ON THE INTEGRATION OF

“ BOLTZM ANN-LIKE” COLLISION

INTEGRALS

\ 5

EP-RR 15

14 DE 18S7

E. L. BYDDER B. S. LILEY

March, 1967

Department of Engineering Physics

Research School of Physical Sciences

HANCOCK

THE AUSTRALIAN NATIONAL UNIVERSITY

rra, A.C.T., Australia.

TJ163.A87 EP-RR15.1 9 2 4 1 3 7f T J 1 6 3

. A87 E P - R R 1 5

A.N.U. LIBRARY

This book was published by ANU Press between 1965–1991.

This republication is part of the digitisation project being carried out by Scholarly Information Services/Library and ANU Press.

This project aims to make past scholarly works published by The Australian National University available to

a global audience under its open-access policy.

*“ <V, 'R S PH '

OBRAR' *

ON THE INTEGRATION OF

"BOLTZMANN-LIKE" COLLISION INTEGRALS DEC 7967

by

E. L. BYDDER

B. S. LILEY

M arch, 1967

Publication EP-R R 15

D epartm ent of Engineering Physics R esearch School of Physical Sciences

THE AUSTRALIAN NATIONAL UNIVERSITY

C anberra , A . C . T . A ustra lia

CONTENTS

SUMMARY

INTRODUCTION

2 .

3.

4.

GENERAL DEFINITIONS

APPENDIX A

REFERENCES

ii

iii

1.1 C ollision Dynamics 1

1.2 The C ollision In teg ra ls 2

1.3 T ransfo rm ations 4

1.4 Special In tegral Functions 6

1.5 The K ronecker S Functions 6

THE t C f i ) IN t e r m s o f 4 , 9 , g ' 7

2.1 The Continuity A je | 7

2 . 2 The Momentum 7

2.3 The T herm al Energy 8

2 .4 The S tre ss Tensor 8

2.5 The Heat Conduction 8

INTEGRATION OVER AZIMUTHAL ANGLE 10

3.1 The Continuity ^ 10

3 .2 The Momentum ^ ^ 10

3.3 The T herm al Energy 10

3.4 The S tre ss 11

3.5 The Heat Conduction ^ ^ £ 12

INTEGRATION OVER X VELOCITY SPACE 13

4.1 The Continuity In tegral 13

4 .2 The Momentum Integral 13

4 .3 The T herm al Energy Integral 15

4 .4 The S tre ss In tegral 17

26

30

i

SUMMARY

Boltzm ann-like collision in tegrals a re reduced

to sum s of the generalised collision ’’frequencies",_TX •

The m athem atical form ulation is such that all possible types

of binary encounters a re considered, including inelastic ,

charge exchange, recom bination and collisions of the second

kind. However, only the relevant m athem atics is presented ,

the relevant physics being given elsew here. In this sense

this rep o rt is m erely an appendix to o ther publications.

ii

INTRODUCTION

In the general kinetic theory of gas dynam ics a num ber of "B oltzm ann-like”

co llision in teg ra ls m ust be determ ined. These in tegrals a re of the general form :

i/>*

w here

(W = jhflT- k'J' r k'i'E I A M r

ktThe and a re velocity d istribution functions fo r p a rtic le s types fe.

and .£ . is the re la tive speed and ^ 6 ^ ^ the d ifferential c ro ss section a sso ­

ciated with the production of p a rtic le s and Jt* . The is the change in some

dynam ical variab le in such a collision. Integration is over all velocity space

W k . anc* over a^ p a ram ete rs of the d ifferential c ro ss section.

The object of this repo rt is to reduce these in teg ra ls , fo r specific \j/j , to

sim ple sum s involving the generalised collision "frequencies":

/M t \-nz,*r(r) - T

oO

e 1w here

ft “ J^17C is the p o la r sca tte ring angle in a cen tre of m ass system , while A

is a function of, among o ther v ariab les , and £ E , £ E being the energy loss in a

collision. The in tegration lim its a re from O to I T and em brace all possible values

of £E. The ’s and (J) 's a re generalisations of functions orig inally introduced

by Chapman and Cowling. ^

In the p astjin teg ra ls of th is type have been determ ined by severa l authors

’ ^for the case of e lastic co llisions. In p rincip le, the essence of th is rep o rt is

to extend such re su lts to include all possib le types of b inary encounters. Only the

i ii

INTRODUCTION (Cont.)

m athem atics however a re p resen ted , the associated physics being given elsew here.

In th is sense th is rep o rt is m erely an appendix to o ther publications, in p a r tic u la r

re fe ren ces 6 and 7. N evertheless, it is com plete in itse lf in that, at le as t as fa r as

the m athem atics a re concerned, all p a ram ete rs introduced a re explicitly defined.

Finally, it is to be noted that fo r notational sim plicity subscrip ts and ce rta in

su p e rsc rip ts , associated with t h e j f l ’s, <j) 's and s im ila r unambiguous functions a re

ignored. It is, however, im portant to rem em ber that such functions only have meaning

when associated with the appropriate subscrip ts .

iv

1. GENERAL DEFINITIONS

1. 1 Collision Dynamics

C onsider a collision between two p a rtic le s fe and Ji such that

— » k ' + J L 'Denote the p a rtic le velocities by Vs/ and define

= 1

\ = M* w* + M^w*w here

yV\ is the p a rtic le m ass. In te rm s of g and

w K = ^ +- M i g

— )?' " "4* + ^ 4' 9*W ^/ = jf - Mk> i'

the p rim e in general denoting quantities afte r the collision.

From the conservation of m ass and momentum

V=4 > =while from the conservation of energy

± *■ + 1 m k .t , o; * = M ke A Ew here

lrT\ h . — ------- ------m 0A E = E h > + E r - E k - Ej + i E

G E N E R A L D E F IN IT IO N S 2

Eu » t u e tc. a re the in te rn a l energ ies o f the re le va n t p a r tic le s and S E is the

loss in energy due to im m ed ia te ra d ia tio n o r any o th e r cause.

D efineW k l / _ 3 - A E \

" V i * 1 ;

then i t fo llo w s fro m the co nse rva tio n cond itions tha t

T h e re fo re

9 ' - A g XgM^XCOrt iL '+AoYiBLz)% •

w here , and L ^ a r e two m u tu a lly p e rp e n d ic u la r u n it ve c to rs both being

p e rp e n d icu la r to (J • That is

| 9 - f - 3 * ^ , 4 , + £ * L l ) =

X being the id e m fa c to r o r u n it second rank ten so r. X - and £. a re p o la r

and az im u tha l angles re s p e c tiv e ly , i t being, how ever, im p o rta n t to note tha t

X = X kJ ^ X j k = TT-XkJt .1. 2 The C o llis io n In te g ra ls

The in te g ra ls to be evaluated a re

w here

The and a re taken to be g iven by G ra d ’ s th ir te e n m om ent 3 4

a pp ro x im a tio ns . ’ That is , ig n o rin g su b sc r ip ts

^ ~ £ E q *• vV ci J v y u / + w "

GENERAL DEFINITIONS 3

where{ ~ V) ( Tr) ^ ( - o<Wz)

fa is Boltzm ann's constant and T" the kinetic tem peratu re , p is the

sca la r p re ssu re , p the s tre s s ten so r and

U( is the mean partic le velocity and T" a heat conduction vector.

By definition

( = Ae' ~ A hj+ t i< ~ V f ki

where the S y*S *s a re the usual K ronecker £ functions. £ is the c ro ss

section, and in p a rticu la r

d <5~ =: <S(q,%^E)AA*.-)Lch/Ldlct tZE) ■It is assum ed that & is independent of £, , while it should be rem em bered

that both su bscrip ts and su p ersc rip ts have been ignored.

The in tegrations a re over all velocity space , W £ the in teg ra ­

tion lim its being from - o o lu + oO for all velocity components, j t goes

from 0 to IT , £. from 0 to £77“., and $ E over all possib le energy

losses.

GENERAL DEFINITIONS 4

w here

1. 3

Defining

w here

The collision in teg ra ls a re to be determ ined fo r

is = •

\U . — rA ■ W .J - j J ~ J

~ = X Wj

= -k vs/j-wf

{ wwj = W W — " 3

T ransform ations

T ransform ing from , Wg to ^ , gf the ja c o b ia n is unity and

x /3 ( £ + f 9)

i = y £

5 yx = ^ **** >P*' - <** +•

= 5

a fu rth e r transfo rm ation is possib le .

It may be verified that- 3A

dJlfolq — d s t o(

GENERAL DEFINITIONS 5

whence

ct(x,y) =■ d 0 + d / * + d Z'L{ *• x x+ - : x < £ + d + ¥ 4 1 ' X ^ " i ) -hdtj'ljx*-

In term s of the U , £pj> and T“ 'Sjthe ö( 's are given by

4* s 1

d , s / s ' r * + ^ - r > ) 3

d x = y C 2 . ( ^ K ' « f ) - H - C y r T k -fy 'T jf) !

d , = J ^ T T

s l » = ^ c 7 ^ { P * i ‘ T ^ P ^

ö (5 -

- 7 =

Ä

dcj~~

r _ * i L r L P* x

<*4 K

*S /S *

>sl r Y- - °<*:S / S * L “ k ~r £S /S *- p * -*- * ' pjt1L r ilL t, - fSLr®lC/2^ Pv> — k ^ - J

P k * P* r i l i L V - . K?L P* r * +

Tj tJ

X ? 1

GENERAL DEFINITIONS 6

4 ,0

4 it = ^ L % r „ - f t r jwhere 'S Y\ Yf\1.4 Special In tegral Functions

Several in teg ra ls of a general nature occur in the subsequent analy­

s is . These a re defined as follows

Tfj j Gr) -where J is any function of 5C and t-/ , and Gr C is any sca la r function

of the sca la r Lj ( r */'*/ ). Again define

It is also convenient to define a general lin ea r o p era to r by the in tegral

while, in p a rticu la r,

n(r)G r = 7T * £

2.r+27 "X e ' i

1.5 The K ronecker £ Functions

F or notational convenience define

Sj> +■ $js + Sjfet-..........= C j O + s + k-t-................

and, in p a rticu la r, define

SO) = CjSOO - E j ( k ' - S ' -i(s) = Ej(k'-r)3 S M = Z j ( k ' i S ' ) l

*In o rd er to m ain ta in a uniform n o ta tio n , a fu n c tio n (£>(- oO, 0) i s a ls o Qdefined by th i s in te g r a l . The a sso c ia te d term in v o lv in g X and y/L i s X~°° cos yC ^ Qt

always being taken to be zero no m a tte r what the m agnitude of )\ .

2 . THE A fc ) IN TERMS OF THE , g , '

In expressing the in te rm s of the ^

to define a reduced mass

" V =•’’ ' j

7

i t is convenient

2.1 The Continu ity

& h i += ■ M j C j O

— M j S O ) .

2.2 The Momentum / \

£jk'

In te rm s of , 0 and ^ » as given in section 1 .1 , th is

becomes

( Mj v J j ) =■ -/■ Mjf' 3 ;) ^jfe'

-

- ^ - H fe i ) 6,7

= mj Tj (k '+l " -k-J)

+ M jr C j ( k ' - S ' ) J 9 '

- M j t Cj ( h - ^ ) ]

8THE A £ 1 ( l / 'j ) IN TERMS OF THE , g , g '

2.3

2.4

= >Yl j S ( I) Xjf

+ U * - ) i

+ > ^ j r ^ 3 ) ( 9 ' - 9 ) .

The Therm al Energy A (^£

In an exactly s im ila r m anner to that used in section 2. 2

a wOc W/)= -kmjU’)($x + CSSpy)+ mjTS(a)^k-9

4 wJTS ( 3 ) ^ * ( 9 - 9)

+ t v d i SOt)(9'3' - 3 ^ .The S tress T ensor A

Afr/rtjf t / j W j } ) = ( 9 9 I )

4 *.w\iTS ö ) ( f (g'-g)}J r

2.

4 £(tf) { - t t ) .

2.5 The Heat Conduction A

A ^ ^ ( - z W j W j V j- ) ~ - z M j ' b ( i ) J y Z J f /

+ Mjr $ ( 3 ) (§'-%)

THE 9A f c f ( IN TERMS OF THE ^ . 9 - 3 '

t- ^ 2'Sff)f('j'9 '-99^

MJ r S ö ) j 2( g ' - 9 )

+ Mjr £(2-) 2.4+ i.^-W g'V-fä)+4^V ')£g*

J

+~ “5- r S ( 2-) ^

+ i ^ r ^ 9 2i

+ 9 * - )

+ ^ r - £ W ^ - | i •

3. INTEGRATION OVER THE AZIMUTHAL ANGLE

From section 1.1 Q 7 may be expressed in te rm s of A , ^ » X . and £ . .

Since the c ro ss section & is independent of in tegration over this p a ram ete r is immedi;

tely possib le , and the (j ) 's may be expressed in te rm s of the A , ^ , and^(

o r to be m ore exact the (j) 's , C) , and .

In carry ing out these in tegra ls it is to be noted that

^ 2.7T r 2TT

Jo ~ Jo S d &r r x /•'XR

J o - J unrUTT

J o

3. 1

3. 2

3 .3

=■0

^ 0

The Continuity

* ~ TTWj SO) $(-<* * ,o) .The Momentum

- z i r M 9+ X1\ I

The Therm al Energy

(j)("x Mj vJj*')Ä TTMj $(i)

-Ä7T kvIjt

-Tr^Viri<jiM9*j

+ 2 J T t ^ j r

\ — cxO> _*Note w e l l th a t / \ = . C/

INTEGRATION OVER THE AZIMUTHAL ANGLE 11

+ TT —r- S2'-

3.4 The Stress I k j tIn determining this integral it is to be noted that

{gg-gy}^ lQ 3}( l -X i'Un''X

- Xxqx AXx {i,i,c#ixL+LxLaM*,Lt}+ term s which are odd in sin £.

or cos .

Furthermore,

t t ? ) = 3 ‘ f l i = o

Using these facts

(KW { ^ W } ) — ZJTrrtj £(i)s.

- t f W t / \ j T S ( 3 ) ( J ) ( i ) i ) { f y q }F 47TMjr [£ g }

F XT U\)j>(-<*,o){q gj

Kir)37rk^j

FIT

INTEGRATION OVER THE AZIMUTHAL ANGLE 12

3.5 The Head Conduction

J (*<£ w . w f s / j ) « "TTM j S ^ l )

3 aJ

- I T h \ j r 9(2)<f>(,>d'fyZ3_

' 3ir^ XS ( ^ M j ' 3 gJ

J

- T T ^ f i 5 ^ 3 )^ 3 ,1 )^ 3vi

- it tg d S(i) /M 4 3 4j

+ Z1T t ^ j r

+■ TTJ

- j -T f T v l ^ 3

+« swf-*<*»4-siJ

4

4. INTEGRATION OVER 2L VELOCITY SPACE 13

Using the transfo rm ations given in section 1. 3 the may be w ritten in

te rm s of the and . Since the (f) (^U ^) a re in no way dependent on 3^ , in te ­

g ra tion over th is variab le is im m ediately possib le. In general, the in teg ra ls a re

in itially exp ressed in te rm s of the as defined in section 1. 4. These in te ­

g ra ls a re then determ ined in the appendix as a function of the !T1 Of") 6 r and a re

finally reduced in the m ain tex t to functions of the ( t ) •

4 .1 The Continuity Integral

J&fi

which from section 3 .1 ^

= r\jtr\kT 0 , a )- % m j S O ) tyC-eo^) .

T(i}A)= 4- f l (o) Awhence

( r * j ) == Y) j Mj £0)1?/ i ' ( 6 ) .

w here AFrom the Appendix

4. 2 The Momentum Integral

S im ilarly , from the expression given fo r

and the transfo rm ations given in section 1 .3 ,

in section 3. 2

HinkT(x;ß) +- r \ / \ T ^ c )w here

A = ZpL SO)■ 0

B =■ - -j-(p(-oo,o)~ 2 ^ r S ( s ) -j~ ( j > ( l , l )

INTEGRATION OVER DC VELOCITY SPACE 14

From the Appendix

CT(x, B) = (*4 t + s4t,)/L(o)B+Udg + %dJ/l(i) B

j(^ c) = (^d^-hf^-h^ncnc +%d„a(*)cwhile from section 1. 3

Äöf, + sdt

‘X . 3 —

1 K +i f - * *

/®_ *** i f w -

~ W_ _/& y V *

4- **— &

Pi

PPk T* + 3 M r i

% d x + Z- dy 3 )( U k 3 ) ( U j

/4JLV3 r MIk !f33 7 f T*

3 2 _ y i X15 0 pk T k

y3 I r<s y

Collecting term s in U » U ji > Tj^ > and

I k l ( 'rA ) = + cff**)]W

+ ^ n KCc,u ) + c ^ W ] V f t

INTEGRATION OVER X VELOCITY SPACE 15

where

Sf ^ r*ij S O ) Cl °°'°(d>

- mjr &(3) XV'(i)- Mj so ) jn ‘’,'(i)]

C » * ¥ SO) ( / T " « ) - I

c “V ) =/ 2 - 0°>0 - o& 0 \

4IY1j t S ( 2 - ) ( ^ / I 7 i) J

4 WjTg(S)(/l',6) - s

< V ) *

=

4. 3 The Thermal Energy Equation

As in the previous two cases this integral may be readily expressed

in term s of X and i f . In particular,

T k J t =

+ f)

INTEGRATION OVER X VELOCITY SPACE 16

w here

D = S O ) ~p- <j)(-00,0)

E = " J y - $(-co,o) - Y^rfijr $>(?,)$(1,1)

+

F = 4

5L^1 ^ S ( 2- ) (j) o& )0 )

y * S O ) (j>(-°O,0)f y | KV)jV £(3)^1)

~ i^i/gO f)-^ (j>(^o)-jl^r^ H o )j

+ - K r VT - ^ 2- S O ) 0) .From the Appendix

T ( x z , O ) = 6 H( 0 ) D

TCZr'H) e ) - o

t ( l/S f ) = if/icoFwhence on collecting te rm s in Q

IM (i »■ w/) = [ Mj M(f (' +

~ ^ T S ( Ä ) x ^ r i ' p o )4- w j V S ^ 3) 2. ? n , , , 0)

»«,0 V i )

INTEGRATION OVER O L VELOCITY SPACE 17

4 .4 The S tress Integral

From section 3 .4

Ifcjg Wj}) = {x x}; (v)

+ H)

where

Gr =

H = -

K -

«*• KV>j SO) ( p C - ^ f O )

yfi»*j SO)

- J 7 V * j T £ ( 3 ) < / > ( l , l ) - h - ^ r t J r S(Z)<

Ä. Ä

- !ti *o J

- ft ( ’W m >.From the Appendix

X ( { X *3C}) G r ) ~ J2. ö l 2 . 0 . (o) Gr

jUsy},«) = f ^ rn o HT ( ( v i i , K ) = I f l , f l ( l ) K

INTEGRATION OVER X VELOCITY SPACE 18

while from section 1. 3

< ' m w W ' w i f !

Therefore, on collecting terms it is found that

I kl (**j fWj Wi 1) =• fyrt* r+ ^ n k [ c M ) - c ^ ) J

where

C(p (V) = t L Wj $(i) (1 ^ To)

v j

- |* j r S(*.)f f / T ^ V )

+ rtii)- in*’?»)]

q,w = $ r - « ; S ( 0 ^ n " ' ' ,(.)cW W .

INTEGRATION OVER X VELOCITY SPACE 19

r . v CLt/. _ LI .- ™jr za) p

+ a ' ’“•1)3.

4. 5 The H eat Conduction In teg ra l

F ro m sec tion 3 .5 ,

(§) = t (A^ z ßty'SJly

+ Dfy\ +E>}*g9+52£ F )

w here in th is sec tio n the A , ß . C e tc . a re given by

A = $0 ) (j)(-eo, o)

ß = - A M J r [ $ ( 3 ) ( f > ( l , l ) -

c = CS(ti(2.fo*,d)-<j>(tt2))-$J

D = - m J t E i ( $ ) - i f o ) $ ( - * , o ) l

V

f r$(3)<j)(3,/) - h(*.)<p(-‘o,o)2.j

INTEGRATION OVER Zr VELOCITY SPACE 20

Expressing the ^ and ^ in term s of X and if the heat conduction integral

is

Ij^ v/j wf) = n* a * 37 * > Or)

4-

+ M * T (^ Ut M)

where

INTEGRATION OVER X VELOCITY SPACE 21

From the Appendix

T ( 5 t a X t Gr) = ( . s d t ^ T d(,)^l(0)Gr +(5dt+-%4,0)O-(')<*

T (x-y x,H) - (% dx f § -h fe{„n.(2)H

j ( f x Ji<) = (* d l+s<Ai,)sio)K-h(fi.dv+§<dJn.(2)K

T ( x l y(L) = ( z d z+i<i1 + s<d<i)si(i)L f idqfMzyL

T (x ‘i{<{)M ) - d d l -h^dt)fL(i)M+

«T ( y2-«/, a/) = ( f <7 v-*6 )/U*)a/ f

while from section 1. 3

5 ^ i +■

Ä.

- ^ o - i p ) -Ya

Y a

5 dr + fd,0 = v° Ya * X Y/J3 4 .^ -h fe j - ] + 3 d<l - % ) f d k r r < d j

3 )T^ ( f p * - ) ~ k / p t

+-

INTEGRATION OVER TG VELOCITY SPACE 22

2z- A 3 ^11 f

x d , * 5 4 b ~ ^ +/» ß A fi

4.2.^§■ 4-i + $d<i

+ S«l l / ( l - f& ) *>/a

- ii f r3 f * i r ,

i i h \ 4 io^0- O t ,, 3 ~ff + $ 2 - > 0

i d i + - I " - -|

- T f r *//V -i-4 y3 Xf/ft

INTEGRATION OVER X VELOCITY SPACE 23

After a considerable amount of tedious algebra, it is found that

4* V)ß C (^Jt)~ clotjti {tijL

+ «, «„Cc,*V*)t^OSs^ + c

INTEGRATION OVER X VELOCITY SPACE 24

r nO |j~ wjT s(3) Lsz* nd)3/1

x xK

c,M<k) == MijSo)[- 0-i£.)nZ)

p+ « , , » * £ f /T Y .) - ? r m > ]

c ” ^) Mj $Ü) f [ ( s ~ l l ß i . ) f l '(i)

C<rvJxA%

INTEGRATION OVER X VELOCITY SPACE 25

+I«.-, $«)C" I jk (s-il o)3 / J « w ,.psx- ( ^ )2) n %

+ f ( s f H ^ r i ^ U , ]+-WjT8(3>r

- 1 <- <2dW f f (/!*'(») **/!*•%>)

(ii^ 3)

26

APPENDIX A

The object of th is appendix is to determ ine the in teg ra ls 3* in te rm s of the

X X ’s. Explicitly, if GrC Oj ) is any sca la r function of lj then the in tegrals

J f ^ G r ) « T T ' ^ j c U x , G r U f ) a l x c U fa re to be reduced to theX X ’s for the following values of g :

1 XX'

i U x i

X Hi]

X‘WH'i H(i'i)X X X(X*X)

It is convenient to sub-divide these in teg ra ls in the following way

j Gr) = TT~ZJdf>(Xr, l { )such that

X T r ( l t Cr) -

Com parison with section 1. 3 shows that the ^ a re given by the following

functions of 7C and .

c t c - I

d, - d,'X d-j d^x x-y

d y - 4.^'X t/ 2'

APPENDIX A 27

ö/ 3 = ö l 3 : x x

d < + - d i f ’ ^ ' i

d 5 s 4 s ' i i

d<j - öL »cf x2-

d u -d , , ^ ya .In all there a re 180 in tegrals \Tp($ to be determined. Of these, however,

inspection soon shows that 142 a re zero since they a re e ither odd functions of X and

Lf o r involve only the t race of divergenceless tensors such as , o lg , and .

The remaining 38 a re readily determ ined by using the following integration aids

(e. g. cf Chapman and Cowling, chapter 1)?

J F(x) K • x x d x = -j K f FU) x xdx,

J* FCx) dzr - f F( x) xxdx.

( FU) K' Xx i x x } dx - {l<\ jRaOxV *

Jo x re ' Xö(x *Sim ilar relationships also apply to integration over (y . Using these aids

the determination of the individual T p is basically tr iv ia l and only two p a rticu la r

in tegra ls will be considered, these illustra ting the method.

%Cr) = IT 1] ^ ( - x ^ ) d i 'Xx.l <{(*l)Gcdxct'i

~ i X X If

~ i J ~ Z d i [ X V 'xja^-y *) f a j i t ]

APPENDIX A 28

- Y -f> P(-i)" ■§■ *d f> x y frUu

= % 4 t > n u ) G r

Ts(l'1!llt Gr)= 7 T * J W f-* a y *)ds :yy \ i i}Grct^

= Tg 4 5 /U 2 -)< t .

The final re su lts for the 3 Y ^ Cfjonly a re given, the individual non-zero

3 p (^ $ )b e in g , however, easily recognisable.

J O , Gr) * ^n(o)Gr

7(X,Gr) = (* ^ , +sdi)fl(o)G‘+(*‘dj! + J*lle)-fl-(,')G‘

jC^Cc) =(%d2 + j d 7 + *-1 n0)6r + * &

T ( x a, £ ) = G*

T (X*y,Gr) = 0

APPENDIX A 29

T ({**},&) = Ä^j/lfoXr

T ( { W j , C c ) = 7 s 4 5 a(2.)Gc

J ( x 2Xi6r)~(Soll +^4i)rL(0)0r^(5dti‘3 ^ ) ^ 6r

T ix ^ C r ) - ( ^ 4 2^^41 +% 4i)A M b+ii«a ( *)6r

T ( L I 7- * . ) < * ■ ) - ( x 4 i i ~ 5 4 l ^ ^ ^ C r + +

J U ' ^ b ) = ( x 4 i + i 4 1+s4<i)n(i)Cc+ ^ 4 , A (Z^

J(x'7^6-) = ( § ^ ,^ 4 ^ ) A O ) ^ + ^ + f ^ ) n W

T((yl y ) 6r) = f ' 3 öfa + Jö/ + Ätf/^)A^)<r+ T^,,/2. 3)

30

REFERENCES

1. CHAPMAN, S. and T. COWLING: "The Mathematical Theory of Non-UniformG ases," Cambridge, 1952.

2. KOLODNOR, J . : "On the Application of Boltzmann Equations to the Theoryof Gas Mixtures," Doctoral Dissertation, New York University, New York, 1950.

3. GRAD, H. : "On the Kinetic Theory of Rarefied Gases, " Comm. Pure andApplied Mathematics, Vol. n, p.331, 1949.

4. HERDAN, R. and B. S. LILEY: Reviews of Modern Physics, Vol. 32, No. 4p. 731, 1960.

5. HERDAN, R. and B. S. LILEY: Associated Electrical Industries ResearchReports, Nos. A. 1002, A. 1005, Aldermaston, United Kingdom,1959.

6. BYDDER, E* L .: "The Theory of Partially Ionised and Non-Uniform Plasm a,"Ph.D. Thesis, The Australian National University, Canberra (to be submitted).

7. BYDDER, E. L. and B. S. LILEY: "The Formal Non-Equilibrium Theory ofPartially Ionised ancj inhomogeneous Gases" (to be published).

y or r»On (Y T P 0 c ,4 i'lbS

No.

E P -R R 1

E P -R R 2

E P -R R .3

E P -R R 4

E P -R R 5

E P -R R 6

E P -R R 7

E P -R R 8

E P-R R 9

R1

P ub lica tions by D epartm en t of E ngineering P h y sics

F ir s tA uthor__________________ T itle___________________ P ub lished R e -issu e d

H ibbard, L .U . C em enting R o to rs fo r the C an b e rra H om opolar G en era to r

May, 1959 A pril, 1967

C arden , P .O . L im ita tions of R ate of R ise of P u lse C u rre n t Im posed by Skin E ffect in R o to rs

Sept., 1962 A pril, 1967

M arshall, R. A. The D esign of B ru sh es fo r the C a n b e rra H om opolar G en era to r

Jan ., 1964 A pril, 1967

M arshall, R. A. The E lec tro ly tic V ariab le R esis ta n ce T e s t L oad/Sw itch fo r the C a n b e rra H om opolar G en era to r

May, 1964 A pril, 1967

Inall, E . K. The M ark II Coupling and R otor C en tering R e g is te rs fo r the C a n b e rra Hom opo- la r G en e ra to r

Oct. , 1964 A pril, 1967

Inall, E . K. A Review of the S pecifica­tions and D esign of the M ark II Oil L ubricated T h ru s t and C en tering B earin g s of the C a n b e rra H om opolar G en era to r

N o v .,1964 A pril, 1967

Inall, E . K. P rov ing T es ts on the C a n b e rra H om opolar G en­e ra to r w ith the Two R o to rs Connected in S e rie s

F e b . ,1966 A pril, 1967

B rady, T.W. Notes on Speed B alance C on tro ls on the C an b e rra H om opolar G en e ra to r

M ar. ,1966 A pril, 1967

Inall, E. K. T e s ts on the C a n b e rra H om opolar G en e ra to r A rran g ed to Supply the 5 M egawatt M agnet

May, 1966 A pril, 1967

P u b lic a tio n s by D ep artm en t of E ngineering P h y s ic s (C o n t.) R2

No. A uthor T itleF i r s t

P ub lished

E P -R R 10 B rady, T .W . A Study of the P e rfo rm a n c e of the 1000 kW Motor G en­e ra to r Set Supplying the C an b e rra H om opolar G en­e r a to r F ield

June, 1966

E P -R R 11 M acleod, I.D .G. In s tru m en ta tio n and C ontrolof the C an b e rra H om opolar G en e ra to r by O n-L ine C om ­p u te r

E P -R R 12 C arden , P .O . M echanical S tre s s e s in an Infin ite ly Long Hom ogeneous B it te r Solenoid w ith F in ite E x te rn a l F ield

O c t . , 1966

J a n . , 1967

E P -R R 13 M acleod, I.D.G. A Survey of Iso la tion A m pli- Feb. , 1967f ie r C irc u its

E P -R R 14 Inall, E. K. The M ark III Coupling fo r the R o to rs of the C an b e rra H om opolar G en e ra to r

E P -R R 15 B ydder, E. L. On the In teg ra tio n of L iley, B .S . ’’B o ltzm ann-L ike"

C ollision In te g ra ls

E P -R R 16 Vance, C . F . Sim ple T h y r is to r C irc u itsto P u ls e -F ir e Ign itrons fo r C ap ac ito r D ischarge

F e b . ,1967

M ar. ,1967

M ar. ,1967

E P -R R 17 B ydder, E. L. On the E valuation of E la s tic Sept. ,1967and In e la s tic C ollision F r e ­quencies fo r H ydrogen ic-L ike P la sm a s

E P -R R 18 Stebbens, A.W ard, H.

The D esign of B ru sh es fo r the H om opolar G en e ra to r a t The A u stra lian N ational U n iversity

M ar. ,1964

R e -issu e d

A pril, 1967

A pril, 1967

S e p t. , 1967

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