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Kinetic theory of transport coefficients of a relativistic binary mixture
PHYSICS LETTERS
Volume 47 A, number 1
25 February 1974
KINETIC THEORY OF TRANSPORT COEFFICIENTS OF A RELATIVISTIC BINARY MIXTURE W.A. van LEEUWEN, A.J. KOX and S.R. de GROOT Institute of Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands Received 2 January 1974 Within the framework of relativistic kinetic theory expressions are derived for the diffusion and thermal diffusion coefficient of a binary mixture.
The diffusion and thermal diffusion coefficient of a relativistic binary mixture are given by 2
D = -(ckTA~/3m,m2c~c2n)~$mi(cl-Gli)Bilp~pi4fi(O~d3pi/p~,
(1)
i=l
DT = -(cpAap/3mlm2clc2n2T)
& smi(Cl-Gli)BiPi*Pi4t;‘“‘d3PilP~,
(2)
i=l
where A3 is the projector gc9 + cw2 UaUP(gaP= diag(-1, 1, 1, I)), U, the hydrodynamic four-velocity, mi the mass of a particle of species i, Ci the mass fraction, pz? the four-momentum,$‘l the zero order distribution function, n the particle number density, p the mass density and where BiI and Bi (i = 1,2) follow from the linearized relativistic transport equations 2 C .@u [B.lAipP] j=l
(3)
= -(kT)-‘mi(C1_G,i)A~P~,
(4)
with Pij the linearized collision operator, and 4 the partial specific enthalpy of component i. (Note that in contrast to earlier work [l] hi occurs instead of the total specific enthalpy h. This is a consequence of the fact that such linear laws are chosen that the traditional transport coefficients (1) and (2) occur in them.) In order to solve (3) and (4), we develop the functionsBil(ri) and Bi(7i) of the variable Tim -(pz?U,+mic2)/kT in terms of associated Laguerre polynomials:
(5) Bi(Ti) = ~ (bs’li + s=l
b_s’,)‘z!i(TJ
(i= 1,2).
(6)
Insertion of these expressions into (3) and (4) yields equations which may be transformed into a set of linear algebraic equations for the unknown coefficients ds and b,: 31
25 February 1974
PHYSICS LETTERS
Volume 47A. number 1
fjf brsbs= p-‘p, s=--m
f+ b,ds = (nkr)%,., pZ_C.3
(r = f 1, * 2, ...).
The primes indicate that zero is excluded from the summations.
where O(r) is the unit step function.
Furthermore,
The quantities
(7>8)
at the right-hand
sides are given by
the expressions
b,,=bb:,tb;,
(11)
which occur in (7) and (8), consist of two contributions:
(12) 2
x1x2 C
b; = n(pkr)-’
[e(~)e(s)6,,6,i
+e(r)e(--s)61i62j
+ e(-r)e(s)62i6,i
i=l
(13) where xi is ni/n (nj is the particle number density of component
[‘(ri)5G(7j)l
i). The brackets are generally defined as
i EZ[zfc/8K~(zi)l (14)
X 7 dx 7 d!P i de f dO 7 d@ sinh2xsinh3\ksin0 0 0 0 0
[F(r : 1’
G(r)] j
s
sin@exp [-(Pc/kT)coshx] o$i [F(T~)]6 [G(Q]
(i=1,2),
tz~z2CI4~2(z~Yr2(Z2>1 (15)
.i,j’,*jdtIjd@ 0 0
0
0
7 d8 sinh2xsinh3\ksin&inQexp b
[-(Pc/kT)coshx]
0~~6~ [F(Q] 6, [G(5)]
(i,j= 1,2),
with the abbreviations 6 [F(q)] -F(q)
•t F(7;) - F($
- F(T;‘),
6, [F(q)] = F(T~)- F($.
(1617)
The primes denote quantities after the collision of a particle without and a particle with an asterisk; furthermore K2(zi) is the modified Bessel function of the second kind of the argument zi s mic2/kT. The differential crosssections are denoted as uii and a12. Finally, P is the total momentum of two colliding particles. The quantities b:s, bl and thus brs may be expressed in terms of so-called relativistic omega-integrals, defined as
32
25 February 1974
PHYSICS LETTERS
Volume 47A, number 1
w;$‘d= bW3/4~Tk2(Z1)Kz(Z2)]
where p is the reduced b,, =(8~c~c~/Mn)[2
mass mlm2/(mI
4&)(or2)
cn =; _(_l)j+k+l),
+m2) and M the total mass ml +m2. In this way one obtains, for instance,
- 3Z-‘4r&o
( r-712 119
(19)
with Z 1 aMc2/kT. From (l), (2), (S), (6) and (9) one gets for the diffusion
and thermal diffusion
coefficient
D = (pk2T2/3m,m2clc2n)
2’ d$$ r=--00
(20)
D, = (p2k/3mlm2c1c2n2)
2’ brSr, p=_OO
(21)
where the coefficients dr and b, may be solved from the eqs. (7) and (8). First approximations to D and D, may be obtained by limiting the summations in (20) and (21) to r = + 1 and r = & 1, + 2 respectively, treating (7) and (8) in a similar way. The authors wish to thank Dr. P.H. Polak for his interest and help. A detailed account of this work, part of the research programme of the “Stichting voor fundamenteel onderzoek der materie (F.O.M.)” which is financially supported by the “Nederlandse organisatie voor zuiver-wetenschappelijk onderzoek (Z.W.O.)“, will be presented to the journal Physica.
Reference (11 S.R.de Groot, C.G. Van Weert, W.Th. Herrnens and W.A. Van Leeuwen, Physica 40 (1968-9)
581.
33
Volume 47 A, number 1
25 February 1974
KINETIC THEORY OF TRANSPORT COEFFICIENTS OF A RELATIVISTIC BINARY MIXTURE W.A. van LEEUWEN, A.J. KOX and S.R. de GROOT Institute of Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands Received 2 January 1974 Within the framework of relativistic kinetic theory expressions are derived for the diffusion and thermal diffusion coefficient of a binary mixture.
The diffusion and thermal diffusion coefficient of a relativistic binary mixture are given by 2
D = -(ckTA~/3m,m2c~c2n)~$mi(cl-Gli)Bilp~pi4fi(O~d3pi/p~,
(1)
i=l
DT = -(cpAap/3mlm2clc2n2T)
& smi(Cl-Gli)BiPi*Pi4t;‘“‘d3PilP~,
(2)
i=l
where A3 is the projector gc9 + cw2 UaUP(gaP= diag(-1, 1, 1, I)), U, the hydrodynamic four-velocity, mi the mass of a particle of species i, Ci the mass fraction, pz? the four-momentum,$‘l the zero order distribution function, n the particle number density, p the mass density and where BiI and Bi (i = 1,2) follow from the linearized relativistic transport equations 2 C .@u [B.lAipP] j=l
(3)
= -(kT)-‘mi(C1_G,i)A~P~,
(4)
with Pij the linearized collision operator, and 4 the partial specific enthalpy of component i. (Note that in contrast to earlier work [l] hi occurs instead of the total specific enthalpy h. This is a consequence of the fact that such linear laws are chosen that the traditional transport coefficients (1) and (2) occur in them.) In order to solve (3) and (4), we develop the functionsBil(ri) and Bi(7i) of the variable Tim -(pz?U,+mic2)/kT in terms of associated Laguerre polynomials:
(5) Bi(Ti) = ~ (bs’li + s=l
b_s’,)‘z!i(TJ
(i= 1,2).
(6)
Insertion of these expressions into (3) and (4) yields equations which may be transformed into a set of linear algebraic equations for the unknown coefficients ds and b,: 31
25 February 1974
PHYSICS LETTERS
Volume 47A. number 1
fjf brsbs= p-‘p, s=--m
f+ b,ds = (nkr)%,., pZ_C.3
(r = f 1, * 2, ...).
The primes indicate that zero is excluded from the summations.
where O(r) is the unit step function.
Furthermore,
The quantities
(7>8)
at the right-hand
sides are given by
the expressions
b,,=bb:,tb;,
(11)
which occur in (7) and (8), consist of two contributions:
(12) 2
x1x2 C
b; = n(pkr)-’
[e(~)e(s)6,,6,i
+e(r)e(--s)61i62j
+ e(-r)e(s)62i6,i
i=l
(13) where xi is ni/n (nj is the particle number density of component
[‘(ri)5G(7j)l
i). The brackets are generally defined as
i EZ[zfc/8K~(zi)l (14)
X 7 dx 7 d!P i de f dO 7 d@ sinh2xsinh3\ksin0 0 0 0 0
[F(r : 1’
G(r)] j
s
sin@exp [-(Pc/kT)coshx] o$i [F(T~)]6 [G(Q]
(i=1,2),
tz~z2CI4~2(z~Yr2(Z2>1 (15)
.i,j’,*jdtIjd@ 0 0
0
0
7 d8 sinh2xsinh3\ksin&inQexp b
[-(Pc/kT)coshx]
0~~6~ [F(Q] 6, [G(5)]
(i,j= 1,2),
with the abbreviations 6 [F(q)] -F(q)
•t F(7;) - F($
- F(T;‘),
6, [F(q)] = F(T~)- F($.
(1617)
The primes denote quantities after the collision of a particle without and a particle with an asterisk; furthermore K2(zi) is the modified Bessel function of the second kind of the argument zi s mic2/kT. The differential crosssections are denoted as uii and a12. Finally, P is the total momentum of two colliding particles. The quantities b:s, bl and thus brs may be expressed in terms of so-called relativistic omega-integrals, defined as
32
25 February 1974
PHYSICS LETTERS
Volume 47A, number 1
w;$‘d= bW3/4~Tk2(Z1)Kz(Z2)]
where p is the reduced b,, =(8~c~c~/Mn)[2
mass mlm2/(mI
4&)(or2)
cn =; _(_l)j+k+l),
+m2) and M the total mass ml +m2. In this way one obtains, for instance,
- 3Z-‘4r&o
( r-712 119
(19)
with Z 1 aMc2/kT. From (l), (2), (S), (6) and (9) one gets for the diffusion
and thermal diffusion
coefficient
D = (pk2T2/3m,m2clc2n)
2’ d$$ r=--00
(20)
D, = (p2k/3mlm2c1c2n2)
2’ brSr, p=_OO
(21)
where the coefficients dr and b, may be solved from the eqs. (7) and (8). First approximations to D and D, may be obtained by limiting the summations in (20) and (21) to r = + 1 and r = & 1, + 2 respectively, treating (7) and (8) in a similar way. The authors wish to thank Dr. P.H. Polak for his interest and help. A detailed account of this work, part of the research programme of the “Stichting voor fundamenteel onderzoek der materie (F.O.M.)” which is financially supported by the “Nederlandse organisatie voor zuiver-wetenschappelijk onderzoek (Z.W.O.)“, will be presented to the journal Physica.
Reference (11 S.R.de Groot, C.G. Van Weert, W.Th. Herrnens and W.A. Van Leeuwen, Physica 40 (1968-9)
581.
33