.greement lvith those ca.lculrtecl in Section 5'

(r) The first tem of (t4) is1 f /ò

\ar )Pp

V Gr(r ) t

anci this gives rise tr¡ the equation

192.! .ll

(6t )

(62)

(6lr )

- dzr(of2)) /ò(ò"

(t )1f v Gl

G1

p

r("r)

(o) I ,

which on sul:stituting

(z) r) / ò.

\us( (r )

)Õ !I

in (62) becomes

ft\

J(1 / + K (t ) 2Jo

(r ) 4? = n"(t). rc3)d'

This is the form of the integral eguation r',¡hich u¡as

rec.lucecl to a diff erential equotion in 2..6. From (2.6,

(32)), we can rmite this equation (65) as

kat"' + k( 2qY - $).r " þat' + crtl

Iu*,(u,) # e#) uu,

Here a¡ is just the relabel-led- equival-ent of )(¡.

On reduci-ng this to climensionless form by ¡:ubs'cituting

q, (c) = ø2a ¡ (v), (r,vhere q(c) is just i(") nelabe 11ec1)

anct substi'butlng the partlo-11ar form nf p(1 ), vre ol¡tain

(,ttt

lCt"J f + c1 kqr"'ÚQt'+6Qlc \

!.d

oo

cle I l^# (- \tr2efll u" | (65)

193.l+' 4

a r' , ,..fø-l=-+ll!-ce)-Tl. (66)¿- L "J

The zubscript 1 on c1 on the R.H.S' of (65) merely

inc-licates c¡ is the d-ummy variable of the integration.

It is nr¡'t t<¡ be confusecl with the srbscriilt 1 on Q¡

lvhich inclicates tn¡e arc c]ealiry{ with the fi::st term of r\'

involving G¡, [t/e want to solve this equaticn su-b ject to

the same bounclary cond-itions TVe irnposecl on the previ-ous

n - 1 equation which r¡¡e clealt with: the thærnal concluction

equ.rt ion (3.1 , (lZ)). Th.ese conci-i tions are (in term-e of

Q vr¡hich is iust f of that equation relabelÌed-).

Qr and. all its d.erivatives ap'proach zeroexponentiallYasc+oot (67)

Ql"(o) = ot (68 )

and.(6e )8r o

Tfoi¡reven ¡y looking at the cquâtion (66) in lowest

oncler in c we find at zera

)( 0 a

k ^ .'1I ql I =v --l e=o

,l+

194.l+. ¿+

Thus

er"(o) ='â,"L (zo)

and we are forcccl to allnnclon one of our conditions at

the origin"

Accorcl-inglvr in flncli.ng the numerica.l- sofu'cic'n of

(66) v¡e use t¡c homogeneous solutions wJrich start l-ilce

c ancl c3 , ancl ¡.n¡hich t¡¡e used- in solving (J',1 , (t z) ), nnd

now vye clevelcp nUmerically a non h,omogeneous sol-ution of

(66) r.n¡hich has bounclary concl-itions Ql (o) = O, Q¡ "(o)

z1='A oà. Thc values of Qr'(o) ancl Ql "'(o) are lrilitrary

'¿

anrl we take them eqr.rat to zero. Then vre cornbine these

three sol-utions as l¡rc d-id. in J.1 to obtain the required

exponen,cial-ly clecreasing solution for" lerge c. Frcm this

we ¡';et ,,1,r? ) ',r*i-1L1

(r) gi:: -g+--, -7t)e|t'' = c c-

ancl find- this is pnoportionnl a" # near zero. f i is

gnaphicalry presiente<t in Fip;. g. That e4tr() is divergcnt

at the origin ancl 1s not proportional to c as riue expeet,

is in fnct a result of relaxing the conclitlon Ql "(o) = O'

195+.+

6.0

5.0

¿.0

lrt

t.0

¿.0

t.0

0

I tc

- t.0¡F10. t

196.l-1. l+

This, from the d-e,f inlti on of Pi, si, al' and Q, (tfre

relabellec-l variables of the general n = 1 eqr-ration of

2.6)r mcrans imrnedintely nl is proportional to + ' a-ncl

ccnseq.ulent]-y ,,¡, r(l) is proportional tt # near zero'

(2)lri/e calculate the contrii¡ution to q fron l;he

solntion (ll7 using (Zl) and e v¡.Iuating the inte.qral

numcrically: the resul1, being

(r )I (r ) gse(c)ùt (c)actJo

oo(2)gt

(r ) +o. 0z)= I 'O5Bh1 !

Thuswefirnltlln-t,byinsistingonanexponentia]ly

clecreasing solution of (66), ,¡rhich is û. consequence of

(r )rcquirecl form of evt. ar, infinity¡ (ancl ,¡'¡hich we urecl

in ¡nrt in d-eriving the L.IT.S. of thc general n ='1

differential- equation) live ¡¡rve prod.ucecl a d-istribution

func tion which diverges nt -r,he .onigin. l'4orcover the

valuc of qr(2) celculeted' by integratin¡i this fl]nc tion

d.ces ,rot r*""c with tÌre value car_culabecl in scctic¡n 2.

the

197 'h.h

The explanation of these results æ.n be found- by

ref erence to the sul:sid-iary conclitions uÀrich have to l¡e

obe5r6d. for the integra]- equation r¡¡hich is prod-uced in

each a ppr oximati çn to have a so Juti on. These c ond.iti. ons ¡

f r'¡m (2.2, (zz)), are

f ¡,u.r-= o , Qs)t-m lnydv=o, (z¿+)

/-ancl

* fnurd.v=o , (lr)¿J

anct thc subclivision of Bcltzmannr s equ.rt ion lr¡hich Enslcog

mad-e j-n cletermining the normal- mlutions ensul:e$ that

the se cord-iti ons arc alwi:ys obeyed-, I'l'otv rntril,s-b tlrc tenms

of  involvinfl G' for n ),2 satisfy (ll) QS) automati-

cally'l:ecause of the form of tl:e tenr:ors Gn, lt is by no

means evldent that the tøms involving G¡ e (tfl) r satisf y

(7h), or thr.rt thc terms involvinp, Gor (1o)r satisfy (l:)

ancl (7E) . f n the sec ond approxlmation the term involvi ng

G¡ is f (o¡(av, - ]>U. ¿1*òïg and (Z¿r) venishe. ajs ib must.

However with ¡ given in (6o)

À1Vd.U=-m /þ-\art

It"l TU !)

(r )

1 98.4.4

c1r Ql)m p

= #('rå'e(")f"(n)o"uo

= _ * . n(,) + o (ze)

Since Enskogts ".rbUrrrrsion of Bolt'zmnnnts equa'tion

does 1n f¡,ct obey (lS) QS) ' there must be a comllens-

ating part in the other terms of (14) ,¡,'hicir enables (Zlr)

to be natif ied when we comþine the terms. Puttin.q this

a:.ro'cher vray, the trreirt of ^t

which inv,ilves 'r-,he d-erivatives

of d_, T and gor i.co tine zl of the gcneral thec-rry of 2.2,

canrrot be incleitcnclent of thc other Vcctors [, t'rhich ap].lear

in thc terms of  ,(t¿l); and crr combining thc cì-epenclcnt

tæms thc firnl resultant set of indcpendent tcrms of ^

musi satisfy (ll) QS).

',Mc now exnmine thc Lt of this term -/\1 . Ílince

n(r) 2,t i,wherc from (3.2, (t t ) )

2- (M\1,n=îF \=-/", (zg)

wi th

199\4.¿+

(Bo)

(81 )

I.oo (z)

L exp (- c')e4ú (c )¿c t

it follows that

o -Àzll _ or(1) ò=2q ò

òTòr

o nvtmr

a

oe

P

The sixth and seventh terms of (th)r the terms of.oÀmo

 involving G¡, contain Za = ã? ' I , and Z, = #' 9

respectlvely, so that in fact ;" fírst term i= -tot

inclepenclent of these, anrl we will see laten that on

comblning the tenns of  the eonditlon (Zl+) 1s in fact

satlsfled for terms lnvol-ving lnclepenclent' !, as it mustbe'

The general theony cf 2r2 gives lntqnal eguatlons

f cr each of the terms of .4. whlch involve inclependent Zi,

so that we can certalnly obtain calculatlons to the heat

f 1ux vector involving tÏrese E¡arne lnclependent åf by solving

the integnal eguatlons to olrtain tJre cllstrilcution f\rrction

ancl then lntegratlng thrat. However if we attenpt to find

solutlons of eguations of the form

drï(o1(2)) = Ài = vi(v)Gr Zi

rnher e zt Ís not lnclepenclent of the vectons !, ln the

a

200.Ll. Ll

other terms of -4.¡ tlren the subsiciiary conclitions must'be

che clcec-l f irst to see that in f ¿:ct a solt]tion exists'

Actually it is not ne ceÊisary to coml:ine al-l the

terms of À involvlnri thc same inclependent Z, if in fac t

,rvh i ch separately sat isfYthere arc turo terms \ and- r\',

[rf-v <lv = o =./ 1-

Z" . Thcn the-K= L

- d2r{or(2)) =

o.(') .,^a o (z)-l

arrt o Q). TnL

^ L

e

th

 V r1v anc.i whi-ch both contain the sameL-

quations - (r2r(ok(2)) = Âu arrcl

c an l¡e so lvecl ¡ and t he co ntr ibuti ons

an ì¡e founcl corre sponrlin¡1 to the ok(2 )

ic; th 1rd a.ppr crximat i on such a s ituat ion

iloes exisb¡ f or the inte¡:ra1 term of ^,

which we have

ì-gnorec1, invo l-ves Zt:¡tt: This is just cne ofòTãr

oe a

the irrl epenc-lcnt ve c tors of the other ter ms (tire fi rst anc]-

the seventh of (rU)). I:Towcver /

nror, ! cig = o, so that

r,/e can in fact solve - d2r(oTNr(2)) = Attm ant-l obtain the

(z)contriÌ¡ution to q frcm Õtnt without combinln0 Âlm-'

with the other terms involvin¡4 All otl¡er terms

2)(

ò!ò¿

oâ=

a

of ^,

uhich ane rlerivcÔ from t].e clifferential tenm of

Boltzmannrs eguation, anc.ì, r¡¡hich are those tl¡e cLeal lvi th

201,!..4

here, must be Civic-[ec-[ into terms which involve inc1e¡en-

clmt 2,. for thc con,;lition (ZLl) to be satisfie¿. Thus-1

rJi/e cannot hope 'to match the results of Section 2 for all

thc incliviclual gi'. Of coul'Se we must get a consisi;ent

ar6\,ver f or g, r;*c.Lless of how rrve calcula.te it, an,-', in

fact \¡E shall- see the result of calculatini: q2'l)y solving

the integral equations - c2r(o. (2)) = Ài, *rrlt" A1 in-

volve inclepenc-lent Zi, is Just that which in¡e obtainecl in

Secti on 2 when v'/e coml-)are the contributi ons r'''¡hich involve

tlrc irrlepcirclcnt' Zr.

A similar pnoltlcm clicl- not exist in this section in

ca l-cul-a tinfl the co ntr ibuti ons to the pre ssure tensor .

Since all thr terms of À involving G, satisfy tl-re suj¡-

sicl|ary conclitj ons because of the form of the tensor Gzt

the inte[tra1 eguations - ù2(*r(') ) = ^i are sofu-lr1e for

ar1 i, ancl the contributions !r('), cal-cLllatecl from th.e

/oìO,\t/ of the solutionsr eirçs r¡rith those of SectionJ.

1

202.4,4

In lookin¿' at the integral ejguation (61 ) u/e '/úere of

course considerin¡1 an egr.ration which h,as nc solution.

The solutlon of the cllfferential equation (66) v¡h j-ch we

obtai necl .lo es not olt cy the implici t bounr-],4r y co nr.Litions

of l,hc intc¡iral cquation - we siaw Ïve Lncl to relax a Con-

clition Q¡':(O) = O in orcler to satisfy the conclition at

infinity-. Ar-so the formal eontributlon q$z) which v¡e

obtainecl from it is not therefore physically meanintlfult

(anr-L oer t ainly cloe s not mat ch ,r(2) = o of se cti on 2. )

\{te will now cleal witþ t]-e inclepenc-lent term,g of (11+)

in turn, anC. f irst shcrri¡ they do in fact satisfy (74).

Then i,ue shalI Élolve the integral equation arisin¡i from the

thircl aliproximation to Bcltzmam'lt s equation for each of

these inclepenclent terrns. This is ilonc'by solvin¡¡ the

cÌiff erentlal f orm of the equation sub ject to the requinec-l

bourrlary eornlitions whj-ch can not¡,/ be enf orcec-L" From the

rlistrl'l¡ution function, f .?), o'l:taincrl from this solution

tfp contril:uti on tc¡ the heat fl-ux r¡ec to¡ involving the

incl epenc-l ent Z . is th en c nl-cula ted '

Combining the first, sixth ancl seventh terms of (ttl)

lve ge t

o.eand

+ v3B', G¡

,

p

?vfl.') rgcg)luTL

2O3.4"'l+

(Bz )

(e¡)

pz5 )

/òt¿r i\.

=/

/òB'(èr--

À1*7 "" the othen.

*?v")/òr[òr a

(a\dr* Ê u""') n,

/ò\¿r

lG,

BømT v 6B'clv

:\=/

o.

"\ , (85)-l

= À1*6 * Ll*7 '

where we c-Lef ine At*6 as the term of (82) involvinq

àÒr

(f ) Then lirniting ourselves to the terms of (1/+) '.uhich

involve ¿òg i. e. À1 *6, lve fi nd

o

=

m

d"r(Õt*6

l¡." - v d.v = ¿ .I 1+b - ¿ dr t

ou lzrÌ *=l_

oo -l

_l,

(84)

ancl on using (lg), (Bo), ancl (l '(lz)) for B', this is

secn to l¡e exactly zcro as lve reguire.

The intqral equation which we havc to solve in this

csse Is4f l,-.)v(2)

and substituting

(z)

)

lrvl (86 )0 ,!1+61 +6

(1)G,

in (85)r ancl using (1, (16)) f or B', and (79) for rlc 1¡ie

2Ol+'4.1¡

get

"(r ) +1(

(1 ) (r )2JoWq2 t=?b (# #.#;

(2" Ð kQr*6

r("r) cl

dcr \æ

(ez )

eQ+ 1+6

c ¡ (BB)

1zc")ã

(eg )

"(t ).

TVritten in clifferential f orm using tlre thcory of

2.6, this is

kQ1*6' + þ8t *6

a,'o'P(1)tlc

co

Tn obtaining (89) ïve have sul¡stitutecl- the explicit

f orm of p(1) given in (87) intr¡ (BB), and pc*fornecl the

integrat i on.

Norry r¡¡e llrove in Appendix 5, (A.SrG))

l"rã" , (zo" sc)ã' + 3

[" 3oeã(c)ac - Êltt, - "e)

+l 9+-ì .'-)) rr2)

t,, o

c+ 3oe tfr(e, )ac,

3o / *"U,c)ac

2L, (go)

2o5.4.4

so that the R,Iï.S. of (gg) has no constant terrn (in fact

it iS proportional- to e near zero, irrid- decrellses exponen-

tially as c - oo jus'ú as the non homogcneous term of

(1.1,(rz)) does)e ancl so !,¡e cnn put Ql*6"(o) = o as \¡/e

require. The methocl- of ol¡tainin¡1 tþe nurneicel solution

of (89) thcn follows the samc patte¡m as outl-inerL forbhe

solution of (3.1 ,(tz)), the heat co¡cLuction cqua-tion. In

this case hor''¡evcr we hnve to supply thc da'ca for'¡he R,11.Îi.'

involving the known sol-ution of (1,1 ,(lS)) vrhereas thc

R..[i. $. of (3 .1 , (lZ)) ulas kno-,r,¡n annlytically'

The result f c* ta*O?) vr¡hich is calculatect front

the solution of (89) us;ini! (Z),

a(r 1+6) (st ¡,þt*6 Ic

is convergent at the origin be ing ¡rroportionnl to C as

ïve expect. It i s plotted- in Fig. 1Or arrl comparison r,ui th

Fig. A1 shov'¡s tþat thls exnct clistributi on fì;,nction is

quite simila.r in form tc the finst tern in the Sonine

polynomlal expansi on 'wh i ch i s proponti onal to ee" \¡Z(1)

,(* " - "').\-/This v'¡e saw to be the casc with eþ

(1),

2A6l+.1+t.c

0.t

0r

n)

0.¿

'4"2

€P¡,

0

I

c

FlG.;.ll

207.l+. L¡

the Cistribution function obtained in the first approxl-

rnation by solvlng the hcat eoncluction equation, although

by looking at thc pcrsiti-ons of zercsancl turning pointst

sce thrr.t ,ç()

is perhaps better reprcsentecl by jr-rst one tæm in the

exlÈ)ansi on.

'rhe rn.lue of ,,*r(') , thc contributi,:rn to the trc¿rt

f rux vector from ãr*U"' , is carculated. via (zl) by

usi ng the numtr ieaf soluti on just proclucccl f or ,þ., *6?)

ancl intcftrrting numtrically usi-np; 'simpsonts rrtle.

+1g

o2c]-m

*" u, (c) þt*6

(t ) (" )u"

(g21----a|-3(a" r)z

1

oqÌd2

1-3a2 à

òr

2òA.'dv-

= .ztaiall (#þ) Ê ; . 0:)

This is exac'cly the result r¡¡e obtainecl- in (Z-,(Ztl-))

for ttre ter.ms in n(2) which invcrve $ Ë r fro that u¡e

have ccnsi"a"n"o It our final- answer.

(z) We now consid.er the terns of (tLr) which involve

ÀTr oË ' €u namely ¡., -,Of' =' " 'l+t'

)l-,+7?5 ;)Ât c4vf (,,)

L_bT:"+ ye /òR'

\ffi/òr|r òr.

(gr+)

The s-lbsic-Liqry conclition (Z+) is

V3 c1V

208.h¿h

(ç¡)

(96 )

et)

(ge)

t r ^r*7 dy =

Ð_òr ; E.% l"\,(#_#)"1 ,m

ò81òTand substituting from (t , (Zo) ) f or alfot¡rs us to

write

9s15 fuu

,lod.cg5c

Also employing (t, (tt ) for L' arrl the au,xiliany

olJffi-o clV =

i î" # dv = t.F]; [.* Zv4I* dv

[*r,o) d(t )5

2d,c2

lJT

the latter using (æ¡ ancl (lg).

co nd it icn (3.1 , (zl ) ) Y,Ie iie t

to solve for tæms involving

Thus comblning (gn)' e6) ancl (gg) llue

(gg)

see

Now thc equatitrn lve hnve

oeis

m I n,*l tr c]g = o as v,e oxllecto

òTòr

(z) *åv"J

(ü _+:\1n,.(Ë.!¡,\òr òv2 )f

(r oo)

- d2r (O )1+7

vrhich on subs titutin¿¡

209,Ll. lt

(r or )(z),lr1+6 G1

(r ) /òT\¿r :\

=/o1+7

-(r )ù +K

in (ror), ancl usi::rs (t'(tg)) and (t,(zo)) for

#, anct (lÐ for rJt is

Ltr2 l- c3 ñ., ..F L;GT L?3o44co

(r ) (1)2Jo

oltñtr

(1 )F a

l/a '+ eQ1+7 1+7

ancl

Vo'c4 +

.+ /i(4) 3i(4) ¡ir.\ BLc

\-æ- -7.+-).#)

"r"o'P(1 )

e(cr)

+

dcr

)c )g

(r oz)

(to¡)

In differential fornt this 1s

KQ 1+7 e+

ooc1

ä"t I+fl"

= "3$r "

3oc tf (c, )ac ¡

z.'ã"

* þ'*tur

3 + 1Bc')ã( 8"" + +

e

Ði"l*4 (,!Jxf

(2." 3e)x"'

c€)

+

(l ze2 ,

vrhere vr¡c have agnin usecl (lO¡ to cancel constant termg.

Once again we have an eqtation of esscntially the same

form as (3.1r(tZ)), ancl jt is solved. in the s{l'me man:Ier

2104.'+

as that eguation, although of course we must supply the

d.ata for the non homogeneous tern. This however can l:e

pneparecl from the solutions i ana ã or (3,1 ,(lz) r(lS)).

As we expect the d.istrlbution f\-rnction €''' (t )v1+7 ,

o'btained by using (2)t is like c near zero ancl 1t is

presentecl graphically in Fig. 11. Conparison with Fig,

A1 shoi¡rs that this is agai-n quite elose to the form of

"rs3/z(1 ) 1"'¡ although not quite so close as ,þ.r*6() .

By using (zl) ancl the numer ical solut ion erþr*t(t ),

the contribution to the heat flux vector from thls tenm

can be founcl by lntegration. The result is

(z) (, l")u"4t +7

(104)

= 2.53877 /k(ñ7

oe

)òTôr a (105)

(2)which is the id-entical ansïver for that Bant of q

òTòg

oinvo lving ' e whi ch was ob tai ned in (2, (27) ) .

2114.4

1.0

0.8

0.0

0.¿

0.2

.0.2

0

II

c

Ff6. il

212.4' l+

3) I{aving srccessfully treatecl the first of the terms

of  which invol-ves G¡ by com'bining it r¡rith thc sixth and.

sevcnth termsr we nollI If,iss to the second- term of (1/r-)

v'¡hich is the only one invclving AÀn'lULòr

Ä¿ = 2YA'/G 1 '' #

ôrf A'V4C1V A

(r o6)

T (107)

a

The su'bsicliary conclition in this case is

Br¡m

r-3f n, uò

m òI

anrL this intcgral- v¡'.nishes by virtue of the nuxilinry

relrtion requirlng correct clef lniti on of -c-^ in the

sec'ond npr-rroximrtion ¡.s we hrve scen in (99). In this

cr.se the equntion to be sclved is

- cr2rlor(z), = 2v:L'aer . # . (1oB)

On using (t, (t¿l)) for .4.', encl subst j-tutini¡1

æztF a" fr

into (toe), we ob'crin

o

2Jo

þ(1)

2 aGl ÒT<)g

(z) (tog)

Lnr2d2 = I''(1).J

(1) +K (1) (r ) c /i.''" l' '\(rro)

213.h.4

Once n5¡ain uslng the theory of 2.6'we can I'unite this in

cliff er entia I f orm:

= ["i"' * (2., 1)i" - 6ci,' + 6i] . (lll)

The soluti on of this equntion proceecls cs in the

other rr = 1 cns es ¡ ¡urd- tve ol¡tai n €Ú z(r ) by usin,3 Q).

It is I1ttle clifferent from t]-e form of the first tern

in tl.e Sonine polynominl- exp¡.nsir¡n as l¡Ic sec by com-

pering Fig. 12 ntr-L Fig. A1 .

Direct lntegrat ion of thc numeri-cal soluti on gives

'che crntributi on to q(2),

Qz2d-m

1

(z)

t(odr)'l2 ð"2 o-47

= I '1711.82

)^Ë /-.(c)þ,

\¡4/dr/-/k[ffi a

(c)csac

(ttz)

(113)

This agnin is exactly the value v.ffiich rvas cnlculr"tecl in

(Zt(ll)) f on the ¡nrt of the he at flux vector r'irÌrich

^òTdcperrls on a ãã .

i).1 | t

l.l . /+

r.ã

tD

2

0.5

- 0.5

1.0

eyr

0

2

C

F tG. t2

(J+) The thircl term of (ttr.) is exactly the s5.me 1s -bhe

sec oncl so f O.r as velocity <lepend.enee Ís concffned:.

,þ

ò8,ävã

/Ø\¿s

215.4.¿[

(111r )

(115)

(1 16)

(117)

(rra)

(11e)

As = \lA'Gl

Thus su-bstituiion of

(k (sä) (È-l (Ë)) a

and-

Qs(z

7-d2

LrFa" o4'r

(1 )þs Gz (k (sË) (&.") Ë)

into_ d2rqou(z), =

^Jprod-uccs (ttt) rdrere Q, replaces Qs.

þz(1 )

I

ConseErentlY

(1)5

O.e\ nrrl taken se'.ra'cel-y clo not satisfy the suJr-=/

(N"\av

Ore\ ==i

\)

(A- vs5- \l*a

v

'587',t,,11/k(æ-"7

*ÐloLöT

(Ë) a

(D) The fourth arx_t fif-bh terms of (tL) both invol-ve

sid.iary concl-itiOn (71¡),but combinecl, v/e fi nd- ihnt

Âh*

sat i sf ies

2 VB, G,mcl Ë)+

om / n,*# dv - W, ["* (f #- + Yal')av (3Ë

a

(tzo)

21 6.¿+. ¿{.

Thls result is obt¡inecl by integrnting by lnrts and-

oo

using (l ,(lz)) f or B' which shows lvoø'I

= O..o

.l

Tlren solving the e quation invorvlng 3Ë

(2) /ùlrar

o'9

/9p\ar

eQ

(tzt)

(tzz)

(123)

a

(124)

d" r ( t4*5 (d;""#.#uu") *,'(z) oê= )

a)

by sub,st ituting

Õ(z)

;)

m,Jì_ ü_-__l _ Eco (-

)

G1L++5

(1).

into (lzl) e arcl using (t , (zt ) ) anci (l , (16)), \Ã/e fi nc'[

rvhich in 1ts cquivnl-en'c clifftrential f orn is

kQL,,+5" -r k(zc - itQ'].*5'-úa,' Ll+2

", 'o'P(1 )

Lr+5

= - f *.(",)

te 4r' C1_q_clc I )"

Vfe solve th i s equat i on nunor ic al-Iy 1n tf:. e 'slnìe

nanner as previcus 11 = 1 eguations, ancl the cxact'

217 .ll.Il

(r )rl istrÍbut ion funct i on €Ql*5 lvhich is thus o-btninccl is

plottecl 1n Fì-g. 13. ,A.gain it shoin¡s quite a sinilarity to

the fonn of lrlg. ^1

.

(2) o

Final-ly the contributi,:n to g involving _æ.òg

gis

calculal,ecl via (Zl) b5r ¡1inect1y inteElrat in¿1 €Ù\+5(r )

(z) 2cln1

d.2

ù.tkr04ònò_q)

O "@c I ,k)euú, ..= ./o ' I--+)

(1) (c)ac

c

(tz6)

9l++5 -1

3(a" rr)

.312105 (#-) èpÒr

o

=t

Ci-rmpar ison wl'¡h (2, (zl)) shi:ws !'¡e hnve c'j:tainecl

ex.ctly the snme value for this Flrt of thc heat fl-ux

vector ar: vüe cl-icl there.

l¡/e have not¡/ c-[e¡.-l-t with r11 the terns of A' irvhich'

invofve G¡ anc-l v¡hich involvc ind"cÞenC'ent Zrr nncl on

comitarison with Sec ti-on 2, \'ue have se en the val-ue of ,Q

cnlcuhtecl lto'ch rhere ard. here is exactl¡r 'r,he s&meg so

thnt lve hnve corrficlenCe in ()l.tr nunerical CalCull'r,iOnS'

n - O [ìqu:'tions

Tn thÍs section Ì,,\re wíII cÌca1 r¡rith the tæns of ¡'

which inv,-rlve only scalars G.r. These are ¡4iven in (tO)

nncl v¡e wil-f cle nI fulty with the terms which involvc

)

lz t

t'ï BïA

E¡ 'gls

9'Z '

0.2 -

9.¡ -

0,1 -

.0.

lD

0

9çl t,tn 3

1)

9.0

219 "

4.t+

inclepenclent gralients of 'che tlrermocl¡mamic vitr iab les

rathæ than just the incÌivÍclual tenms rre hove v¡ritten in

(t o). This of course meâns the subsicÌian5r ccnclitl-ons

will- be sat i.f ied. f orfüeoo'úerms, encl althou¡4h lt is 1:y' no

menns ol¡vious that (ll) n-ncl (lf) nre o'beyed-, 1¡/e t¡'rill

shov¡ in Append-ix I thnt such is the clrse c

(t ) tr'irst u/e c1e¡rl vrith thc tæms r¡f (tO) v,¡hich lnvolve

proclucts of spati:r1 d-erivrtives u¡ 9o, 'Il:tc equation to 'be

solvecl 1n this ca,s c is

ô.2r(o,(2)) = fea,{o) (1 *, j\ _ Ç v,ø'\-¡ / - Lp - V / t

(127)

which cn usi ng(o)

ftv" 3#l;

qtlt t

,O

e

I

oe=

ce

l---It3I

I

(r za)

(t zg)

?or4d2 o'4

and. the values (f ,(f e)) arrL (l ,(zl)) for B' nncl pu#

,

neciuee s tott'

c,(o) )o( (o) l+K 2Jo

I )+ #

220,4,4

l¡ile reducccL the integral equation of this form to a.

d.l-fferential equation in (2,5r(25) ) ron ¿rrl¡itrary F

+ 1ç'\¡/' - k"1Ä/)

(o) ¡

zc;(kw" [äru)#ffi¡"u.

(n)Sul:stituting the ex¡1l-ieit fc¡rm of F in this

(t to)

(ttt)

eqrr"tion from (lZg) ancl mdeing the eguaticn d-imslsf cnless

by using c2 = aY2 nna ',ï, (c) = ø'\,tIl (V)r ïve finci

Ixf

Lce+ 6ã- It

,)

Tc solvc this equrtion v'¡c nust first find- the con-

d.ltion v'¡hich Wr arr]- its clerivttives must se'cisf y, In

proclucing (l\o) rue usecl W'(o) = o, nnd. !V nnrl all its

c-[erivat ivcs O exponentlnlly as V ¡ ooo l¡urthen,

frorn the clef initi on of 14/, Vl¡' ' ' ( o) = Or anÖ

oo oo ( o )u3\ry"',(u)au = (r.I ) cru, (t lz)uae(u)/

But f rom (2,5, ( 6 ) ) trre R.l-i.s. of (llz) i* zero, c

consequcnee of the a¡xiliary rel-ntion lvhich ensu¡es

correct c-Lef lnition of 'I in erclr approximatign. llhen

221 .l+.¿+

since on integra'cing lry irnnts ancl using these abovc

b ound.a ry co ndi t i ons r ue fi nil

oo

tJO

ool'

J.u31v"'(u)c1u = - 6 vr/(u ) du, (1 T)

(ß2) slror¡rs

and.

ffí'(o) = eL6

ñî"(o) = o

tJ.

oo

\ry(u)du = oo

Thus the bounclary conditions on 'Vt are

Wr ard n1l- its clerivatives tend- to zero

(t 3t.¡

(136)

(t tt)(1lB)

(13e)

exponentiallyasc¿o"t

fr,(") = Bs ,

.[**,(c)ac = oe U3D)

oí (o)W ,

v¡here a5 arrl o6 ârG constants to bc cletenr'iined-.

A stucly of the incl1cÍal equation of (lZl) with

R.I1.f;. = O shovrrs that thc hOmogøreous solutions stnrt

lil<c eo, ca ancl c2. Onl-y the soluti ons proportional to

co rnd ez at the origin satisf 5r the conditions (136)

(llÐ, and we develop thesc numericall-y from c = O uslng

the Run¡4e-I(utta step by step techniq-rc. A non homo¿leneous

222,l+,4

solution of (lll) is also obtainccl nümenicall-ye (r,ue

sLlrt j-t tike c2) and- it is founcl tlrat for large c this

solution and- 'cLre honogenecus solution vr¡Lrich starts like

c2 nre propol:tlonal, (lrotn going like e) so that i'c is

possible to o'l¡tain n comblnntion of then ,r.rhich is exì)onen-

tially cleereasing. The othe r homo¡leneous s;olution is in

fact the analy-bic f\:.nction e(c) ¡ so that a.c-[d.in¡; n

multi pl-e of -i:h is to the exponentially clecreas ing soluti on

procluces illothen such mlution; ancl ìty choosin¡; -bhc

mu.lbip1c correctly wc can satisfSr the aclcLi'ui<¡na1 conc-liti on

/-\ N

iil¡e heve plottecl the value of uú r\o) = - V+: ,

vr¡hich is obt¡.inecl fron thir.: sol_ution in Fig,'14. The

polynonial of least cle¡lrce in the Sonine polynomial

(o)exp arrsi on of þ t , rrvhich also satisfie s thc auxiliary

(t3s).

co nc-L it i onr¡ (2.2 , (l D) , (17 ) ) , i,s a2$i2

-.r Q)Þö1

(z)

in Fi ¡1. I-'3,

_uftz2 \4 5ct'

+ e4 , ancì. urc ha ve plottecl

Compar ison \¡ri th th is, lookinÉl at the 'positions of

'curning points ancl zcros ancl the ratios of successive/ -,\

meximn nncl minima, shows our exact .ú, \'/ i* of quite

)

similar fonm.

22t"k.tr

2.4

. ?.0

-.¿'0

' t'0

.f'0

- lc.a

0

I2c I

ev:o'

Ft9. tj

(Z) The next ta.m of i\. which lue consi¿er involves

second. d.e rivatives of T ancl comcs from the fourtht ancÌ

part of .¡1re finst term of (tO). First thoughe fnom

(3.2, (z), (¡))òTÖY

(r )

22Lt'.4.lL

( t r-ro)g

wher e

¿rnd-

l-'.n[2 /U3f3q2 \øm

(tLrt )),=

oo

J.

)'* 9

tr/r exp (- c")ø (r ) (c)csac. (ttrz)["

Thus

the tcrms invcÌving *, (#) r we

f , \_LO_Lò

ãr

òr-lãrl

cornbining the first term of (llr3) with tlrc fourth

of (f O) ancl vvr iting the ai)prorriate intqgral equation f or

)p:

(r e aV2 1

)

þ òrò

(z)

/Èr.\\òsl

+ _h2T

Òf:òr

(tttl)

o- c12I ( Oa/òr\ar

_Ò_òr \

( t t+4)

This is sc¡lvecl by pr,r'bting

"( o) (o) N,.

225-,414

(t lr¡ )

(1¿r6)

(|¡)

o,(z) - T#^&"(ßä)in (tt-¡L¡), ana 'chen enploying (t , (tl+) ) fon .A', (tt+¿l)

red.uees to

+K c

The equivalent d.iffcrential cgu.ation is

Efrr"' + zç,.ñr"

¡OO

/ e(cr)ie

+ k"iN 2' k' 'fr r)

d c , o"p( o)

clc I l+tr2 ).c1

cÅ+ (2"' 1)X', 'dN

a --a /-,bCjr + {l^

+¿r

T,Mce a

Ï\/e solve this e quction in the sane manner in r.¡ririch

\ilie solved. (lSl)¡ and. the so},rtion for ,,¡rr(o) which is

ol¡tained. from i?re solution 'b¡r using (f ) is shown in lri.g.

15. Again, rvc ,see 1t is not very cliffment ft'om

(2)1",¡.e$r

2

?.26h.l+

1.0

- t.0

- 2.0

- ¡.0

. t.0

- t.0

0

2 tc

r0tev,

Ft0, t3

'3¡

227.4,,¿r

3) The thircl term of .4. involves the pnoclucts of first

clenivatives of T" It is

(rUa)

(rt'g)

(tst)

(t oz)

and- so l-ution of(z)

ltd2r(o5

is mad-e by surbsi; ituting

which 1n d-iffenential form is

)

o,(2)= *-q(i)*:#.# (uo)* 5 24Ftz É o'¿ dg

in (thg). Using (tr(te)) alsop ,üe find- (tLlg) red.uces to

,(n) * r<(o) 2ro(o) =,"# Þå_, ("t{,*l * 3i..._ +,)

*+M (+- '' \l=u(o),'lr' \. - -l

l+ 6x- )-t

#Mce a

228'-U.l-t.

Sol-ution of this e quation by numæ icnl meansg using

the rjame technique os with (tll) givec . (o) itr ."'€tlt =' ' =-r o c

which is plottecl in Fig. 16. In this case the exact

(z)solution is not quite a*s r¡¡e1l re'presente d- b¡r eS Iz

,1, ^(o) ÀÐ 9J:Òr

a

( t 5rr.)

(¿r) The remaining terrns of (r o) lnvorv" $f,-òTòr ancl the

equation r¡le hnvc bc¡ sr-¡lve in this instance is

(2) òT- d2I(-aro

Sub stituti r¡n of

a" (z)

g!,òr

Ò,A. '^)ãvã

]

2)=6. v2cl-m3 Òr' (1fi)

in (lSS), arcl use of

z\tFa" fut(r , (rlr ), (19)) re rluccs (153) tu

Ò n

o (t ny)

llsinlt tkre cLifferenti¡.1- f cnm ç1syç]1¡r¡ccl i:i 2.1 this is

L4æ

rr )(a

¡@

/ .(" r )JLdcr a2F

( o -)

ì

)

_ x(4) + Zcx, . , zx, .. ( 156)

Ttre solution f or ffo gives e,¡,o(o) through ,þn(o) =

lll. . '- !L:* ai:rl solvlng (lS6) "" for the previours I'l = o equn-

c

t ions, we ob tai n the cll s tr ibutl on func ti on llre s entccl in¡/o\

Fig. 17, This is vcr5r silnil¡.r in form to .nrt" onee r'-gain'

z^g4.4

2.0

- 2.0

-¿.0

- 6.0

- 4.0

'S'0

0

3Ic

(0,

€t/3

-n.û'

F. tG. ,!¡

3l@.h.b

t2. 0

m.0

t.0

¿.0

2.0

t0,:V,

0.0

0

2

c

Flc. t

231 "4.4

In the 1:revious part of this section where \¡ie

dealt with equations f ormecl from terms of r\. which

involve G¡ r we solved- a c1lff erential equat ion (66) wh ich

arose from a tøm of .4., (6O), vdrich did" not satisfy the

subsicliary condit ions. ilhis soluti on for the d_is.bri-

bution f\rnction r;r¡as cì.ivergent at the origín; and of

course it is nc¡t a solution of the integ;ral equation

from vr¡hich the dlifferential equation was ol¡baÍned, f or the

intqgral eqr-rat icn is not sol-ublc. Correspon.dingly, 'r,he

cl-iff erentinl- equat i on is not so luble su'b jec'c t o the

bounclary cond.itic,ns riuhich we shoul_d apllfyr ancl in fact,

the soluti on which we proclucecl was only ob'bainecl by

relaxing one of these ccnd.itir:ns. The same sort of thing

can be observecl with the terms of r\, involving G wheno

tl-t cy c-lc not sat isf y the sul¡si,liany concli t io.ns ¡ ûricl

l¡efore closing this section on the rr = O equationsr vúe

t¡¡il1 d-emonstrate this. Tn analogy ¡nrith ihe n - '1 case

we conside r thre tæm whieh is 3¡¿l.r7tic

(r1^pD

) ò

-edf -oÒ

òr+ )a g

(157)

(r )

232.¿+.4,.

This must 'be inccrporated. r¡vith the terms of (tO) r,vhich

invo lvc (3ä) and ÒEòg in orclcr that the

o

=

odg'ãî a OJ

òr'

subsidlary condition (ln) be satisfiecl, I¡.Ie v¡ill- however

consid.er (157) alone arrl look at the equatlon

c(1)1^= - - Iz rp (o/

))

(1 )qòÒr

C+-o

trD(1 )L-

.t-'(",) # (3 .," ^ o) u",

- d2r(oo

Þo

Substitution of

) (â crV2 I(2) 0_òg o

Ò

õË(r (1 58)

(1 5e)

(160)

(161)

+

òòr

:

in (t fA) gives

so that the corresponcì.ing clifferential equalbion is

lKl.l,l-' ' '

*- - z(r<.fio + k'ffo'- k-.flrro)

The bounclary cond-itions r¡,¡hich should be satisf iecl ¡V fib

are "fro

and. a1l- its clerivatives tend to zero exponentially

A$C¡oor

tJO

oo

233. ,

¿+.1+

(162)

(163)

fio(")ac = o ,

ancl

fro'(o) = o, fir"'{o) = o

eún

o

Then foll-owing the method. in the other þ = O equa-

tions r¡re d.evelop a numer¡ieal, solution of (161) stanting

like c2 at the origÍn, ¿urcl atternpt to combine i-b vvlth

th€ homogeneous solution $tarting l-ike e2 to o'btain an

exponentially decaying solution for large c. This how-

ever io not pcssibley ad th¿rt such a solution cloes not

exint correslond-s to the fact that (t ¡g) nns no solution.

Only by relaxing the conclltion (163) anrl the reb¡,r al]-ow1ng

the thircl homogeneous solution, r¡¡hich starts l-ike c at

zero, to contrij¡u'be c¿ìn we fincL a combination ro¡hich has

the requirecl ex, c¡nential Ì¡et'raviour for 1ar54e co Flnally

!Ì/e a.cLcl a multiple of 6 to this ox1.rç¡s¡tial solution to

ensure (t6Z) i" sat isf iecl, The distrilruti on f\rnction

(n)

which u¡e cri:tain from the solution is p].ottecl in Flg. 18,

and. it is clivergent at the origin being prrrpontional to

2rt+l+.1+

r.0

2,0

I

€

- t.0

- 2.0

' 3.0

0

Ot

0

0

v

2I

c

. 1.0 Ftû. t!

1c o

235,4.4

From thc clef initi on of \,1/ (2,5, (tg) (zt)) v'/e see

t

in f act that this is an imnecliatc consequence of relaxing

the bonnclary co ndit ion at zayo, (163) .

Tt i- s not at once ev iclent thal, the concliti on (163)

should" hnve to be re]axecl, for lty looking at the equration

near zero we fincl f or lowest ord-er in c

ffo"'(o) = zfi'ro'(o)

whi chr i s cer tai n1y sat isf ied bv (1 63) . Horvever (l 6l ) cân

l¡e lntegrated. to give

[kfi¡' + (- k' * 2kc)ffo' + (k" 2k)fiD]

.L* - ct'€(c¡)ac1 = ilr , (t 64)

r',rhene v¡e have insistecL tfrat ",.Ï^ ancl its cterivat ives go toJ)

zero aS c + oo e ard. rrue see immed-iate]y that tþis forccs1

'tio'(o) = - t. This in fact is the bcundar¡¡ condi-r,ion

vuhich we producecì. when we obtained- the exponaltially

clecreasing soluti cn of (161) a$ just clir:cussed.

\¡Ve coul-d- Ctr COurSe haVe sol-ved second c¡rcler eguatitlns

with the L,T,.I,S. as in (tgh) for the four II = O equations

236,l+,1+

which anise from the terms of -4. involving inclepend-ent Zf

There is howeven little 'uo 'be gainecl by doing this ¿ìs to

obtai n the di. strll:uti on funct ion fr,' ' ' mus'b b e founcl.

Accorc-ling1y we solvc;c1 the trird ordlen equations lvhich we

cliscussecl ancl so obtainecl \i."' dir,ectly. Tf vre hacta

Lone to second ord-er equaticns we v'¡ould- have fc¡uncÌ that

the inhono¡leneous tenril is of orc'ler e at zene, ,so that

we could e nforce (163) ." lve stroulcl-. fThis of course is

just egrrivale nt to the integ;ral eguation having a sclu-

tion ¡,vhen lt involves inclepenclent ,i.l

At this point vre shoulcl point out tlnb in producing

the general n = O th ird oncler differential egu-ntion in

2"5, u/e m¿rcle use of the fact thnt the c]-istribution

f\rnction cLecreases exponortially fo r large c ; ancl th is

together with the condition r,4/' = O and- thc auxiliary

relations was sufficient 'r,o ensure tV arrl all its cleriva-

tives were e).pcnentia1ly c1æreasing. In sclving (161)

'!ve hrrve again 1or¡ked. f or exponential-ly decrensing solu-

tions, and. in fact the form of the equation alJor¡¡s such

237.4.4.

Eoluiionsr However the sorution \vhich we obtainecl for

øo ( o ) ," u iven ge nt at the oni gin, ancl fu rthen, the

auxiliar¡¡ conc-lition r¡Írich ensures correct ctefiniti on of

d in all- approximations is no longçø satisfiecl, a conße-

quence of the relaxation of (j$). In fact r¡e have fnom

(z "5, (zB) , (zg))

t,/o

oo

e(c)c2þD

(c)oc =¡OO

/ "tff^"'(c)d.c.l g L'

(o)

= ËE"(c) - fi^'(")l-tuL D Jo

= - ñ 'lo)"D \v'l

1

=É_2'On integratfng the sc¡lution u¡hich we founcl for

(o)e/n th i s re sult is se en to be ver if i erl.

Finallyr ït¡e pofnt out that the equation

ffr"' +z(uñ" +k,r,¡i,-r.,,fi) = ¡(e) (165)

hase when ¡(c) = O, two analytic solutionso In ac-ld-ition

to',ü = € ¡ which we have uscd-, the other soJution is

,.ï = k'. (166)

238.4.4

The theory of cllff øential equations tells u's then

that (165) can ì:e solved- nnalyticall-y. Proceecling

f ormally vre ned.uce (165) to a second. cncler eq;.a'blon in

À' i¡y substltuting

iV=g'À'

The eguatÍcn is

(167)

(16e)

(r zo)

(tzt )

E ¡... - rkÀ.. + r[(r" Z)x + k.l r. = *{€+. (168)

t4Ìhen the non homogeneous tcrm in (teg) i" neglec-becl it

has a solution X'= å (a coneeguence of thc sol-ution

V\¡ = k' of (1 65) rnri th J - O ) . The n sub sit ituti ng

h' =f; u e

(168) is reclucecl to a first rrcler equation in ¡J'

kþ" + 2k'pt' =Jck

Tntroclucing the lntegrating factor k2 i¡'/e find 'che

solution of thls equation isoo

c¡(c) + t-t'(*)lr

ancl f rom tlrisr vin (169) ancl (167) trre solution of (165)

ccrr be ol:tained.

239.4,¿r

The eguatlons we have cÌcalt with all Lrave J(c) a

numerical f\rrct ion i-n the câses lvhrere the clifferential

cguation ls clerived. from an integral equation which is

soluble, arrd there is nothing to be gainecl- from this

form of the sr¡lution as the integrations nece,s,sany to

obtain 'ff have to be d-one nunericnLly any\{/a}r. I'ü is

ilossible to gi ve the so luti on of (1 61 ) anal¡¡'bicn I1y f or

there ¡(c) = Q€o

If we d.ef ine

or(c) = cxp (- c2)acï"

oo

9 (ttz)

(tts)

then in this case* xe (x) + a.,(x)pr(e) = cl-x t

2]K2

a solution rryhich appnoaches zero exponentinlly es c +@t

a consequence of the e)cponential behaviour of WO nncl its

d-erivatives as e - oo which we enforce" Then using (169) t

c * xe(x) + c¿(x) d.x dy + À(o), (171t)2]K2

antl- the constant X(o) j-s obtainecl by u,sing (q ) nrtl

(162) which ensrlre

2tl9oo

À(x) e (x)c1x =

1

rr2

240.il.l+

ar(x)À'(*)c-[x = o. (175)[*I

'ol(o) +

Thus

^(n) = -+ tT- lo

* ø(x)tr(x) ('tt6)ã(") ¡;(x)ax .

Substitutins ?lA) i.t (171+) and usins ?q) trre

solution ih is then nnnlytical-l-y

fip= €?#Å*'"uì#l'' L

* xe(¡r) + a;(xfk" (x) clx c1y

c * xe(x) + ar(x) I)

a1

analytic soluti on"

Åcì-x d.y (ttt)k'(x)

By stuclyÌng the relationships between lþr he ancl p we

- 2rin' I o)can sholv that p' is ltkc neÐr zero. But using

(lll) \Me finc-[

1

Thus near zero þL' is rike $ " +

t ¡@

= i+ I *, (x) ¿x.1!t /c

p

ancl vue verifY

ergain the re sult fi!t"l'l

Tr'.T t this tirne f'rom the

rr = õ lllquations

0_B--ò\r2

241.l+.4

(178)

(ttg)

(rao)

The terrns of Â, (16), ancL o , which involve Gg

are the next we consicLer'. Vlle solve the cquatlons whlch

they give rise to by making use of the resul-ts of 2.8

where we red-ucecL 'the lntegral eguatlon

(z)

G5:

G5:

3),ß) + K(;) _

" ß) __ ,(S) ,o

to d.iffenential form for arbitrary F

(t ) Firstly, wc have to solve

- d2r(O¡ -1r-5,im

(2) ys

ancl subst itution of

.R Q)S/1 =

1

ia2 3)( -oODöY=

t-tox€ú futÀtr

in (179), and then usinr4 ?,Ql)) ror ff; ,

,l)

v (t)

gr-ve s

6e''c"

1+# ry) = r(f). (rer)

"', T(3)", Y(3)', ,3))+

(4)If we ]-et u("(¡)

be the L.I{.S. of (e.Br(zB)) we can u¡rite (131 ) in

d.ifferential- form as

,

(L)\ (3)' ' ', ß))

242.4' l+

(i82)

s(vr (¡) , \3)

=e(v)ù

dVçLLv

a

Here the subscript 1 in ,,G) incU-cates we are

dc.aling with vß) cefincclotrúrß), i.". lve are consiclering

the first tmm involving Gs.

This is recluced- to an equation in the d"lmensionless

variable c by ur.itlng c2

'Ihen if S' represents the L,I-T.Í:!. of ' (2,8, (Zg)) with cx = 1t

ancl we su'bstitute from (tAt) for the explicit value of

u(l) ard. penf orm the inclleatecL differ entiations, (182)

l¡ec ome s

Ê'(i1 ( j) (4) , it (r)"' , it (s) , it (3)' , î, (r, ,

lc,ã(5) + (4c" c¡fr(4) * Q+ca 1zc2,g(f)

+ (tzc 24co )ã" + 6oc'& - 6ocãJ (1Bl)

The bound.ar¡r concl-ltions rvhich \(l) "trl 1ts

dæivatives must o1:ey, are, from the d-efinition of yß)

ancr 13çrye (z,l-t, ?D),(¡)),

= a\ï"-and. \ßy(c) = otrr(1,, (v)'

,r G) tnd all its d-eriva;bives approach zeroexponentiallyasc+ooe

2l+3.4.h

(t a¿r )

(185)

Yt (s) (n) = Ð.7 ,

Yt (t) '(o) = o ,

Yt (z) "(o) = â6 ,

it(l¡"'(o) = o 2

and. (4)vt (t) (n) = âe

where a7, a6 and- &e cì.FG constatrts to be cleterminecÌ.

A stud-y of the inclicial- equ¿ì.tion of the homo5leneous

eguation formed- by nqlcctingthe R.l{oij. of (1Bi) s}rot'is that

'GYE

e -3

homogleneous solutions near zero ene prcporbional tcr

-1 cc ' , c ând c2; anl f rom (185) only the last ti¡¡o ofI

the se can co ntri Ìlute to 'b hc re gr-r irecL so luti on of ( t S3 ) .

Accorclingl¡r we d-evelop these homogeneor-ts solutions

numerically from z.eyo (-firc 'nourrlcrS¡ condi ti lns bei ng

cleterminecl by looking at a l?ou/er series solution of the

homogeneous equation). et:c'l likewise, an arbitrany non

Tromogeneous solution of (t g¡) is obtai rredl which satisf ies

( )Yt (s) o = o ancl \(S; '"(o)

llt.l+.4

we fi.nd thc rrnigue eombination of ther-;e numerical

solutions which clec'neases eq1onentially 1n the Same

menner as we d-icl in solving (3.1 ,(lz)) for i. Fnom

th 1s so luti on the cli st r ibu ti on func t i on i s fo uncl u si rg

,t,(3) = cît (3)'' - ,i1 (3)' . (r e6)

This is plotted. in Fig. 19,

(Z) The se.:onc:t term of -4. involvin¡¡ G" gives rise to the

equat ion- c.rzI ( Oe

;ò3'òr(2) oU' (187)

(r lo)

) = g v3B'G'5

/-\ - o

þr\5)e"; ;i g

wh-ich we solve by usin¡i (tr(te)) for B', anc-L substitutin¡i

r!') (t ee)

in (t A7), This ¡;ives

"(:) *,.(r) - zto(3) = p-fð (F #. #¡ = "(3).(1Be)

Uslng 2.8 t¡re fi ncl the equivatent cLifferential equation is

s(î2(j) (4) , îz(3) ' ' îz(j)" ' îz(Ð" iz(Ð)

"a o,r(3) [esã(L) + s,.,lLtc,4fl2

+ \eol + g"hcs + 4c" - l+c] + g'l- 24e' 12eaf

+ el24e + tzcs )\ '

= e(c) clcic

c1

clc3Lc

-z4glt.b

€vt

0.6

0.6

0:4

Itl

0.1

0-2

0.1

0

I 2

c

FtG. It

246.4.4

r¡¡hich is solved. numerically as ïue solved- the previous

egr:ation. The rer+u1t for ,rl ,(3) o¡tainecl from the

solution iei Ilresented graphicall¡r in lrig. 20,

ß) Finally,\mo

the term in (16) which involves *å q givesdn=

rise to thre equat ion

- d2r(0. /ò:B' 2òA'\(ãõ- - ãË"=)*

òqÒq

using (t , (t 9), (2o) ) ro" ffi: ana ffi and putting1

(

t

oe=

3

òTo.ìn

ir1)"', is1) , ir(Ð', il (¡l )

@ cx,p('r) = .a¿$) + (r¡cs + ec")õ(4)

+ (t.c6 + 2oca + 12e2)å"' - ã"[12e3 + 12e3]

(r\ = ? v=,5 (r gt )

(1 e2)3ot

LrOøzaz o4t,þ 3

ïue t eci-uce th is -bo

v,¡hich i¡. iLifferential form is

cl

,3) +Kß) - zros) =y:db W # .#.+(# %..#)]= n(:) ,çtst)

3) G5:

S(i¡(r) (")

d3ê Ê lq

CLC ICL

,

d.c

2474.4

0

- 0.1

t)

- 0.2

. 0.3FrG. 20

+ i(5) lu.o + Ltc,l + i(4) [¿+"u + l+c" - trc]

(o)

2L¡8.4.'"1

(1e5)

- i"'llzca + 214c2 ] * i" llzcs + 2L+cl . (19tt)

We have p]-ottecl the numerical- value of u,lt"(3) ,,,,,rrich

is obtained. f'rorn thc nuntrical- solution of (19h) in l'ig.

21 ,

In all three cas es, Vr3) .: tarts proportionnl to cs

near zero as e.xpected, and by ccmplr.ing Fig. 19t 2Ot 21

r¡rith Fig. Al¡ rrue see that fon ,,¡r3) and ,,þr(3) *nu exact

prcìportional to the first term in the Soni-ne poÌynomial

expansion of uf r(t) . Ho,¡,¡ever Fig. 21 ind.icates that

,,lr13) n*n consicler.able contribution fnom tri¡;her tcrms in

solution is similnr in f cnm to €c3 s- 1nI/ L = ec3 r r¡ftrich is

-bhe expansion.

ri = l+ Equation

The equation we have tc solve fr:r the only term of

 vnhich involves Gae Ul), is

16(z) ò.Lòv2)

oo0a::e s I

5- d-2r(Q

3ya

?¡;9;l+.h

¡.0

2.S

¿.0

(3t

EP'

r.5

1.0

0.5

0

c

FrG. 2r

and., substittrtion of

-(z\ 3 4(tL eQ" =-iDrzd2ú

into (lgS), and use of (l ,(zl)) f or OTJ

ov-

250.4,h

(1 e6)

(1e7)

(r lB)

oo¿i""ê I

(* '(u)

, gives

,(h) *(u) _ 2J (! ) = W+a çfr' " 6;''r'" -FK'.' -uo crre(ãTL*"-7--ãr

.# #l-F(4)'This is just the integral equation u¡e reduced to

d-ifferential form in A.9 so using (2.9, (¿lZ) ) \',/e have

i (6) (¡) (¿+)(ø (lt) ' ct(h) ' tt(L

) ,a t a aa

"(r*) 'o(rr) '

"(l-,-))=Dcx

4) o( p

where is the short hand representation of the L.H'S,t-

of (2.9, (47 ) ) o As r¡/e lcf t ín 2.9 it contajned- many

known funct lons u¡hose clerivat ives were requirecì..

The algebna invofveil in calculating ttresederivatives

and- the R,I{.si. of (t gg) although straightf orlvard, is

quite exhau-stive¡ and. we have teft Ít and the subseciucnt

numerical solution of (rgg) from vrrhieh ,,¡,(Lt') ," obtainecL

t

fon another time.

251.L+,5

4 " 5 o qn:p?_r_i_:: q 4 gl.LegtJJ:?

be wnitten

Av1 A

o

ñ2

d.mT

òT ^ n2 fDoãE * øz lmr* Lbr

/ò|t¿r

The thircl apl_troximation to thc heat f lux vector cnn

)

-q-

2()

o

ÒTlòn(c

-o(Ë)

o' I + osÒ

ï,Êclm

É_d-mT

Òf:-

òr

a

' e, (1 )+ ork 9.pòr 9 + 04

ò

whcrc fr om (3,.2, (l S)) ' n' . I OOB122lcTrnn7 a

1j.1/e \\rifl no\,r/ comllare the values of thc numcrical-

coef ficie nts U j cal-cul-a-becl exactly in Sec'ci ons 2 nniÌ l4t

ancl_ the approximnte values obtninec-l in chapman ancl

Covrling¡ (15À, (¿l ), (5) r (6) ). In this apLrr,)xim3'ue calcu-

lation the inte¡4rnls of Section 2 .ÌrG cc"Iculatecl; but

insteacl of using ihc ftrl] seconc.l âpproximation to the

d-istributi on fLrnct ion as u/e have rlo ne, the expan$i ons of

t(t ) "na

p(2), made in tslrns of Ílonine polSrne¡11ials, are

usecl, and -bLren onl3¡ 1|c first terms of these exps'nsiohs-.

The intettral tæms of (l ,(f) r¡¡hich we have ne¡5electecl

clo of course contribute to the fifth tenm of (t ), but in

252.4.4

the allproxinate calculation using the first terms of the

Sonine polynomial ex.rlansi on,s, the ir contrillntion to 0s

is zeyo? so thnt the approximate values ancl tþe exact

values, We have calCulated f rir 0s, d-e¡ive from 'bhe Same

'rlef irls ,

The valucs of the tÌrree calcul-ations arc give n in

the tabl-e belr¡trt¡:

at Appr oximate

11.6536

5;82679

3.09590

2¿\1527

25.183?-

1 I .6536

5.82679

3.09590

2,1-t1527

25,1 832

11.25

- a-5 'Ô)

3

1

27,75

There is a cons:ideral¡le d.evi.a'c1on in -if:e ex¿rct

results calcul-otecl here and- the apllroximate values of

col-umn three of the tal¡le, This compares r,l¡i-bh 'r,he sma1l

cleviation of on]y 2r5')o vÍricir oc curs when the sec ond-

0t

02

03

04

05

q3

Section 2 Via Õ

approxiina.,cion to tlrc he nt fl-u:l vector is calcula.becl-

u,sir¡g jrr-rit the first terrn of the Sonine pol5inomial-

expcnrion of ø(t ).

In a simil¿rn mÍlnner bhe thircl ûpilroximation to the

253.4.5

ß)

pressure te nsor can be expre ssed. ._*,o____

(2\ ñP o ^^2 ,D- o ö ;--E"'=d1 To=+r'f (Uf (S) 29'utn"

*-o* *9-: --o:n2 ;-ãT n2 dp-ìT n: äf-fl+ '" är*E ur Ë . ', ffi'pT Ë E . @s õF är. "ÒÏ

---'=:

)

+@g É\!

a

oe=

oe

C¿rlculabions of the integrals of Sec'oion 3 are once

n¿¡ain given in Cþal?mnn nncl Cor,vlin¡; using <¡nl¡¡ the fir¡:t

terms 1n lhe expansionsof t(t ) anrr fQ). T-'he terms

involving intqçral-c 1n (t, (6) ) which in 3e n.eral contri-

bute to the fif th ancl sixth terms of (S) d-o nc¡t virittr this

simplification give any contribution. Rltrnc'¿t has calcu-

letecÌ the coefficients @1 . o. ctl6 for rigicl spheres

nssumirrs: four 'r;erms in tJre exllansi on of ø(1 ) a.ncl ø

(2)and

therefore inclucling the effects of the inte¡1ra1 tærns

,¡rhich y,¡e havc neglec-becl. In the 'cabIe Overleaf rr¡e have

ta'ltula.i;ed. oïr1¡r ilrose at, trrrhich come from the same terms of

À.

(z) 2Ìl

Section 5 Via O

L,,jJJ2J

2.02862

2 .lú 527

.68o63L¡

¡¡proximaíe t

1 Term

254,4.5

(2)Ð Br-rrnett

Lr.056

2,O29

2.L,,18

,681

()( I!)

2)

(z

4

2

3

U

.Jjl-¡O2O 1.5

7.62952 B

Because the series of f cur terms rr'vhich he olltained-

yras auite ra¡¡iclly d.c creasing, Burne tt claimecl his ailpr'¡xi-

mcrte resul-tg are ltprobab ly corrccttt tr¡ .01;,1' for a7 anc-L

úr, '1;, f or ds, ancl .J¡ì for (Ð4, Comparision v'lith 't'he

exact values calcula-tecl- in this ther:is sho\iv this estinrnte

to be reasonabÌe.

t,)6

Øg

ú)n

@3

@2

@1 )+.O5723

2.oz\6z

2 "l+1527

.68o63Lt

.394021

7.62952

255.5

CHAPTER 5

CONCLI'SI ON

In this thesis we ha.ve used. the successive approxl-

mation scheme to obtain normal solutions of BoltzmanntS

equation for a di 1ute, simple gas of ri gid- spheres.

Ëloruever we have not evaluated the transi:ort cr:efficients

by tre usua] method in which an expansion of the d.istri-

bution functlon is made in terms of Sonine polynonials.

Insteacl we have reclucecL the integral equation which

oceurs in each approximation to a set of ordinary

d.ifferential equat i ons. The C'Ístribut ion f qnction is 1n

effect cì,eveloped. in q,lherical harmonics, ancl the rcd.uction

to clifferential form of the integral eguation, containing

a given orcler spherical harmonic, hÐ.s'þcen carriecl out

in partleurar for ord-ers rl = otle2e3 and' 4' rn fact the

method. presentecÌ, in which räre have used certain auxiliary

fr¡nc tions¡ ancÌ the clynamics of a collision betvrreen two

rigicl spheres to perform the integr"a'tions irnrolved. in

the collision integnal, is applicable to any ord.er in

256.5

generali although the al¡4ebraic clifficulties associatecl

with hì-gher orflers increases rapiclly as we have seen in

2.9. The c-[ 1ff e rentia]- equat ions obtai nc,1 in the s econcl

anrl thircl aBlrroxima.tion have been solved by numerical

techniques, anrl tLre clistri'bution f\rnction obtained. from

these so lutions hes been int%ratecl numæ i-cally to tji ve

the exact corrections to the pressure tensor p and- heat

f1ux g vector in each approximation.

Tn the seconcl approrimation the distribution

fu¡ction d.epend-s only on the graclients of 1oca1 tempera-

trre, T, and- mean velocity¡ 9o, the twc thermoclyner:ric

vari¿bles ,¡frrich \¡rlûh number density, d-, clef j-ne the local

equilibrium state; and_ the graphs ,¡r¡irich ,rtre have pre-

sented- show the velocity clepenclence of the distribution

function, which vras f irst exactl-y knorlrn only in 'dre case

of Maxnrellian molecules. BV calculating the pressure

tensor ancl heat flux vector in this approximation v¿e

obtaineC the exprcssi-ons of linear macroscopic transpcrt

theory at 1on clensity because of the dependence of the

257.5

d lstribution fl-rnction on the grad.ients. The exact

values of the coefficients of shear viscosity ancl thermal

conduction which r¡rere founcL are i.n good. agreement wlth

the rapiclly convergent series which is oþtained- for the

transport coefficients when the clistnibution f\rnction

i,s expanclecl in an infinite series of Sonine polynomials.

Up to this point our results conf irm the exact calcula-

tions of Cotter, and- Pekeris ancl Alterman.

wíth each stlccessive approximatlon the ord-er of the

spatial gradients of cle T and co 1n the d.istributj-on

fur:c tion, ancl so the pressure tensor and heat fl-ux

vector, is increased. In solving the integral equatlcns

of the third. al?proximation, b3r makinP, use of the clifferen-

tÍal- equations previousl-y clevel-oped, r,ve have therefore

obtalnecl ref inemerrt of the linear transport equaticns.

Tt must be recognised- however that the values of the

pressure tensor and- beat f lux vector So obtainecl come

from the theory of normal solutions, and vrre woul-d I jke to

stress that any results clerivecl 1n such a theory are

rea1Iy only applicable tc systems near local cguiliJcriurn'

258.5

The exac t expressions obtainecl f or the distril¡ution

function, (which we have presented graphically), the

pressure tensor and the heat flux vector in this approxi-

mation are neïu. Prev lously Burrnett, ¡s ing the expansion

of the clistributi on f\rnc tion in Sonine po]ynomlalsy cal-

culated the third. approximation to the llressure tensort

but he macl-e use of onl-y the fi rst four ter ms in. the

expansion of the seconcl approximation to the cl-istribution

function (which ctr course enters into the d-etermination

of the thirrl approximat ion). His re sultsr which are

ol¡tainecl as rapid.ly convergcnt seriese agree with our

exact calcu-lation to the accuracy he statecl in his iÐ'per'

îhe evaluation of the heat flu.x vector has not apparently

been cÌone to the Same accuracy 'oy this methocl, ancl our

exact answer i-s rather c-lifferent from that calcu]-atecl

vrhen only the f irst term in the Sonine polynomial expan-

sj-on of the seconcl approxirnati-on is used.

Wehavenotsolved.thcthird-approximationto

Boltzmann?s equation for all the terms in 'A', the in-

259.5

homogeneous term of the integral equation. There ere

in acld.ition several terms ,¡lhich involvc íntqrals of

proclucts of the sol-ution of the sec ond approximatlon.

In princì.pIe these oulcl bc eyaluated by techniques 'similan

to those ugecl in reducing the col-lision integral of the

unknourn f\rnction, but we have left this problem f or

ano the r ti me,

Finally we stress that 'bef one obtalning the solu-

tion of the clif ferential eguation arlsing from the

integral equatlon for each of the terms of r\. d.c¡nnclent

on a given spher ical harmonic, the integral equ.ntion

itsel-f shoulcl be examined, to cnsure that it does in

fact have a sol'ution. This is r]one simpty by calculating

certain integrals; ancÌ in fact it is a conseguence of

tfp sgbd.ivÍsion of Boltzmannt S equation whích Enskog

made that if one d.eals lvith terms of .4, involving ind-epen-

clsrt tensors wlrich clepencl on the grad.ie nts of c1r T ancl

9o, then this is sufficient to ensure that the lntegral

eguation will have a sol-ution. Because v¡e spent corl-

260.tr).

sid.eral¡Ie time prod.ucing exponentially d.ecaying soluti ons

of d.ifferential equations which arise f rom insolu'bl-e

integral equations, before recognizing that ln fact the

solutions were meanin6Sless¡ \M€ sound- this as a wanning

f or othere treating similerr 1æob1ems..

A1 .1

æÈEÐA-L

The tensor Gn(g) is clefined

= (- .1 )" Yii' ÒiÒvn

for rr >o, (t)LVGn

tnwhere \ i."

òv"Ò

òVapplled n times: i.o, 1t is a tensor

thd.ifferential operaton of ihe n ord.en ¡

voöt G2 = (+)l

GnòãiG

v,V

ug v /--l-\12 2 \

VVGe=f--å

ys ò rÒ2 òV lòV:L-

a

Alternatively, G, can be d-efined. y -bhe recurrence

re 1at i ons hip

\T (z)

(¿r )

n+'l n n+1

withGo= 1, 3)

'Ihe values of G' for fr = 1r2ri and- lr wil-I nol be

ca.lculat ed. usi ryj Q) and- (l) .

(t ) Su'l:stltutlrig n = o in (2), we 6ptv

G¡ =T(z) SirnÍla rIy

òaÒV \V /\,/

VV+L-ye

ô

Ð

VV

_!__òV

1

+

fur2)

Ã1 .2

(¡)

(6)

(7)

(B)

_22

1"-x0.F"

.Vt-

Thls is clearly a synmetric non d-ivergent

tensor¡ its component form'being

(c")i r = *rä j -+6. .- r-Jv

ß) It ls easiest to work with components of the

tenecngG^ for n > 2. From (z) we get

V.(o")iJk = +

= # uruJuo

= # urujuo

- l--. rr2V '1

-*(v

/: Y-Jrc - ålt - ¡¿-\2 v2 2) 3

J.2V

uv.v -* tvgl

d (drrvo + drouJ

d..),r_ J'

jk

idjt * v¡dit

In vector form this is

Gg=6

îffi

V.K

( 3SYUI )

whe¡:e [U fJ*"r,r, is the sum of the th¡'ee possible ten-

sors formecl f rom V anct { ' It is c}early synmetnie¡ its

iikth elernent being urdjo * uJdito * uodíJ. Contraction

of any paln of ind_ices of the symmetnic tensor G" gives

Z lFOo

V q.

(v ç

+

+V V

A1,.3

d.. d1K

(4) Irinally sul¡stituting n = 3 in (z), vue get

V.(co)ijkL = + (# ojuou, - *o (v¡dr, * uodJü * vrorn))

Iv+ V. d.KJL

. $ rdr¡{o,+ V.V d.. )1L JK' i¿+ V.V. d.L L K JLj.Vt_

dlJlr

+ V.V d..ILJKJ¿V.V .V. VlJKL

G¿ = #" y.u Y u - ëæ tg v 4ls¡nn(e) . å.,tg 4lsvur(¡).

)

jI(i L

)( jÒv1-

2\TV.V.V

J.K L

/-L-\2v"

òJt4 k

6

jLLi

+ V.V. d.lKJLLj.V1

1_2V2v.v .v. vlJKL

tr

2V*

tr

BV2

-l_Bv2

7Ê,ffi

+).v.JK

8. v1¿

d.,v.v +]-KJL+( d, juou,

(v t.K

+

)+ ,('

L

(vrv Jt- + V.V. dKL 1K

1tJ

+

+ di ¿ d¡¡)'

6

BV2

d. .v. v1.'l K Ld.. v.v +lKJL t t u,. J6.t + frodi,)kî. v1L+ +

(g)

In væton notation this is

(r o)

G4 is thus symmetric, ancl contraction of any

pain of inctices gives zer1, as cloes contnaction oven the

A1 ,4

f inal renaining pai¡. to fonm the tnace of Oa"

In tLe ,sarûe marulen Gge G6 .. . could j:e f ouncl

but in ûiíe thesis we use only tenns up to fourth orclerr

ITB e where b is the coefficient of then n

tq:m of G of degree n in [r then using the values ofn

Go ... Ça cal-culated. kere ancl also writing

U ,

we have

Gn;-n

UU

co (u-) Bo (g) = Go (u ) ao (I)

=1

_T) (g )t v (r t )o t

cr (g) nr (!) = Gr (U,) er (!)

=Û.9= p,(ü . ü) ,

G, (u ) : B, (Y,) = å n" (g) : ca (Y,)

7.pd

= É (g . g), - +

= P"(i.' ü) ,

(tz)

(13)

(sB5e"(U) =ân,(g) ic,(-v-)

=+ (g'f)o-#€'ü)".â

)

L1.D

çt t+)

and-

=Z G'ü)" -â (û'i).N N.

= Ps(U . y)

s¿(U) :: e¿(y,) = # e¿(E) :: Gr(Y)

irP a(U a 1u)

In ,lencna1 lt ean bc pnoved thåt

orrQ) I n,,,(u) = nr.G' ü)so that

err(u) I urr(u) = l,r(1 ) = 1' (16)

These pnoperties are usecl when we obtain the

e qunt ion satisf iecl by t(") (2.2r(ttr-) ).

A2,1

LElj,q|ID--Irx eThe exp€lilsion of 'bhe corrcc tion to the clis'crillution

the¡lis he lrm ote

(") (r ),, (") = c ifn+L

- lrr)functionr t(o)0\'''', in terr,ls of Sonine polynornials'ovas

finst macl-e by Burne -bt (Z). Tn the notation of tiris

(r)1""¡,a

l'L

oo\---'ì

\L-tL=o

L

wlpre e is the climensÍonJes,s peculiar speeclt

^ (n) rq (m+n)s_ r--, (x) = \ (- x)n *-n-(rr'+îfm /-t

Il=O

and-

r¡rith

(**n) (* * n)(ni + n - 1)(nr + n - 2),..(tn + p + 1),

(z)

ß)

n-p

a 14 ocluet of rr - p ter rns "

In this thcsis we have obtainccl tire ftrnction ,(") (")

for a gas of rigicl spheres clirectl¡r, by nr-rmerical solution

of the c'liff erential eqrrations lvhich we derivecÌ from

Doltznatïrf S equat ion. Ilorievcr it is in'be re sting to colll-

pare th c f orm. of ,þ k) (t)

, the co rre cti on to the

L2.2

distrilrutíon funct ion olttained 1n our exac'c methoc]r arrrcl

the fonn of the -l}irs'b tcrn in the expanslon, (1 )r muJ'ci-

pliecl by € c

Belov¡ ,¡',e give the first tern of the expansion f or

values of n rruLrich occur in second and. thincl approv'l-ma-

ti ons 'úo the d,is tniiruti on ftrne ti on.

(r ) 1 a It = 1o2.

For n - 1 the e)$ ansion isoo

ù,(r ) tL-t

(¿) (r)(c') a

L 3/2L=0

(i )The conclltion imposecÌ on the solutj.on c¡ fron the

auxiliary nela'bion (3.2, (16) ) is

ï"

oc¡ (r )

(c') = 1

exp (- e'-) þ (c)e"¿c = o .

Substituting from (4) into (¡) gives

exp (- c" ) o ts j/z

(¿) (c")c3dc" = o. (6)co

$)

(z)

oo

Tl-t¿=0

(o)ñr) ,

Bu-b for aI1 rn

m

"L t*u*n (- ez)s3/z

i---1)

/-l

L=o

so tl'lat (6) c¿rn ìre lrrritten( ¿)

A2.3

(c2 )c3d.c' = oo

(B)

(e)

(.i o)

(e" )s1/2(o)

Then using tìre orthogonality of the Sonj-ne polynomials;

oo

exp (- ")s(n)1x)."(o)1r¡**u"=o ron p/ q em

l-(tn+p+ t)/ú forP= et

-bhis re:l.uces to

ø(r )

o

Thus the f irst non zero term in thre exÐanßion of

(1)is pro)or'¡iona I -bo c ît3/Z (c')

ÙLo

r-,42

c c3 a llhe

corresironcLing correction to tÌre clis'cributi on fì:.nction is

pro rortional to

ancL this is ploticcÌ in Fi3'. Ä1 .

V,/hen n = 2, the exjlansion is

2) i-)

(¿)

(t r )

co

(arS (c').

Lþ =c2

L=O

5/2(12)

42.4

0.t

c".6

il''ecs_tl2

0.t

0.2

' 0.2

0

I

c

FIG. AI

112.5

(2)The tenson a.-js oeia-terl v¡iih þ in the ex1)ansion of

*(r ) i Gzr so tlrat the auxiliary rel-ations (2.2t (15),

(16), (17)) are autom.atically satisf ied. Thus -bhe f irst(c\

(o)5/2 = e2i ancl the corres'ttoncling cornection to-bl'lec2S

cl i s tr il¡ut i on fìr nc ti on i s llrop or ti onaf t o

f= €c2

which is Plotted- in Fi3;" L2.

(2) -LlhfåQ,{p:Ëg=:f.:q-Lo*n¿- ][:=å. *å*=-!-,,1 ¿2' "j,,}L"-

(13)

llhe ex'l?ans ion

u,(n) 1") = ",\-----ì

)/_l

( ¿) (c" ) (ttr )aS {L n+à

L

is: usect in thi,s approxination, following Chcpber 4 ';r/he re

we rn¡rote ,(n) insteacl of *(n) to ¿rvlid. confusion. The finst

(r )terns in the expansion etþ a ncl etþ are proportional

to ,os3¡z?) ancl u.'s5¡zG) rvhich are given ln Fig. A1

ancl Fig. A2 respectivcly,

Fonn=Otheexlnnsionis

(z)

( )"-s,

(u)1",¡a" ,

L2\'L,L

(15),!o (") =

0.1

0.¡

o.,2

0.1

0

¡aIG

Ft6. A2t

I&t6

/-2.7

this eaße ar isi n¡1 fror,r the auxili-any relat ions (2.2, (15) ,

(t z)).

The first reguires

There are tv¡o aurciliarSr conclit ions on ø(o) (") ir.

I

oo

exp (- c") (¿) (e')ag=o, (16)'\-.1

\L,L=o

aS"L -:2

s o that

tloî/_,

¿=o

exp (- c2 )SL

\-)

1 ,tL=1

(c2)cd-c2 = o. (17)( )L (c')s (o)1

2.1

i)

Then using the ortho64onality retationship (g), (17)

imposes tir.e co ndi'b i on

a =O (ra)

(1e)

o

The sec onc-l- auxÍ1iary conclit ion neguires

exp (- c')./\

a s.(L)(.2)czd.c = o .1!t2

exp (- c" )sr (o)

1r" )s,, ( l) 1"t )c3dc2 = o . (2o)

22

Tltus

Ioooo

aL

L=1

A2, B

li-quation (zO) imposes a res-trictÍon 1n the choice of aL

for al-I L - 1r2rJoo, Thcn since r,,¡e are interer:-becla

here in only tire contnibr-r'bion to thc cxllansion from 'cltc

lro1y:rorrricrl of leas'b cle¡;rce, r/r/e arc f orcccl 'co consiclcr

at feas'c two torrns, ¿ = 1, ancL L = 2 in (lS). Fron (Z)

s.,(1)1""¡ =A-c2tr1(zt )

(zz)ancl

Now consiclcring jue-b these first tlvo tæns of

expeno l-on, (zo) is

=+(')(",) =*(l - 5c'* "")

f *

u*, (- c") G c')caac/o

f *"*o (- c')LÆ ' 5c' + "'\"lo -\4 i

â¡

+ D.2 ad.e = 0 .

(23)

The first in'be¿;ra1 is finite, 'ìrr.r'i; the sccond is

icì-entically zero so that

A¡ = O" (zL )

Thus b¡r consi<lcring only the f inst trnro terns in 'che

exJlansion wc lrave l¡ecn reclucecl to just the sec oncì., nrmely

The graph of

âesr (z)1"r)

= & n5 - ¡"- + c'\2 2 \¿'

2'v /

t = 8. ('O - 5c2 + "')

L2o9

(zr)

(26)

Ís shor¡¡n in lrig. .[\3"

ir¡hcn h = 3 thc expan,si-on is

3)þ =e3

i-

\L.¡

S(¿ )

7/2¿=o

anci thcre is no resltriction on the a iniposed. by the

a (c') IL

L

auxilÍany rel-ations, '-lhuo the f irnt term in the exl?an-

11 to .=sr¡z(o) = c3 r ancÌ thc

corresponcLing eorrection to the disfribution is protrlor-

tional to

f- 6cg (zt)

r¡,¡hich is graphec'L in Fig. /\):,

The contrilrution to erlt(4) is proportional to €c4t

lcu'r,;:'e have not'botherecl to plot thl,s as v/e have not

solveci thc equa-bion fon rr = l.¡, anci so have no comparisons

-i;o m¡k e ,

A2.10

2.0

t.0

ESr2l

h

0

I I2

c

FIG. AI

tY 'ÐtJ

3¿

0

1.0

¿,0

t.0

r.0

9.0

l L!av

33

$:EpJeåIn this aptrenclix we prove certain m¡lthematical

L3.1

(1)

r esults,

(r) Finst we prove (h"lr,(So¡;: r', = 15 / *"fi1"¡4"'

rO

Usi ng the d ef init ion of th e vaniables usecl in obt ai n-

ing the shear viscosity equation (h.1, (¡¡)) , namely

0(r), T(r-), g and- ã or (?.t't'r(l),(15)), (3.1r(32),(3tt)),

ancl integrat ing by partse vre fi ncl

.L* uu(v)¿v = | î' ,e)(v)¿v

= F r,',]:- [-ç ve)'(v)¿v

l-v4= ./" i P(z¡(v)av

= F r,r) (u,] "

- #,[ o* uuulz¡' (rr).rv

+.t* u'u(v) r(2) lrr¡av a

In dimensionless f orm -ultis is

[ *

"oe (c) ø(t) (c)ac

/o15

oo

"ã(")a"(z)

L

by def initi on (l+.-h, (Bo) ) of L.

3)

L3,.2

(Z) Simil-ar1y uning the variables which were neeclecl to

get the thenm¿r} concluct ion equat ion (3.1 ; (lZ)); p (t )'x ancl ,t or (2.t+,(¡)r(15)), '(2',6, (zs)) ancl, (1,1,, ,Y(t)'

(t I r. ) )

Å-u*(v)¿v = F -*,];- ¿-ç x'(v)av

v){"2

(v)¿v

oo

o

yab

(v) avp

oo

+(v)13

(t )

oo

[v3[_T / . € Yu)'(v)avv(r )

(v)

l"

l +

oo

(r )

l*vs

Lî,i loo yj

3o 13 (r '(v)¿v(r ) o )

a(r1_l

30 Jovse (v) ø dV

co) (lr )

In dimensionless fonm (¿r ) ic:

.t* "n, c )ac t

JO

oo( )1

3o c5e(c)Ø (c)ac

14, $)

by definÍ.bion of I{ (b.h,(ti¡z)).

A3,.3

3) \¡/e noïv prove the s¡u'bsicliary cond.itions fon -bhe n = 0

egr.rntions of l¡"IJ.,

(")

I fiv"fl;; L""ffid G ^u" -,,,)

rOo ¡

I + v4e(v)avJo )

? v2B',3

o

9

( 6 )

- Zvan. - frv" ffl av

=O,

(7)

(B)

(e)

This nesuJt oceurs læcau,se on inte¡-ira.tin54 the first

and- -bhircL terms by parts th c fi rst is seen to be zero, and-

oo

the thlrd- cancels the secon<l, Also .¡'¡e use [vsg'I = o--o

which follows froro (4.1r(tz)).

Als oo

t ÂlV'd.v = 4o g

^= l4tr e

1

/z\a aVo V'ênr ( o\Lp

o ¡ooel=l jo

- ?u"o' - +v" ffil av

tJo

I)

+srÈ'J/2 æ

oo(10)

h fBL-*ir-*T l-Tjrrz oQ az Lrf

1

::4X Io* 'o'¿(")u"]

!oõ

q\Ta - \12-

L3.l+

(11 )

(t z)

(t t)

(r¿+)

çt s)

In proclucin¡g tJrls re sult v'¡e have usecl (Ll.)1, (gO) ) f or

(h.1, (16), (zr ) ) for B' ""o $.lr , inte¡lrated. bY Partst

arrl tiren used- (l),

n anÔ.

(¡)

llz = ot\'12 1V2A'

+3

/ol,\\òål

aaa Âe dI = r*ott

\

o e

The f ir,st tæm is ze?o on e)ællci-t integration a.nd

the seconcl vanishes on ¡rrp13rl¡g (4.1, (tt )) f or A' and

'r,hen the auxili-ary conclition (2.2r(16)).

Condition (4.1-r.

ÄrV2c1g=¿*zr* '

, Qn)) ir:

(3Ë) [-[ry(ä aV6 - Va dV+V 6A'I*3*

)

Àòr

(16)

(1)¿voo

l(tt)

(3Ë) H ["* ?vae(v)av - [. v5e(v) ø

43.5

g , /gs\ t-ktl * É-= _ u) + èF \òe,r frrå 43d/^fro'2

[.oo (r )cse(c) 6 (c )acl

U a

(ra)

çr l)

FIere vrre have usecl (l+,4r (t¡É), (ltt-Z) ) for the def ini-

tion of X and I{¡ and (4.t, (11)) for A',

(c)

^

/\s c1J, = 4n Ò_l

òr

aYz 15

VT

\V2lJ.-l)

(20 )

a

I tJO

-Ò.'r.òq

oo Ttol2W

'#)l *4

(to+ + v- (zt )

çzz)

= o. (zz)

Eo-lation (4.1, (zz)) has been used. f or ff ^rro tiren

integna'bin¡¡ Vt # by parts gives a fur"bher term of the

f orm VaA'clV which as v'e have seen is zero, In ad-clition

t#" G"u"-u')àT r*J-= IÒrl ro

V3

6+

A3.6

òTI *= 4?r òr

1ðT ( /^-ttònl/_ L "JO

èpòn

òA'læl

? uoe (v)av + *6,¿7tr"þ/r-*

/ *

("s+(tr-) * 3e4i,., 3e3i,,)crclJo j

rLMlõIJT

'l

È4

oo

+ 3OcXclc öI . _Òlr.lòr òrX ']

(ztr)

(zs)

(26)

(27)

o a

Ilere v¡e have used the cle finiti ons of I'l ancl À and

(h.1 , (18) ) ro" S: a

(¿)

0'r'òr

ÒT rv3 n,-l *tr L j " J o

Ë Å- (u'o' *Ç #) * ctl2e1", - k èPuvc,Ln Or

CA'

L.lr.À,= + aV2

hn òpd.m òn

JcLm

kPJ. 9Iclm Òg òr

.vP . V #=)"u

a

'Ihus

r d.v =L,1

0 I

(za)

(2Ð"

3o)

using (4.1, (tl ) ) for" A' and- integratinc by parts.

Irinal1y,

r ^.rv

[*ÒTòq

lrrr òp-.-G.- dln Òr

a z3

V4A'dV 3z)

= o, 3s)on using (4,1 , (t t ) ) ana the auxiliar'}r condi tion (2.2, (16)).

Brooker, P. I. & Green, H. S. (1968). An exact solution of Boltzmann's

equation for a rigid sphere gas, Australian Journal of Physics. 21(5), 543-561.

NOTE:

This publication is included in the print copy

of the thesis held in the University of Adelaide Library.

It is also available online to authorised users at:

http://dx.doi.org/10.1071/PH680543

-l- i r j. S P f- ¡, ¡'' r- ,, ,',,

i.r¡ -l (,1-l ';.l ìi i: .¡)¡i\¡i .-, li-t '\=l i\il.J/;i¡rìj\l (..i..¿,(.1 ¿-'+))€ v:ì,

-Ï-¡, f. i7i\[- úi; ..-]l-' l:-' r 11., !¡ r-;r.'+l_ irÈ JI i: 1,i'ir I,r.r ¡ r1i(.1¡,1''ii-,)it1,', r'f fl?r, I vqr.!1'j:lrrr i!,r,iL;,i r-rtl:)Jl.t ir i;l'ri':,,)',r 'r ,.,.1 r o

C.i-ri',,¡l(,),\ ./\i;¡ rr, ì r/i t - i,|.1 -:,, L ei: eiii- e/' ¡rrlI or\l_tj;ri\¡ 1r lii,'l i,,,F r,lS f t:,.' i-i- (/ r,t\, )

,ii,i\i;i4cl'-tI 4 l'-i,,!"'',rr T ( ¿+ r. ¿-.''' !,1.)

frli: i,il-(,i l,'\i¡, i j l'r 1,,i.. i(èr-r€ )d ill (",,,.,.(l;l')).I I ii'rl/i)L v'l: ' f ,;\, r] .L r l.,r.,r ,,t l'1i 'ri,i,,(ii lL.;, !tr ( I, r -(-f:.)) .'\i V,,l 1i, i..I !_; t-)ì/,1-,.r it.'1ll;.!,,,r_:. ,,1- ù{j!r.-,,T,l[" !;;l*

l- i,,i::- / o / J.'l ¡) i .,, 1 j¡::,j :_-. Iiir,"l =-'. J¡ll ir,' ì¡ ;),-i,.tt,,ç.';-.

;( ) .l = * )3,,, l¡ j \) ,' ,-) ¡- tt;1 i,, r; :_r

!,51 I =- " ìif )U ir íI'',it a /' j .'l i-.'r r':)/i.) i/!- I =i),, rJr).f trr) i,-/.1,ìi rr¿+

i'L l=. .t¡ ),!¡,1 J!,t,I l. L¿ i.

Itrn ¡'r-,1 1.¡,',[1,r.ì r,,!', ì-i.r 'rrlijr.] )L-I -:llt_.r/r rl |1.,!,1.t,!i-,irl-l-f,uil¡r rli-li.'1.t ¡, ;t (tit,t,lllLj, :.1 (Vi:,,6c.!.,=t;;u,nJ_.. llr; j..r:;r{ìi 1::Iù)" li'rr. ,ìi, ,{rr ir,,i-l.i '-, t-rri,irlji,rt! . tr': r'i !>!r :. l,t,r,,j',-,rÌr;'¡.1 i',¡ ¡ '?J.(-rii_i.v f',)t.i l¡,i irri_: l¡i)L_lrlt,Lr,!-, -¡,-, j ,,r,.1 r.tr¡tri(.i-j (-t-,Jt ;i i,í),\1., l,ltii: u i.1 '--lrr'{r It,l-'¡f' (içì | r,ij(,tr, .'

Èi_.Pli4 ,ltlr/ ,,i-ì¡¡r (¡.i.l.rll-,-\lr:r/ 1.i¡:r-rjrrr i)i:il.i,,rj,*i i,,,i.tút lliJ[,f.-: 'll- .¡,-:I iii\ili r'L r,ri1¡.'-i-'i !1'!\i l fr. i i- \r;,,r.r]I't-l Trrl-: irt;ir iìiii.li.jirì.,¡,r,r) \ -rrr[_lJ I I rrl,r ì t; r' j\rt: li,i:llL^-(ìtllrii'_1,, t:.nl-,il,':irrl l/rtIi.y i.)r-Lr.t._,r..rl],,t, :r{rl r:ll(,r,o

¡)j.'(ì(':'(1'\i,,j riirrl-:., ( i :t t.rl sr.)i;l;,i-,rl )

Ai F, l-1 à=- (n¡q-i¡i-rl t.-i-,i'.,i-;i¡ ) ) I (Ä,¡lìi|,1_ I*^:. I i')ii,i )

irf--[ l.r,:- (|. ir*l.ii r;''1 tj'i: i.'..1 ) /l'L

i1= 4l,r I\i = t1

"

)Å )=i't.t-i-,i¡¿=00X J=l " +,:t- I A * I .''it.i.l.,i ¡.,

1. -i¡ ø

V V=Vl. x='^).¡l:,1¡)\,^'ì -

^ /-

i: (1;: fi o

i 1¡,:: ti .,

i:¡,i,1::,¡o'r,/',-rl: P:r) u

r=l

7 (,Iit =11ii x t Ifrir l r,: i tt'l-,t,,t ¡ r ç n l r'. ( .a 1\_lv, i,r. I ¡ \¡'.ii..i' ^ i ti'+ì [-'t-i<i.r 'rT (l-;.o I'i i-/"ç).J=.-,1 + !.

ll¿1i)=tl-)i¡,A¡.(l,tiI I ]('¡ i lri'1.r(,.i¡r-irìil,iV';='Vf{1")iil/',)t=¡ + a5-)i(,.j ¡

Ä ¡ l, -.1 L + .':r-;¡1..2.¡ ¡,/-.r.¿'+ o -)';i|it¡

Cjl=rì^;i.4¡1(¿ i:rl-;¡X/i¿C'ìt L iii¡lt;r-tC.J I =Ir-;iiji-l. ì\- Á + ,,i )-)'. (, 1 IÀ^.1 =i. 1.+,-i:¡.¿1/,.\. ¿= l, / + . t',")r l. -l l

Cl¿:=r-l'iiÀÅ J

( ¡' l::i1')i ¡. Á ,'(,ål-L Å>[irr';i(- -l r- -- i '')i i "i-.\¡i¡=r7+¡),a.\.-t.+(.-I Iii;(l:tI+t.-¡'tl:.

^.: - \ /. + t.'. -\,:.

(., j.j=r¡-)i,t¡ r

(.,.'.'.j:: i'lji i 4,:if_'1.r1._L

^r'' ttir li,

C ,i.:l =,r';i tj L-_

'J'¿

,<:1.+ (Llrr+ /.:r\,1 1,,, ,-jl(.,i,1 +L l-.t ),/;,./. l::f, | + (Lr'\i+,' n'.i ( ,:'I + c o')i'e a.Í.+r,,a. i) 1,,

"

ì ¿= ^rl+

( r, Jt; + ¡ *-)i(. j I + i .,)i.r.. --r¿+\.-j-J ) / a,

i¡.=ÀI o j =,\ lA.'tl-¡¿.(.,,ri-l- .,()li,i ì,

f' ¡-r.I:: ( Ã -1 / \, '' .r.,-. / (\i 1r | ) )

\l lj) ir F'= r/ -ii ¡r -)i \l -;i \l -)i y ;i ¡: ;- " ; i

fF (J-:i) r'.tl ¿lt),:*),¡-'l-¡-,=\ir')trJi:,i,r 'ij)

'+,)í.:

i F ( J- ,t-) i i;.-1 .r l- ir"j :r I i; "rl- j. --\,-:ti-.;tírí-; lt' ICtJt;l tr\tlifr.'i if =ll;( +i iìi' ( I r: +,+,',:¡ I t r ,:.) / 1,.) -- |

F'ù r- v t', i, P

ll (V-'t ot')-. .,) I o t r rt- il i',r I .t r\ t-r r1:.,'lil

iI)4

)3l(ì

:ìi.-rlll((lrU'l I Jt {.::) [iji ri.

¡ t) Û l) 0 ll it li J'j rJ ij rj -j rl (j, tr ij

C(lirll.rl)t\i Vll r.r¡ +,,izi.tt o ^l-t:

q.i,.:I;i' +r,i: ç,/- r i.lI e¡:\l f,t1J\eilr I r)

i,l li',Ì-t!:i lr,l'o, | (,jr'iti, )

if(vrr-{i")r, cIigli-? C,(Ji.iI il'¡t-rr-

-J -.:

\.7 ì¡ ,:l- ?, t, ',) * + L ..,

./=f-tl. (J)(., l\ l.- | ¡':- i1 ¡: ( ri \u c r- e )r i, - ,. :, c i-,:, )

f,.=l ¡,.i- l. o I i ) *'-.> .t, ir,i! i, .;.; ! {; ii 6 l1-.

X5 J." i.:r';í'l e / i c!..':t-ir 1., Li.;,il L)-r i ,;i.'. ',(.,AY-

"?_a-;l ( (rr r -)i-V rTiiu \.r t' i " )

-;;-Å ) [ r \i v';.!¡-_ )

f.il=-Ë iiÀ \ /i-i'tY ì-l\ l-;i^ ^

I il,r'f - ( ¿l o';iV v- ! " t \t ,t )'): ¡,/r:.I*L/CÄYi.'Ì: TtJ,<i'r

] l.'l--=i *',ì[- I A.-,/ ui'rjil I,i:Alfil.f l-rtti.t

h: l'.i i ì

i :,,ti ùf l.lLLr.r¡.L-i

Iri1: i':¡iN¡'i. rr:,1 i'/l.-lt.r t,t,L ':) ì l,f- i\r (..i i\] ni.r:'r()rih lrr i (r¡-r'. lr':';,.;()l- (¿+o¿i(ll+)) ,¡, li.l(-, 1, rL,rìt) ,rl-Ltjl,iJ 1,., lili: ¡.-,-:i '/i'r/.t)Px()trxAt"r.

TfiÈr i-'ii.)t;i.1.,\r", j I :;r-i ¡r ',r1i, r;¡-'¡ I t¡t: t't:a t iìlil: | [i-.;r\(,i.i n (.J..r) )' liif':'¡r'i- rìi: V j:r{,,U5r I f i:,.it.¡¡'i 1i,,.1ç l-r)r<jl"j'f b'iVr+[-¡ ijf ,i=¡ i]{r';o

i-.'tlil(:t(jr:4 ¡.t-,;1t)\i l ( i.¡ii l iJl.r I ¡i,urrL:¡)

(,t.jrt,l'1i)i'l ^,,i

à ^'i, l.e ^^ t', i ¡ ttrj V ciii- s/,'.¡ | sl-

7."t\1 . i-'i';'r ",'r'. I I

rì1", r \! 'r1.,,', (::)i-l: u (l ì)'lt\A= n I I l- I r'.r.'',-:t., l, ;:¡ l';li,',; Ò i+f: i 1't','tii,t J. i.)r'(lL.=

" ii.:ri: / ":') t¡ la t1ir, ". -,

[.rt,:- i n )rìi'rilt,-,¿t'-¡.'.]l'.. II it i A=- ( ( ir¡¡ i,r(,(- +L1r.i¡ - r.\:'trl.;\ | ) :-:,':! 1!: ! 1,.) /

I ( iil.)-)itiC+r1,1.;r a ì,'t.r,: i, :i(. .i.l J ri,¡,,+ ) )

Al_i.il-1{: (;rrl+il¡j)ì,Jr-r.- [ l. ) /,tt.

r.::Íi "I I *--]ii_Pi-r,

lj=l " +_ir')i (/\i lr il_\+f | ì l.)

¡,1.,l=\I\ ¡ 1.= t, z-.

À A.:i = r, ,ii11:u *l,; I

/' ( l.i ) :ì"i Il")ir [], l¿+i sv e,i rr l- n 44.,: t.Jc^.i,..rII"úiilo, q I ir lo 1ri,i- c'.:ì, ).¿)C l_ (.t =tr-)i ¡: ¡ I(. ¿- lt =ri:Å' ¡ I t;[ ,1 r) - t-'1';i X, .¡\ i(; it¡ ¡ ¡-'¡,li-,i(.... ,+ l.'t - i-l ')í 1 ; ¡'r,i i,l:: ¡¡ + e ¡'jl ! i

X,(,-)i+uif-)il.j rì

å^ J =X [ + o:rìi¡ ,;¡,',4,v¡/.=',tr',1 + o t-;it.,.jr;Ä¡ l=..( l+ c )';ir-.¿+rr

t?.

7

Lll.=riìix^lCr'] =tl;i¡ n2L -i 1 = l-r ')l',Ã ,{ .,i

C¡"[-L X:, ii.;t:,i(l+ l. = ri';i l)'t--

Xl.=.X + 6 r)-;ii.-, I I

X¡l=,¡,1+" j¿ií.rlI.\.¡. l= / ) I, 'l1rr. i r

,/i ¡. .'t = l, )+ . ìì?i.,1+ l\. L,2= il 1r,{ Å [

l" ¿7 = i-r -)l'^ ¡ ¿1...7í= rììi \{.JC,.,, [ ¡- { :r i i.r i.-,.

l, -.i ;) :r .1 -ìi i.J F'

V'l:\1 +r¡t¡=X +e l¿al'!=/tI+i, r.'.c.

l'.i,i:= a,:i ¡L..j,lÅi,'\={,J+1.+¿(,.I-1=i-l-)iXl{ I

( 2 ,r'.. i-l -)! ,r 'r z:

(-..,j j: r l'ü A )\ 1

(-.j\l_l- r.1)ii)l i\(---4 :;;ri")i¡ t¡-

!i:.! + (CJ il+ ( ."|\.. 1 i + r "':i..l l+u l.i) r'r,¡i i.=X I + ( -r't1 r,) .,,i \ ¡ ì 't r. *'":(,r;r1 +i-,1 \) /'¡ ,Å/..=x/+(í-, >i.j f r,'"-;!(.ii+¿ "-;i(. itrL j))/ ).r',J ::

^ -J + ( ú. + r) + /ì u')i ¡.,r1 I +,r' e -ii lL+ a, + (, tt 1 ) /':t "

V=VVX¡.'^ÀÀ'l=¡'lx.t,l=t,r,xi ì=¡-iC¡l-L- /i:rii,,f.ji,\+=iìl-_irItrjV';¡4 j+ ((-*-):-,.t -;¡\,.-j â )'):-'¿:-"io-;i\;-ir11 i+ "ri;lllf li.l f +lçVìÅel. lc)\.!'tt,-1 ç,(¿!-r¡li'i=ì"t + I/(r'1):þLÍf' (i.i,l-l .r )rrrl I tj';rll:ii.;¡',(.ìììi(.ì " ( t ( 1 ) e L:t :- )

1(l [-!i.:iri;i-[ (1r a¡] " ) / )

t\i_{t

l(.t Í¡t (r/-').\t,i't t) I "r' *t1'¿ ('.rtt.iJ i ilr.lt

i f ( i.i, i..rl. r.r ),,'1t.,¡,lrt,,r.,L¡-.ìfJ!¡ i ( r: ( I ) r j -. ¡. .i,: )

I I'rI

Ctil'ri"lUl\ ^.4r

¡ Ä Lç "1.1 ç i r -i¡ Vrl çlii i /''' | 'i.?. q,i\1. lrr1i\ ì r{i i ¡!

ìr (t/V*tl " ) I,:l s | ) c I,'2 rlüt'¡l i.rriJi

CAI L- t-lrl- ( V ú e ¡i s ¡il,c i-::s (-:> )

t-:=Lt S-;r i . / 7 ¡,+rj i::1 :r i; ! iiir j-l r_. )i ,'¡XSJ=.\:j';i [" t It!+ ¡.Jr.!-'rr¡ìir':''.r[1;';in

"'

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1 It{axwe11a

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