.. 'K.‘ . . J . . .. ... . t I h ; , . ‘ 1 . I‘ . IQ | u . ‘ ‘ A. l \.n.‘hl1lli‘| ILJI 1.11. 'l“ n 2': ill”! ‘ 1.4 ABSTRACT GENERALIZATION OF GRAD'5 THIRTEEN-MOMENT METHOD TO MAGNETOGASDYNAMICS BY Lawrence Tsi-kong Wong The present work is primarily concerned with the general- ization of thirteen-moment method deve10ped by Grad valid in neutral gases to one-component charged gases. A distribution function is defined as the mass density of particles in a one-component, homogeneous, uniformly charged gas. Several moments of distribution function are defined as symmetrical tensors. The Boltzmann equation is multiplied by a summational invariant and integrated over the velocity Space. By setting the summational invariant to equal to unity, velocity and velocity square, the Boltzmann equation yields the continuity, momentum and energy equations respectively. Using Hermite poly- nomial approximation, the equations of time, physical and velocity space variation of the second and third order tensors are obtained from the Boltzmann equation. These equations, along with the con- servation equations constitute the system of thirteen-moment equa- tions. By considering a one-dimensional heat flow in a gas at rest and a plane Couette flow, the thermal conductivity and the coefficient of viscosity are deduced from the system of thirteen- moment equations. GENERALIZATION OF GRAD'S THIRTEEN-MOMENT METHOD TO MAGNETOGASDYNAMICS By Lawrence Tsi-kong'Wong A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR 0F PHIIDSOPHY Department of Mechanical Engineering 1970 To Agnes, my wife ii ACKNOWLEDGEMENTS The author is deeply indebted to his major professor, Dr. Maria Z. v. Krzywoblocki, for his guidance and assistance throughout the course of this study. The author also wishes to thank the other members of his guidance committee for their interest in this work: Dr. George E. Mase, Dr. Mahlon C. Smith, and Dr. David H. Y. Yen. Thanks are due to the Mechanical Engineering Department and the Division of Engineering Research for financial support during graduate study and research. In addition, the author expresses his gratitude to the management of Ford Motor Company for granting him educational leave and scholarship. Special thanks are due to Mr. O. D. Dillman and Mr. B. T. Howes for making these arrangements possible. To his wife Agnes, the author dedicates this work for her understanding and encouragement throughout this study. iii TABLE OF CONTENTS Page ACKNOWIEDGEMENTS iii LIST OF TABLES vi LIST OF FIGURES vii LIST OF APPENDICES viii NOMENCIATURE ix INTRODUCTION 1 Chapter I BRIEF REVIEW OF BOLTZMANN'S EQUATION 2 1.1 Solutions to Boltzmann's Equation in Neutral-Gases OOOOOOOOOUCCOCOODOOOOOO0.0.. 2 1.2 Solutions to Boltzmann's Equation in Ionized Gases OOOOOOOCOOOOOOOOOOOO00...... 4 II SOME DISTRIBUTION FUNCTIONS ................... 6 2.1 General Remarks on Distribution Function . 6 2.2 Discrete Distribution Functions .......... 7 2.3 Continuous Distribution Functions ........ 8 2.4 Distribution Functions in Neutral Gases .. 10 2.5 Distribution Functions in Ionized Gases .. 11 III THE THIRTEEN-MOMENT EQUATIONS FOR ONE- COMmNENT CIiARGED GASES .0..................... 13 3.1 Moments of Distribution Function and Beltzmnn's Equation COOOOOOOOOIOOOOOOOOOO 13 3.2 The Maxwell TranSport Equation ........... 15 3.3 Conservation Equations ................... 18 3.4 Generalized n'th Moment Equation ......... 21 3.5 The Second and Third Moment Equations .... 27 3.6 Third-Order Approximation in Hermite Palynomials OOOOOOOCOOOOOCCIOOOOOCOOOOOOOO 28 3.7 The Twenty-Moment Equations .............. 31 3.8 The Thirteen-Moment Equations ............ 36 iv IV TRANSmRT mEFFICIENTS OOOIOOOOOOOOOOOOOOOOOOOO 4.1 Thermal condUCtiVity OOOOOOOOOOOOOOOOOOOOO 4.2 Coefficient of Viscosity ................. 4.3 Tables and Graphs of TranSport coeffiCients O.CCOOOOOOOOOOOOOO00.0.0.0... LIST OF REFERENCES 42 42 44 46 54 Table Table l a l b Table 1. Table 2. LIST OF TABLES Ratios Between the "Apparent" Coefficient of Viscosity and the Coefficient of Viscosity .... Ratios Between the "Apparent" Thermal Conductivity and the Thermal Conductivity ........ vi Page 47 51 Figure Figure Figure ‘ LIST OF FIGURES Figure l. Ratios Between the "Apparent" Coefficient of Viscosity and the Coefficient of Viscosity ...... Figure 2. Ratios Between the "Apparent" Thermal Conductivity and the Thermal Conductivity ....... Figure B.l Unit Vector a in Spherical Coordinates ........ vii Page 48 52 63 Appendix A. Appendix B. Appendix C. LIST OF APPENDICES sumatiorlal Invariant 0......OOOOOOOOOOOOCOOOOO Collision Integrals ........................... The Equilibrium Solution of the Boltzmann Equation 00....OOOCIOOOCOOOOOOOOOOO0....0...... viii Page 58 59 66 (n) (‘31 out: 9: 0.1 2:21 La NOMENCLATURE Hermite coefficients Magnetic induction Geometric collision parameter Intrinsic velocity = E - 3 Intrinsic velocity = El - 3 Electric charge Electric field Distribution function (mass density) Maxwellian distribution Distribution function (number density) m Mass ratio = 2 _g Magnetic flux density Hermite polynomials Electric current density Collision integral Boltzmann's constant Characteristic dimension 551 Particle acceleration = SE— Mass of charged particle Mass of electron Mass of neutral particle Number density of neutral particles ix pij P.. 1] U #1 <1 Q! 11 '13“ *1 <1 51 C1 Q1 Pressure Stress tensor Second moment of d.f. f Fourth moment of d.f. f Radius in spherical coordinate system Gas constant Heat flow Third moment of d.f. f Time Temperature Mean velocity Relative velocity = § - §1 Dimensionless velocity ='ég: [RT Position vector Unit vector Collision frequency Kronecker delta Azimuthal angle Dielectric coefficient Alternating unit tensor Polar angle Thermal conductivity Mean free path "Apparent" thermal conductivity Coefficient of viscosity "Apparent" coefficient of viscosity Magnetic permeability X d5. d§1 Kinematic viscosity = % Particle velocity Density Excess electric charge (1de dx 123 dg1d§2d§3 Element area = rdrde xi INTRODUCTION In neutral gases, the methods of approximate solution of the Boltzmann equation had been developed by many authors, namely, Maxwell [35], Enskog [12], Hilbert [l9], Chapman [9], Wang Chang and Uhlenbeck [8]. Grad [13] developed the thirteen-moment method with the aim of obtaining phenomenological equations. He derived the equations for the successive moments of the distribution function expressed as a series of Hermitean tensors. In the present work, we extend the use of the thirteen- moment method to an one-component charged gas (i.e., all particles are either electrons or identical ions). Two transport coefficients are obtained from the thirteen-moment equations. Chapter I gives a brief review of approximate solution of Boltzmann's equation in both neutral and ionized gases. In Chapter II, various distribution functions which include continuous dis- tributions, discrete distributions and some distribution functions in plasma dynamics have been collected. Chapter III contains the derivation of the system of twenty-moment equations. With the use of the distribution function expressed in terms of Hermitean tensors, we reduce the system of twenty-moment equations to the system of thirteen-moment equations. In Chapter IV, the thermal conductivity and coefficient of viscosity of the gas are obtained by considering a one-dimensional heat flow and a plane Couette flow reapectively. In addition, the transport coefficients are calculated and plotted for an electron gas. CHAPTER I BRIEF REVIEW OF BOLTZMANN'S EQUATION In most gases, where the departures from the local thermo- dynamic equilibrium are not too large and the flow speed does not exceed Mach number three approximately, the Navier-Stokes equations prove to be valid, according to our present knowledge in gas- dynamics. The Navier-Stokes equations are derivable from the Boltzmann equation, although they are generally derived by con- sidering the elastic deformation in continuum mechanics. In many cases, the Navier-Stokes equations are no longer valid and one must return to the Boltzmann equation to obtain a general solution for the distribution function. 1.1 Solutions to Boltzmann's Equation in Neutral Gases: Before Boltzmann established his integro-differential equation satisfied by the particle velocity distribution function, Maxwell [35] established tranSport equations with the assumption of Maxwellian molecules. He obtained approximate solutions to his equations by means of a method of successive iterations. It was Boltzmann's [2] merit to establish an integro-differential equation which describes the variation with time of the distribution function f, the state of gas, the molecular interactions and the external force. However, Boltzmann did not find a general solution to his equation. Lorentz [33] associated a nonhomogeneous gas with the theory of electrons in a metal and sought a solution of the form f 8 f0 +-vxm(v) to Boltzmann's equation. This method also failed to reach the general solution of the integro-differ- ential equation. Hilbert [l9] pr0posed a solution of Boltzmann's equation by solving a linear integral equation of second kind. His approximate method was obtained purely from mathematical view- point. Chapman [9] calculated the coefficient of viscosity and thermal conductivity by means of second approximation to f. Enskog [12] modified Hilbert's method and obtained general formulas for the viscosity and thermal conductivity in gases. The Enskog- Chapman method is valid only if the mean free paths 'i are small with respect to the characteristic dimension L. Distribution function f was expanded in the form of a power series of 'i/L. Burnett [5] expressed the distribution function in the form of expansions with reapect to the product of Sonine polynomials and spherical tensors. He successfully calcluated the complete second- order approximation. Grad [13] obtained the celebrated thirteen-moment equations for the successive moments of the distribution function which was expressed as a series in Hermitian tensors. It is a very effective method and probably more general than Enskog-Chapman method. Wang Chang and Uhlenbeck [8] treated the case of rarefied gases (§ 2 1) by means of a linearization of Boltzmann's equation. Jaffé [26], Ikenberry and Truesdell [25] developed the theory for highly rarefied gases (é >> 1) by the method of linearizing the Boltzmann's equation. 1.2 Solutions to Boltzmann's Equation in Ionized Gases: Spitzer and Harm [43] solved the Boltzmann equation by direct numerical solution in the absence of magnetic field. Cross [17] considered the problem of plasma oscillations in a static magnetic field. He linearized the Boltzmann equation by neglecting the collision term and assuming f 8 f0 +f1 with f1 << f0. Howard [24] attacked the problem of hydrodynamic properties in electron gas. He obtained two coefficients of viscosity for both shearing and normal stresses by assuming Lorentz forces existed between individual particles. Unfortunately, there are no known experimental method to verify his results. Krzywoblocki [31] applied Howard's results to the problem of boundary layer in electron gas. A different approach to the ionized gases is provided by the Fokker-Planck equation which is derived from the Boltzmann equation. Chandrasekhar [7], Rosenbluth, MacDonald and Judd [39] employed this equation to obtain the transport coefficients. Krzywoblocki and Wadhwa [32] proposed to extend Grad's method to magnetogasdynamics. Kolodner [30], Burgers [4], Herdan and Liley [18], and Yen [48] applied Grad's method for the Boltzmann equation to ionized gases. Kelleher and Everett [29] extended the Grad- Everett method to partially ionized gases. Hochstim [21] expressed the distribution function in the form of Laguerre polynomial expansion. Meador and Staton [36] applied the concept of simultaneous many-body interactions in solving the Boltzmann equation. Shkarofsky [41] used a different approach to solve the Boltzmann collision integral. He employed the concept of the supposition of many successive binary encounters. The results from both models proved to be in good agreement. Marshall [34] used the variational principle for ionized gases. He considered the trial function of two polynomials. Robinson and Bernstein [38] employed a different variational method with a trial function of six polynomials. CHAPTER II SOME DISTRIBUTION FUNCTIONS 2.1 General Remarks on Distribution Function: A random variable (denoted by r.v.) is defined in the usual manner. Namely, a r.v. X is a real valued function whose domain is S, and whose range is a set of real numbers. For every real number x, the set of elementary events 8 for which X(s) s x is an event [11]. The event 8 belongs to a probability set S (i.e., s is an element of S). The distribution function (denoted by d.f.) of a r.v. X is defined by [47] Fx(x) = P[x s. x] , (2.1.1) for every real number x. This d.f. Fx(x) satisfies the follow- ing conditions: (i) Fx(xl) s Fx(x2), if x1 < x2 ; (2.1.2) (ii) 1im.Fx(x) = l ; (2.1.3) xqa and (iii) iimaFXQ) = O . (2.1.4) There are two kinds of distributions known as the discrete and continuous kinds. For the discrete kind, the total mass of the distribution is concentrated in discrete mass points. Thus, the discrete d.f. is given by Fx(x) = 2 pi . (2.1.5) xiix F (x+h)-Fx(x) If the distribution is continuous, the ratio X h represents the mean density within the interval (x, x+h). The derivative Fx(x,x+h)-Fx(h) h 9 Fi(x) = f(x) = lim haO (2.1.6) if it exist, gives the probability density or frequency function f(x), and the continuous d.f. is given by x 13,00 =31” f(c)dc . (2.1.7) Absolutely continuous Fx(x) is a necessary and sufficient con- dition for the existence of the frequency function f(x). Since Fx(x) is monotone non-decreasing, the frequency function is non- negative and fin f(x)dx - 1 . (2.1.8) We may also be interested in studying simultaneously several r.v.‘s X1,X2,...,Xn of the distribution. In such a circumstance, we can choose points of the n-dimensional space to represent all possible values of the r.v.‘s. The definitions of distribution and frequency functions are defined in the same manner. 2.2 Discrete Distribution Functions: The Binomial Distribution. A trial of a random experiment with the outcome either "success" or "failure" is called a Bernoulli trial. Let us consider a sequence of n Bernoulli trials, where the probability of 8 (success) in each trial is p, and O < p < l. There are (a) ways of selecting k trials at which S occurs. Thus the discrete distribution Fx(x) is written as x Fx = [2:] (1:)Pk(1-p)m-k . (2-2-1) k=0 for every real number k. The Poisson Distribution. The Poisson distribution may be obtained as a limit of the binomial distribution. Let Xfi denote the number of successes that occur in the n Bernoulli trials, and let pn be the probability of success. Then the d.f. is [x] "A x Fx (x) = z: 5—2;}- , x = o,1,2,..., (2.2.2) n x=0 where A = npn. 2.3 Continuous Distribution Functions: The Gamma Function. Its density function has the follow- ing prOperty 1 xae-x/B f(x) = I‘ (0+1) 9““ for x > 0 ; (2.3.1) o forxéO, provided a > -1 and B > O. The Normal Distribution. Many references cite this dis- tribution as the Gaussian, Laplace or bell-shaped distribution. The d.f. is defined by the relation: 2 Fx(x) - -L_ J"; exp(- §—)dc . (2.3.2) ¢Ch1 The corresponding normal frequency function is 2 -5. fx(x) = é; e 2 . (2.3.3) /2n 2 . 2 The x Distribution. The x frequency is r 9. 1 a f ( ) - 1 2- -2 f o - x2 x < 2. x e or x > , 22mg) (2.3.4) k 0 for x § 0 . The correSponding x2 d.f. is ( .r1_1 _1:. F2(x)= 1 j'xtz ezdt for x>O; n 0 x 2m 2 F(§) (2.3.5) Lo forxso. The parameter n is often denoted as the number of degrees of freedom in the distribution. Fisher's z-Distribution. Two independent r.v.'s X and Y have a xz-distribution with m and n degrees of freedom, respectively. The z-distribution with (m,n) degrees of freedom is X t ‘77 10W“ f°r ”03 ]F('2')1"(§) '2— FX’Y(x) - (t+1) (2.3.6) 0 for x § 0 . K Student's Distribution. Again we consider two independent r.v.‘s. X and Y. The distribution of X being N(0,l) and “1.12:. 4h 3 5|... _ My. 1") .‘ 10 2 the distribution of Y being x -distribution with n degrees of freedom. Thus the Student's d.f. with n degrees of freedom is expressed as n+1 F(--) . 1 2 dt Fx,Y(x) _ n I: ———-n+1 , (2.3.7) fun“? .27 (1+?) for all x. 2.4 Distribution Functions in Neutral Gases: Maxwell [35] proposed that the d.f. f be expressed as f = f0(1-+F) , (2.4.1) where fo is the solution of equilibrium state and F is a rational function of particle velocity. Hilbert [19] and Enskog [12] used an asymptotic expansion in a power series of a small parameter s for d.f. f (0) f + 13(1) 6 + 61‘” +..... (2.4.2) f3 f(o) <1) (2) where ,f ,... are is locally Maxwellian, and f obtained by solving a series of integral equations derived from the Boltzmann equation. Burnett [5] introduced the d.f. f expanded in an in- finite Sum of Sonine polynomials S's. The d.f. f takes the form (0) (p) 2 (p) 2 f = S + S . . rm {2 Ap ,5 (e > 2 Yw was )1 (2 4 3) p to where Y are general spherical functions. LP fil’U" ll! (1... p u s. P- 3.3.... v Ada View? 11 2.5 Distribution Functions in Ionized Gases: Considering the forces acting on electrons by electric and magnetic fields, the Boltzmann equation is 5-: af_ 2 a_f_ _ Bf. at + 51 3x1 + m(Ei + eijkéjBk)agi (at)coll. . (2,5,1) The d.f. f could be expressed as f = £(°) + ¢(§i) , (2.5.2) where f(o) is an isotropic distribution, and ¢<§i) is a small perturbation which causes f to be anisotrOpic. The Maxwellian Distribution. For a weakly ionized plasma, the d.f. f(o) is often assumed as Maxwellian: (0) __L___ c2 (2nRT) The Margenau Distribution. For a plasma in an alternating electric field, the d.f. f(o) is known as the Margenau distribu- tion [23]: f(°) = 0 exp if: “‘ng 2 . (2.5.4) 2 2 3Gm(r fin ) The Druyvesteyn Distribution. For a plasma with a constant cross section under the influence of a strong direct current or low frequency electric field, the d.f. f(o) is known as the Druyvesteyn Distribution [23]: f(0) = C exp(-3Gm?N2Q2Cé/4e2A2) . (2.5.5) N 12 The approach to the distribution function in terms of Hermite polynomials, used by Grad, will be thoroughly discussed and used in the present work. CHAPTER III THE THIRTEEN-MOMENT EQUATIONS FOR ONE-COMPONENT, CHARGED GASES 3.1 Moments of Distribution Function and Boltzmann's Equation: In a one-component, homogeneous, uniformly charged gas, the d.f. F which is defined as the number density of charged particle, is a function of seven variables, namely, velocity g1, position xi and time t. The d.f. f is defined as the mass density f®£o=w@&o . aLn where m is the constant mass of charged particle. The moments of d.f. f with respect to velocity g1 and the intrinsic velocity are defined as follows: Zeroth moment: p(§,t) =.ff(E,;,t)d§ . (3.1.2) Mean velocity: U(§,c) = g-J‘Ef d§ . (3.1.3) Intrinsic velocity: 3(§,;,t) = E - U(;,t). (3.1.4) First moment: I? f d§ = O . (3.1.5) Second moment: Pij(§,t) =‘fcicjf d; . (3.1.6) Third moment: 51115;") = J‘eiejekf dg . (3.1.7) Fourth moment: QijkL(§,t) =‘fcicjckcbf d§ . (3.1.8) 13 14 The second, third and fourth moments are symmetrical tensors. Contraction of tensors by using a dummy index and summing over all yields P,, = 3p ; (3.1.9) 11 Sijj = Si ; (3.1.10) Qijkk 3 Q1] , (3.1.11) and Qii = 3q . (3.1.12) The divergenceless tensors can be formed as pij = 1j - 136ij , (3.1.13) together with pii = Q11 = 0 , (3.1.15) where 611 is the Kronecker delta. The Boltzmann equation is taken to be df(§,§,c) a; 31? 3x1 a_f__a§i f = + —+ —- 5—) , ,(3.l.16) dt at axi at agi at at collision where 32—. is the particle velocity §1 and S?‘ is the particle acceleration. Equation (3.1.6) can be written as (B.1) a£+ af_+L_a£.a£ 11 at §i 3x1 1 551 (at)c . (3. . 7) The symbol L denotes the sum of all electromagnetic forces: H- --0 La 94%: -fi (if +3 x E) , (3.1.18) 15 where E and B are independent of 3. Since the gas is assumed to consist of only one kind of particle, it is fully justified that the governing Maxwell's equa- tions are taken in their simplest forms: {7'Xiis3-I-afl 8t ; (3.1.19) # a = _ RE. . v x E at , (3.1.20) 3 x E = 3 - nefi = o , (3.1.21) and 3 - EE = pe . (3.1.22) The equation of conservation of electric charge is EES- * 7 o 3 1 23 at+VJ - (°°) The Ohm's law gives the equation of electric current as 3 = o[E +-ne(fi x E)] + peu . (3.1.24) 3.2 The Maxwell Transport Equation: Let U denote the relative velocity of the two colliding identical particles. Here E, 6' represent the velocities of the first particle before and after the collision, reapectively, and El, Ei the velocities of the second particle. The symbol dw is given as the element of area in the plane passed through the fixed first particle and perpendicular to the relative velocity u [Appendix A]. The notations f - f(E), f' = fdz'), f1 = f(El). fi = f(Ei) are used here. The collision term [Appendix B] is taken as 16 a£=l II_ (at)c In.fu(f f1 ff1)dwd§1 . (3.2.1) Thus, equation (3.1.17) becomes +§ +L &f__=_].-.J"U(f'f' m a: 1 aii 1 551 - ff1)dwd§1 . (3.2.2) 1 Multiplying (3.2.2) by a summational invariant m(§) [Appendix A] and integrating over the entire velocity space with the integration limit from 4m to +m, we have wad§+ 11512—51: d§+jchigEId§=o , (3.2.3) where the collision integral vanishes [Appendix B]. The first term of equation (3.2.3) could be written as ' BE ii. - ET .jo at d§= f“ (of)d§ If at dg . (3.2.4) Similarly, the second term of equation (3.2.3) is hi. 3 e__ _ 21.. Isfii 5x1 d5 faxi(¢§if)d§ If axi(aging . (3.2.5) Since ¢(§) and gi are independent of xi, equation (3.2.5) becomes BE. g e__ fqgi 5x1 dg faxi (mgif)d§ . (3.2.6) The third term of equation (3.2.3) could also be written as . - a__ joLiL 351 d; I? (oLif)dg [f agi(oLi)d§ . (3.2.7) The first term of the right hand side of (3.2.7) is ISE:'(¢Lif)d§ = oLi£\f: . (3.2.8) 17 As the velocity tends to infinity, d.f. f tends to zero faster than any velocity-dependent function, because of the exponential character of d.f. f. Hence chiflz = o . (3.2.9) The second term of the right hand side of (3.2.7) could be split into two terms: 5L L. 3.99.. 1 - f L. d = - fL, d - f --'d . 3.2.10 f 85169,) e 1851 e “’85, 5 ( ) BL. a The gradient SEA. could be written in vector form as V; - L.. i We recall equation (3.1.18), where E and E are independent of E. Thus —0 -o e -o —o v °L=-V - XB . 3.2.11 g m g (E > < > Since 3:, - (E x 8’) = o , (3.2.12) we get 3g . '13 = o ; (3.2.13) or 8L1 -—- = 0 . (3.2.14) as, Then, equation (3.2.10) gives -]'f 23; ((pLi)d§ a {£11 gsgL dg ; (3.2.15) 1 or af_ asp. ch dg = - £1. dg . (3.2.16) I i agi I i agi 18 Using the relations (3.2.4), (3.2.6) and (3.2.16), equation (3.2.3) yields jgzflpffllé - If 3'? a: +I§q 2 3 3 11 853 gj . ( . . ) Hence, equation (3.3.9) can be written as 2L. 8t f(uiu1 + 2ui C1.. + cici)fd§ +-§;;-I(uiui +2uici + cici)(cj + uj)fd§ = iffljéjdé . (3.3.12) 20 By use of relations (3.1.2), (3.1.4), (3.1.5), (3.1.6), (3.1.7), (3.1.9), (3.1.10) and (3.3.6), equation (3.3.12) becomes §--(pu2 + 3p) + gx—‘(Zui P. + S + puzu + 311 p) ij 1.1 J J .1 e O 2 4 m{pquJ + ejy sz(pujuy + Pjy)} , (3.3.13) 01" 5.. ___. he. a__ ._.i ulat(pu.)+(pu)::-l'-.'Jat'+-ZaxP.1(uiij)+axj a. 2 a. . _ + axj (pu uj) + 3 axj (ujp) m pquj + $9 ejyz Buz(p juy + Pj y) . (3.3-14) all Eliminating Semi) and Sti- by use of (3.3.7) and (3.3.8) reapectively, we have an as e2+a_ +_p ..1. .1._i.4e p . 3.3.15 or ex xjmjp) 3 11 xj + 3 axj PJysz 1y ( ) This is the equation of conservation of energy. By use of (3.1.13), we get 80183 ER... 5— +_2. _+l—_.1 a: at ax (ujp) (p. .j +96. j)ax 3 8X J J J 9-9 (p +p6 ) , (3.3.16) m eBJYZ 2 JV JY where on, 3_“1 P511 3;"' P BX - (3.3.17) J .1 Hence, 6“ an as an 3.. - _i .2. 1 ,3 3H,). (“3?) +3 pij x *3 P§l+3§l J J J J 4e ;- ejysz(pjy + péjy) . (3.3.18) 21 3.4 Generalized n'th Moment Equation: Let us define = c. °c ’c. ..... c1 = c1c2c3....cn , (3.4.1) and 2‘“) = jantog . (3.4.2) Multiplying the Boltzmann equation (3.2.2) by Zn and integrating, we get ‘n hi hi. 3!. = (n) je (at + g1 ox, + Li agimg J . (3.4.3) From (B.5), we write 30“) = %I5"8(9,U)(f'fi - ff1)deded§d§1 . (3.4.4) Evaluating the first term of the left hand side of (3.4.3), we have J‘e g-t- dg jg?“ not; - If g-t— dg . (3.4.5) If the function (Enf) is uniformly convergent, equation (3.4.5) takes the form -n -n if. d a L -n - C ([9 at g at J‘c f dg If Lat d§ . (3.4.6) an The term ‘3;" could be written as -n n -n ac ac = E ‘g_ 8 ; (3.4.7) at 8-1 G8 at or 22 -n n -n ag au .EE. 3 2 E—-(——5 ..——3) , (3.4.8) at 3-1 G, at at Following (3.1.18), we write ags e e 5'2" " Ls = :1: Es + ; esyz‘ész ° (“'9’ Also from (3.3.8), we have Bu a“ DP ___.—S = -u __S - l—Si +2'SE +29. 6 U B 0 (3’4'10) at i- 6x1 p 8X1 111 8 m syz y z o§ Using the values obtained from (3.4.9), (3.4.10) for S-t—‘l and au 3:3 reSpectively, equation (3.4.8) gives a“. 2 31m _a“8+..1._aia" -9-E 42. u. at s=l c8 1 5x1 p 3x1 m s m syz y z .2 B - 3411 m esyzgy z} ’ ( O . ) or -n n -n an ap c C s I si e e a z -—-u ———-+---——-- —-E - -e u B at s=l c8 1 axi p axi m s m syz y z e . ,...—6 B} . (3.4012) c m syz y 2 By use of (3.4.2) and (3.4.12), equation (3.4.6) becomes - (n) n -n on 31’ ~renaizdg..af___ we.“ ._8.+l__8_i_§E at at s 1 s iaxi paxi s e e - m 63,22“sz + m esyzcsz}fd§ , (3.4.13) or 23 (n) n au 5? Ianaidg-L_ 2“, +l_§.1_£E at at 881 iaxi 9 3x1 u: s e E“ - E esyzu yB 2 }‘f— fdg - 321; esyszI c—s- cyfdg. (3.4.14) Let = [cicjfdé ; (3.5.3) and = ye c B(e.u)(f 'f' - ff1)dedsd§d§1 - 3f? . (3.5.4) Similarly, we write fr =1” ——E-:— fdg , (3.5.5) <2/s) , “1°: . fr 1’5;- fdg , (3.5.6) and fy -f -—c-:—1 :3; . (3.5.7) By use of (3.1.5), (3.1.6) and (3.1.7), equation (3.5.1) furnishes 8P an au1 __HL _1 at +axr (Sijr +|urPij) +'Pir axr +|P Pjr axr 33. a (2) - B z(6:1sz Jy+ ejyz Piy) J. ij . (3.5.8) 28 For n = 3, equation (3.4.42) yields (3) n an n 3Psr at a {f(4) f(3) f(4/8) 1 f(3/s) at +'axr r +u } +'82145xr fr 521-9 ext 2: 39 e B fws) ._.. J(3) m syz z y . (3.5.9) Using definitions (3.4.1) and (3.4.2) and also the relations (3.4.4),(3.1.5), (3.1.6), (3.1.7) and (3.1.8), equation (3.5.9) gives 58, au an an, __llIS+a_—{Q +113. }+S __15.+S __l+s __1; at 5x ijkr r ijk ijr ax irk ax rjk ax r r r r 3P 6P 6P --l(P kr +’P, it + it) ij axr 1k axr jk axr 25-13 s + s + s - m 2(eiyz yjk ejyz iyk ekyz ijy) _ (3) - Jijk , (3.5.10) 3.6 Third-Order Approximation in Hermite Polynomials: We assume that the d.f. f is not too far from the state of equilibrium" We can express the d.f. f in the neighborhood (0) of Maxwellian distribution f by use of Hermite polynomials. Thus, the d.f. f is taken in the form f = 15(0) “20 i,— af“)(§€,t)21£“)(3) - 0-6-1) From (C.13), we have c2 (2nRT) 29 The Hermite polynomial Vin) is introduced as a tensor with n subscripts, i = il,i2,...,in, as well as a polynomial of nfdldegree yin) = [2,2] (-l)sn! vn-28 ’ (3.6.3) s=0 28(n-28)!s! where v is the dimensionless velocity 41 ll llol (3.6.4) 29 T The first few polynomials are 4‘0) = 1 1v?” = 1 <2) ”13 11 U (3) g _ , ”ijk. i j k (v16jk +~vj6ik +-vk61j) , and (4) .. ijkL vivjkaL' (v ivjb k6 +-v1v kbjL +'vivL5jk +vjvk6LL (3.6.5) ll < 4 I 01 fl' +vv6.H+VV5 )+(5 .1 1 111111113 13 Mk4, 5115’); + 5166119 ' (n) The formula for the coefficients ai is a(n) i a: fi‘ff azimdg . (3-6-6) These are dimensionless polynomials. The first few coefficients are (0) a = l ; <1) , 0 ; .3? . .1). . (3....) a(3) "‘ , aijk Sijk/R/RT , and 30 8(4) g _ _1_ 811M Qua/par page“ + Pike.“ + Put)jk + ij6M + PjLéfl PkL6 ij) + (bij6k6+ MGJL + fiicbjk)° Taking only three terms in Hermite polynomial expansion, we have = (0) ‘51(2) (2) +al1(3) (3)) . f f (1+ -21- J K ”61.1)!!in , (3.6.8) or (0) 31.1.. 111; 31.. 3 f = f (1 +12pRT cicj +16pR2T2 cicjck - 2pRT c1). ( .6.9) The variables which define the state of gas are twenty in number, namely, 9(1). 3(3), (311(6) and s (10). ijk Following Grad [13], we use the contracted Hermite coef- ficients instead of the full set aéii. ‘We write “(3) (3) aijj fijf ”111 61; . (3.6.10) Using Grad's derivation, the correSponding contracted Hermite polynomials are introduced as ”13> = 1 vi(v2 - 5) . (3.6.11) The contraction is obtained by letting j = k and summing over all J's in adj; to obtain ”1(3). We write f= £(°)(1 +-1a(§)y (i) + b ism) . (3.6.12) Multiplying (3.6.12) by’ kg?) and integrating, we find a(3) 61- To“ . (3.6.13) 31 Hence, (3.6.10) becomes (0) a(2) (2)+ ”((3) (3) . f (1+ 2 1111-11111 To 111 ”1 ) _, (3.6.14) or S c g (0) p1 _ 1 10 f f {1 +——-1—ZPRT cicj ZpRTa -5:T)} . (3.6.15) The number of variables in the expansion of d.f. f is reduced to thirteen - namely, p(1), 3(3), p (6) and 81(3). Integrating 13 (3.6.14) and taking account of (3.6.8), we obtain 8(3) (3)6 (3)6 8(3) aijk =5(ai 6jk + ajb “k ak 6ij) , (3.6.16) 1.e., s(3)=1-(S6 +86 +36 ) . (3.6.17) Sijk 5 i jk j ik k ij 3.7 The Twenty-Moment Equations: Letting 8(4) = 0 aijkL , (3.7.1) equation (3.6.7) gives = RT(P. 5 + Pik +P Qijkr 1j kr kéjr irsjk +'ijbir +Pjrbik + Pkrsij) - pRT<5ij6kr + bikbjr + sirajk) . (3.7.2) Use of (3.1.13) yields = RT(p 6 + + + + Qijkr 13 kr pikbjr pirbjk pjkfiir pjrbik + pkrbij) + pRT(6 6 r + 6 “6119 . (3.7.3) ijbkr + 6ik j Substituting the value of Qijkr obtained from (3.7.3), equation 32 (3.5.10) gives as -1115. h— at +'axr{“rs1jk +'RT(p1j5kr + pikajr + pirbjk + pjkbir + pjr6 ik +'pkr6 ij) +pRTwig]6 kr +5ikajr +I6ir6jk)} u P ijr ax irk ax rjk ax ij axr ik ax r r r r aPir 2e ij axr ) - m Bz(€iyzS yjk+ ejyzSiyk +iekyzsijy) = (3) . Jijk , (3.7.4) or as an. an an __111 3....(113 ,+(_.1.s +—-is +-——ks.) at axr r ijk ax rjk axr irk axr ijr W(ij T) +5 “(P kRT) + BK L(PinT) axj k h__+ a__ + (piréjk + pjrbik +9 M13)a RT axr(pir6jk + pjrbik a... 5.. +352. + pkrs ij) + p( 531. + axj 61k +axk 6U) +RT(aE_5 4.5.2.5 +BL5 )--1-{(p +135 >353: a i jk axj ik axk ij p ij ij axr P 3P 8 jr it + (91k + p61k)BXr + (pjk + péjk)ax } - 39-3 (e - (3) (3.7.5) m z iyzsyjk +’€jyzs1yk + ekyzsijy) ' Jijk Rearranging it, we have 33 as an an an _UJS a— __L _1. __k. at + (ursijk) + (axr Srjk + axr Sirk + axr sijr) + L(pj kRT) + E5-—(p RT) + a—-(pi J,RT) ax ik ax 13x1 k 3(1) + 96 ) a__’ it it + (911-531: + pjr51k+ pkr 61j)axr + RT{ axr 51k a(pjr +’p51r) + a(pkr + pbkr) 5 } 1{ aPkr +| anr + 6. ' " P "_ P axr 1k axr ij p ij axr 1k axr P 3P 3? P 1r 1r jr kr + pjk ax } RT{5xr 6jk + axr 61k axr bij} 1111; LRT; am; + 6 6 6 p(axi jk axj 1k axk 1j) 22 a (3) . - m z(€1yzsyjk jyz 1yk kyz 1jy) Jijk ’ (3.7.6) or as an. an 6“ 415+1—(us.)+(—¥s +—1$ +43 ) at axr r ljk axr rjk axr irk axr ijr + BL-—(pijT)a + Egg-(pi kRT) + 3:03.13“) (13 + 136 ) a(p +96 ) a3: _ l, a kr kr Ajr 1: +'(pir6jk +ijrbik + pkrbij)axr pipij axr + pik axr 3(1). + p6. r) +12.k u u }+ p(a'x—Tb +3316 +5335) J axr xi jk 3x1 1k axk ij - 23 ) {1(3) (3.7.7) m B2(eiyzsyjk +6jyzsiyk +’ekyzsijy ijk By use of (3.1.13), we can write (3.5.8) in the following form: 34 BL_{ + +1 + at +'axr Sijr ur(p1j p611)} (pir+ p6119:): + +p 6)a—ui-Z-e-B ( + 6 )+ ( + 6 >} j .—_i ,—— A _. 1LT _ £2 +.axk éij) 5 ijz{eiyz(Sy5jk + sjbyk +'Sk6yj) +8jyz(Sibyk +'Syéik m W (3) + S + - . o O jaiy Sybijn Jijk (3 8 4) + Skbiy) + ekyz (3161), The terms are numbered here, so reader may follow through easily. Taking ®,®,@,@ and C5) of (3.8.4), we have Bui 1 58i 5(urS1) +—-——-—-+s 1531. @+@+@+@+@~..—.— .1, 3S 5(u S ) au 53 a(u S ) au +31"al+—E-L+S 4151R'slf'zh—r—L” J”- B axr r axr a axr r axr 1j au an au au au an ... 51(_i + .531 + 51( i + ——1-‘-)s + £44 + —-i)sk . (3.8.5) axk ij axk 5x1 j 5x1 3x1 Equation Contraction of tensors is performed by letting j = k. (3.8.5) yields as a(u s as a(u s 31-{:71+—-1)- + —-l}6u'51“:—;1{+-gx—-L)- + s 43511 r axr r axr _2_ 5“r 2 3% 2 5“j +— 5 Si er 4-3'Sj -—3'+-§'Sj 3x1 . (3 8 6) Now we replace dummy subscript j with r and get a_S__ +1.a_______.(uS)+ an (9+ @+ ++® @ <9 2+— +£s Bur 2 bur _..+_ — (3.8.7) 5 r 5x1 5 i axr Similarly, we let j = k and obtain @+@+.+®+.+ ®= 2;:(ijRT)+§'x-'(pinT) J B— 5.. + axj(pi-1RT) + (piréjj + pjrbu pjr 5,198x _ l“) 8(pir + 9611,) + p “fir + Pit) P ij axr ij axr 8(1). +136 ) +p 1r 111’}+p (L5 +E’)1:——6i +flé..) . (3.8.8) 3.1 axr 3X1 Jj axj ij axj 13 From (3.1.15), we write = 0 . (3.8.9) 911 Summing the Kronecker delta with repeated subscripts and re- arranging, we have @+@+oo+@++@ ima— ruling». 5th.: bran-“Em j. _ . ,1 fr . w. 39 2pir a(prs + pérs) P 6 6x8 ax, Letting j = k, we also have 2. 2 ® B ' 5 m BzhiW-(Sybjj + Sjéyj + Sjbyj) +3 (3.5 + s . + s jyz 1 VJ 5 y 11 Jéiy) + S. + . +S ; Jyz 1 1? j 1y or 2 e = - _.—.B . 3s + 23 6 + 2 s (:> 5 m» z£€1yz< y j yj) ejyz( ibyj +-sy6ij + 836iy)} Using the following relations of alternating unit tensor: eiyzbyj a eijz ; e1jz B ' e312 3 and 6112 = 0 ’ equation (3.8.12) furnishes 2e ® - m Bzeiyzsy ° Again, letting j = k, ® takes the form (3) B (3) Jijj Ji From (3.33), the collision integral is given as (3.8.10) (3.8.11) (3.8.12) (3.8.13) (3.8.14) (3.8.15) (3.8.16) (3.8.17) 40 43 J(3)= ___.; p 5. (3.8.18) i m 1 Using relations (3.8.7), (3.8.10), (3.8.12), (3.8.16) and (3.8.18), equation (3.8.4) gives as. a(u S.) au an 6“ BP 1 ——r—-1—-+Z- —-1 Zs ——‘-'-+-2-s —§+2RT-1-5 at ext 5 r er 5 r 3x1 5 i axr r + + 7 p ART _ 2pir a(prs pars) p it axr 9 6X8 8x1 2e 1 —_ = . . .1 m Bzeiyzsy + m p 81 0 (3 8 9) Adding the conservation equations (3.3.2), (3.3.8) and (3.3.16) to (3.8.3) and (3.8.19), we obtain the system of thirteen- moment equations: h2.+.h... = . at ax (put) 0 . r an an a

mo ucmHonmooo on» was muHmoumH> wo uaoHUwaooo=ucouuaduronu soozuon moHuuu mo Luann may o o.H NHuonqw.N NHuoncw.N NH-OHxNN.N NHuOHxN.H NHaonaH.H mHuonmc.m O O O X 0 x O x O x O X 0 H a NHuOonmm N NHuOH can N NHnOH no N NHcoH mm H NHuoH No H mHuoH HH m . . x . x N .N o XNw. onom.H o xwo.m o x . N m NH-oH NNN N NHuoH N N NH: H H NH- MHu H mHu H mm a I O O x O O x O x O x O m N NHnonwwa H NHuOH mam H NHnonmm H NHuoH mH H mHuoH mm N nHuoH mm m 0 O x O x O x O x O x O x O a o NHuOH eon H NHnoH «as H NHuOH on H NHuOH No H mHnOH Nm o mHuoH Ha m m. n. NH-0HXN¢.H NHuoHXN¢.H NH-0HX¢H.H nHuOHxn.m MHuOmeo.m mHuon¢Q.N O O x O x O x I x O x O x 0 e a NHuOH oMH H NHuOH oMH H mHuOH H m mHuoH Na 0 mHuoH mm a mHaoH NN N N. m. mHuoHXNm.m mHnoHXNm.m mHuoHXNm.o mHuOHxH.m mHuOHXH¢.m mHuOHxN.H m. N. anonmo.m MHuonwo.m mHuonm.¢ mHuonH¢.m mHuonNN.N mHuOonMHH m. H. mH-OHx¢w.N mHuonawN mHuOHxNN.N mHuOHan mHnOonMHH aHuonmo.n «an we «as ' 8 dl.‘ OOH w. c. #0 umm\H N. “ fll : Ham M am :0 Id mm a com .3303 4m. * MM. Ns\uono3 mm .mmo souuuon u0w muHmoomH> mo udoHuHmmooo ago new munoomH> mo ucoHonmooo.maouunqs.onu cowsuom moHuwm .H mHan Figure 1. 48 ALLA]. I L I l l I ALJ \‘l-7awo 2 3 4 sovawo’ 33 x 1013 WEBER/M2 Ratios Between the'kpparent"Coefficient of Viscosity and the Coefficient of Viscosity for Electron Gas. 49 We can write (4.2.11) in the following ways: * an. (a). p.. = -u(——i> . (4.3.2) 1] axj where * — ui ui - U1 , (4.3.3) and 3 is. (4.3.4) 1 = m ijkkaj * The symbol 11 i denotes a new velocity coordinate due to the magnetic induction effect. all i * . , = - -- , 4.3.5 (b) p1j u < > where an B“- i * 1 1 . (6x1) 8(xj) (axj) . (4.3.6) and 2e B -‘ e i k k = 1-m . 4.3. 803) < an > < 7) ij The function i ) may be considered as a coordinate shift due .1 to the magnetic induction effect. Thermal Conductivity: We can write (4.1.17) as 2 L:. 2e283 X l l - -f:§-E . (4.3.8) 45 Blp Using (4.2.7), we have 50 2 2 * 2e B 3 2 i:— = 1 - 2 ($1.) , (4.3.9) 45 p We define the kinematic viscosity v as v =4: . (4.3.10) Thus, equation (4.3.9) becomes i=1 .1‘°:(-(-a-B—3)2>’—)2 4311) 1 45 111 (RT ° (' ° The table of ratios between the"apparen€'therma1 conductivity and thermal conductivity for electron gas is listed as follows: 51 .vouuoHd mH mum couuuoHo now muH>Huonvcoo HmEHonu ago new muH>Huosvaoo Hgauonuzuaouaqaa:u£u cmosumn moHuuu mo seen» any o c.H Hnonm.¢ Hnonon.N n.H mo.¢ mm.m onmN.N N. m. Huonmo.¢ Huonon.o am.H mo.¢ mo.m cHxHo.N a. o. HuOme¢.m H-0onm.m 0H.H w¢.m mm.o 0Hx¢N.H o. a. HuOmew.N Haonms.¢ Hnonn.m mw.N we.m oHXN¢.H m. N. HuonNo.N HuOmem.m H-0HxN.c No.N Ho.¢ OHXHo.H Na\uonos m x. x. x. x. x. .Im A Hmv 8 .MN NHuoH o H NH10H o o NHuoH o m NHnOH o H mHuoH o n mHnono N 1H TmNAumlvfi .. N as am .oom ll 9 .m«@ souuoon you muH>Huo=wsoo Husuosk map was huH>Huosvaou Hmauonh.mcouuma<:onu amo3uom moHuum .N anwH 52 .mmu «.030on no“ >uH>Huo=HEoo Hannah. 93 was 3H>Huo=ucou Hauo£H_wcouwma<..onu ~503an moHumm .N mustm 3 fl . / _ ,... , _ i ,. , ,7 *KIK 53 We may also rewrite (4.1.18) in the following ways: * (a) S. = ~21 BL , (4.3.12) 1 axi where * ZeZB: T = (1 -'-f:§-§)T . (4.3.13) 45 Blp * The symbol T may be treated as a new temperature scale which is caused by the magnetic induction effect. .-. - 51. * (b) 81 21(axi) , (4.3.14) where (a--T >* = 1 3—T ; (4.3.15) and 1 2623: 4 3 16) -——-—- = (1 — -——_-—— . ( . . h(Bk) 45 Bip2 The function h(; ) may be considered as the effect of magnetic k induction on the temperature gradient of the gas. LIST OF REFERENCES 10. 11. 12. 13. 14. LIST OF REFERENCES Bird, R.B., Stewart, W.E. and Lightfoot, E.N. TranSport Phenomena. John Wiley and Sons, Inc., New York, 1960. Boltzmann, L. Vorlesungen fiber Gastheorie. 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Physics of Fully Ionized Gases. Interscience Publishers Inc., New York, 1956. Tanenbaum, B.S. Plasma Physics. McGraw-Hill Co., New York, 1967. Thompson, W.B. An Introduction to Plasma Physics. Pergamon Press, London, 1962. 47. 48. 57 Tucker, H.G. An Introduction to Probability and Mathematical Statistics. Academic Press Inc., New York, 1962. Yen, J.T. Kinetic Theory of Partially Ionized Gases. of Fluids, Vol. 11, 1968, pp. 309. Physics APPENDICES an?! .t...h¢.uuu mwfllunwfiuflml . L APPENDIX A SUMMATIONAL INVARIANT By the conservation of momentum and energy, one gets §+§1=§'+§i ; and (A.l) §2+§§=§'2+§f. Grad [15] stated that a collisional invariant is a point function m defined in the six-dimensional space (E261) as cp(§'.§i) = cp(§.§1) . (11.2) A collisional invariant which split into a sum of functions of E and of E1 is called a summational invariant. Thus we have «3.51) = 1(6) + “31) . (A.3) If a continuous function satisfies the relation 4(3 + 31.52 + 5?) = 1(6) + “‘61) . (AA) then it follows that 1'63) = a'§2 + B'-§’ + c' . (4.5) 58 m N s APPENDIX B COLLISION INTEGRALS The rate of change of d.f. f(E,;,t) is given as §§-3§+§i§§1+L1§-§f—i=(§f)c . (3.1) where (§§)c is the rate of change of f(§,;,t) due to forces between particles. The evaluation of the Boltzmann collision term is taken from Grad's paper [15]. The following assumptions are used in evaluating the collision term (3%)c: (a) Point Particles: This assumption provides the justification in writing equation (B.1). (b) Complete Collision: This assumption states that the time of collision is small. Hence, the energy of the gas is al- most entirely translational kinetic energy. (c) Slowly Varying f: This assumption reveals that “6,2,: + (it), 136,; + d;,t) and f(E + d'g’,x,t) do not differ appreciably from f(E,;,t). (d) Molecular Chaos: This assumption reduces to the fact that the joint distribution function of two particles which are exerting forces on each other is equal to the product of the two individual distribution functions. 59 60 We can also write dt at c ’ ' (B 2) where F = f/m is the number density of particles. The symbols 9,3 and r are polar angle, azimuthal angle and radius respectively in a Spherical coordinate system with axis along the relative velocity U between the two particles (0) and (1). Let dw = rdrde as the element of area. We use symbols dg and dx to denote d§1d§2d§3 and dxldxzdx3 reSpectively. The probability that particle (1) approaching dw collides with particle (0) in the time dt is the probability that particle (1) lies in a cylinder of volume dbUdt. By the assumption (d), this probability is [F(§)dgdx][p(El)dg1wadt], where 6' - E - El. The probability that in time dt particles 3' and 3i will collide and become E and El reSpectively, is [F(§')d§'dx][F(€i)d§iU'dwdt]. The notations (E,E1) and (€‘,§i) designate the velocities of particles before and after collision respectively. The Jacobian of transformation from (3,31) to (€‘,§i) is proved to be unity, i.e. a(§',§i)/a(§,§l) = 1. Thus dg'dgi = dgdgl and U' -=U. For fixed dw, the net increase due to collisions of the number of particles (0) in the product Space dxd§ during the time dt i,F, and F1 imply F(E'), F(§i), F(§) and F(§1) respectively. Integrating is (F'Fi - FF1)wad§1dxd§dt. Here the notations F',F over all orientation, dw, and over all colliding particles dgl, *we obtain the rate of change of the particle density F due to collision, 61 £1.11 a 1 . _ . dt J‘U(F F1 FF1)du)d§1 , (3.3) or 9£.. l t I _ dt m [00: £1 ff1)d(0d§1 . (3.4) Equating (8.1) and (B.4), we get (3%,: = iij'fi - ff1)du)d§1 ; (3.5) or (~21?)c = fiffleflMf'fi - ffpdededsl ; (3.6) where 3(e,U) =-U r(e,V):—;- . (3.7) We define (n) s: l- \- I t _ Jcp mjcpéUUa f1 ffpdwdglds . (3.8) where m(§) is an arbitrary function of velocity. A change of variables from ( , ) to (E',E') has Jacobian unity, implies 1 1 that (5'51) becomes (6.31). so (n) a; I 1 1 _ . Jcp m [ME )U(£ f1 ff1)du)d§d§1 , (3.9) or (n) l "1 I I Jcp = -m j‘cp(§1)U(f £1 - ff1)du.)d§d§1 . (3.10) From these relations, we write 3:9“) a fife); - cp')U(f'fi - ff1)dmd§d§1 ; (3.11) 62 or (n) B L _ 1 _ 1 1 _ JCP 4mj(cp+6pl (p cp1)U(f'f1 ff1)dwd§d§1. (3.12) The term Jé?) = 0, if m is a summational invariant, since by definition m +.¢1 = m' +-mi. Applying the same transformation of variables, equation (8.12) gives (11) _ 1— 1 1 _ _ . J6 - 2m ((9 + cpl cp cplwffldsdgdgl . (3.13) or (n) E L Jcp 2m Ilcpffldgdgl , (3.14) where I.) = f(cp' + cpi - 9 - cpl)B(e.U)dede . (3.15) - . "’ '9 d2 —0 42 U31ng notations c,c = c c and c c a c c , we obtain 1 j i from (8.14) (3) g L . Ji 2m fliffldgdgl , and (3.16) (2) e l. d (1 J11 2m Ilijffl 1; 51 ’ where I = [c 2'2 3(6 U)ded - and 1 .1 1 1 ’ 3 ’ (8.17) Iij =J‘icicj]§(e.U)dede . The symbols [3'3] and [Z 32] represent 3'3' +-d'd' - 3 3 - d d and 3'3'2 +~d'd'2 - 3'32 - 3'32 respectively. The following notations are introduced 63 ’ \ Z'=§'-3=3+3(§-8) , 3 = 31 - 3 . a" . '51" z; = a - 3mm ; > (”'18) fi=§1-'§’=3-2 ; and 1’ = g1 + a = a + c . Using notations (3.18), we have J [2' 2'] = 2(6) 52-15) (3 3-8) - (3 63-13% - 13(3 62-8) ; (3.19) and [2' 32] 417513) (3 217-15) - (3 67-me 11' + V 8') + 2(5) 3.11) (6; 3.13.1 + (3.1321 . (3.20) Ql Figure 8.1 Unit Vector 3 in Spherical Coordinates Defining the unit vector 3 (Fig. 8.1) with reapect to the polar direction U as a-U = U C08 6, we obtain 64 2 _. .. 2 fon(a'U) dc = 2n UZCosze ; (8.21) 211 -* -0 2 U , f0 (0, (1.336.; = 211 C03 6 , (3.22) and 2-0—0-9 —-1-o—o 2 Io"(a a‘U)(a a°U)de = n U200820 sin 0 6i .1 2 +-n Cosze(2 Cos 0 - sinze)UiU (8.23) j . Using relations (8.19), (8.20), (8.21), (8.22) and (8.23), we obtain 2n 2 . 2 2 . Io [cicjjde = 2n Cos e Sln 9(U 5ij - 3Uin) , (8.24) and 2 H2 2 , 2 2 IOHEcic ]de a 2n Cos 9 81m 9(U 513 - 3U1Uj)vj (3.25) Let -' _ - 2 , 2 81(U) - qf8(e,U)Cos 9 Sin 0 d9 . (8.26) Equations (8.17) can be written as -— 2 Ii - 281(U)(U 6ij - 3Uin)Vj , (8.27) and — 2 Iij - 281(U)(U 6ij - 30101) . (3.28) Assuming that all particles are Maxwellian (i.e. the potential energy function varies as l: , also the cross section for r momentum transfer varies as E; where g is the gravitational acceleration, and the collision frequency for momentum transfer is independent of the particle energy.) and using the universe 65 fifth power law, we retain only the highest order terms in Z and d in the following expansions: 2 #2 «2 U 5ij - 3Uin c bij - 3cicj +-d bij - 3didj + terms linear in ci or di ; (8.29) and 2 «2 32 (U 6, - 3U,U )V = —2c,c - 2d. + terms linear in 13 1 J J 1 1 c1 or d . (8.30) Using relations (8.27), (8.28), (8.29) and (8.30), equations (8.16) become: 3 (3) 1 42 42 . Ji = “TIGZCic - 2did )ff1d§d§1 , (3.31) and J(2)= 2 Integrating dg and d§1 separately and using definitions (3.1.6), (3.1.7), (3.1.9) and (3.1.10), we obtain 4'3 (3) = 1 J1 " m p Si 9 (B 033) and 63' (2) 1 1;) -—m p pij . (3.34) APPENDIX C THE EQUILIBRIUM SOIUTION OF THE BOIIZMANN EQUATION Following Grad [15], we define that state of equilibrium is a state in which f is independent of K and t, and Li B 0. Thus, the Boltzmann equation reduces to (35). = o ; (ml) or J'U(f'fi - ff1)du>d§1 = o , (c.2) where f is a function of E only. We can also write fcp U(f'fi - ff1)dwd§d§1 = o ; (c.3) or _ v _ v I I _ a f(gp + (p1 (p cp1)U(f £1 ff1)dwd§d§1 o . (0.4) Letting m 8 log f and inserting this in (C.4), we get f(log f + log f - log f' - log fi)U(f'fi - ff1)du>d§d§1 = o; (c.5) l or ff ju log fiwq - ff1)dwd§d§1 o . (C.6) 1 The integrand of (0.6) is never positive, since U 2 O and ff 1°8‘—Tl7 ‘has the opposite sign as (f'f' - ff ). If f is f f1 1 1 continuous, the integrand must vanish identically. Hence 66 , 1, 9. knight. '.|.W ‘- . 67 or 103 f + log f = log f' + log f' l 1 We see that log f is a summational invariant. follows that I2 "n I log f = a g +-b .6 + c ; or —o -02 f = a exp{-b(§ - u) } . The parameters and c'. and (3.1.6), we obtain (C.7) (C.8) 8y (A.5), it (C.9) (0.10) a, b, and 5' are introduced instead of a', b', Integrating (0.10) and using relations (3.1.2), (3.1.3) p HEP/2 ; \ II = E ; and F (c.11) 2;.E.3/2 .3 23(3) ) After rearranging and using the equation of state (3.8.21), we get . .___JL___. . a 3/2 ’ \ (ZHRT) E .‘3 ; and ? (0-12) b 3 SET. . ‘) Hence, equation (C.10) can be written as (0) 2 f 3 ——L372- exp{-c IZRT} o (C.13) (ZnRT) This is the well known Maxwellian distribution. IV E3 MICHIGAN STQTE UN 1|WWIH111||||||11|||1||H| 31293009 . LIBRARIES 11111111111111 11815