Determination of exact solutions for relativistic gas mixtures at high effective temperature

Chiu, H. H.

English

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https://ntrs.nasa.gov/search.jsp?R=19690012185 2020-03-12T03:55:22+00:00Z
NEW YORK UNIVERSTTY 
New York, N.Y. 
CLASS OF EXACT SOLUTION OF RELATIVISTIC GAS 
H.H. Chiu 
Associate Professor 
Department of Aeronautics and Astrgnauties 
Prepared for the Office of Grants and Research Contracts 
Office of Space Sciences and Space Administration 
National Aeronautics and Space Administration 
under Research Grant NGR-33-016-067 
January 1969 
ABSTRACT 
A method of obtaining exact solutions for the relativistic gas 
mixtures at high effective temperatures, subjected tq a body force or 
no body force, is presented. The method yields the solution of the ' 
conservation equation by linear superposition of the energy momentum 
tensor of the electromagnetic field with or without electric charge 
and current. The nature of the flow field of high temperature gas is 
similar to that of the photon gas within the first two approximations. 
I. INTRODUCTION 
There has been a r e c e n t  i n t e r e s t  i n  t h e  f l u i d  dynamics of relativist ic 
gas i n  t h e  r e a l m  of a s t r o p h y s i c s ,  p a r t i c u l a r l y  i n  those  problems a s s o c i a t e d  
w i t h  h igh  temperature gas i n  a s te l lar  i n t e r i o r .  
Various fundamental aspects of t h e  re la t ivis t ic  gas have been ex- 
c l u s i v e l y  t r e a t e d  by Synge"), Taub (2-5), Lichnerowicz(6),  M c V i t t e  (7) , 
Guess(8), and most r e c e n t l y  by T r u i t t " ) .  
has been made i n  o b t a i n i n g  t h e  exact s o l u t i o n s  of t h e  ogveroing equa t ions ,  
presumably because t h e s e  equat ions are non l i n e a r .  Undoubtedly o b t ~ i n i i ~ ~  
e x a c t  s o l u t i o n s  i s  of practical importance i n  view of t h e  inadequqcy of 
l i n e a r i z e d  theory p r e v a i l i n g  a t  h igh  v e l o c i t y  and h igh  temperqture. 
Y e t  v i r t u a l l y  no i n v e s t i g a t i o n  
It i s  t h e  purpose of t h i s  p a p e r  t o  p re sen t  t h e  method of o b t a i n i n g  
a class of e x a c t  s o l u t i o n s  a p p l i c a b l e  f o r  (1) photon gas ,  (11) material 
gas  a t  high e f f e c t i v e  temperatures and (111) a mixture  of t he  photon gag 
and t h e  material gas a t  high e f f e c t i v e  temperatures.  
The governing equat ions and t h e  n o t a t i o n s  adopted he re  fol lqw c l o s e l y  
t h a t  o f  Synge"). 
and t h e  L a t i n  a f f i x e s  ab t ake  va lues  1 , 2 , 3 .  c i s  t h e  s p e c i a l  r e l a t i v i t y  
speed of l i g h t .  
The Greek a f f i x e s  pvo . . . t akes  t h e  v a l u e  1,2,3,4, 
1 
11. DEDUCTION OF THE ASYMPTOTIC PERTURBATION EQUATION OF THE 
RELATIVISTIC GAS 
The equat ion  of motion of  t h e  re la t iv i s t ic  gases  i s  t h e  conserva t ion  
o f  t h e  energy momentum t enso r  T 
PV 
a~ = o  
a a a a  
IJ. lcat ax ay aZ 
P V P  
where a s t ands  f o r  t h e  ope ra to r  -+- + -  + - . 
The energy t enso r  i s  given by 
E i s  t h e  mean d e n s i t y  of t h e  gas o r  gas mixture ,  p i s  t h e  h y d r o s t a t i c  
p re s su re ,  1 is  a 4 -ve loc i ty  v e c t o r  s a t i s f y i n g  t h e  fol lowing r e l a t i o n  
P 
1 1  = - 1  
P P  
The equat ions  of  c o n t i n u i t y ,  and t h e  equat ions  of s t a t e  of t h e  
photon gas,  t h e  gas  composed material p a r t i c l e s  of proper mass m,  and 
t h e  gas mixture  of t h e  photon and t h e  material p a r t i c l e s  are assumed 
t o  be those  given by Synge 
* 
(1 1 
Photon gas a $14 = o (4a 1 v 
- 
E + c e 2 p  = 43/5 , C - ~ P  = N / F  
Material gas  a (NX ) = 0 
P P  
Gas mixture  a (XIp)  = 0 3 aPmP)  = 0 @a) v 
E + c-’p = mNG($) + 4 3 / 5  , c-’p = ( N + S ) / S  (6b) 
.cy 
where N ,  and N are t h e  number d e n s i t i e s  of t h e  photon, and t h e  material 
p a r t i c l e s  r e spec t ive ly .  
~~ 
* It w i l l  be  assumed t h a t  t h e  photons and t h e  material par t ic les  are 
i n  dynamic and thermal equi l ibr ium.  
2 
5 i s  t h e  r e c i p r o c a l  temperature  r e l a t e d  w i t h  t h e  absolute? temper- 
a t u r e  T as fol lows 
kT = ea /5 (7 1 
where k i s  t h e  Baltzmann's cons tan t .  
and l / m <  
e is  a non-dimensional number, 
i s  t h e  e f f e c t i v e  temperature.  
The func t ion  G(x) i s  def ined  i n  terms of Bessel func t ions  
For a gas composed of  material particles a t  high e f f e c t i v e  t e m p e r -  
a t u r e  i . e .  % << 1, t h e  equat ion  of  s ta te  i s  approximated") t o  be  
. . .  mz E + C - ~ P  = mNG(m5) + + 
c -2 p = N/< , . . .  
Note t h a t  t h e  l e f t  hand s i d e  of Eq. (9) reduced t o  t h a t  of t he  photon 
gas  when m vanishes .  
The asymptot ic  behavior  of t h e  equat ion  of s ta te  sugges ts  the  
p l a u s i b i l i t y  of  developing t h e  asymptot ic  theory of t h e  material gas 
which approaches t h a t  of t he  photon gas i n  t h e  l i m i t  of vanish ing  rest 
mass. 
To begin with,  t h e  r e c i p r o c a l  temperature  i s  non-dimensionalized 
by some re fe rence  r e c i p r o c a l  temperature  < r e f .  
m 5  = (m5ref>5 = 32.5 (10) 
where n w i l l  b e  regarded as a small parameter i n  t h e  ensuing p e r t u r b a t i v e  
a n a l y s i s .  
All t h e  f l u i d  p r o p e r t i e s  are expanded i n  terms of  x 
3 
4 
The traces of T ( O )  and T(2)  are found to be 
PV ' PV PV 
5 
111. SOLUTION OF THE PHOTON GAS 
The conserva t ion  equat ions  of  t h e  photon gas ,  and t h e  z e r o t h  a r d e r  
equat ibn  of  motion of  t h e  material gas  are given by Eqs. (1 
The energy t e n s o r  is given by 
The energy t enso r  T ( O )  given above is an  exact express ion  f o r  t h e  
P 
photon gas ,  and is  a ze ro th  o r d e r  approximation f o r  t he  material particle.  
I t  i s  t h e r e f o r e  expected t h a t  t h e  behavior  of  t h e  gas composed of  t h e  
material particles is  similar t o  t h a t  of  t h e  photon gas wi th in  t h i s  
a p  p r oxima t ion. 
The equat ions governing t h e  motion of  t he  photons are non-lJnear ,  
and d e s p i t e  t h e  f a c t  t h a t  Eq. (13a) admits  t h e  " F i r s t  Tntegral"  (1 1 
r ep resen t ing  t h e  a d i a b a t i c  l a w ,  t h e  genera l  s o l u t i o n  i s  d i f f i c u l t  t o  
ob ta in .  
The approach adopted i n  t h i s  pape r  i s  t o  seek  a s o l u t i o n  of  t he  
flow f i e l d  i n  terms of  e lec t romagnet ic  f i e l d s  which are proper ly  
superposed s o  t h a t  t h e  flow v a r i a b l e s  cons t ruc ted  from such e l e c t r o -  
magnetic f i e l d s  s a t i s f y  t h e  phys ica l  boundary cond i t ion  imposed on the  
f low f i e l d .  The p l a u s i b i l i t y  Qf t h e  present  approach i s  suggested by 
t h e  f a c t s  t h a t  t h e  energy momentum t enso r s  of  t h e  photon l i k e  gas and 
t h e  e lec t romagnet ic  f i e l d  are bo th  traceless, and t h a t  they s a t i s f y  the  
s a m e  conserva t ion  e 
Acc t o  t h e  above, one writes 
6 
T (photon gas)  = 
tJv 
n 
i s  an  energy momentum t enso r  
n 
where An i s  an  a r b i t r a r y  cons t an t ,  
of  a n  e lec t romagnet ic  f i e l d .  
G,) is  given (nPv/ are t o  be  proper ly  s e l e c t e d .  The elementary t enso r  
i n  terms of  f i e l d  t enso r s  F and F as fol low 
The cons t an t s  An and t h e  elementary t enso r  
1 
n n * 
PV tLv 
o r  
(npv) = 3 c-'(F pT F VT + F * F * )  rJ.7 VT 
n 
-IC 
F i s  t h e  t enso r  dual  t o  F 
PV P 
1 i f  t h e  set of number pvq3 is t h e  set  ( 1 , 2 , 3 , 4 )  
i n  o rde r  o r  i n  an  even permutation of t h a t  o rde r  
- E pvag -1 i f  pv@ i s  an odd permutation of (1 ,2 ,3 ,4 )  
0 o therwise  
* 
The f i e l d  t enso r  F and i t s  dual  F s a t i s f y  Maxwell equat ion.  
PJ PV 
S u b s t i t u t i n g  (17)  and (19a) i n t o  Eq. (18), w e  have 
The above t e n s o r  equat ions  con ta in  5 unknowns i.e. N ( o ) / 5 ( o ) ,  and 
7 
A ('I, which are presumably t o  be  so lved  i n  terms of  t h e  components of 
t h e  energy momentum t enso r .  
I-L 
By mul t ip ly ing  both  s i d e s  of  Eq. (22) by AV(O), and n o t i c i n g  t h a t  
LV (O) x, = -1, the  fo l lowing  equat ion  is  obtained 
w e  rewrite t h e  above equat ion  i n  t h e  fol lowing e igenvalue  problem 
The energy d e n s i t y  cf t h e  photon gas is thereby given by t h e  eigenvalue 
o f  t h e  energy momentum t enso r  (n,) of t h e  e lec t romagnet ic  f i e l d ,  
n 
n 
whereas t h e  four  v e l o c i t y  v e c t o r s  are t h e  corresponding e igenvec tor .  
For n = 1, the  fo l lowing  f o u r  e igenvalues  are obtained 
Since  E(') i s  g r e a t e r  than o r  equal  t o  zero,  t h e  nega t ive  va lue  should 
be  d iscarded .  
The p res su re  of  t h e  photon gas is thus given by 
p ( O )  = c-'{(F F >" + (F F *,"}' 
12 P* PV tJv ELv 
Of t h e  f o u r  a l g e b r a i c  equat ions  contained i n  ( 2 4 ) ,  only t h r e e  of them 
are independent by v i r t u e  of t h e  r e l a t i o n  ( 0 ) p  = -1. 
P P  
(O) f o r  A = 1 , 2 , 3  xa By t ak ing  t h e  f i r s t  t h r e e  a l g e b r a i c  equat ions ,  
are obta ined  as fol lows 
8 
+ 6 E('))-' is  t h e  inve r se  of t h e  matrix fl 
where (nkA k,c kk? MER - + 6 (0 1 
Note t h a t  t h e  r e s u l t s  (25a,b),  (26) and (27) are only a p p l i c a b l e  f o r  
n = 1. When n > 2 ,  t h e  s o l u t i o n s  need be  proper ly  modified.  Af t e r  t h e  
energy d e n s i t y ,  .p ressure  and t h e  fou r  v e l o c i t y  v e c t o r s  are found, and 
t h e  cons t an t s  A's  p roper ly  chosen, t h e  numerical  d e n s i t y  Z ( O )  is  
c a l c u l a t e d  from t h e  c o n t i n u i t y  equat ion  (14a), which becomes e f f e c t i v e l y  
a l i n e a r  par t ia l  d i f f e r e n t i a l  equat ion  by v i r t u e  t h a t  A") i s  a known 
vec to r .  The r e c i p r o c a l  temperature  i s  subsequent ly  obtained frow 
n 
V 
t h e  equat ion  of s ta te .  
The p resen t  a n a l y s i s  cam be  extended t o  treat t h e  case where t h e  
flow f i e l d  i s  s u b j e c t  t o  t h e  four  f o r c e  f . .  L e t  t h e  body fo rce  a c t i n g  
on t h e  f l u i d  be  f i ,  t h e  conserva t ion  equat ion  reads  
1 
L e t  f be expressed by 
P 
f = Bn(fJ 
P n n 
where B i s  an  a r b i t r a r y  cons tan t .  n 
/ \  
Assuming f u r t h e r  t h a t  ( f J  i s  expressed by 
n 
where J is  t h e  e lectr ic  f o u r  c u r r e n t ,  i .e .  (Ja, io,). 
case previous ly  considered we w r i t e  t h e  energy momentum t enso r  of t he  
In  analogy t o  the  a 
gas as 
T (photon gas)  = Dn (n,) 
P* n n 
9 
where (npv) s a t i s f i e s  t h e  conserva t ion  equat ion  
n 
The t e n s o r  (nP,) is  traceless and symmetric. 
n 
The e lec t romagnet ic  f i e l d  w i t h  charge and c u r r e n t  i s  obta ined  by 
s o l v i n g  t h e  system of  l i n e a r  equa t ions ,  i . e .  
The flow v a r i a b l e s  a s s o c i a t e d  wi th  t h e  photon l i k e  gas can t h e r e f o r e  be 
expressed by s u p e r p o s i t i o n  of t he  e lec t romagnet ic  f i e l d s .  
For t h e  gas mixture  of  photon and t h e  material particles a t  high 
temperature ,  t h e  ze ro th  o rde r  energy t e n s o r  i s  Given by 
The r e s u l t s  ob ta ined  previous ly  f o r  t h e  energy d e n s i t y  (25+), 
pres su re  (26), and t h e  mean 4 -ve loc i ty  (27) are a l s o  a p p l i c a b l e  f o r  t h e  
p re s  en t cas e. 
10 
I V .  EFFECT OF FINITE MASS 
Since trace of t h e  f i r s t  o r d e r  energy t n e s o r  (Eq. 12c) i s  n u l l ,  and 
t h e  divergence of t h e  energy t e n s o r  is ze ro ,  t h e  s o l u t i o n  of t h e  f i r s t  
o r d e r  approximation is  a l s o  photon gas l i k e .  Hence without  loss of 
g e n e r a l i t y  t h e  f i r s t  o r d e r  s o l u t i o n  may be taken t o  be zero.  
The e f f e c t  of t h e  f i n i t e  mass appears  i n  t h e  second o r d e r  approxi- 
mation. Consequently t h e  flow f i e l d  o f  t h e  high energy gas d e v i a t e s  
from t h a t  of t h e  photon gas i n  t h e  second and the h ighe r  o r d e r  theory,  
which w i l l  no t  be t r e a t e d  i n  t h e  present  a n a l y s i s .  
11 
V. APPLICATION OF THE EQUIVAT.,ENCE PRINCIPLE 
I n  t h i s  s e c t i o n ,  t h e  flow p a t t e r n  of a gas a t  high e f f e c t i v e  temper- 
a t u r e  w i l l  be  examined under t h e  equivalence p r i n c i p l e .  
For s i m p l i c i t y ,  t h e  case of one-dimensional f low w i l l  be assumed. 
L e t  II c o n s i s t  o f  t h r e e  ind iv idua l  f i e l d s  S R and V Each 
PV PJ’ PJ PJ 
energy momentum t e n s o r ‘ w i l l  be assumed t o  be a s s o c i a t e d  wi th  plane 
e l ec t romagne t i c  waves given by 
V E , = O ,  E2 = 0 ,  % = a ( t )  
H I  = 0 ,  H 2  = 0, H3 = b ( t )  
PV ‘ 
It can be shown t h a t  t h e  components of each energy momentum t e n s o r  are 
given by 
SI1 = s22  = s23 = = SI2 = q4 = $4 = 0 
-2 f 2  %3 = cm2fZ, s~~ = % 4  = i c  
1 1 
Vll = 5 c - ~  (a2+b2 ), V2, = 7 c - ~  (a2+b2) 
c - ~  (a2+b2), V44 = - - c - ~  (a2 +b2 ) 2 v,, = - - 2 
By e q u a l i t y  T = , and s o l v i n g  f o r  t h e  2 components hl and 
tJv =PJ 
of t h e  4 - v e l o c i t y  i n  terms of f ,  g, a and b,  y i e l d s  
12 
i 2f2  + (a2+b2 1 A* = -  
From the  c o n t i n u i t y  equat ion,  one o b t a i n s  t h e  fol lowing equat ion 
governing N 
2 f ”  + a” + b2 dL + 2 p  - (a2 +b2 ) - -  &It - - at  {in a t  2Ed + (ad+bd) ax 
where t = JnN. 
The above equa t ion  i s  a l i n e a r  par t ia l  d i f f e r e n t i a l  equat ion of 
t h e  f i r s t  o r d e r  and can be solved i n  p r i n c i p l e ,  by t h e  s tandard mathe- 
m a t  i c a l  technique. 
13 
V I. CONCLUDING REMARKS 
Since t h e  fou r  divergence of t h e  electric fou r  c u r r e n t  Ja appears  
i n  t h e  body f o r c e  expres s ion  (30) van i shes ,  t h e  type of body f o r c e  
occur r ing  i s  sub jec t ed  t o  the  fol lowing scalar equa t ion  
-1 
tJ,c 
where F i s  t h e  t e n s o r  element o f  t h e  i n v e r s e  of m a t r i x  F. D e s p i t e  
t h e  above l i m i t a t i o n ,  t h e  case of flow without  body f o r c e  i s  sub jec t ed  
t o  no c o n s t r a i n i n g  cond i t ion ,  and a p p e a r s  t o  have general  a p p l i c a b i l i t y  
i n  t r e a t i n g  r e l a t i v i s t i c  flow wi th  v a r i o u s  boundary and i n i t i a l  cond i t ions ,  
14 
c 11 
C23 
r31 
c 4 i  
c 51 
f 6 1  
c 71 
c 91 
REFERENCES 
J. L. Synge, The Relativistic Gas (North-Holland Publishing Co. , 
Amsterdam, 1957). 
A.H.  Taub, Phys. Rev. 74, 328 (1948). 
A. H. Taub, Phys. Rev. 103, 454(1956). 
A.H.  Taub, Phys. Rev. 107, 884 (1957). 
A . H .  Taub, Arch. Ratl. Mech. Anal. 3, 312 (1959). 
A. Lichnerowicz, Relativistic Hydrodynamics And Magnetohydrodynamics 
(W.A. Benjamin Inc., New York, 1967). 
G. C. McVitte, General Relativity and Cosmology (The University 
of Illinois Press, Urbana, 1965). 
A. W. Guess, in Advances in Astronomy and Astronautics, Vol. 4 
(Academic Press Inc., New York, 1966, pp. 153-232). 
R.W. Truitt, Phys. Fluids 7, 2089 (1968). 
1 5  


Class of exact solution of relativistic gas

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