Matrix elements of the Boltzmann collision operator for gas mixtures

Chemical Physics 41 (1979) 81-95 0 North-Holland Publishing Company

MATRIX ELEMENTS OF THE BOLTZMANN COLLISION OPERATOR FOR GAS MIXTURES * Michael J. LINDENFELD and Bernard SHIZGAL Department of Chemistry, University of British Columbia, Vancouver, British Columblb, Canada V6T I IV5 Received 25 January

1979

The matrix elements of the collision operators that arise in the linearization of the Boltzmann equations for gas mixtures are calculated. The matrix representatives of the operators with the Burnett functions (products OYIagucrre polynomials and spherical harmonics) are evaluated for power law potentials. The final expressions for the matrix elements involve a smat1 number of summations and for the hard sphere cross section, analytic espressions in terms of the mass ratios are obtained.

1. Introduction Almost all gas dynamical problems require the solution of one or more Boltzmann equations for the distribution functions of the species involved. The method of solution often used in the solution of the Boltzmann equation involves the expansion of the distribution functions in a suitable set of orthogonal functions and the reduction of the integral equations involved to a set of linear equations. This procedure requires the matrix elements of the integral collision operators that result from the linearization of the Boltzmann equation. Various schemes for the calculation of these matrix elements and related quantities have been reported in the literature .[l-151 . They are in some instances applicable to one component [l-7] or to multicomponent systems [8-14]_ Many of these calculations apply to the hard sphere interaction potential [4,6,12], inverse fourth power potentials (Maxwell molecules) [4-61, general power law potentials [2,7,8] or are unrestricted as to the form of the interaction potential [9]. For certain energy relaxation problems [16] a restricted set of the matrix elements’ which pertain only to the isotropic portion of the collision operator is required_ In other problems [17] involving multicomponent systems, it is important to consider expansions about equilibrium distributions characterized by different temperatures, so that the matrix elements depend on the temperatures of the various components [lo12,181. These matrix elements also find application to problems involving the sdution of the nonlinear Boltzmann equation [10,15]. An ingenious method for calculating the matrix elements for muhicomponent systems with arbitrary collision cross sections was presented recently by Aisbett et al. [9]. This was extended by Viehland and Mason [18] to components with different temperatures and applied to their detailed study of the calculation of ionic mobilities. Weinert and Suchy [IO] and later Weinert [l I] have presented similar though not identical schemes for the calculation of these matrix elements. The present paper is an outgrowth of the authors’ study of the non-maxwellian effects associated with the escape of a planetary atmosphere [19]. Work is in progress to extend this earlier calculation to include departures of the distribution function from isotropy. This problem requires matrix elements of rather high order since the perturbation of the distribution function from maxwellian occurs at high energy and many terms in the expansion of the distribution function are required for convergence. * Research supported

by a grant from the National Science and Engineering Research Council of Canada.

M.J. LindmfeId, B. SI~izgalfBaltznmnncollision operator forgasmhnrres

82

The purpose of the present paper is to report the calculation of the matrix elements of the linearized Boltzmann collision operators that arise in mixtures. The basis functions used in this calculation are the Burnett

functions (products of Laguerre polynomials and spherical harmonics). For the most part, a general power law potential is retained, although our principal interest is in the computation of the matrix elements for the hard sphere potential. The present work is an extension of an earlier calculation [I2], although in the present work the components are taken to be at the same temperature. Although the general results by Aisbett et al. [9] are applicable to this case, the approach in this paper is quite different and leads to a form which we feel is better suited to the numerical computation of the higher order matrix elements. In particular, we present for the hard sphere potential usefu1 analytic expressions for the matrix elements in terms of the mass ratios.

2. Calculation of the matrix elements Linearization of the nonlinear Boltzmamr equation encountered in the study of gas mixtures leads to a consideration of the operator 1,, defied by

(1) In eq. (l), o is the differential cross section for collisions between species 1 and 2,g is the relative velocity and 51 is the scattering solid angle. The notation in this paper is entirely consistent with that introduced in an earlier paper [12]. Here, the maxwellian distributionsf$‘) are characterized by the same temperature, T. Although the previous paper [12] considered the operator I12 characterized by different temperatures, only the isotropic portion of the operator was considered. In this paper, we employ the Burnett functions, $,*rm(&) = [2n3’%!/r(n

+z + 3/2)] 1’2C~L~i’2(S~)Y,(6,4)

,

(‘-1

as a basis set for the calculation of the matrix elements ofI12. In eq. (2),5, = Q7~,/2IiT)*/~c, is the reduced velocity of component y; Lf;‘1/2(@ is the associated Laguerre polynomial defined by $1/2(~2) 7

(-1)“W + I + 3/Z) = 5 ,=Om!r(m+I+3/2)(n-rn)!~~

2m

(3)

’

normalized in accordance with

Jexp(-~~)~~2,j:1’2(~~)Lfr:1’2(~~)

r?r+lt3/2)6 d.E,r= ( 2nl

0

nm -

Y,(B, 4) are the usual spherical harmonic functions defmed by (2z+ 1)(1-m)! yI,(B, 9)= 4rr(I + m)! [

1

112

(-l)“eimQP~(cos

O),

where Pf” are the associated Legendre functions and Pp : P,. With these defmitions, the basis functions satisfy the normaIization

We now consider the explicit calculation of the matrix elements written in bracket notation

MI. Lindenfeld. B. Shizgal/BoMzmatmcollision operator for gas mixtures

a3

where y takes on the values 1 and 2. There are thus two sets of matrix elements. Since the linearized collision operator is a scalar operator, the matrix elements of II2 with respect to the Burnett functions are diagonal in 1and independent of m. Therefore

where

X-I

- t&i ’ ‘*1’2@f’l(~1 -&>I

w dQde,d&.

The quantities defined in eq. (6) are real and symmetric in the sense that MC’), =M$r,M~~~&zl, h2) (7) = M$!rL 0. In eq. (S), the factor in square =M$$m,, ml). Also, due to conservation of number density Moon brackets is related to the normalization of the Laguerre polynomials. The quantitiesM$$ are determined by substituting eq. (2) into eq. (4), summing over nz and making use of the addition theorem,

where & and & are unit vectors in the& ande, directions, respectively. .. (r) are evaluated employing the generating function methods introduced by Mot&Smith [8] The quantltles Mln,,, and later employed by Ford [2] and Shizgal and Fitzpatrick [12]. The Laguerre polynomials can be generated as given by G(t, $) = (1 -.t)-‘-3”exp[.$2f/(t

=,5u Lfr+1/2(&t”,

- l)]

(7a)

ItP (. 1.

(7b)

The basic methodology of the generating function method was discussed in the previous paper 1121 in the calculation of the matrix elements of the isotropic portion of112_ For the case 1= 0, the quantity G(fl, !$ll,, IC(t,, tt)) is easily evaluated since the velocity variable in G appears in an exponential function. The desired matrix elements are then recovered as the coefficients of the expansion in tl and t2_ For the case with 2+ 0, there is no suitable generating function to use to represent the Legendre polynomials. Instead, an integral representation for the Legendre functions, introduced by Ford [2] ,

is employed. The relation, eq. (8), is used to construct the quantity

given by

MY. Lindenfeld. B. Si&gal/EaKtzmann collision operaror for gas mixtures

84

With this procedure, the velocity variables are all in anexponential function which facilitates the evaluation of the integrals. The quantitiesM/$?t. are the coefficients in the series expansion, that is, m

B

Mp,

r2,21) = ii n=O

c CM;;~‘t;*;‘u]ll! rr’=O

(11)

I=0

In subs_equent manipulations, much simpJ.ification results in eq. (10) if we write l& X E&lcos @= i- (& X Sk),>, where I is a unit vector whose direction in the plane of & and 6: is still arbitrary. With the use of the operator 6, defined by 6%;)

= F(&) - Q,),

eq. (10) can be written in the compact form, J!f(Yl(t,, t,,u)=$x-s[(l

- r,)(1 - tJ

-3’2

W)

With the transformation to reduced relative velocity 5 and center of mass velocity& that is E=(M2p2~,

5= @fl)1”S2 - (~2)1’%1 > where Mr = mr/(m,

+m,) andM2 =m2/(m,

M(-!r{“-Ct,, t2, u) = -&

+&p~

(13)

1'

t m,), we find that eq. (12) can be written in the form

~~~~6expC-[a~)~Z+=~).~+,~)]la3}o~dndBdSda,

(14)

3

where Q = (2k~T/fi)r/~ and fl is the reduced mass of species I and 2. In eq. (14), o(r)= 1 -M2(fI 0

ff2)-(Ml

-M2)f1f2

n~1’(~.~‘)=-(~~M2)1’2[2(1 #~,~‘)=Q[l

-(M,

-MIU,

-t2)trG+2(1

-Mr)tlt2

-M,(tr

-t,)f2~‘-~(St~‘)-iuiX + r,)] -M2u[~*~+i1.(~X

(S-S’)], $)I, (1%

Li$Q= 1 -A!&

- M2fl - (MrM$‘%,

~~“‘(&4’) = -[2@flM,P2(l Qf’({, s’>= t2 [l -AfIt a3 = (1 - -t,)(l - f2)_

- f2)f15 - 2(M11M,)1’2(1 - fr)L&‘-U(M2f - M2f2] + (M$f2)%

[Set’ + ii-(5 X e’)] ,

- Mle’) - iuiX (M& +M25)],

MJ. Lindenfeld, 8. Shizgal/Bolt:mam~ collision operator for gas mixtures

a.5

In the transformation to center of mass and relative velocity, eq. (13) with the primed velocities has also been employed together with the conservation of momentum, that is, 5’ =E. The form of eq. (14) is somewhat different than that employed in the earlier paper [12]. Eq. (14) has been written in a form convenient to show the integrations over 8. It is important to point out that the vector coefficient @I and the scalar quantity a$‘) depend on the reduced relative velocities 5 and <‘, given explicitly in eq. (15). The integration over E can be easily performed and with the result -312 a3

Sexp{-[@W

+ ap)-)-6] /a3} dl=

[+$):‘I 3”exp[(a~))2/4a$3

];

(16)

eq. (14) can be written in the form

where b(‘)(B, cr) = 1 -(MI - M2)2fI t2 - ~IW~M~‘~r2cos2(S/2) -(MI -M2)u - 2M2u cos(S/2)[cos(19/i) + i sin@/?) cos o], lP(tl,

a) = 1 - 4M1M2~lf2sin2(Q/2) - 2(M1M2) l/2, si@/2)[sin(0/2)

+ i cos(e12)

cos a].

(18)

In eq. (18), 0 is the scattering angle, that is, the angle between 5 and 4’. It is important to mention that there is a considerable number of algebraic manipulations involved in the substitution of eq. (16) in eq. (14). In particular, considerable simplification occurs in calculating [a?)] 2 in eq. (16) by choosing isuch that I-({-5’)=0,

y= 1;

and

i-(MIS’ f M2@ = 0,

y = 2.

It is useful to point out at this stage that the integration over OLin eq. (17) can be performed in a manner similar to the calculation of the matrix elements of the one-component collision operator by Foch and Ford [7]. The rebaI, quantities can be expressed in terms of the familiar Wntegrals [l] involving arbitrary cross sult is that the MC7) sections. This to our knowledge has not been done before for mixtures. However, in the present paper, we proceed in a different way by restricting the calculation to power law potentials for which the integration over the reduced relative velocity scan be performed. For a repulsive power law potential V(r) = V&/r)“;

~uJ?~O,

it can be shown that 1201 the differential cross section can be written as a product of functions of g and 8 as given explicitly by

a(g,.e) = d2(2v,/~g2)‘l”rV(e).

(19)

In eq. (19), the function Z,(e) which depends only on 0 is given by an expression similar to the usual formula for the cross section [21]. Z,(O) is given by Z,(0) = @/Sin

0) Ida/de

I,

where in place of the impact parameter one has the dimensionless quantity o which is an implicit function of 0 given by 20

e=n-2

1

a

dz[l -z2 -(z/@~]-~‘~,

86

MY. Lindenfeld, B_ ShizgaI/Boltzmann collision operator for gas mixtures

where z. is the smallest positive solution to 1 - z 2 -(z/o)” = 0. Consequently, for power law potentials the integration over relative velocity can be carried out independently of the integration over 0 and permits a further reduction of eq. (17). With eq. (19) in eq. (17), the result of the integration over 5 yields A4’Y’(t,,t2,u)=Fd 1 2n -%(2

- 2/V)(VO/XT)~~VQ[a~~]1/2-2lv j-J d[b(7)(0, o)] 2’V-21V(c9)da da.

(20)

The gamma function r(2 - 2/v) in eq. (20) results from the integration over the relative velocity, that is, s

exp[-b%$/@]

.$-4/Vde= 2n[@)/@]

2/r’-2T’(2 - 2/v).

It is useful to point out that with u = 0 and the hard sphere potential (V+ 03,I, =$) the results eq. (20) reduce to eqs. (26) and (27) of the earlier work by Shizgal and Fitzpatrick [12] , provided that the temperatures of the two components (T1 and T2) are equal. The matrix elementsM/~~, can be calculated if the integration over 01and 0 can be carried out and the coefficient of tyt$u”‘/l! in the power series expansion ofM(Y)(tl, t2, u) can be extracted in accordance with eq. (11). The basic procedure is to expand the quantities [b(Y)@,o)]%-2 (eq. (18)) with application of the multinomial expansion, that is,

ryn, + .._+nnr +p) - a2 - . . . - ~2,n)-~ =

(1 -aI

Zk ,I1 ;..n,n=O

nl!

-.-nm! l--(P)

(UJ

... (amp.

(21)

This permits the integration over 01and 19to be carried out. The details of the expansion and integration are presented in appendix A. Since eq. (20) is to be written in the form eq. (1 l), the quantities [@‘)] ‘I2 -2iV in front of the integrals in eq. (20) are also expanded and the various summations that arise must be reordered. These tedious but straightforward manipulations are summarized in appendix A. The final results are &I;!;, = d2r-1’21! [r(2 - 2/~)/r(2/u -$)] ii Fi-pii-p-s x ,Fo

sFo

I

m?.

l-q

Jz2n-1h!

[r(2

- 2/v)/r(2/v

&

~(~t~+ptq+2-2/f)iY(n+n’-2s-2p-m+I-r-q+2/v-$)

qFo rFo 4p

r(p+q+2-2/n)r!s!

XB(l)(v)Ml+p-r-qM;+n’+q-Im-2s-p(~ 1 M$,

(Vo/kT)2’u

_M 12 - +)]

(n-m-s-p)!(n’-m-s-p)!(l-r-q)!m! )m+r+2s

’

W)

(VolkT)2ivQM;‘f112M~~1f2

(23) In eqs. (22) and (23), i?= mti(n, PZ’)and the quantitiesBE fmed, following Ford [2] by,

which depend on the power law exponent Yare de-

(244 (24b)

M.J. Lindenfeld, B. Sidzgal/Boltzmann collision operator for gns mixtures

81

It is important to notice that the L?$$v) quantities defined by eq. (24) are independent of the particle masses. Therefore, for a given power law they need be evaluated only once for all mass ratios. A different expansion of M(l)(t, , f2, u) (eq. (20)) was developed which yields au expression for Mi$ which involves fewer summations than in eq_ (22) but with a set of quantities CPs which depend on the mass ratios in a complicated way. This work arose in the comparison of our results with the results of other authors [13,14] for the special case u = 4, discussed in section 3.1. The details of this alternate expansion are presented in appendix B. Eqs. (22) and (23) are the desired final results. The matrix elements whh the normalized basis functions can be calculated with eq. (5). In section 3, we consider certain special cases and provide the results of extensive calculations of the mass dependence of the matrix elements for the hard sphere interaction.

3. Special cases 3.1. Inverse fourth power lmv potential; v = 4, Maxwellmolecules The inverse fourth power law potential has been traditionally employed in kinetic theory as the Burnett functions are the eigenfunctions of the one-component linearized Boltzmann collision operator [1,4,5]. This potential is useful as an approximation to the long range intermolecular potential between an ion and a mo!ecule. With u = 4 in eqs. (22) and (23), 2/v - $ = 0 and I’(2/v - +) that occurs in the denominators is infinite. For M$:;‘,,to have a finite value, the arguments of the r functions in the numerators must be zero for some values of the summation Indices so as to cancel I’(2/v -i). In eq. (22), this occurs for Y= 1 - q, m = 0, s = n - p and n = 12’. In eq. (23), this occurs for q = 1, p = n and n = n’. As anticipated M/77, are diagonal and we find that 40(l)(4) MIst,‘,l= s,,,2,(d*/2)(2v,/~)‘/21! X

M;M$+q(Ml

P=Oq=Or(~+q+~)~-q)!()t-~)!

r(,r + I +$) fl:

;;

_M2)2n-2?‘+1-q,

M,f;r = S,~,.(rl”/2)(2V,/~)‘pI! 4~~~~~)(4)(MlM2)““/“.

(254 (2W

Since the matrices M$$ are diagonal, the following operator equations may be written 4, Cd&&N

= d*o~~l~)‘l’~~,~,*~~(5,),

(264

112 [&&)I

= d2(~Vo/~)1’2v,~1~rr~~(~~).

(26b)

It is important to note that although eq. (26a) is in the form of an eigenvalue equation, eq. (26b) is not. The eigenvalues ~~1 calculated with eqs. (5) and (25a) do not appear to be equivalent with the previous results of Sirodch [ 131 and Naze [ 141, whereas the result for vnl calculated with eqs. (5) and (25b) does agree with their expressions.The reason for this is that the expansion of Mi,$ and hence pll, can be done in several ways whereas the corresponding expansion of M!ii, and hence vnl in the particle masses is of a very simple form. It is important to mention that the procedure that lead to eq. (25a) was adopted in this paper so as to have the result expressed in terms of the mass independent B$v) quantities. To show the equivalence of our results with those of Sirovich 1131 and Naze [141, a different expansion ofM(I)(tI , r2, u) (eq. (20)) was carried out. Although the work was motivated by the comparison discussed above for v = 4, the new results are valid for all power law potentials. The details of this calculation are.presented in appendix B and the equivalence with the previous calculations [13,14] is also demonstrated.

M.J. Lindenfeid. B. Shizgal/Bolrzmann collisionoperatorfor gas mixtures

88

3.2. Hard sphere potential For the hard sphere potentia!, Y+ 00,I,(6) = $ and. (2p f4 + l)!

24-1 (P +4 + I)! p?q!

B3m)=2q!(2p*l)!-

-’

B(2)(,) P4

=

(2P f 4 + W

1

2q!(2p + I)! -~$)~qlJ-

(27)

These resuits follow from the defmitions, eq. (24). With eq. (27) en d v + m in eqs. (22) and (23) we find that ii iT--pz-p-s

1 I-q

r(n+n’-2s-2p-m+Z-r-q-$) ’ (n-m-s-p)!(n’-m-s-p)!(Z-‘r-q)!m!

B(l)(~)M:fp-r-qM~+,z’tq-2n2 P9

-2s~p(M1 _ M2)m+r+2s,

Eq. (28) was employed to determine analytic expressions for some of the lower order matrix elements. The results of these calculations are shown in tables 1 and 2. In the tables, we have listed the matrix elements with the normalized basis functions; that is, the matrix elements given by eq. (5) denoted by J,$‘$. 3.3. EquaZ mass When the masses of the two components are equal (m, = m2), the matrix elements of the linearized Boltzmann collision operator for one-component systems, 1, can be calculated with the present results since, 1041) =&Ml)

+&u&

l/2 III 9&J =r(n (Qin’ . tI+~)r(n’tl+~) 1 Mlwi61,‘6mm”

In this limit, the matrix elements of1 are given by 4rf3,!n’!

[

where

MInnr=M;;;a(M1 =M2 =+) +M$,(M1 lMz =+)_

(29)

With eqs. (22), (23) and (29), we find that for a general power law potential MI,rn, = d2(2kT/rm)‘12 x 5

f;

p=o q=o

[I’(2 - 2/v)/r(2/v - +)] (V~/~~)~IU2-(“t~‘+Z-1/2)Z!

(30)

4pl?(n+n’-2p+Z-q+2/v-~) (n - p)! (n’ - p)! (1- q)!

Bt(v)y

where BP(V) = #l(v) +Bf21(v) is defmed by eq. (4.2) of ref. [2] . Eq. (30) agrees with the matrix elements calculated by%ord [fl(eq. (3%)) for the one-component operator. The result eq. (30) can further be specialized to the hard sphere potential (V+ m) so that a comparison with the previous computations [6] for this case can be carried out. This is equivalent to setting Ml = M2 =4 intheexpressions in tables 1 and 2 and calculatingJlnn~ =.T/i$ +.$$I_ The numerical values obtained in this way for Jmnaagree with those in table 2 of Sirovich and Thurber [6] provided that our result is multiplied by 5/S so that the matrix elements are expressed in the units employed in ref. [6].

89

MJ. Lindenfeld, B. Sixizgal/Bolt~nzm~n coliisioli operator for gas mixtures Table 1 (1) a) Hard sphere matrix elements: Jm,I, n

I=1

11’ 1 = 0

0 0 0 1

0 0

-8M2/3 4(2/5)“*M;/3

0

2

0

2(2/35)‘%&3

0

3

0

2(3/35)“%f;/9

0

4

0

5(6/385)1’2M;/18

1

1

-161Vf,fif*/3

-4M*(27fif:

- 1OlcIl + 13)/15

1

2

16(1/5)%f&/3

2(1/7)‘%;(75M:

1

3

4(6/35)“%,M;/3

(6/7)‘“M:(49Y:

1

4

8(3/35)‘%f1M$/9

(3/77)‘!*M;(243M:

2

2

-16MxMz(15M:

2

3

4(6/7)“%fIhf;(35M:

- 30MI +23)/U

-M2(2625Mj - 402OM; + 3006M; - 644M1 + 433)/105 61’2M;(3675M;’ - 434OM; + 3270&f: - 444M,

2

4

8(3/7)“%f&(211;

- 14M1 f 11)/45

(3/11)‘%&10395~f;

3

3

-2M,M5(945M;

3

4

2(2)1’2M&(1155M~

- 18M1 + 13)/15

- 14M1 + 23)/15

- 6M, t 11)/45 - 22M1 t 43)/90

+ 359)/630

-ZOOM:

+219OM:

- 1188M,+433)/35

- 2lOOM: +2030&f:

- 94OMl

- 9996M: + 7602M: - 748Ml t 667)/1260 -M2(56595M: - 14679OM: + 173775M;’ - 11126OM: + 42945Mf - 6462M, + 2957)/630

(2/11)‘%&315315M;

f 359)/l 05 4

4

- 43078OM:

-4M,M2(1501511f~ + 34485M;

- 4851OM: +72135M1:

- 623OOM:

-M2(18243225M;

- 12102M1 t 2957)/315

0 0 1

-16M*(2M*

+ 3)/15

-8M2(9M;

4(2)1”2M:(25M:

f 1,115

2(2/11)“2M1:(49M:

2

4(2/7)‘%f:(2M,

3

4(3/77)‘“@(8MI

0

4

(6/1001)“?&(10M,

1

1

-8Mz(l2oM:

- 63M: f 32Ml f 51)/105

1

2

4M:(420M:

- 135M: + 126MI + 93)/315

1

3

2(6/11)1nM:(336M:

t 3)/45 + 3)/9

(3/143)“*M;(l980M: 2

-2M2(17010.M:

- 77M: + 92Ml + 45)/315 - 351M: + 482MI + 177)/315

- 29925M: f 234OOM: - 6318M; + 1278MI

3

(6/ll)1’2M:(27720M;

4

(3/143)“2M~(90090M:

+ 34M1 t :6)/105 +46M1 t 16)/105

(2/715)*nM:(121M:

t 58M1 t 16)142

- 54OM: f 177M: + 202Ml + 336)/315

2(1/11)1nM~(3675& + 624)/315

- 154OM: 7 995&f; + 866M1

(6/143)lnM:(346SM; 304)/315

- 1092&i: + 959M;

t 65JM1

(1/715)‘“M$4719M: - 1188M: + 1251M: + 240)/126 c .e 1 -M2(169785?!; - 32697OMi + 26495OMi

+698M1

- 7788OM: t 8385111; t 9754MI + 16656)/3465

- 39375M’: + 3024OM: - 6270M;

- 107415Mj

t 7158M1 t 3441)/1890

(6/13)‘“M;(315315M~

- 50589OM:

+ 39795OM’:

- 98280&Z; t 23815&f; + 20554M1 + 14704)/6930

t 2496Ml+ 1821)/945 2

+ 22MI + 16)/105

2(3/143)1RM;(81fif;

-4M2(875M;

+ 2115)/945 2

+ 288473)/55440

+ 10Ml f 16)/35

+ 3)/15

0

2

- 828872Ml

8(2/7)‘“M;(4MI

0

14

- 66186120M: + 110053020M~

1=3

n’ 1=2

0

+ 752325Ml:

- 17666MI + 9419)/2520

- 10732428OM: + 67246550M; - 2731436OM: + 7152924M;

n

- 690690&f:

+ 165565M:

+ 82152M:

- 1365OM:

(1/65)‘nM:(231231M: - 51912M;

- 318318M:

+ 1815lM;

+ 24717Oti’:

+ 12206MI + 5616)/2772

(continued

on r.ext page)

M.J. Lindenfeld. B. Shizgal/BoItzmam~ collisiorl operator for gas mixtures

90 Table 1 (continued) n

n’ I=2

3

3

1=3

-M2(480480M:

- 1382535M$

+508560~:

- 91245M:

t 1782900M:

+ 10204M,

- 1251775Ml:

-Mz(6081075M,s

+ 17331)/3465

+ 77433M; 3

4

(2/13)LnM~(3003000M: -5691825M;

- 7462455M:

t 886809OM:

f 226391OM: - 342745M:

- 1885884OM:

t 25863915M;

+ 8121225M;

- 153842OM;

- 19276ilOM:

+ 91386M1

(2/15)‘%;(8423415Mf

+ 79370M1

+ 29114085M:

t 57393)/13860

+ 159056)/30030 - 2294292OM:

-2013627OM:

- 1418060111: + 177555M:

+ 8251425M’: f 152226M1

+ 108272)/24024 4

4

-Mz(191441250M’:

- 756080325M;

- 144297846OM;

t 136264128OM:

+98000154OM;

+ 11926584OM:

-hZ2(117366249hZ:o

-43185835OM;

- 15938364M:

- 1065416352M:.+

+ 1140586M1

t 10179918OM;’

+ 1974603)/360360

0 0

-32M2(20M:

t 469098M1

+ 2lM:

0

1

16(2/11)‘%&7OM:

0

2

8(2/143)‘nM:(168M:

0

3

8(1/715)1nM$66M;

0

4

2(2/12155)‘1”M;(572M:

1

1

-16M2(2835M:

1

2

8(1/13)“%;(1386OM;

1

3

4(2/65)*“M;(3003M:

t 45M:

- 1155M:

3

2;(2/5)‘nM;(300300M:

2

4

(J85)1J2M:(1225224M:

3

3

G2M2(1276275M; ; t 14184Ml

3

4

4

+ 6204M:

+ 1232280M:

-525525&W:

+4167M:

t 1880)/3465

t 2200MI + 728)/693 t 25880M:

- 11445OM;

+ 1477476M:

+6018012&f:

t 2648M,

-I.408M~ + 18411693

- 37765OM;

t42273OM:

- 1870869M;

- 4189185M;

(1j17)‘nM;(9699690M:

-M2(733296564M;’

= [47r3n!n’!/r(n

- 28333305M:

+ 124056MI

t 815869656&f; a) &

+ 798M: f 623M:

- 7293&

- 1479555M;

t 584MI t 1000)/3465

+ 336OM: + 3095M:

t 27909M: + 33016M1 + 57640)[45045

t 1883OM:

t 17235M:

- 35343OM4 + 88788M;

- 4677981M:

+ 2023119M:

t 14664Ml

+ 68523M: - 392175M’:

+ 10360)/9009

+44488Ml

t 19928)/9009

+ 1089OM: + 11853.M:

+ 25128)/9009

+ 146475M; 4

t 152Ml t40)/63

- 6825M’:

2

t 828432)/144144

t 24&f, t 8)/63

- 1925M’: t 47OM: t 5OlM:

$h%f2(720?2OM:

- 355731516M: + 392985M:

+ 88Ml t 40)/315

+ 341M:

2

765769158M: - 14131440M:

+ 56h1, +40)/315

+ 133M:

2(1/1105)L’2M~(23166M:

14

+ 94756662OM;

+ 24MI +40)/315 + 65M:

2

-49468419OM;

t 378378001: -27558531&Z: t 11580954M:

- 2111655M’:

t 16086OM:

+ 87288)/18018

- 3254245995M;‘t - 11487546OM;

6538612080&f; t 1549980M:

- 769566798OM;

t 1699893M:

(l).-entries , + I + 3/2) r(n’ + I + 3/Z)] IR MInn

t 5768979216M:

+ 205308OMl+

-2781346722M;

3679272)/612612

are in units of d2(2nkT/@)‘n.

91

MJ. Lindenfeld,B. ShizgalfBoltzmann collisionoperatorfor gas mixtwes Table 2 Hard spherematrix elements:JzA, a) n

Ii

00 01 02 03 04

1 3 1 4 2 2 2 3 2 4 33 3 4 4 4

I=1

I=0

0 0 0 0 0

8’3 -(4’3)(2’5)“’ -(2’3)(2/35fn

4;;8Mi385) -(2’9)(3/35)‘n,n

-(4’3)(6’35)‘” -(8/9)(3/35)“* -16 -(28/3)(6/7)ln -(56/15)(3/7)“’ 54 -22(2p2 572’3

[=3

1=2

-10(1/7)‘” -(49’45)(6’7)“’ -(27’10)(3’77)“2 -(::,6)(6)“2 -(33/4)(3/11Y 53916 -(1001’8)(2’11)‘” 5265’16

32115 -(32’15)(2/7f2 -(8/15)(2/7)1n -(32/45)(3’77)“* -(10/9)(6’1001)“2 6417 -16/3 -(32/15)(6/11)‘n -(44/7)(3’143)‘n 36 -(88/3)(6’11)“2 -(1/3)(429Y 41613 (650/3)(2’13)“’ 2125’4

----

1=4

72135 -(20’21) (2)“2 -(14/151(2/11)‘R -T::;;::;::;:;‘:;” 100/g -(70/3)(1’ll)‘n -11(6/143)‘” -(1573/42)(1’715p 49 -(91/2)(6/13)“’ -(1001/12,(1/65,‘n 40512 -(2805/8)(2’15)‘” 39083’48

128’63

-(32’9)(2/11)1R -(64/15)(2/143)‘n -(176’21)(1/715)‘” -(1144/63)(2’12155)‘R 144/u -32(1’13)‘” -(52/3)(2/65)“’

-(468’7N1/1105)‘” 64 -(200’3)(2/5,‘R -136(1/85fn 85013 -(1615/3)(2/17)‘” 1197

a)J/;;, = [4rr3n!n’!‘r(n +I++)r(n’ +l+ $1 I” IIQ~,~ “),*entrics , arein unitsof d'(2~kT/~)1~2M~+ii2M~~'i~

Appendix A In this appendix, the details of the expansion of eq. (20) in a power series will be shown. Using the multinomif expansion, eq. (21), we find that the quantities b[b(T)(0, CY)] 2’V-2 can be expanded as a)] w-2

(j[$l)(&

=

W+s +P +4 + 2 - 2/v) 22p+4MTM:+9cM1 2 r!s!p!q! r(2 - 21~) rJ,p,q=o

x { [coS@/2)] 2Pi9 [cos(U2) + isin(O/2)

&$2’(,3

I

1y>,2’~--2= 2

COS aI4

-

_M2)1+2~

l)(tlt2)S+pPq,

r(p *q + 2 - 21”)22P+9(~l~2)P+9/2

p,q+-J p!q!ry2 -2/v)

X { [sin(0/2)] ?P+9 [sin(0/2) + icoS(O/2)~0s (~19 -6poOSqO)(tlf2)p~*.

(A-1)

With eq. (A.l)in eq. (20) and the integrals P9(cosP,=(2n)-’

Jf~(cosfl+isin/3cos~~)qdpi, 0

P,(sin@)=(2n)-1

jn(sinp+icosficosr@ 0

da,

the integration over OL in eq. (20) can be performed and we fmd that,

(A.21

92

u6 [b”‘(B ,

a)]

M.J. LindenfeId, f3. Shizgal/~oItzmann

collision operator for

2 21v-21 .( t?) dS-2do = 4?r

4pr(r++s+piq+2-2/v)

5

r(p

r,s,P,q=O

gasrnljctures

B”)(V) Pq

f q f 2 - Z/v)ris!

X MfM$+4(M1 - M2)r’2s(tl t2)s+?d+q, es

./I-

O[d2’(t?, a)] ‘/“-2Iv(O)

where the@(v) eq. (21), pe%ts [#‘]

da da = 4i7 2 c

4PB@) Pq (v)(MIM,)P+qe

(tlt,)p~~~>

(A.31

p.9 =0

quantities, which involve the integration over 0, are defined in eq.(24). Similarly, the expansion, the factors [a(T)] 0 1~2-2~Vappearing in front of the integrals in eq. (20) to be expanded as

1/2-2/v

k M&W1

I’(i+j+k+mt2/v-$)

= i,j,k,m=~

i!j!k!m!F(2/v-i)

-M*)

,n i+m J+lnu k t]

2

1

1

-r),I+k/~~~li/2t:*~~k_

~(~+i*k+2~u

(A -4)

i!j!k!I’(2/v-+) The substitution of eqs. (A.3) and (A.4) into eq. (20) yields the power series expansions k!F)(t,,t*,U)=d

i0.m. r,s,p,q=O

271-%y2

- 2/v)(YO/kT)2’vQ

r(p+q +2 - 2/v)r!s!

X B(‘)(v)M;+kM;+i+p+q(Ml P4

M’2’(fl, r2, r() = &&r(2

i!j!k!m!r(2/u-3)

- M2) m+r+2syit,n+s+ptj+llll+s+pILk+r+q 1 7

_ 2&) (Vofk@‘Q

>

i.$k 4P r(i ii + li f 2!’ -+) i!j! k! l’(2/v - f) ;,;=‘o

i+pt(k+qIl~Mi+p*(k+q)/2tf+pr~Uk+q_ x B”‘(u)M 2 Pq 1

(A.3

To calculate the quantities, Mgi,, eq. (AS) must be rewritten in the form of eq. (11). Therefore in@)(tl ,t2,~) wereplacei,jandkwithn=i+m+s+p,n ’ =i f m f s f p and I = k f r + 4, respectively. Similarly, in Mt*)(tr, t2, u), we replace i, j and k with n = i +p, n’ = j + p and I= k + q, respectively. This change of summation variables gives the expressions Mt’)(f,, r2, 14)= d2n-“2r(2

’ P.9’0 rnzs,

X

- 2/v)(Vo/kZ’)2’“Q r(i+s+p+q+2-2/v)

R

n?n+s+p ’ l=r+q

4p

IYp+q+2-2/p)r!g!

I$$@)M,i+p-r-qM2n+n’+q-2m-2s-p(Ml

r(,*+n’-2s-2p-m+I-r-ql2/v-~) (I-r-q)!(n-m-s-p)!(n’-m-s-p)!m!r(2/v-+)

_ M2)m+r+2st;fr,z’Ui

,

MJ. Lirrdenfeld, B. Si~izgal~Bolt~l~mtlr2~ collision operator for gas mirtwcs co

M(‘)(t,, r2 >u)

=d 28’2q2

- 2/v)(V,/kT)2~“Q

93

m

c c 4p (~~p~~~,;_2,4;;:r,4,~~~~v-~~) p,q=o n,n’=p i=q

2

(A.61 A final reordering of the summations is required such that the sums over II, >I’and 1 are from zero to infinity. This involves a calculation similar to the manipulations in the appendix in ref. [12]. The matrix elements,M$$ given in eqs. (22) and (23) are the coefficients of ryti r&/l! in these expansions.

Appendix B In this appendix, the details of the alternate expansion ofM(L)(tl, f2, u) (eq. (20)) are shown. In addition, the results for v = 4, the Maxwell molecule limit, are obtained so as to compare with the previous results [ 13,141. The expression for b(l)@, ol) in eq. (18) is rewritten in the form b(‘)(B, (w)= I - [I - 41cr,M,sin2(0/2)] tlfZ --II [I - 2M,sin2(f3/2) + 2iM2sin(8/2) cos(8/2)cos a],

(B.1)

where the terms in tr tz and 11are collected together. The application of eq. (21) leads to the expansion zlr

(j[b(l)((j 0

,

a)]

2h’-2da=

5

p,4’0

p!q! I?(2 -- 2/v)

2r

XJ

[I-2M2

sin*(0/2) + 2iM2sin(8/2) cos(0/2) cos cr]qdcu (rIt2)Puq.

(B.2)

0 With the result, ?a s

(a t b CDSI$ da = 2rr(a’ - @“Pq

[a/@’ - b2)*‘2] ,

a2 > b?-,

(B-3)

0

the integral over ocin the summation in eq. (B.2) is evaluated and after some simplifications, this integral is given by 27-r s

[1 -

2&f2sin2(6r/2) t 2iM2sin(e/2) cos(0/2) cos o)] 4 da

0 = 2n[l - 4M,M,sin2(0/2)]q/2%

{ [I - 2M,sin’(0/2)]

[l - 4M,M,sin2(0/2)]

-‘12].

(B-4)

With eq. (B-2) in eq. (20) and using eq. (B.4), we get

M(l)& , t2, U) = d27+“r(2

- 2/v)(VO/kT)2’“Q(u~))*‘2-2’v

p$zO 2-4 CPJv)(rl r2)%q.

In analogy to the quantities B:;(D) of eq. (24) we defme the integrals

(B.5)

M.J. Lindenfeld. B. Shizgal/BoItzmann collision operator for gas IniXhlres

94

X Pq [El - 2M,sin2(~/2)}(

1 - 4M,M,si11~(0/2)}-~‘~] - I} sin 0 de.

(B-6)

It is important to notice that Cpq(u) is a complicated function of the mass ratios, but reduces to E$:(v) forMI = M2_ The matrix elementsM(l), can be obtained in a manner analogous to the calculation described in appendix A by expanding (@)1/2-21V inn and redefming the summation variables in the appropriate way. The result is

x C

P4

(v)M;-qM;*n’-2s-2p(Ml

- M$.

03.7)

The result

eq. (B.7) is valid for all power law potentials. It differs from the result given by eq. (22) in that there are two fewer sums involved and the dependence on the mass ratios occurs in a complicated way in the C q(~) integrals in addition to the explicit mass dependence in eq. (B.7). When Ml = M2, the two results, eqs. (B-75 and (22), coincide. For the special case, v = 4, the eigenvalue /+ in eq. (26a) is calculated following the arguments used in section 3.1. For v = 4, the only finite term in eq. (B.7) is the one for which q = 2, s = 0 and p = n = tz’ and we find that M$

= d2Z!(2V&)“‘2-‘-

1Cn1(4)6,zti_

(B-8)

The eigenvalue p,,r is evaluated with eqs. (5) and (B.8) and is given by pn, = 2n jIh(0){ 0 X PI[(l

[l - 4M,M2sin2(e/2)] ‘+I/’

- 2M2sin2(O/2)}/{1

- 4M,M2sin”(tJ/2)}‘/2]

- I}sin 0 d0.

(B-9)

For completeness we list the result for vnr that appears in eq. (26b), Vnl =

27~j14(0){

[4M1M2sin2(O/2)] ““/2P,[sin(O/2)]

- Sno610} sin f? d0.

(B.lO)

0 Eqs. (B.9) and (B.10) agree with the previous results for the quantities pn’nland vnl [13,14] _

Rekmnces

[l] S. Chapmanand T.G. Cowling, Mathematicaltheory of nonuniform gases (Cambridge University Press, London, 1970). [21 G.W. Ford, Phys. Fluids 11 (1968) 515. [3] K. Kumar, Au&al. J. Phys. 20 (1967) 205. [4] Z. Alterman, K. Frankowski and CL. Pekeris, Astrophys. J. Suppl. 7 (1962) 291. [S] C-S. WangChang and G.E. Uhlenbeck. in: Studies in statistical mechanics, Vol. 5, eds. J. De Boer and G.E. Uhlenbeck (North Holland, Amsterdam, 1970). [6] L. Sirovich and J.K. Thurber, Rarefied gas dynamics, Vol. 1, ed. J.H. Leeuw (Academic, New York, 1965). [7] J. Foch and G.W_Ford, in: Studies in statistical mechanics, Vol. 5, eds. J. De Boer and GE. Uhlenbeck (North Holland, Amsterdam, 1970). [B] H.F. Mott-Smith, MIT Lincoln Lab Report V-2 (1954).

M.J. Lindenfeid, 8. Shizgal/Boltzmann collision operator for gas rniuttues [9] .I. Aisbett, J.M. Blatt and A.H. Opie, J. Stat. Phys. 11 (1974) 441. (101 U. Weinert and K. Suchy, Z. Naturforsch 32a (1977) 390. [II J U. Weinert, Z. Naturforsch 33a (1978) 480. [12] 0. Shizgal and J.M. Fitzpatrick, Chem. Phys. 6 (1974) 54. [13] L. Sirorich, Phys. Plaids 9 (1966) 2323. [14] J. Naze, Comet. Rend. 251 (1960) 651,854. [l-5] K. Abe, Phys. Fluids 14 (1971) 492. [I61 B. Shizgal, Chem. Phys. 5 (1974) 129. [17] B. Shizgal and J.M. Fitzpatrick, I. Chem. Phys. 63 (1975) 138. [lSj L.A. Viehland and E.A. Mason, Ann. Phys. 110 (1978) 287. [19] M.J. Lindenfeld and B. Shizgal, Planet. Space Sci.27 (1979) 739. [20] J.C. MawelI, The scientific papers of James Clerk Maxwell, Vol. 2, cd. W.D. Niven (Dover, New York, 1965) p. 40. [21] L.D. Landau and EM. Lifshitz, Mechanics (Addison-Wesley, Reading, 1960).

95