THE UNIVERSITY OF MICHIGAN 2756-3-T 2756-3-T = RL-2070 GENERALIZED MAGNETOHYDRODYNAMIC EQUATIONS FOR PLASMA SYSTEMS WITH LARGE CURRENTS by Willis Lynn Everett January 1962 Contract AF 33(616)-5585 Aeronautical Systems Division Wright Patterson Air Force Base Ohio

GENERALIZED MAGNETOHYDRODYNAMIC EQUATIONS FOR PLASMA SYSTEMS WITH LARGE CURRENTS Willis L. Everett ABSTRACT The differential equations for number densities, mean velocities, temperatures, stresses, and heat fluxes of the electrons and ions, obtained by taking velocity moments of the Boltzmann equations, are closed out by utilizing Grad 13-moment approximations for the distribution functions. These equations, coupled with Maxwell's equations, constitute a closed set. The approximate distribution function, for a given type particle, is such that the first term is Maxwellian in a coordinate system moving with the mean velocity of those particles; the higher order terms are nonsymmetric in velocity space and have for coefficients the non-hydrostatic (traceless) stress tensor and the heat flux vector defined relative to the same mean velocity. The range of validity of the resulting magnetohydrodynamic equations is anticipated to extend to systems in which the difference in flow velocity of plasma constituents is not negligible relative to the thermal velocity of the electrons. By limiting attention to systems with slowly varying flows, it is found possible to exhibit transport properties to third order in the ratio of the difference in constituent flow velocity to the electron thermal velocity. Transport properties may also be exhibited for systems in which the above ratio is large.

ACKNOWLEDGEMENT The author wishes to express his gratitude to Professor Richard K. Osborn, Doctoral Committee Chairman, for his many suggestions and his guidance throughout the course of this research. This work was supported in part by the Aeronautical Systems Division, Wright Patterson Air Force Base under Contract AF 33(616)-5585, while the author was a member of The University of Michigan Radiation Laboratory. For this support the author is very grateful. ii

TABLE OF CONTENTS Page Acknowledgment ii I Introduction 1 II General Theory and Method of Approach 5 III Collision Transfer 12 IV Generalized Magnetohydrodynamic Equations 29 V Reduced Magnetohydrodynamic Equations and Transport Properties. 33 A. Plasma Systems in which E < 1 33 B. Plasma Systems in which 2 1l 46 VI Summary 51 Appendix A - Integrations overAngles of Scatter 53 Appendix B - Integrations over Velocity Spaces 56 Appendix C - Comments with Regard to Definitions of Temperatures, Stresses and Heat Fluxes in Plasma Systems. 62 Bibliography 64 iii

I. INTRODUCTION Accepting the Boltzmann equation as providing an adequate description for the behavior of gas constituents, continuum equations for macroscopic properties of gases have been developed by many authors. The ChapmanEnskog method has been developed In detail In the classical treatise by Chapman and Cowling, "The Mathematical Theory of Non-Uniform Gases(l). A different method, which utilizes a Hermite 'polynomial expansion for the distribution function, has been developed for a one constituent gas by Grad, and generalized to include gas mixtures by Kolodner(. These methods have been found successful In providing descriptions for systems of neutral particle gases close to equilibrium. With the Increased interest, In recent years, In plasma systems, attempts have been made to generalize these developments to Include systems in which the gas constituents are Ions and electrons ( 5 '. Although these developments provide closed sets of equations for macroscopic properties for systems close to equilibrium, their value is questionable for systems away from equilibrium. Of particular interest are systems in which the difference In flow velocity of the constituents is appreciable relative to the random (thermal) velocity of electrons. (See, for example, comments by Tuck, Allis, Burgers, and others (9) In the Marshall development, which utilizes the Chapman-Enskog method, attention is restricted to systems In which collision effects and external magnetic field effects dominate the behavior of the plasma, giving rise to a first order velocity space solution, for each constituent which is Maxwellian relative to the 1

2 plasma flow velocity. To this order, the difference in flow velocity of the constituents is zero. With the second order velocity space solution, a first approximation for this difference can be obtained, however, it is not anticipated (10) to be adequate for systems in which this difference is appreciable(l With the realization that a Maxwellian distribution, relative to the plasma flow velocity, is not an adequate first approximation for systems in which the difference in constituent flow velocity is apprec iable, an attempt was made to modify the Chapman-Enskog approach In such a way that the resulting distributlon functions would be appropriate for this wider class of systems. In this attempt, only like-particle collisions and magnetic field effects were taken to be of first order, giving rise to a first order solution, for each constituent, Maxwellian relative to the constituent flow velocity. Since like-particle collisions are more important than unlike-particle collisions for transfer of ion momentum and energy, and electron energy, the unlike-particle collisions were taken to be of second order. The second order velocity space solutions were obtained, the moment equations closed out, and transport properties exhibited. However, one shortcoming was in evidence. With regard to collision transfer of momentum, the Important coupling with heat flux did not enter until third order; and unfortunately the third order solution is very difficult to obtain as a result of mathematical complexities. The above difficulty is a result of the fact that, for momentum transfer of electrons, unlike-particle collisions are of the same order of importance as llke-particle collisions. As a result of this difficulty, and for reasons of lesser importance, attention was diverted to an adaptation of the Grad 13-moment method.

The Grad 13-moment method has been utilized for plasma systems by Kolodner(6) and by Burgers(). In these developments, the distribution functions assumed are based on Hermite polynomial velocity space expansions. As In the Marshall distributions, the first term, for each constituent, is Maxwelllan relative to the plasma flow velocity. The non-symmetric higher order terms have for coefficients the constituent flow velocity, stresses, and heat flux. Closed sets of equations for macroscopic properties have been exhibited, as well as transport coefficients to first order in the ratio of the difference in constituent flow velocity to the random electron velocity. In the method developed herein, the Grad method Is modified such that the results are expected to have validity over the extended range of systems In which the above ratio Is not small. The Hermite polynomial velocity space expansion Is utilized; however, the Independent variable is taken to be the particle velocity relative to the flow velocity of the constituent. The first term of the expansion is then Maxwelllan relative to the constituent flow velocity, and the non-symmetric higher order terms have for coefficients the stresses and heat flux defined relative to the constituent flow velocity, With regard to the Boltzmann equation, which Is assumed to provide an adequate description for plasma systems, the point of view is taken that the binary collision model Is appropriate for collisions with Impact parameter less than the Debye shielding distance, and that distant encounters (those with impact parameter greater than the Debye shielding distance) are accounted for by the use of the Inhomogeneous Maxwell's equations(4'.

4 In the next section the basis of the general theory is outlined and the method of approach presented. The method of computing the collision transfer terms is given in section III, followed in section IV by a presentation of the resulting set of magnetohydrodynamic equations, which coupled with Maxwell's field equations, constitutes a closed set. In section V, attention is restricted to systems with slowly varying flows. The equations are simplified and transport parameters are exhibited to third order in the ratio of the difference in component flow velocity to the random electron velocity. Comparison Is made with the results obtained from the Kolodner method, as (8) exhibited by Herdan and Liley ) Attention is also given to systems in which the ratio of the difference In component flow velocity to the random electron velocity Is very large. Here again, the equations are reduced and transport coefficients exhibited. Finally In section VI, a general discussion Is given on the method of development and the results obtained. Gaussian units are used throughout, and all symbols are defined where they first appear.

II. GENERAL THEORY AND METHOD OF APPROACH The Boltzmann equation for the distribution function of particles of type a may be written, i r (E Ei vj Hk) f = I(f (II-1) I c ijk 3 k v at ar M where e and mr are respectively the charge and mass of the a-type particle. I(f f ) expresses the change In f per unit time due to binary collisions of the a-type particles with 3-type particles. These collision terms are considered in detail In section III. The electric and magn etic fields are governed by Maxwell's field equations, a Hk 1 b E 41r (a) + J ijk B r c t c 1 Hi a E a HI (b) a c at ijk b rj (n-2) H (c) = 0 a ri & E (d) - = 47r q, dr, I with appropriate boundary conditions to account for externally applied fields, The current density and charge density, Jk and q, are defined by (a) Jk n Zew -neewk (11-3) (b) q= n[ Ze -ne e. 5

6 The number densities n and the constituent flow velocities wk are velocity moments of the distribution function f, that is, (a) n = fc d3v (b) w 1 f d3 (11-4) b) W = Transport equations for these and higher order moments of the distribution functions (temperatures, stresses, heat fluxes, etc. ) are obtained by taking velocity moments of Boltzmann's equations (II-1). For this purpose It is convenient to transform independent variables In equation (II-1) from the actual particle velocity vi to the random velocity u[ = v. -w. With these transformations, equations (II-1) become, D U L+ Hk O H - 6 ] _r Dt ar a. ijk uj k au au i br I I (11-5) where D (a) D =at +w a a D w (II-6) D w. a ca 1 (b) A ( e E.+ r w H i m 1 m c ijk j k Dt On multiplication of equation (II-5) by an arbitrary function yr(uk), and integrating over the infinite space of uk, we obtain,

7 D fla <n >i ~a ui a,31Jz Dt + — i n <(au. 3r. 1 e a a U Pau. a wr /1 1 1 ~ If fa ) -) 3u where, ) =fa d u. (11-8).a 1 a 2 ao 1 a 2 Taking / equal to 1, m uk, 2 m u, m uk, and m Uk; transport equations are obtained for the number densities, flow velocities, temperatures, non-hydrostatic (traceless) stress tensors, and heat fluxes ot the plasma constituents. a a D n 3w. aO at 1 (a) + n a = Dt 3 r. a D a e 1 +a HJ 1 (b) (E +w- E w Hk -a r. Dt m i c ij k j a J p DaQ ar (II-9) 3nak 3ri. D Oa w Da a p U. U + a rjk Pis k rs r _ _ o a a 3 ws ~c 2 + Lrs - 2 P. Jr 3rj

8 D a (e) ar P / 2 5 a a Oa a e (e) u u. + PA A. + Dt r/ 2 r ir 1 mc 1 m (1II-9) cont. aw. a w a3w a cr 1 a r ( U.U. 1 Ma Q. H-Q -Q. u.u.u j +M rjk j k Qr ar. j ar ijr ar. r a, a a Oa a p is the mass density, S., R L and M are the collision transfer of 1 rs r momentum, energy, traceless stress, and heat flux respectively; that is, /3 (b) Ra= (m u ) I(fa3) du (II-10) (c) a u 3 (c) L = (m u u)I(f f3) d u rs J r s 32 (d) M =7 (i m u u) I(f T3) d u. r.2 r /1 A circle above a second rank tensor is used to designate a traceless tensor, and a double bar is used to designate a symmetrized tensor. That is, o0 (a) T =T -- T.. rs rs 3 rs ii (II-11) (b) T = (T +T ) rs 2 rs sr

9 The temperature, stress, and heat flow of a given constituent are defined relative to the flow velocity of that constituent: () Ta 2 /ma 2> 3k 2 (b) P =n m u (II-12) rs r \ a a/m 2 \ (c) Qr = n \ u ur The temperature and pressure tensor are related by the expression, a a 1 a nk Ta P (II-13) 3 ii as may be seen from (II-12, a and b.) Also p =n kT = P. (1-14) defines the hydrostatic pressure of the a constituent relative to the constituent flow velocity. The set of equations (II-9) coupled with Maxwell's equations may be closed out if a representation of the distribution functions f can be found in terms of the macroscopic properties, n, wk, T, P, and Q; permitting the determination /2 a Oa a fu u u, u u Sa R, L, and M in terms of these properties. r s r r rs r A representation of fa meeting these requirements is the Grad 13-moment (2 3) approximation, obtained by expanding the distribution functions in three dimensional Hermite polynomials and retaining terms up to and including a contraction of the third rank polynomial. The three dimensional Hermite polynomials are defined(3) The three dimensional Hermite polynomials are defined

10 (a) r/(n) (-1) U_ i1i2'*in W c. a c... ac. 11 2 1n where (II-15) 1 -c/2 (b) = (27r) 3/2 e The first few polynomials are, (a) ( =1 (b) = c. 1 (c) (2) = c. c.- (I-) ij 1 J 1J (d) (3) = c. c. c-(c.. + c. +c C ) ) i j k i jk j ik k ij An arbitrary scalar function g( c ) may be represented by the expansion, g( ) =W a(n) (n) il i2.. in il i2... in ' (- ) n=O and the coefficients determined by multiplying both sides of (II-17) by /2 (n) i and integrating over the infinite space of ck. That is (n) 1 1/2 (n) (c) d3 (1-18) a. d. 11 12.. n Letting ck = m /kTa uk the distribution functions may be represented by the expansions 0o -1/2 f 1/2.7 (n) (n) (II-19) w f w a-, (II-9) ~n=i i2-.~. in i i2- ~- in

11 (3) (3) 2 Terminating the expansions with a contraction of. (i.e.. =c ck5c ) ijk i ijk k k and evaluating coefficients, we obtain, 2 2 f= n (/27) e 1 + P. u.+ a Qi ( -, (II-20) 2p 3 I J 5pP p where 3 = - kTa It may be noted that, for each constituent, the coefficient of the polynomial of rank n involves the nth velocity moment of the distribution function; hence, the use of these solutions implies a restriction to systems in which moments of the distribution functions higher than those which define the heat fluxes are small relative to the traceless pressure tensors and heat fluxes. To my knowledge, no velocity moments beyond the heat flux have been found to be ot physical importance in hydrodynamics or in magnetohydrodynamics, hence it is anticipated that the solutions have validity over a wide range of systems. More detailed arguments (2) in behalf of these solutions are given by Grad As pointed out previously, the approximate velocity space solutions (11-20) 0 2 a Oa a make it possible to determine (uu u, u uru ), S, Rs L and M in a a a Oa a terms of n, wk, T P, and Q. Then the transport equations (II-20) rs r coupled with Maxwell's equations (II-2) constitute a closed set. The next section is devoted to the determination of the collision transfer of momentum, energy, stress, and heat flux. In section IV, the expressions for (uu u ) and (u uiu are given, and the resulting closed set ot equations exhibitedr explicitly. exhibited explicitly.

III. COLLISION TRANSFER The collision transfer terms may be written in the conventional form, (see, for example, chapter 3 of reference 1), / (ay a'f d3 3 f a I(f af)d3 a - f -)v bdb d0 d d u. (III-1) r The prime is used to designate post collision variables. The range of integration on the azimuthal angle 0 is from 0 to 2 7r, and on the impact parameter b, from 0 to the Debye shielding distance h. Also, v = | - v the relative velocity r between interacting particles. The Greek symbols a and Y can correspond to either electrons or ions. For example, if a corresponds to electrons, g to a e me 2 e e ions, and g to b = 2 ue ue, expression (III-1) is the transfer of electron 2 j heat flow resulting from electron-ion collisions. Similarly, if a corresponds to ions, Y to ions, and iP to i = m u u, expression (III-1) is the transfer of ion r s stress resulting from ion-ion collisions For the Coulomb interactions involved in a fully ionized plasma, the collision cross section is given by the well-known Rutherford formula, '0( 4( ) 4e ' (III-2) v sin (111-2) r where X is the angle through which v is deflected, and r a y a Y m m m =m (III-3) m +m 12

13 Hence, 7X /eey 2 1 i d(sin 2 ) do b db d0= crd-L= od sin7(dd0 =p - -. (III-4) m V sin - r 2 2 a Y \2 The limits on sin - are from 1 to 1/ i 1+(- c -) which correspond to the limits on the impact parameter bfrom 0 to the Debye distance h. To facilitate the integration over scattering angles, ( -_/ ) are expressed in terms of a collision invariant vector ck and the relative velocity between interacting particles which rotates through the angle X on collision. To simplify the notation we let (a) Ma m a J m +m my (b) M: - m +m a (c) 3r= na (11-5) k Ta aY (d) 3 = /3+/ (e),' = (f) 3~= = 3e+3i

14 Utilizing the relations for conservation of energy and momentum, it is easily shown that, (a) uk = c + VM (b) Uk= ck + M vrk The vector ck is equal to the velocity of the center of mass minus the mean flow velocity for particles of type a. Then, for e = m u; m ukuL; ma a2 a' 2- uu; the expressions for ( _-g~ )may be written a a a t' aY (a) m (u - )= m (vr-v ) k k rk (b) m (uku-uk u )=2m c) +m M (v v -v vr) (-7) kik r kVrk Vr k a! 9Y' 2 a] 2 m a2 a 2 at m a a2 m Xv 2 v) mk 2 (vr v (c) u-(u u-u uk)2 2 c VrkM k + a Ya a i' '+\ a~Y' a I I +m ck c (vr-vr)+m M c1 (Vvr -vr vr) * Perhaps it should be pointed out that the collision exhange term in which c/ = m uk uX contains the information required for both the energy balance equation (II-9, c) and the traceless stress equation (II-9, d) since,

15 a/2 ( m a2 dkJ a a a (a) -u = m U u 2 2 k," (III-8) ( a a ak a ( k- a2 (b) m uk u m (uku - u It is now possible to carry out the integrations over angles of scatter. On substitution of expressions (III-7) and (III-4) into (III-1), we see that only two types of integrals are involved. These have been carried out in Appendix A, where itis found that, (1) ' d(sin 2 ( 2) r r rj rk i rk (b)|(VrkYrR~rk~2d i3= 12 X vrk~Vr~n sin3 2 _5 2 where, (+ 2 ai 2 e aei rj (ar-/ h ee1 r (III-10) (b )( ) me hh2 ____4 2 2 /m lh 4 (2)L1 { 1+ (ma h 4 1 e2 e+l V( 1+a e r Inasmuch as the value that should be used for the upper limit of the impact parameter is not fixed precisely, and the results are not very sensitive to the 2 relative velocity, we replace v by an approximate average obtained by assuming the distribution functions of the interacting particles to be given by the Maxwellian parts only of distributions (11-20). This approximate average

16 is found to be, V2= (3+62), where ~01/2 (3+), where F ' r /3? — ak W IWkW- wk (III-11) m h Also, em e 2, v 7 1, so we take, r (a) (1) = j(3+~2)m h, a \ ea e (III-12) (b) f'(2)=], (3+~2) p()j m h oe e e I 1 2 The collision integrals (III-1) may now be written in the form, 4r (eye 2)2(1) - ma ) Ma Y me d Y pa 3 3 ff -d u d dUy 3 a y v r (III-13) where, for momentum transfer, stress transfer, and heat flux transfer, jf is equal respectively to, (a) Vr k (b) 2 ck Vr 1 c2 1 (c) c v +2 rk 2 ~'L(2) i a o +3M a- v M (1) k r M2 2 a a M Vr Vr +Ck cVr (III-14) (2) +3Md a a, +3M - Ca a y 0 Vr Vr,

17 To facilitate the integrations over velocity spaces, we define new dimensionless variables, gk and Gk, by the relations (a) Uk u b a Gk+gk- k (III-15) (b) b aGk+gk-, where 1/2 (a) a (b) b jA 2 ( (III-16) - ) (c) aJ oa (d) b and o 1/2 (a W (III-17) ~k (Wkvwk) With this change of variables, it may be shown that: 1 (a) vrk=- 1 gk f' 2 ( i2 (III-18) so that the expressions (III-14) for - a may be written, in terms of the new variables,

18 (a) 1b 5d() g d(4)G (C) (1 (2) 9 +e(4)g2 gkh(3) GG (4) G g \h~rsG gS+ h~r tr~s~j where (a cia=(2) 0r (3)~dkfr (1) (1) a (b) h e, ( 2d1 +& 8 kr a~ kr k r (C hkrs ea~' ks r ea rs k (3) e(5) 1 (d) \rth -f c 5'0 rsta y2 kr st ks rt (111-19) (4) \e ~rs - (6) -kr +ksr rsk (f ) (5) \rst e(7) ~ S+e (8) cP l e~ G rs kt aY ySt kr 3 -t(2) (h (2) ~'L \.~a 2a (h) =3 ( (1120 OIII-20)

19,-(2) (i) d(3) -M' a? /:, (1) I \ (4) id 6' <(j) d 4 2 l ) 2 (III-20) (cont'd) e(1) (k) e -v [ t (2) (1) e (3) (m) e = - f3~ V~ (-, (2) (2 +3M )' P a f,(2) (~ -M /(1) ) a3 (nI) e 3 2 ac 1 + 2 t M 2 (2) a() (5),L (o) e: (p) e = 2 a<' /3 <p>(~-) (q) =ea ( (,(2) (2 )a+3My -. ) ag (8) 1/2 j (2) (r) e =()_= (y~a_My a ----Y ay 0 l(1) 'LI a a)Y a ^

20 a y Making the change of variables (111-15) in the approximate distribution functions (I1-20), it may be shown after considerable manipulation that (III-20) (cont'd) faf'd3ua3uy n n (27r)3 -G 2 -.3. _,*- ) 2 1 -+ (-K - 6 2 2 IZk 11 I I 1 1+ p I V, 1) v..(A..+B..) v =a, xij 1] 13 I. + zi: ' D 1 i d 3g d 3G (111I-2 1) where VV2 (a) A~.= a G. 13 0 0 0 +a a G + g. g+. 6. 21 12 1 J 0 -2 g a gie. (111-22) (b) B. 0 0 = 2a (Gg.g -G. 6 ) 12 i j c) Cy'= a2 (G g+2G.G.g.-G2 &2G.G. Ea-5g +5 ~.+gg-~+E g -2. gg.+2gE a a5 g+5 F-a (d) D. a3(G2 G. - 5G)+a (g2G +2G. g.g.. -2 g. 6 a G. -2 G. g. E -2 G.. g. V 1V J1 J 1 21211 21211 +~G.+2 G.E' -5 G.) 1 Jji 1 ea24 i

21 V 2 ov b (e) ~P - (III-22) ij v (cont'd) 2 p Av3/2 Qz bV () QtV i 5p In taking the product of the approximate distribution functions, terms involving the products of pressure tensors and heat fluxes have been assumed negligible. (The integrals involving the larger of these product terms were carried out and found to be less than five percent of the terms retained.) As may be seen by substituting expressions (III-18, a; III-19; III-21, and III-22) into (III-13), hundreds of terms are involved in the collision transfer expressions. However, all the velocity space integrals are of the form, 1 11 G2 (g - )2 2 2 GG..G..G. g. i m 3 3 312T e n i 12 dm dGd, 4-r(27r)3/2 3 (III-23) which may be carried out exactly. The method used in carrying out these integrations is outlined and the results tabulated in Appendix B. Finally, after considerable manipulation, the collision transfer of momentum, energy, traceless stress, and heat flux may be expressed as,

22 (a a~la'(2) (3 ) V I aC a:a(H24 ki j k jV 1 ~(4)7 vE+ (5) 2i a i a (c) Lg (cZ (2a k' Ra7a kj i Ee ayv aY 1Jl J? al~~V aI Lka~,/+ a +~ Qa EF k a d k7 aQ" k Y P.. v' Is I

23 (d) (1) +? (2).a (3) v +X a Y (III-24) (d) k a..C I/ V a kj (cont'd) ffi 1. kj p ' +~7 a Y k r k v =a j' Y....=a (4) +7' 2 v' aC (5) a2 V (a) a a ij i j k / ay 7 P'/ k 1 Grea sa1aod c v=c, / V =C.Y V= (6), a (7) ' +(8) Q +7a Q a QV <, +,3, > a 2QV'a a (12) - v i a k a ( -) i i iaJ v, J v=a, =/ Whereas in the notation used for the individual collision transfer integrals, the Greek symbols a and ) could independently correspond to either electrons or ions; in the above expressions (III-24) the symbols a and Y are to correspond to particles of different type, that is, if a corresponds to electrons, then / corresponds to ions and vice versa. The coefficients in (III-24) are given by; (a) e4(1) Jei (III-25) ( 72

24 (C) ("-e - V1 /2 F;A /43&e-( (d),e /3 A.01/ (e) (5 =Ae7j, /3 1 5 1 -t ) (III-25) (contfd) (m[I- 26) A. (a) (1) _ 3 -eld(1) a 2 0 ay A ( 1 6 2 4 )+ d (2 1 3 al + 2 )+ d(3$( (/ +&2 1 ) 2)3A.e (b) 0I a/2 3 (1) 1 1 s 3 1 (3) 1 n, 3 -.4 ) +- + -1 4)+ d - E2 + -P j d Y (- - PE 2 ay 9 (3) 3 2 __ d(4) O0da y 2 - E2 4 ) (d) =,a (e) 3 2 A 'e 0 /3 + &32)+ d 3 a') ' I~) 3 -"ei[ d (1) (a Y If I

25 'ai) (1) ed(l) +d(2) 1 2 - (a) (A - /3 d a(7)+d (+-2 ) 2. a j 0 a~, %. (III-27),( (2) (b) C, ^ -. ei /3 [ d(2) 26-;' )+ dy (2) 8 d+ ____ de (3) ei -z o 2) I ( 22 <+2,J1 - ) ' (d) (Aei id 4 (2. ) f ay (e) c/ a a~: _ei (1) 2 d(2) (_ 20 r' 12,+ af E2- E2 E 4 -^ p aIy 60 ) E4 ) (f ) 16 ei W ay o (4) 2 6 "P d ( 7 -,:2 ll-t ) a f g (7) (g) oay Aei (1) - 1 2+ 1'5 d(2) (. 35),+ 15 105 QId a j -".a i,- 4 - z/(8)9) ei (i) ( / = = o 0 C~~ a /3 L d(1) ( )+d(2)( 4(- -12).v,)" d (Y 2 Al/ (4) 4% fV 1 C) ( (10) l a ei d(1) ~. (2) 20 10( l 3010 3 ( ay 0 a y 'j'a - y 7d L (k). (I1) Ai d(4) 2 q 1 0 30Jh a/3 a a,(-( ---4 + 3I

26 (III-28) 1 2.4 J (1) A ei ,j (1) 3 6 (2) -11 (3) (J+F2,- 4)+ e - i -fl?, e - 11 _ _: (a) oil, ( " 2 %! +- )+e 91 a I a -. a f (11 I I A I +eay( (5 5 -~ 2 (b) 2/ (2 a2Y Aei (l '4 " (5 e e,Q( -3 l)+ e~?( 13' 11 (T 1 5 - (2) TF I )+ e %-, a t.2 j~-4-, 2 4 7) I (4) 26 I + e (- 6~~ a?- E2J_. -6 t (3) (C) P o A. el 1003/2 (~6) ' (7) 4 f 2 ey(2 ~-2 )+ e (- /+2 V a d o a~ ~, )+ e(8) (-2~ S) (d) (ei) 1j03 2 (I a I (1) e ) 6 3 C,,, 1 1 (j + 33 (5) 5 O..- "4)+,a (- - 2 ` 2 ,- - 2 2 2 F 2 a? 2 8 - 25 - T,-c4 + 75 2 E4 + (2) 1 +2 9 0 )e 3 1 (4) 5 ~\ + e ( T-'~~ ad~' 3 -,? 15 E2 V - 844) (f) rr QE a 1 32 e (6) ( 4 a 1 E2 12 +3 7 0 -1- 6 +/3 ) F-: \, (5) 1 (7 +2( - e 7 3 a ez 5 " (2) )+e (2 I-6~ Cax (4) V )+e (-2~ ) 5 A8 __ -- e() ) e 3 a a A. ei -J

27 (111-28) (cont'd) (h) n'- (8) aY -Yyl(I0) aY~ A. eW( 2 (h) 5 " (2) ~ 1(4) )e (2P- 6 -I)+ e (-2. i) 2~~ a " (7) (, 7 )+ ~(- + -) e( —24a Aei / (6) e I/ a 'Y 0 2 0 I'A: i, (_ p 2 14 " , + 12 " '. I 13t (1~ (5) \ e ( ) (ii1) Aei a! = p3/2 (1) 3/ f-, E 2 (5) ( 5 \?" )+ e I, a y 2 a2/6f8~ (3) r1. (4) + e a,( )+e (-Y, 2 ~) 6 -)3 () (12) I____ (13) = ei/ (13)Ae e (63 (-8~ - 36)+ e~ 126t 8 2 6 4 2 I-/ ) / (5) 9 27-i. e9 where (a) A. el = 4\/ 77r neni (Ze 2) mei 03/2 (1 JIei (III-29) (b) -A aa - (n aeC2 ) ma a3/2 J- 1 0 eaa (C) (1= e 6/

28 (III-29) (cont'd) () 1 x2!/2 (d) e = x2 dx 0 The above collision transfer expressions are obviously extremely complicated. Considerable simplification has, however, been found possible 4 for systems in which E < 1 (terms of order E negligible), and for systems in which ~ >)1. These reduced expressions are exhibited in section V.

IV. GENERALIZED MAGNETOHYDRODYNAMIC EQUATIONS Utilizing the approximate distribution (II-20), it may be shown that, (a) \iU.UU i r s b 2 (b) Ur sI a ( is ki 5 a2 ( kr k is +ki rs + ks ) p (IV-1) b - 2 5rs 7 + tpa rs On substitution of these expressions into the transport equations (11-9), we obtain Dn a (a) D Dt D w. a! i (b) Dt Dt D Ta (c) Dt D P (d) rs (d) Dt Dt a w ~w. a W +n i =0 ar. 1 a e ma (E.+-. w H ) 1 c ijk j k aPa a ar al i P" j P" 2 3nak 4 Qs 5 ar r I -- po, ar. ij 1 aw. ar. 1 + Ra (IV-2) a e +2 -m c aw. Oa Oa 1 E P H -P rjk is k js a r. 1 aM 0 a, -2P - jr al ZQ s +L S+ La r. rs aw 4 a ws 5Qs ar r D Qa a r (e) Dt a aw. 7 a 1a =- - Q5 r ar. i aw" a r Qr j ar. ] Oa P. ir P ap. ar i kTa a m oar aP rs ar s 7 k Oa 2 ma rs aTa ar s 5 2 ma a e - +. a r me Ek QJ Hk 2 P i+P rjk Qj Hk 2 lr lr 1 p + M r 29

30 Alternate forms ot the equations for conservation of particles (IV-2, a), and conservation of momentum (IV-2, b), are of interest. These alternate forms express conservation of the properties of the mixture, as opposed to conservation ot properties of the constituents, that is, equations for total mass density p, total charge density q, total momentum density p wk, and a generalized kL Ohm's law for the conduction current density jk. The latter macroscopic properties o f the plasma are defined by: i e i i e e (a) p=p +p n m + n m i e i e (b) q=q +q =nZe-n e i e (IV-3) o i i e e (c) pwk=p k+p wk ', i i e e o o (d) Jk =(ql w - q w ) - qwk J - q w On multiplying (IV-2, a) by m and adding the ion equation to that of electrons, the total mass conservation equation is obtained, namely, ap a t + p w. =0 at ar. 1 (IV-4) 1 Multiplication by e and addition of the constituent equations leads to the charge conservation equation: aq aJ. at + a' = 0 (IV-5) 1 Similarly, the equation tor conservation of total momentum and the generalized Ohm's law equation are obtained using the multiplicative factors p and ma/ea.

31 These equations may be written, o D w. e o D a e 1 1 o 1 (a) p Dt )+ - c JkHi. r — 0 0 Dt arni e ij 3j c jk Wk c jki e D. w.~ aw e m o i 1 1 a n (b) ne2 Dt j ar. i r. e ar. (P j+ nee2 n e i 1 ne 1 ne (IV-6) +(E.+- w0 H )- 1 i H j c jkQ k kH e jkk k n e j nec ne where D o a o a - + Dt at k ark - 3t (IV-7) In obtaining the equations for total momentum and current, use has been rade of the relations, i \ e (a) m Z m b) i i e e (b) nm n m (c) e i \Z ni e (c) n m )> Z n m, which are, of course, valid for most systems of interest. Equations (IV-4), (IV-5), (IV-6), (IV-2, c), (IV-2, d), and (IV-2, e); with a a 0Oa a expressions (III-24) for the collision transfer terms, S., R, L, and M rs r constitute a generalized set of magnetohydrodynamic equations. The electric a nd magnetic fields which appear are governed by Maxwell's equations (II-2).

32 Since the magnetohydrodynamic equations exhibited by Spitzer, "Physics of Fully Ionized Gases" have received wide attention; it is worthy of note that, making allowance for the difference in notation, the Spitzer equations for particle conservation (2-14) and (2-15), total momentum conservation (2-11) and the current equation (2-12) with definition (2-13), are the same as the set of equations consisting of; the ion conservation equation (IV-2, a) above with a corresponding to ions; the equation for electron number den sity obtained by eliminating the ion flow velocity from equations (IV-3, c) and (IV-3, d) and substituting the resulting expression for electron flow velocity into (IV-2, a) with a corresponding to electrons; and the reduced forms of (IV-6, a, b) obtained by using the approximations specified by Spitzer on page 20 ot "Physics of Fully Ionized Gases." The need herein for the equations for traceless stress and heat flux is a result of the dependence of the momentum transfer on these properties [expression (III-24, a). Under certain limiting conditions, the momentum transfer may be expressed independent of heat flux and traceless stress as indicated by Spitzer in section 5. 4 of "Physics of Fully Ionized Gases".

V. REDUCED MAGNETOHYDRODYNAMIC EQUATIONS AND TRANSPORT PROPERTIES. The magnetohydrodynamic equations in their general form are seen to be extremely complicated. Fortunately, however, considerable simplification is possible. For systems with slowly varying flows in which 5I 1, the collision transfer terms (III-24) are reduced to a manageable form by expanding the E -dependent functions ~ andir in powers of E and retaining terms to order. (Recall from section III that 1/2 e3 m a 0 (wa 130 a a ek= k- k e kT a= ik ' 3 e xdx.) On the other hand, for slowly varying systems in which?>K1, the expansion is made in inverse powers of e and only the highest order terms are retained. Further, the slowly varying flow restriction is such that the differential equations for pressure tensors and heat fluxes reduce to algebraic transport relations for these properties. A. Plasma Systems in which E~ 1. To obtain reduced forms of the collision transfer terms, use is made of the assumptions, (a) e <<~1 i b e -=ee( (V-l) Te <4~ 1 33

34 which are anticipated to be valid for most systems. Further, for order of magnitude purposes, it is assumed that, 3/2 e (a) e Qk(a) Q k Oe 5 p p.. (b) pe = Lj ij 2pe i 3/2Q (V-2) (c) Qi A L << 'Qe (c) ' 5p o k 0* (d) P ij i 2pil ij The validity of these latter assumptions may be verified in retrospect. Representing the collision transfer terms in powers of 6, ignoring terms of higher 3 order than ~, and utilizing the above assumptions; it may be shown that the collision transfer terms may be put in tne form (ak Sk 3 10 k 5 kj 2 k Qi j k e 2 2 (b) Re P 'e,e e 3/2e (1 +Qi i -3 +Q iei (V-3) i e e 2 e e

35 e (d) L - kfi 3/2 3e Xe -(2) 2! 2 J, ei 0 _.qei -t(2)e 4 ei n 5 A +e F Z2 n (2) (2) 7 ILee 12 ~jl) Q e pie+(2 - -I te( ei e (e) Lki / kQe/2f (2 e2 me (-3 i -T o 3 (2) eT. e e el-4 K ) e e + 4 e +~(3M. m5 (2) ~ei e -2i( - 20 J1(1) ~ei 13e k 4 me +- (3i m (2) Q'e e ei - e ei 131 0 t +4 1+ 3 + -(1+ ' Z~n e _ _ m e i n m i3/2 0 e3/2 0 (2) 11) ei 1 I

36 (2) e 3 7 e4 ei 2 e (f) Mk - 7 e t k 2 3 15 15 (1) ) k e e A.ei e(2) (2) 4 ei 2 e e 4 n ee 5.1 15 kj 3 Z ni e) ei ei (V-3) cont. (2) (2) 26 ei 21 e' 4 2 lei e e + (-+ + k + 5 5 (1) ) i k ei e 2 i e.3/2 I(2) / v i 0 J 4 Z n m fii,i (g) M =- 5+ m e K e3 m 3/2 n() m In the above relations and those to follow, it has been found convenient to introee duce the electron mean free path Xe, and collision time s c, defined by e( (k Te)2 (a) X = - n2 e4 () (V-4) e re e1/2 (b) 7= 3 With the exception of ion heat flux transfer, terms have been retained to order 6 3 The ion heat flux is sufficiently small relative to electron heat flux that inclusion of terms of order f3 is unnecessary. To within the limits imposed by (IV-8, a, b)

37 1/2 e _ and (V-1, a), 86 = - e jk; hence, the collision transfer terms could n-e equally as well be expressed in terms of the conduction current instead of 6. k' With regard to the momentum transfer term S, it is of interest to note that, in addition to the usual heat flux coupling (-Qe), (see, for example, 62 reference 6), there is higher order heat flux coupling ( 2 Qk + Q.% j k ), and 2 e2 e pressure tensor coupling (- 5 P e' ). Attention will be given to the influence of this coupling on conductivity later in this section. To obtain transport relations for the traceless stress and heat flux of electrons, attention is restricted to systems in which space and time variations are small. Explicitly, it is required that << < 1; 1; e 1; (V-5) 4e (b) _ _ ~3 r e e where 7' and X are the electron collision time and mean free path respectively; and t and r are respectively the characteristic time and characteristic distance for macroscopic changes in the system. For electron densities in the range 10 to 10 particles/cm and a temperature of 10 K (i. e. 8. 6 ev), the mean free path ranges from 8 x 10 to 8 cm., and the electron collision time ranges from 6 x 10 to 6 x 10 seconds. The limitation on characteristic distance is seen to be too severe for many laboratory plasmas.

38 With these restrictions, the differential equations for electron traceless stress and heat flux (IV-2, d, e) reduce to coupled algebraic equations. These equations may be decoupled, however, since the traceless stress to second order requires the heat flux to first order only in 6, and the latter is independent of the stress. Hence, the procedure is to solve for the heat flux to first order in. The first order heat flux is then substituted into the traceless stress equation and the traceless stress determined to second order in E. Finally, the second order stress is substituted into the heat flux equation, and the heat flux determined to third order in E. The relations which result are exhibited below. The heat flux, to first order in $, is given by e e p e e e (V-6) Qr 3/2 ri (V-6) Be where e 15 e (2) (v-7) 8 n ei (1+ 1) 13JZ- z2 n ei With the choice of a local coordinate system in which the magnetic field is in the z-direction, ) () o (a) a 2i) ( 0 (V-8) (1) 1 \ 0 0 1 (b) X 2 2 1 + e 71) (1)

39 e me (2) (1) (c) X = 2 2 1+ e) (V-8) cont. T() is the collision time for electron heat flux transfer, and Ielis the cyclotron frequency for electrons: e e e (a) e =2 7e (1) (V-9) e eH (b) o= - mec Representation of the above expression for heat flux in the local coordinate system specified does not imply that the direction of the magnetic field is known. The traceless stress, to second order in, is given by e e e A pee pr -p2 H:0T rs 2 (2) rmn ms n rs (V-10) A where H is a unit vector in the direction of the magnetic field, and 7(2 is the n (2) collision time for electron stress transfer given by, e =5 0e Te (V-1) (2) 2 The dimensionless quantities 0 and T are given by, rs (a) e (a) (2) e (V-12) ei n ee i 2 e ei el

40 (1) ~.+a(2) (a. + ri (b) T = rs e e e a.) 1 s ri i s (V-12) cont. where (2) (1) 5 ei_ (a) a =( - 3.(1) ei (2) b (2) = (- 1 + 6 (b) 5 ei ei (V-13) With the choice of a local coordinate system in which the magnetic field is in the z-direction and ~k in the x-z plane, the traceless stress components may be written, (a) e = e e (l) e zz ~b(1) ~z ~z - (b) P 2 e b(3) xx 2 e e (2) (2) ) -e pe 0e ( yy P 2 2 1 + 4 we 2) (d) e =pe pe e xy yx e2 e2 1 + 4 (2) bb(2) e Ee x x le,e b(4) e e~ e e b(4) e ~e x + z z X X Z Z) (V-14) {b(6) e Ee X X

41 (e) e = = P= 0 2 l ' xz zx 2 2 b x z + e e 1+((2) e Oe ((V-14) (f ge = P b(8) e e zy yz 2 2 w )'-Z + oe e (+( ( T(2) where, (a) b(1) 2 (a (1) + a(2) 3 (2) 1 (1) a(2) W (1) (b) b( )- (a + a) 3 ( (3) 2 a(1) (2) (1) e e (2) + (2) e e (c) b -(a + ao Xs e r(1 +) - )-a a w 7 3 (2) )(2) 2 2 (d) be 4)=(a +as(e 2))(1+4we we (V-15) (e) b (a +a X )(l - 2 T ) - a W Y 3 (2) (2) (ge) 1) (2) 7u(2) e+1) 1 a(2 2) 2e e (h ) (1) 1 (2) (1)+ ee 1 (2) (2) -h) b La - a X+a ) w 3(2)+ a For the class of systems considered here, there does not appear to be a reliable expression for the electron stress tensor with wlich this result could be compared. With regard to the result exhibited by Herdan and Liley, the restrictions are in one respect less severe in that the influence of space variations

42 of plasma flow velocity is retained and the stress tensor is then a function of this variation. On the other hand, the influence of the space variation of heat flux has been ignored, which implies that the difference between electron flow velocity and the ion flow velocity is much less than the plasma flow velocity. (The latter assertion may be verified by substituting the relation which they obtain for electron heat flux into their equations for electron stress). In view of the difference in the classes of systems considered, a comparison of results is not meaningful. If, in the reduction of the transport equations, we were to relieve the restriction on space variations, that is, take X /r to be of the order of C; an expression for the electron stress could be exhibited which would reflect the influence of both the space variation of electron flow velocity and the space variation of electron heat flux. However, this relation would be very complex by virtue of the tensorial coefficient in the expression for heat flux. The heat flux, to third order in ~, is given by e aTe ~ e ee -6 +e re e (-16) r 3/2 ri i reiij (V e 1 where e (a) e =5 e T (T (2) (2) () e e [( 43 8 ei 2 33 4 ei e e +30 15 2 ) ) 2525 +((1) j l ei ei (V-17).(2) Oe Oe (2) (ce r. 1 4 ei ij 2 eSe ik e +33 12 ei 2 e 5 5 f(1 ) ) ee kjp 50 25 (1) ij ei ei

43 Expressed in the notation used here, Herdan and Liley obtain, e i Hence, the latter two terms in (V-16) constitute a correction to the Herdan and Liley result. It may be noted that these terms are of order 5. The thermal conductivity is given by, e' knXe e-19) 3e - (V-19) 5 1/2 e Attention is now directed to the current equation (generalized Ohm's law). To third order in, the equation may be written, ape 1 ap 11 ee ark +(Ek+ km H) E H anec kk m e e, kTe e ki ar. Y7 kk 5 kkp (v20 ^ (V-20) 0 3 32 3 ee e e 3 3 e 2 e + I- E - i a u E +- ~ K i ij i i kb 5 10 wite e o m 3 en e

44 The use of both and jk is a matter of convenience. As pointed out previously, 1/2 k k nee k f-ee Jk and ~ =J~ With the exception of the terms inside the bracket this relation is the same as that exhibited by Herdan and Liley. The terms inside the bracket constitute correction terms of third order in,. To first order in E, the resistivity is, of course, the same as that given by Herdan and Liley. To exhibit an explicit relationship for the resistivity, to third order in ~, in the presence of a magnetic field, is a formidable task as a result of the complex coupling with the electron stress tensor. However, in the absence of a magnetic field, considerable simplification results. For this special case, the current equation reduces to, ape e kTe + +E. e ar. e a r. 1 n e 1 1 1 3 e 3 36 e 201 e 14 e- 6 ee 5 - - (-~+ o 25 2-50 -- -25 (V-22) 22 e2 e 4 e3 ee.2 + 375 + 125e ) Ji It may be noted that the coefficient of the current vector is a function of the current. The resistivity (assuming this nomenclature appropriate) is given by 3 =e 30 36 e -201 e 2 4 e 6 e 0e 0 25 250 - 25 (V-23) 22 e2 e 3 e 2) 375 1254 I

45 By comparison, Herdan and Liley obtain, (1 ) (V-24) whereas Chen, who has approximated the distribution functions by using only the symmetric parts of the distribution functions used herein (i. e. the Maxwellian portion), obtains ~(1 --. 2). (V-25) For a numerical comparison for the cases where Z = 1 and where Z is large we find, Herdan and Liley Chen Results Herein Z=i ~o (.53) ~(1- 3 2) ~(.53-.44 &2) Z (large) Y~ (.31) ~(- (.31-.12~2) (V-26) The Chen result differs from the result herein by a factor of from two to three (depending on Z), even for systems in which 2<- ~1. The source of this difference is the use of the Maxwellian part only of the distribution functions. As may be seen herein, the heat flux coupling in the momentum transfer, which gives 3e o rise to the term (- jk) in (V-22), is very significant even to first order in As pointed out previously, to third order in ~, additional heat flux coupling and stress coupling becomes evident. This coupling accounts for the corrections of order E3 to the results exhibited by Herdan and Liley. If for example, 6 = 1/3, the correction ranges from 13 per cent for large Z to 27 per cent for Z = 1. The correction is such that the resistance is reduced.

46 2 B. Plasma Systems in which S >> 1. To reduce the general magnetohydrodynamic equations of Section IV to 2 a manageable form for systems in which 2 >>1, we again restrict attention to systems with slowly varying flows. Explicitly, it is required that (a) {(m )2 ( )3/21 << mi)2 ( 3/2 1 - i << 1 Z (V-27)w e e Im2 3/2 1;7 _~1;ni Z2 <. m 3 Z riw r rw [Since relativistic effects have not been accounted for, a limitation is imposed on the difference in constituent flow velocity. If, for example, we require that, | w w w | e c/5, where c is the velocity of light; then for kT = 100 ev., 1/2 e -i e /2t (bpe Iwk wkl..3 ~l w 15 The ~ dependent functions in the collision transfer expressions (III-24) are expanded in inverse powers of ~, and only the highest order terms in the collision transfer expressions are retained. After considerable manipulation the traceless stress equation for particles of type a reduces to the form, 0~ a ia / -2a =-2p - (V-29) rs n (3) rmn ms n VIr v -24)

47 where 3a is the collision time for traceless stress transfer, for particles of (3) type a given by, 2 a m a 5 (3) 8/J a ea i/a (2) n e, -L. OfO (V-29) and, is the viscosity coefficient for a type particles, given by, a a a p = P j (3) (V-30) Again choosing a coordinate system in which the magnetic field is in the Z-direction, the components of P may be written, rs (a): =-2 u zz zz 2a (b) = - 2 xx 1+4 (3) 2 2 n + - (u +U ) 4i&i Y +U 2W T xx 2 xx yy (3) xy (3 ()a 2 2c (cu) r 12 2 I3) (c) = - Uyy+ (U + U ) 4 73 - 2 yy a a yy 2 xx yy ( 3 ) 1+4u r (3) (V-31) C1 - - xy y2+2 xy 2 yy xx (3) 1+4 7(3) of &a (e) - xz zx 2 2 +w (3) U +a U L zx (3) zyj

48 (f) P = zy yz 2 2 1 + a a (3) Uy zy a 3) - (3) u zx j 9X (V-31) cont. where, awC v,,. 1 Ui ari The transport relation for heat flux for particles of type a reduces to the form, (V-32) where a ar f f l l! i I I 11 \r a a Ta Q =a <-a 2 2 / r ri,i 1 / - (4) a a 1+w 1 aTa ar. 1 w a (4) 2 2 a + a a T a (4) 2 2 i + a 1+ 4 (4) 1 (V-33) 1 1 /

49 in a local coordinate system in which the z direction is taken to be along the magnetic field Hk. a(4) is the collision time for heat transfer of the a type k' (4) particles, and V is the thermal conductivity: 2 2 a a (a) I' n In (a) - 4 3/2 (4) 16a a (2) n ea (V-35) b a 5 kp a 2 m (4) The generalized Ohm's law equation for conduction current reduces to, e D w~ aw + j ++j - e e i + J k e 2 e2 Dt k r. i ar. n e ar2 i k ne i i ne (V-36) o 1H (2) +(Ek+l<kmW0 H ).. LH - ]k (k c kjmm e k m m n k n ec (2) where the resistivity 1 is given by, (2) n Ze (1) 1 (V-37) 7 (41r ne me i (V-37) m 8 It may be noted that the expressions for stress and heat flux have the same form as would be obtained for a one constituent gas. (See, for example, sections 18. 43 and 18.44 of Reference 1). That is, for systems in which tl1, the unlike particle collisions are ineffective relative to like particle collisions in changing

50 either the stress or the heat flux. This decoupling of the plasma constituents results from the decrease in the unlike-particle interaction cross section as the difference in constituent velocity, and hence the relative velocity of interacting particles, becomes large. With regard to momentum transfer, the like particle collisions have no influence. Hence, although small, the influence of unlike particle collisions must be retained. The resistivity which reflects the influence of unlike particle collisions is seen to be proportional to 1/ 3. Although the above expressions for stress and heat flux constitute new results' the expression for resistivity has been previously obtained by Driecer(13) (7) and by Burgers

VI. SUMMARY Grad 13-moment approximation distribution functions (expressions II-12) have been assumed to provide adequate velocity space solutions for the Boltzmann equations of the plasma constituents. These solutions are anticipated to have validity over a wide class of systems which includes those in which the difference in constituent flow velocity is appreciable relative to the electron thermal velocity, in addition to those close to equilibrium. With the use of these distribution functions it has been found possible to derive generalized magnetohydrodynamic equations, which coupled with Maxwell's equations, constitute a closed set [Equations (IV-4, 5, 6,2c, 2d, 2e) and (11-2) 1. In their general form, these equations are so extremely complicated that their usefulness is obscure. Hence, to demonstrate their usefulness, transport relations (V-14) and (V-16)] to third order in ~ (ratio of the difference in constituent flow velocity to random electron velocity) have been exhibited for the electron heat flux and stress, for systems with slowly varying flows in which ~ is less than one, [restrictions (V-5)]. For this class of systems, the generalized Ohm's law is also exhibited [equation (V-20)] and the resistivity is determined to third order in E for systems in which the magnetic field is zero [relation (V-23)J. With the exception of the relation for electron stress, the results are found to be in agreement with those of Herdan and Liley to order E, giving corrections to 3 order ~. With regard to electron stress, there does not appear to be a reliable expression for the class of systems under consideration, with which to compare. The numerical comparison of the resistivity for the case where the magnetic 51

52 field is zero [(V-26)] indicates that, for systems in which t 2 1/3, the resistivity is less than that exhibited by Herdan and Liley by from 13 per cent to 27 per cent, depending on Z. Further, the inadequacy of the Maxwellian part only of distribution (11-12), which has been used by some authors (see, for example, references (12) and (13)), has been made evident. Simplified transport relations [(V-31), (V-33), and (V-36)] have also been exhibited for systems with slowly varying flows in which 2 2> 1 [restrictions (V-27). The expressions for heat flux and stress are found to have the same form as those for a one constituent gas. The resistivity [relation (V-37)iis seen to vary (12) (7) inversely with the current cubed as previously noted by Driecer and Burgers.

54 Hence, integral (A-l, a) becomes, 4iT vr f = 2 v 1 + (d(S) 2 v x sin rk ee r sin k ^j ^ - -^4.<2 (A-4) The tensor (v v rk r~ - v v ) may be represented by, rk r 2.2 2 -v sin x cos 0 r 2 2 -v sin x cos 0 sin 0 r 2 -v cos x sin x cos 0 r 2.2 -v sin x cos 0 sin 0 r 22 22 2 -v sin x sin r -v cos x sin x sin 0 r 2 -v cos x sin x cos 0 r 2 -v cos x sin x sin 0 r 2 2 V (1 - cos x) (A-5) (A-5) which on integration over 2. 2 -v sin x r d 0 becomes, 0 0 0 -v sin x 0 (A-6) r- r 22 2 2 2 O 0 3 v sinx - v sin x r r In index notation, with the direction of v unspecified, (A-6) may be written r 2 _x 2 2x 2x 2 3r sin x(v v - v )= 12 sin cos (v v (A-7) rkr r 3 r 2 2 r 3 r Hence, expression (A-l, b) may be written, (v 2 x cos 12 (v v - x v 2 d (sin X (A-8) rkr )3 r f sin 2 2

55 1 Finally, on integration over the range from 1 to... on I 2 am ho2 4 xi +( - sin x we obtain, \ ' ( ) vr 2 eae e r m hh 2 4 I ) v r e e 2m h 2 4 6ir(vrv - 2 l v rk r 3 e e (A-9) a 1Y e+ ( ee 4 v r

APPENDIX A INTEGRATIONS OVER ANGLES OF SCATTER. The integrals over angles of scatter were found in section II to be of the form, ( d (sin ) (a ) (v - ) d0 (A-l) rk rk sin3 2 and d(sinx (b) /(v v'- V V ) 2 r k r, rk rp_ 3 x sin - 2 The limits of integration are from 0 to 2ir on 0; and 1 to m 2 4 x I ' v on sin +( r 2 ee Choosing a coordinate system in which v is in the z-direction, the rk vector (v - vr ) may be represented by k k - v sin x cos - v sin x sin 0 (A-2) v (1 - cos x) r which on integration over d0 becomes, 2x 27r v (1 - cos x) =47 v sin - (A-3) rk rk 2 53

APPENDIX B INTEGRATION OVER VELOCITY SPACES All terms in the collision transport expressions were found in section II to involve integrals of the form, r [G2+ (_-9 )2 1 2 2 GilGi2' GingJlgj2''gj m 3 3 e 3 d Gd 3g. 47r(2r) 3/2 (B-l) Before proceedingto evaluate the required integrals, it is found convenient to introduce a special notation. The third order tensor { c ijk is formed by taking the sum of all distinct products of the Kronecker delta and the vector ~ which arise on permutation of the subscripts, each term occuring r once. That is, i Jd ik=, G + 6+c ~. (B-2) - ijk j k ik jk Similarly, 1'4A = 5 +& c ( + s +9 <F8 + K + ~.. ijkf k ik J k k i i k i j k k (B-3) Other tensorial expressions of this type, tCJ Iijk. k 1S 6E ijkm' etc., are formed in an analogous manner. Turning attention to integrals (B-l), the G-space integrations have been carried out by Grad 3are tabulated below. carried out by Grad '. Those required are tabulated below. 56

57 (B-3) (a) 1 (2r)3/2 (b) 132 (27r)3/2 fe d3G=l e-G2/2 i G.G. d G =C. 1 j iJ (c) 1 (2r )3/2 (2r ) (d) -1 (2r) 3/2 (e) 1 (27)3/2 1 (f)32 (2~r)3/2 J e C -G2/2 e -G2/2 e -G2/2 e e G2 d 3G = 3 G.G. GkG d 3G= c \ 1 j k G2G.G. d G = 5 d.. i j iJ G d3 G - 15 To facilitate the g-space integrations, we define, 1 (a) g ' ' 'g lg i2' 1 47 - (9)2 d3 gilgi2"' gi g 3 g (B -4) (b) Ik- 4T e gk d3 g3 and observe that,

58 n - _2/2 a I. gilgi2.. gi. e In i1 ~i2. ai in_1 1 '2 n-l (B-5) That is, all of the g-space integrals of interest may be expressed in terms of derivatives of the integral Ik with respect to the vector. To evaluate Ik, a change of variables is made from gk to xk=gk- 6k' which transforms Ik to the form e 2/2 -xk2 +~ ) 3 I- e e dx (B-6) I ~ I Choosing a coordinate system in which the vector Ek is in the z-direction, and noting that the components perpendicular to &k are zero, we write, k 2 /2 Ik= -e -x2/2 -x/2 (x cos + )x2 sine dv de dx (E k e ~2co 3/2 V ~ ] (x2+2x E cos 0+2) (B-7) where (x,, v) are spherical coordinate variables. Integrating over v from 0 to 2r, and letting u = -cos 0, we obtain, 21 /2(- x2 + ) = - /e e x2 3/2x d.p dx k 2 e (x2-2xb4+ ) 2) ck.2/2t x2/2 F = k e dx e x1 - d p Z e dx e x (x2x +2)/2 0 -1 (B-8)

59 Expressing the integrand in terms of associated Legendre polynomials, the ji integration is easily carried out as follows,, 1 1) 5 V()+ d= t 1 I (x2-2xEu+E 2) 1 -14 (P) +- d2=1; K.. x "2L (B-9) x >E The integral k may now be written, E -!22 2 1=k 2 k e x e 0 dx (B-10) As expressed by relation (B-5), the required g-space integrations may be obtained by differentiations of Ik with respect to the vector E. The operations are straightforward, though lengthy, hence it will suffice to simply tabulate the required expressions. Defining, -_x2/ 12 (a) a / e x2dx (b) e - 2/2 (C) E 2 ) (d) p(2) (1- ( 2 - L (2) 1 2 64 (B-1l)

60 (B-11) (cont'd) (3)- 1 8 35 45 105- ) 14 -(1- + - +(l- -2 -(e) Y - 4 — 2 4 -..I' — -7 E II,4i I — 6 (4) 1 11 315 M + U )ill:= -2 ( 1 - -2 :. 4 - - 4' 1 -<1 12 +90 420 + ~. A -. ~~6 7w) (g) (32 (h) +4(6) 2 _3+3 the g-space integrals required are given by, (a) gk k' ~l (b) 9j + (B-12) (C) gggk (1) =Z S ijk' +4"C- a 0,40 ) ( 2) --- 1 (e) 9 i gj gk g,,,' =,4. I &, F I ijke ij k~t ' i ' (f sp#. ~(2 2)k -' 74(2)+)i3)F2

61 (B-12) (c ont Id) i + kj ij~k<~m ()= (2) + 14EY( 3) (4)(2) 1 /4~~ 45),' ~6)r

APPENDIX C COMMENTS WITH REGARD TO DEFINITIONS OF TEMPERATURE, STRESSES, AND HEAT FLUXES IN PLASMA SYSTEMS. Definitions of constituent temperature, stress, and heat flux relative to the plasma flow velocity, and definitions relative to the constituent flow velocity, are both in common usage. Hence, some comments with regard to the relationships between these definitions seem appropriate. We let, (a) vk wk + ck =w + + uk w + u (particle velocity in laboratory coordinate system) o p ca (b) wk= i P wk (plasma flow velocity) a= e,i (c) wk- =a f Vk d v (constituent flow velocity) k (c) wk, k n -a 1 (d) c - / f c d c (constituent flow velocity relative to n plasma flow velocity). In the development herein, the constituent temperature, stress, and heat flux are defined relative to the constituent flow velocity. That is, ( 3 a, a 1 a /a23 (a) 3n kT = m f u du 2) 2 (b) P.. =m. u. u d u (C-2) 1 1 (c) Qa 1 m f2 u d3U r 2 r 62

63 These definitions, brought forward from section I, are consistent with those (11) of Spitzer By contrast, Chapman and Cowling) and Herdan and Liley) define these properties relative to the plasma flow velocity. That is, 3 a' 1 23 2 2 I/ (b) a.. d3 (b) P. =m c.c. dc (C-3) ii 1, _! la fi 2 d3 (c) maa Focc d c rc 2 r It may easily be shown that definitions (C-5) and definitions (C-6) are related by 3 a a' 3 a a 1 a a -2 (a) n kT = n kT + n m c 2 2 2 (b) P. =P.. +n m. (C-4) 1J 1J 1 ] a' a 1 a a- 2 3 a kTa) (c) Q = Q +- n m + n k a + P a r r 2 r 2 r rs s 2 e i e i For systems in which = e ( w )(w - w ) 1, these relations reduce to (a) T = T at a (b) P i Pij (C-5) a a 5 ar Q (c) Q =Qr +- p r r 2 r

BIBLIOGRAPHY 1. S. Chapman and T. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, New York 1952. 2. H. Grad, "On the Kinetic Theory of Rarefied Gases", Comm. on Pure and Appl. Math, Vol. II, p. 331, 1949. 3. H. Grad, "Notes on N-dimensional Hermite Polynomials", Comm. on Pure and Appl. Math., Vol. II., p. 325, 1949. 4. I. Kolodner, Doctoral Dissertation, New York University, 1950. 5. W. Marshall, "Kinetic Theory of an Ionized Gas, Part III". Atomic Energy Research Establishment, AERE, T/R-2419, November 1957. (Reprinted March 1960). 6. I. Kolodner, "Moment Description of Gas Mixtures - I", Institute of Mathematical Sciences, New York University, NYU-7980, September, 1957. 7. J. Burgers, "Applications of Transfer Equations", Inst. for Fluid Dynamics and Applied Mathematics, University of Maryland, Technical Notes BN124 a/b, May 1958. 8. R. Herdan and B. Liley, "Dynamical Equations and Transport Relationships for a Thermal Plasma ", Rev.of Mod.Phys Vol. 32, No. 4, p. 731, October 1960. 9. F. Clauser, Plasma Dynamics, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960. 10. W. Everett, "Velocity Space Solutions of the Boltzmann Equations for a Plasma with Internally Generated Electric and Magnetic Field", University of Michigan Radiation Laboratory Memorandum, September 1960. 11. L. Spitzer, Jr., Physics of Fully Ionized Gases, Interscience Publishers, Inc., New York, 1956. 64

65 12. K. Chen, "Non-Linear Electrical Conductivity of a Fully Ionized Gas", University of Michigan Radiation Laboratory Memorandum, May 1961. 13. H. Driecer, "Electron and Ion Runaway in a Fully Ionized Gas - I" Phys. Rev., Vol. 115, No. 2, July 1959.