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Diffusion and thermal diffusion coefficients of a relativistic gas mixture
Volume 47A, number 3
PHYSICS LETTERS
25 March 1974
D I F F U S I O N AND T H E R M A L D I F F U S I O N C O E F F I C I E N T S O F A R E L A T I V I S T I C GAS M I X T U R E A.J. KOX, W.A. van LEEUWEN and S.R. de GROOT
Institute of theoreticalphysics, Universityof Amsterdam, Amsterdam, The Netherlands R e c e i v e d 10 J a n u a r y 1 9 7 4
Values up to order c -2 are givenfor the diffusion and thermal diffusion coefficients of binary relativisticgas mixtures of hard spheres and Israel particles. In the first approximation to the first Enskog order the diffusion and thermal diffusion coefficients are given by [1] 1
2
[D]l=(pk2T2/3mlm2ClC2n)r=~-i d~l)~r'
[DT]l=(p2k/3mlm2ClC2n2) r=-~£ b(1)~r'
(1,2)
Here p = O1 + 02 is the total mass density, n = n 1 + n 2 the total particle number density, m i (i = 1,2) the mass of a particle of component i and ¢i the mass fraction. The prime indicates that the value zero is excluded from the sum. The coefficients d~ 1) and b~1) are found from the equations [I] 1
2
~J" brsd}l)=(nkT)-15 r
(r=+l),
S=-I
s~'2 =_
brsb}l)=p-13r ( r = + l , + 2 ) .
(3,4)
The quantities ~r and 3r are given by: 5 1 = - 5 _ 1 = 3 C l C 2, 32=-3x13'1/(')'1--1), with 7i = bl 1 =
(5,6)
31 =3 1 = 0 ,
(7)
3_2=--3x2~'2/(3'2--1 ),
Cp,i/Co,i, the quotient of the specific heats of component i. The bracket expressions brs are found to be
(8pClC2/Mn)[260~l)o0(O12)-- 3Z -1 ¢o1200 (1) ( o 1 2 ) ] ,
(8)
s (i) o 12)_2z160~1)0(o12)_ bl 2 = (8pClC2/Mn)[2(Zl+i)¢OllO0( s (1) _ 3Z -1 (z1+i')~1200(°12)
+ s 2 (1)
+s
2zlZ
-1
(1)
o
6°1210( 12)], (9)
(1)
b22=(8pClC2/Mn)[2(z1 ~)601100(o12)-4Zl(Z 1 ~)601110(a12 ) 2 (1) a 12) - 3 Z -1 (Zl + ~) s 2 601200( (1) o 12) - 4z1(z1 + ~)~ sx~,-1 w1210" ,(1) ¢o12"a+2z12Z -1 601220 (1) ( o 12) +2z1601120( (2) (012) +~Z-260 (2) "~l+8elXl[CO(2)(Oll)+-1 + 1 0 Z 2l Z -2 60(1) 1320(012) + ~'Z -1 ~2200 s 2300¢0 x 12JJ Zl
(2) 0 11)] , 60230(
(10)
and similar expressions for b-1,1, b_ 1,=-1, b_ 1,2, b_ 1,-2, b_ 2,2 and b_2,_ 2. The quantities/a and M are the reduced mass m 1 m2/(m 1+ m 2) and the total mass m 1 + m 2 ; the parameters z i, Z and ~ are def'med as m i c2/kT, ~Mc2/kT and 21ac2/kT, respectively. The quantities omega are certain integrals [ 1 ], which may be developed, for arbitrary differential cross sections Omn (m,n = 1,2), into power series in c - 2 :
221
Volume 47A, number 3
PHYSICS LETTERS
°d(S)[Otlkl" mn .l'-Ya(*)q'+~-1ran' ' [-- ~ g ) n ( i ) ]
+ z - I ([ 2s~_ S(_l)l+k+l]~g)n(i
25 March 1974
)
(II)
+ {}( } -/)+ [km2=+tm -(m 1-m2)2]/2 M } e
-1
),
/3_2=_~x2(
l+~z2a-1),
(12)
with x i = nJn; the coefficients 8 2 and 8_2 turn out to be O(c-4). The formulae ( 8 ) - ( 1 1 ) still contain the unspecified differential cross sections. With the help of the results obtained one may solve dr(1) and br(1) from (3) and (4) and thus find the coefficients (1) and (2) as power series in c - 2 . For the case of hard spheres of diameters a 1 and a 2 the relevant differential cross sections are. a l l = g1a l2, o22 = g1a2 2 and o12 = 021 = gi a l22 , with a12 = 1 ~(a 1 +a2). Using these expressions for the differential cross sections we f'md for the diffusion and thermal diffusion coefficient of a binary mixture of hard spheres of equal masses m:
_ a [ k r ~ V2 1 ( 1 - "
z-l),
is ~mk~ ½ (P2-Pl)al2+Plal-P2a2 2 2 2 [ z-1 2864P+5075Q+7286R'], [DT]I = -~-~Y} 16P+29Q+42R 1+ ~ j
(14)
with the abbreviations p=
22
22
-plP2ala2,
22
2
Q - (Plal+P2a2)al2 , and where z = mc2/kT.
R =plP2a42 ,
(15)
For a binary mixture of Israel particles of equal masses m, for which the differential cross sections are % = mPt/(O)/2g(l+g2/m2c2 ), (i,] = 1,2), with g the relative m o m e n t u m and Pi/(O) an arbitrary function of the scattering angle O, the diffusion and thermal diffusion coefficient are found to be: [D] I
_ kn'p T F(I~ 1 (l+~.z_l) '
(16)
--12 r,~ _,~ ~F(2)+,, F(2) ,~ F(2) [DT] 1 = z _ 1 5_k_k v~'2 ~'1 j 12 ~1 1 1 - ~ ' 2 22 4rr.~n ^ w(2)F(2)., t ~ 2 ~ ( 2 ) + ~ 2 ~ ( 2 ) W g ~ ( 1 ) . p ( 2 ) ,~(l),t;.(2)' ~ 1 ~ 2 * 11 22 -*-Vl--ll /"2--22 J~--12 ---12 ) + ~lPl'°2r12--12
(17)
with the abbreviations /r
Fff) j = f d ® sin o r o
0 (o)(1 -
cosSO).
(lS)
The expressions (16) and (17), which follow here as special cases of general formulae, are in accordance with earher work [3]. Higher order approximations can be obtained by suitably extending the sums in ( 1 - 4 ) .
222
Volume 47A, number 3
PHYSICS LETTERS
25 March 1974
The authors wish to thank Dr. P.H. Polak for his help and interest. A detailed account o f this work, part of the research programme of the "Stichting voor fundamenteel onderzoek der materie (F.O.M.)" which is financially supported by the "Nedeflandse organisatie voor zuiver-wetenschappelijk onderzoek (Z.W.O.)", will be presented to the journal Physica.
References {I] W.A. Van Leeuwen, A.J. Kox and S.R. de Groot, Phys. Left. 47A (1974) 31. [2] S. C~apman and T.G. Cowling, The mathematical theory of non-uniform gases, 3rd Ed. (Cambridge, 1970) p. 155. [3] W.A. Van Leeuwen and S.R. de Groot, Pbysica 51 (1971) 16.
223
PHYSICS LETTERS
25 March 1974
D I F F U S I O N AND T H E R M A L D I F F U S I O N C O E F F I C I E N T S O F A R E L A T I V I S T I C GAS M I X T U R E A.J. KOX, W.A. van LEEUWEN and S.R. de GROOT
Institute of theoreticalphysics, Universityof Amsterdam, Amsterdam, The Netherlands R e c e i v e d 10 J a n u a r y 1 9 7 4
Values up to order c -2 are givenfor the diffusion and thermal diffusion coefficients of binary relativisticgas mixtures of hard spheres and Israel particles. In the first approximation to the first Enskog order the diffusion and thermal diffusion coefficients are given by [1] 1
2
[D]l=(pk2T2/3mlm2ClC2n)r=~-i d~l)~r'
[DT]l=(p2k/3mlm2ClC2n2) r=-~£ b(1)~r'
(1,2)
Here p = O1 + 02 is the total mass density, n = n 1 + n 2 the total particle number density, m i (i = 1,2) the mass of a particle of component i and ¢i the mass fraction. The prime indicates that the value zero is excluded from the sum. The coefficients d~ 1) and b~1) are found from the equations [I] 1
2
~J" brsd}l)=(nkT)-15 r
(r=+l),
S=-I
s~'2 =_
brsb}l)=p-13r ( r = + l , + 2 ) .
(3,4)
The quantities ~r and 3r are given by: 5 1 = - 5 _ 1 = 3 C l C 2, 32=-3x13'1/(')'1--1), with 7i = bl 1 =
(5,6)
31 =3 1 = 0 ,
(7)
3_2=--3x2~'2/(3'2--1 ),
Cp,i/Co,i, the quotient of the specific heats of component i. The bracket expressions brs are found to be
(8pClC2/Mn)[260~l)o0(O12)-- 3Z -1 ¢o1200 (1) ( o 1 2 ) ] ,
(8)
s (i) o 12)_2z160~1)0(o12)_ bl 2 = (8pClC2/Mn)[2(Zl+i)¢OllO0( s (1) _ 3Z -1 (z1+i')~1200(°12)
+ s 2 (1)
+s
2zlZ
-1
(1)
o
6°1210( 12)], (9)
(1)
b22=(8pClC2/Mn)[2(z1 ~)601100(o12)-4Zl(Z 1 ~)601110(a12 ) 2 (1) a 12) - 3 Z -1 (Zl + ~) s 2 601200( (1) o 12) - 4z1(z1 + ~)~ sx~,-1 w1210" ,(1) ¢o12"a+2z12Z -1 601220 (1) ( o 12) +2z1601120( (2) (012) +~Z-260 (2) "~l+8elXl[CO(2)(Oll)+-1 + 1 0 Z 2l Z -2 60(1) 1320(012) + ~'Z -1 ~2200 s 2300¢0 x 12JJ Zl
(2) 0 11)] , 60230(
(10)
and similar expressions for b-1,1, b_ 1,=-1, b_ 1,2, b_ 1,-2, b_ 2,2 and b_2,_ 2. The quantities/a and M are the reduced mass m 1 m2/(m 1+ m 2) and the total mass m 1 + m 2 ; the parameters z i, Z and ~ are def'med as m i c2/kT, ~Mc2/kT and 21ac2/kT, respectively. The quantities omega are certain integrals [ 1 ], which may be developed, for arbitrary differential cross sections Omn (m,n = 1,2), into power series in c - 2 :
221
Volume 47A, number 3
PHYSICS LETTERS
°d(S)[Otlkl" mn .l'-Ya(*)q'+~-1ran' ' [-- ~ g ) n ( i ) ]
+ z - I ([ 2s~_ S(_l)l+k+l]~g)n(i
25 March 1974
)
(II)
+ {}( } -/)+ [km2=+tm -(m 1-m2)2]/2 M } e
-1
),
/3_2=_~x2(
l+~z2a-1),
(12)
with x i = nJn; the coefficients 8 2 and 8_2 turn out to be O(c-4). The formulae ( 8 ) - ( 1 1 ) still contain the unspecified differential cross sections. With the help of the results obtained one may solve dr(1) and br(1) from (3) and (4) and thus find the coefficients (1) and (2) as power series in c - 2 . For the case of hard spheres of diameters a 1 and a 2 the relevant differential cross sections are. a l l = g1a l2, o22 = g1a2 2 and o12 = 021 = gi a l22 , with a12 = 1 ~(a 1 +a2). Using these expressions for the differential cross sections we f'md for the diffusion and thermal diffusion coefficient of a binary mixture of hard spheres of equal masses m:
_ a [ k r ~ V2 1 ( 1 - "
z-l),
is ~mk~ ½ (P2-Pl)al2+Plal-P2a2 2 2 2 [ z-1 2864P+5075Q+7286R'], [DT]I = -~-~Y} 16P+29Q+42R 1+ ~ j
(14)
with the abbreviations p=
22
22
-plP2ala2,
22
2
Q - (Plal+P2a2)al2 , and where z = mc2/kT.
R =plP2a42 ,
(15)
For a binary mixture of Israel particles of equal masses m, for which the differential cross sections are % = mPt/(O)/2g(l+g2/m2c2 ), (i,] = 1,2), with g the relative m o m e n t u m and Pi/(O) an arbitrary function of the scattering angle O, the diffusion and thermal diffusion coefficient are found to be: [D] I
_ kn'p T F(I~ 1 (l+~.z_l) '
(16)
--12 r,~ _,~ ~F(2)+,, F(2) ,~ F(2) [DT] 1 = z _ 1 5_k_k v~'2 ~'1 j 12 ~1 1 1 - ~ ' 2 22 4rr.~n ^ w(2)F(2)., t ~ 2 ~ ( 2 ) + ~ 2 ~ ( 2 ) W g ~ ( 1 ) . p ( 2 ) ,~(l),t;.(2)' ~ 1 ~ 2 * 11 22 -*-Vl--ll /"2--22 J~--12 ---12 ) + ~lPl'°2r12--12
(17)
with the abbreviations /r
Fff) j = f d ® sin o r o
0 (o)(1 -
cosSO).
(lS)
The expressions (16) and (17), which follow here as special cases of general formulae, are in accordance with earher work [3]. Higher order approximations can be obtained by suitably extending the sums in ( 1 - 4 ) .
222
Volume 47A, number 3
PHYSICS LETTERS
25 March 1974
The authors wish to thank Dr. P.H. Polak for his help and interest. A detailed account o f this work, part of the research programme of the "Stichting voor fundamenteel onderzoek der materie (F.O.M.)" which is financially supported by the "Nedeflandse organisatie voor zuiver-wetenschappelijk onderzoek (Z.W.O.)", will be presented to the journal Physica.
References {I] W.A. Van Leeuwen, A.J. Kox and S.R. de Groot, Phys. Left. 47A (1974) 31. [2] S. C~apman and T.G. Cowling, The mathematical theory of non-uniform gases, 3rd Ed. (Cambridge, 1970) p. 155. [3] W.A. Van Leeuwen and S.R. de Groot, Pbysica 51 (1971) 16.
223