The Kinetic Theory of Inert Dilute Plasmas (Springer Series on Atomic, Optical, and Plasma Physics)

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Springer Series on

ATOMIC, OPTICAL, AND PLASMA PHYSICS

53

Springer Series on

ATOMIC, OPTICAL, AND PLASMA PHYSICS The Springer Series on Atomic, Optical, and Plasma Physics covers in a comprehensive manner theory and experiment in the entire field of atoms and molecules and their interaction with electromagnetic radiation. Books in the series provide a rich source of new ideas and techniques with wide applications in fields such as chemistry, materials science, astrophysics, surface science, plasma technology, advanced optics, aeronomy, and engineering. Laser physics is a particular connecting theme that has provided much of the continuing impetus for new developments in the field. The purpose of the series is to cover the gap between standard undergraduate textbooks and the research literature with emphasis on the fundamental ideas, methods, techniques, and results in the field.

47 Semiclassical Dynamics and Relaxation By D.S.F. Crothers 48 Theoretical Femtosecond Physics Atoms and Molecules in Strong Laser Fields By F. Großmann 49 Relativistic Collisions of Structured Atomic Particles By A. Voitkiv and J. Ullrich 50 Cathodic Arcs From Fractal Spots to Energetic Condensation By A. Anders 51 Reference Data on Atomic Physics and Atomic Processes By B.M. Smirnov 52 Relativistic Transitions in the Hydrogenic Atoms Elementary Theory By R. Boudet 53 The Kinetic Theory of a Dilute Ionized Plasma By L.S. García-Colín and L. Dagdug

For other titles published in this series, go to www.springer.com/series/411

Leopoldo S. García-Colín Leonardo Dagdug

The Kinetic Theory of a Dilute Ionized Plasma

Leopoldo S. García-Colín Universidad Autónoma Metropolitana-Iztapalapa Depto. Física Av. San Rafael Atlixco 186 09340 México, D.F. México

Leonardo Dagdug Universidad Autónoma Metropolitana-Iztapalapa Depto. Física Av. San Rafael Atlixco 186 09340 México, D.F. México

Springer Series on Atomic, Optical, and Plasma Physics ISBN 978-1-4020-9329-6

ISSN 1615-5653

e-ISBN 978-1-4020-9330-2

Library of Congress Control Number: 2008938718

© 2009 Springer Science + Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper. 987654321 springer.com

Acknowledgement The authors are indebted to Alfredo Sandoval, Ana Laura Garc´ıa Perciante, and Valdemar Moratto for a careful reading of the manuscript, and for a number of useful suggestions which have been incorporated into it.

v

Contents Introduction Part I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Vector Transport Processes

1

Non-equilibrium Thermodynamics . . . . . . . . . . . . . . .

2

The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . 14 2.2 The H Theorem and Local Equilibrium . . . . . . . . . . . . . 18

3

Solution of the Boltzmann Equation . . . . . . . . . . . . . . 25

4

Calculation of the Currents . . . . . . . . . . . . . . . . . . . . 41 4.1 Diffusion Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Flow of Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5

Solution of the Integral Equations . . . . . . . . . . . . . . . . 51

6

The Transport Coefficients . . . . . . . . . . . . . . . . . . . . 61

7

Discussion of the Results . . . . . . . . . . . . . . . . . . . . . 73

Part II 8

5

Tensorial Transport Processes

Viscomagnetism . . . . . . . . . . . . 8.1 The Integral Equation . . . . . . . . 8.2 The Stress Tensor . . . . . . . . . . 8.3 The Integral Equation . . . . . . . . 8.4 Comparison with Thermodynamics

. . . . .

. . . . .

. . . . .

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83 83 93 99 102 vii

viii

9

Contents

Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . 107

Appendix A Calculation of M . . . . . . . . . . . . . . . . . . . . 125 Appendix B Linearized Boltzmann Collision Kernels . . . . . . 129 = 0 . . . . . . . . . . . . . . . . . 133 Appendix C The Case when B Appendix D The Collision Integrals . . . . . . . . . . . . . . . . 145 (0)

(1)

(0)

Appendix E Calculation of the Coefficients ai , ai , di (1) and di . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Appendix F

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Appendix G

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Appendix H

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Appendix I List of Marshall’s Equations and Notation . . . . . 161 I.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . 161 I.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . 162 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Introduction The contents of this book are the result of work performed in the past three years to provide some answers to questions raised by several colleagues working in astrophysics. Examining several transport processes in plasmas related to dissipative effects in phenomena such as cooling flows, propagation of sound waves, thermal conduction in the presence of magnetic fields, angular momentum transfer in accretion disks, among many, one finds a rather common pattern. Indeed when values for transport coefficients are required the overwhelming majority of authors refer to the classical results obtained by L. Spitzer and S. Braginski over forty years ago. Further, it is also often mentioned that under the prescribed working conditions the values of such coefficients are usually insufficient to provide agreement with observations. The methodology followed by these authors is based upon Landau’s pioneering idea that collisions in plasmas may be substantially accounted for when viewed as a diffusive process. Consequently the ensuing basic kinetic equation is the Fokker-Planck version of Boltzmann’s equation as essentially proposed by Landau himself nearly 70 years ago. Curiously enough the magnificent work of the late R. Balescu in both Classical and Non-Classical transport in plasmas published in 1988 and also based on the Fokker-Planck equation is hardly known in the astrophysical audience. The previous work of Spitzer and Braginski is analyzed with much more rigorous vision in his two books on the subject. With this background in hand the question that came to our minds is why, if true, the full Boltzmann’s equation had never been used in dealing at least with the kinetic theory of dilute plasmas. In their well known and comprehensive treatment on the kinetic theory of non-uniform gases, Chapman and Cowling never developed the theory as they did with ordinary gases. A further attempt was made in 1960 by W. Marshall in three unpublished reports issued by the Harwell Atomic Energy Establishment in L.S. Garc´ıa-Col´ın, L. Dagdug, The Kinetic Theory of a Dilute Ionized Plasma, Springer Series on Atomic, Optical, and Plasma Physics 53 © Springer Science + Business Media B.V. 2009

1

2

Introduction

Harwell, England. And also, none of all the authors in this field with the sole exception of Balescu who did it partially, took the kinetic equation of their choice to provide the microscopic basis of linear irreversible thermodynamics therefore, providing, among many other results, a microscopic basis of magnetohydrodynamics. This is the main objective of this book. Starting from the full Boltzmann equation for an inert dilute plasma and using the Hilbert-Chapman-Enskog method to solve the first two approximations in Knudsen’s parameter we construct all the transport properties of the system within the framework of linear irreversible thermodynamics. This includes a systematic study of all possible cross effects which except for a few cases dealt with by Balescu, today to our knowledge, have never been mentioned in the literature. The equations of magnetohydrodynamics, including the rather surprising results here obtained for the viscomagnetic effects, for dilute plasmas may be then fully assessed. We expect that this material will thus be useful to graduate students and researchers involved in work with non-confined plasmas specially in astrophysical problems. July 2008

L.S. Garc´ıa-Col´ın L. Dagdug

Part I Vector Transport Processes

Chapter 1 Non-equilibrium Thermodynamics The main objective of this book is to place the kinetic theory of a dilute plasma within the tenets of what is known as Classical (Linear) Irreversible Thermodynamics (CIT). Since this subject is quite often beyond the average knowledge of the younger generation of physicists and physical chemists we feel that it is useful to give a brief review of its basic concepts so that the reader appreciates better how and why we are seeking the results to be presented in the main text. CIT, being a phenomenological theory is based essentially on four basic assumptions, namely, 1. The local equilibrium assumption (LEA) 2. The validity of the conservation equations 3. The linear constitutive equations and positive definiteness of the uncompensated heat (entropy production) 4. Onsagers’ reciprocity theorem In what follows we shall discuss as thoroughly as possible the basic ideas behind each assumption, leaving the reader to pursue more details in the standard texts on the subject [1]-[7]. Let us start with the LEA. Consider any arbitrary system which is not in thermodynamic equilibrium. For purely didactical reasons the reader may think of a fluid enclosed in a volume V . L.S. Garc´ıa-Col´ın, L. Dagdug, The Kinetic Theory of a Dilute Ionized Plasma, Springer Series on Atomic, Optical, and Plasma Physics 53 © Springer Science + Business Media B.V. 2009

5

6

1 Non-equilibrium Thermodynamics

Let us now partition this volume in small cells such that the number of particles in each cell with coordinates r, r + dr at time t contains enough particles to be considered as a continuum but small compared with the total number of particles in the system, say N . The LEA asserts that within each cell a thermodynamic equilibrium state prevails. For instance, if n(r, t) is the particle density in the cell characterized by its position r at time t and T (r, t) the temperature inside the cell, any other thermodynamic quantity, for instance the entropy s(r, t) will be related to n(r, t) and T (r, t) as s(r, t) = s[n(r, t), T (r, t)]

(1.1)

precisely by the same relationship that holds for these variables in the equilibrium state. The local equation of state for an ideal gas would read p(r, t) = n(r, t)kB T (r, t)

(1.2)

kB being Boltzmann’s constant. And so on. These equations bring us in a natural way to the second assumption. Think of a monatomic fluid for the moment in the absence of sources and sinks. If we chose to describe the states of this fluid by the “natural” variables, the local particles density n(r, t), the local hydrodynamic velocity u(r, t) (or mu(r, t) its momentum) and the local energy density e(r, t) these variables will satisfy clearly, conservation equations. Use of this fact and Eq. (1.1) with e(r, t) instead of T (r, t) plus the standard techniques of ordinary calculus lead us in a straightforward fashion to an equation describing the evolution of the local entropy s(r, t). In fact if ρ(r, t) = mn(r, t), m being the mass of the particles, one finds that, ∂(ρs) + div Js = σ (1.3) ∂t which is a balance type equation for ρs. Js , the entropy flux, gives the amount of entropy flowing through the boundaries to the system and σ, the uncompensated heat or entropy production, measures the entropy generated inside the system due to the dissipative processes. Its existence goes back to Clausius who indeed identified it with the uncompensated heat which should arise from dissipation. Its analytical expression was first identified by T. de Donder in chemical reactions and later brought into its present form by Meixner. Indeed, in the derivation of Eq. (1.3) one finds that σ=

◦ ← ◦ ← → → Jq 1 → 1 τ : (grad u)s − τ div u J i X i = − · grad T − ← T T T

(1.4)


7

← → ← → where J i and X i denote the fluxes and their corresponding forces respectively, and the contraction of tensors of equal rank. The second equality illustrates its nature for an ordinary monatomic fluid. Jq is the heat flow → vector and the momentum flow ← τ is split into its symmetric traceless part ◦ ← → τ and its trace τ . Eq. (1.4) clearly fulfills Clausius’ predictions. These results bring us to the third assumption. The conservation equations for a monatomic fluid are the set of five differential equations for the state variables ρ, u and e but contain fourteen unknowns, these variables plus the three components of Jq plus the six independent components of the stress → tensor ← τ assumed to be symmetric. We thus need nine additional equations → to express Jq and ← τ in terms of the independent variables. Notice that T (r, t) may be introduced through the LEA since e(r, t) = e(n(r, t), T (r, t)). These additional equations known in the literature as the “constitutive equations” are completely foreign to thermodynamics. They may be extracted from experiment or from a microscopic theory. If we now assume (assumption 3) that the relationship between fluxes and forces is linear so that in general, ← → ← → Lik X k , (1.5) Ji= we may obtain a complete set for the time evolution equations of the local state variables. For a monatomic fluid, Eq. (1.5) reduces to Jq = −κgrad T ◦ s

◦ ← → τ = −η(grad u)s τ = −ζdiv u

Fourier

(1.6)

Naviere-Newton

As it is shown in any standard text on the subject, when Eqs. (1.6) are substituted into the conservation equations one gets a set of non-linear, second order in space, first order in time differential equations for n, u and T known as the Navier-Stokes-Fourier equations of hydrodynamics. These equations require the knowledge of a local equation of state (c.f. Eq. (1.1)), of the transport coefficients κ, the thermal conductivity, η, the shear viscosity, and ζ, the bulk viscosity in addition to well defined boundary and initial conditions to seek for a solution. In spite of its centennial age these equations still pose immense problems to mathematical physicists and hydrodynamicists in finding stable solutions [8].

8


There is however an additional feature brought by the assumption written in Eq. (1.6). For the case of a fluid when substituted into Eq. (1.4) yields σ=

◦ ◦ κ η ζ 2 s s ( (div T )2 , (grad T ) + grad u ) : ( grad u ) + 2 T T T

(1.7)

a quadratic form for σ which is positive definite if κ, η and ζ are positive, a fact drawn from experiment. This means that σ>0

(1.8)

a quite strong statement implying complete consistency with the second law of thermodynamics. Its extension to open systems is not trivial but we shall not discuss this here, it is treated in references [3] and [6]. There is an additional feature about Eq. (1.5) which deserves mention before going into the last assumption behind CIT. Why a summation in this equation is necessary. This occurs in systems where two or more thermodynamics forces are present and are of the same tensorial rank. There are many examples of this nature in many physical systems, let us mention here two of the most frequent ones. Suppose we have a mixture of two monatomic fluids subject to the same temperature gradient and to a concentration gradient of one of the species (c1 + c2 = 1) so only one concentration is independent. This implies that the total heat flow Jq and the mass flux Jm are Jq = −Lqq grad T − Lqm grad c1 Jm = −Lmq grad T − Lmm grad c1

(1.9)

A concentration gradient may give rise to a heat flow an effect called thermal diffusion or Dufour effect, whereas a thermal gradient gives rise to a mass flux or diffusion termo effect (Soret effect). These “cross effects” are very important in multicomponent systems. In the case of a conducting solid = subject to both thermal gradients and an electric field one has that (E −grad φ) Jq = −Lqq grad T − Lqe grad φ (1.10) Je = −Leq grad T − Lee grad φ where Je is the charge flux or electrical current. Leq is known as the electrophoresis or Benedicks effect, Lee is the usual electrical conductivity, and Lqe is Thomsons’ thermoelectric effect. As we shall see both of these cases play a very important roll in the physics of a ionized plasma.


9

The fourth assumption arises in some way from the structure of Eq. (1.5). ← → Indeed viewing Lik as the element of a matrix L it turns out that if it is diagonal, the constitutive equations lead to a complete set of “hydrodynamic ← → equations”, but if L is not diagonal, then more information is required for such purpose. This condition, buried in the very old approach of Lord Kelvin to the study of the thermoelectric effect was later identified by L. Onsager in 1931 in his study on chemical reactions and proved by the same author ← → twenty years later for an arbitrary system. The condition is that L must be a symmetrical matrix whenever microscopic reversibility holds true. This requirement reflects the invariance of the microscopic equations of motion of ← → the particle composing the system under time reversal. Thus if L † is the ← → transposed of L then, ← → ← → (1.11) L = L† is Onsagers’ reciprocity theorem (ORT). In words, it is the ultimate way in which microscopic reversibility is exhibited on a macroscopic level. No thermodynamical theory based on Eq. (1.5) may be considered complete if Eq. (1.11) is not fulfilled. Regretfully the literature is plagued with results which arise from misconceptions about the nature of (1.11). One thing is to obtain transport coefficients which are consistent with Eq. (1.11) and another thing is to be able to rigorously prove Eq. (1.11) from the basic kinetic equation from ← → which L is calculated. This is not the place to enter into a profound discussion on the nature ← → of ORT but it is worth mentioning one example. If one computes L from a Boltzmann kinetic equation for a multicomponent mixture of dilute inert gases, the proof that Eq. (1.11) holds true has been a headache for years. Indeed, this equation may be shown to hold true if one selects the chemical potential as the thermodynamic force in Eqs. (1.9) provided the system is isothermic [3]. However if a temperature gradient is present this is no longer correct and Eq. (1.11) can be shown to hold if, and only if, the thermodynamic forces are represented by the diffusive force, to be thoroughly discussed in this book, and not by chemical potentials [9]. When, like in plasmas, magnetic fields are present the phenomenological form of Eq. (1.11) may be readily inferred but its microscopic derivation is far from trivial. In fact, when using the Landau-Fokker Planck version of Boltzmann’ equation, the proof

10


is impossible since the approximations introduced into the scheme erases the source of microscopic reversibility from the resulting kinetic equation. Thus, Eq. (1.11) may be checked to hold but not derived from the kinetic equation itself [10]. Consistency with Onsager’s reciprocity theorem is often claimed but, we repeat, an air tight proof of its validity may not be so easy to obtain. These general ideas provide the necessary background to understand what we mean by placing the derivation of the transport equations of dilute inert plasma within the framework of CIT. For more details we recommend the reader to seek more information in the vast literature on the subject.

Bibliography [1] S. R. de Groot; Thermodynamics of Irreversible Processes; NorthHolland Publ. Co., Amsterdam (1952). [2] J. Meixner and H. Reik; Thermodynamik der Irreversiblen Prozeses, in Handbuch der Physik; S. Fl¨ ugge, ed. Springer-Verlag, Berlin (1959), vol. 3. [3] S. R. de Groot and P. Mazur; Non-equilibrium Thermodynamics; Dover Publications Inc., Mineola, N. Y. (1984). [4] R. Hasse; Thermodynamics of Irreversible Processes; Addison-Wesley Publ. Co., Reading, Mass (1971). [5] I. Prigogine; Thermodynamics of Irreversible Processes; Interscience, New York (1967), 3rd ed.

Wiley-

[6] L. Garc´ıa-Col´ın and P. Goldstein; La F´ısica de Procesos Irreversibles; El Colegio Nacional, México D. F. (2003), Vols. 1 and 2 (in Spanish) [7] L. Garc´ıa-Col´ın and F. J. Uribe; J. Non Equilib. Thermodyn. 16, 89 (1991). [8] C. L. Fefferman, Existence and Smoothness of the Navier-Stokes equations http://www.claymath.org/prizeproblems/navierstokes.htm, 89 (1991). [9] P. Goldstein and L. Garc´ıa-Col´ın; J. Non Equilib. Thermodyn. 30, 173 (2005). [10] R. Balescu; Transport Processes in Plasmas, North-Holland Publ. Co., Amsterdam (1988), Vol. 1: Classical Transport.

11

Chapter 2 The Problem The system we wish to study is a binary mixture of non reactive dilute, electrically charged system of particles. Their masses will be labelled ma and mb with charges ea and eb where ea = −eb = e. The ions could have a positive charge Ze but we shall keep Z = 1 for simplicity. The number densities of the species are na and nb where na + nb = n so that the total mass density ρ is given by ρ = ρa + ρb = ma na + mb nb (2.1) Following the standard notation of the kinetic theory of gases, the single particle distribution functions for each species is denoted by fi (r, vi , t) where vi is the velocity of the particle of species i, i = a, b. If we now assume that in measured in volts m−1 general the system is acted upon by an electric field E in teslas, the Boltzmann equation determining and a magnetic induction B the time evolution of the distribution function fi is given by b ∂f 1 ∂fi ∂fi i + vi · + Fi + ei vi × B · = J (fi fj ) (2.2) ∂t ∂r mi ∂ vi i,j=a Here, (e) Fi = Fi + ei E

for i, j = a, b

(2.3)

(e) Fi denoting an external conservative force, the electric and magnetic fields, and B respectively, are the self consistent fields generated by the plasma E as determined from Maxwell’s equations, and J (fi fj ) = · · · f (v )f (v ) − f ( vi ) f ( vj )

i

j

vj dvi dvj σ vi vj → vi vj gij d L.S. Garc´ıa-Col´ın, L. Dagdug, The Kinetic Theory of a Dilute Ionized Plasma, Springer Series on Atomic, Optical, and Plasma Physics 53 © Springer Science + Business Media B.V. 2009

(2.4) 13

14

2 The Problem

In Eq. (2.4) we recall the reader that the r, t dependence of the fi s has been omitted. The primes denote the values of vi after the binary collision takes place, σ vi vj → vi vj dvi dvj is the cross section, namely, the number of molecules per unit time of species i colliding with a molecule of species j such that after the collision the molecules have velocities vi in the range dvi and vj in the range dvj ; gij ≡ | vi − vj | = |vi − vj |. For collisions between molecules of the same species vi → v and vj → v1 to distinguish the two velocities. A caution note has to be mentioned with respect of Eq. (2.2). is taken to be the average magnetic field, deterThe magnetic induction B mined from Maxwell’s equations where the current density will depend on =B av + B e where the distribution functions fi . In fact one should write B Be is an external field which may or may not be taken as a constant field.1 We also recall the reader that the cross section σ satisfies the principle of microscopic reversibility, namely, it is invariant upon spatial and temporal reflections, so that, (2.5) σ vi vj → vi vj = σ vi vj → vi vj for i, j = a, b thus guaranteeing the existence of inverse collisions. As well known in kinetic theory, two general results may be derived from Eq. (2.4) regardless of the specific form of the cross section that is, without specifying the details of the interaction potential between the particles. Such results are the conservation equations and the H theorem. In our case this will require particular care since collisions do not exist for Coulomb interactions which as well known is a long range repulsive potential. Advancing the fact that this will be appropriately taken care of using the Debye-H¨ uckel approximation we assume that σ is well defined and finite. We proceed to discuss the first of two general results namely, the conservation equations. Section 2.2 will be devoted to the H-theorem.

2.1

Conservation Equations

As usual, we define the local particle densities as, vi ni (r, t) = fi (r, vi , t)d 1

For a thorough discussion of this question see [7].

(2.6)

2.1. Conservation Equations

15

and denote by ψi (r, vi , t) any dynamical variable whose local value is given by 1 ψi ≡ ψi (r, t) = vi (2.7) ψi (r, vi , t)fi (r, vi , t)d ni Moreover, we define the thermal or chaotic velocity ci as ci = vi − u(r, t) and u is the barycentric velocity given by ρu(r, t) = ρi ui (r, t)

(2.8)

(2.9)

i

where,

1 vi d vi (2.10) ui (r, t) = fi (r, vi , t) ni is the local hydrodynamic velocity for species i. Notice here that contrary to what occurs in the case of a single species, ci = 0 whereas ca + mb nb cb = ρa ua + ρb ub − ρu = 0 ma na or

ρi ci = 0

(2.11)

i

This expression is important because the mass diffusion flux of the ith species is defined as vi = mi ni ci (2.12) Ji = mi ci fi (r, vi , t)d so that by Eq. (2.11)

Ji = 0

(2.13)

i

or Ja = −Jb . With these definitions, the flow of charge is readily expressed in a convenient way. In fact, the numerical charge density Q is defined as Q = na ea + nb eb = (na − nb )e and the charge current JT =

i

ni ei vi

(2.14a) (2.14b)

16

2 The Problem

which, with the aid of Eq. (2.8) reads JT = Qu + Jc , Jc the conduction current being given by Jc = ni ei ci (2.14c) i

which in turn can be written with the aid of Eqs. (2.12) and (2.13) as ma + mb Jc = eJa ma mb

(2.15)

a result often ignored by authors of this subject. Returning to our quest, we now derive the equivalent of Maxwell-Enskog’s transport equation by taking ψi = (mi , mi vi and 12 mi vi2 ). We first notice that from Eq. (2.4) b vi = 0 (2.16) ψi J(fi fj )d i,j=a

a result which follows from the standard transformation of the collision kernels using Eq. (2.5) and the fact that i and j are dummy indices in Eq. (2.16). So let ψi = mi . Multiplying (2.2) by mi and integrating over d vi using (2.16) one gets ∂ρi · ∂fi d + div (ρi ui ) = ( vi × B) vi ∂t ∂ vi ∂fi the cross product ( vi × B) ∂ vi does not contain such component so that the integration by parts yields zero whence ∂ρi + div (ρi ui ) = 0 (2.17a) ∂t and summation over i yields In the right hand term, for any component

∂ρ + div (ρu) = 0 ∂t

(2.17b)

Using Eqs. (2.8) and (2.12) in Eq. (2.17a) we may also write that ∂ρi + div (ρiu) = −div Ji ∂t

(2.17c)

Eqs. (2.17a)-(2.17c) are thus the several alternative expressions for mass conservation.

2.1. Conservation Equations

17

Take now ψi = mi vi = mi (ci + u). Multiply Eq. (2.2) by it using Eq. (2.11), after integrating a couple of terms by parts and summing over i, one readily gets that ∂ ← → k · ∂fi d (ρu) + div ( τ + ρuu) = ni F i − ei vi ( vi × B) vi ∂t ∂ vi i i → where the kinetic part of the stress tensor ← τ k is defined as b ← → k τ = mi fi ci ci d vi

(2.18)

i=a

so that we Integration by parts of the last term reduces to i ei ni vi × B reach the result that ∂ → (ρu) + div (← τ k + ρuu) = ni Fi + (JT × B) (2.19) ∂t i the conservation equation for momentum. If the external force is zero using the definition of JT we readily find that ∂ → + (u × B)) + Jc × B (ρu) + div (← τ k + ρuu) = Q(E ∂t

(2.20)

+ u × B can be interpreted as the effective electric field as = E Here E viewed by an observer moving in the mixture with the barycentric velocity u. Also, it should be pointed out that often Eqs. (2.17a)-(2.17c) and (2.20) are referred to as the equations of magnetohydrodynamics for isothermal fluids in the absence of external fields F e = 0. We finally take ψi = 12 mi vi2 and repeat the procedure as in the previous case. After summation over i and use of Eq. (2.11) we get that, 1 ∂ 1 ∂fi 2 1∂ 2 2 (ρu ) + mi fi d mi vi · v d vi ci + vi + 2 ∂t ∂t i 2 2 ∂r i i 1 ∂f 1 i 2 · ∂fi vi2 d Fi · vi dvi + ei ( vi = 0 (2.21) vi × B) 2 ∂ v 2 ∂ v i i i i We define the internal energy density of the mixture as, 1 ρi c2i ρe(r, t) = 2 i

(2.22)

18

2 The Problem

In the third term we set vi = ci + u, expand, use Eq. (2.11) and find that it reduces to 1 → div(Jq + u · ← τ k + ρeu + ρuu2 ) 2 where 1 ρi ci c2i (2.23) Jq = 2 i is the total heat flux in the mixture. After a first integration by parts, use of (2.14c), the definition of JT and assuming F e = 0, the fourth term simply Finally integration by parts clearly shows that the last reduces to −JT · E. term vanishes, so that collecting all terms we find that, 1∂ ∂ρe 1 → =0 (ρu2 ) + + div (Jq + u · ← τ k + ρue + ρuu2 ) − JT · E 2 ∂t ∂t 2 Using Eq. (2.20) and following the standard steps to combine the first three terms in this equation we are finally lead to the balance equation for the internal energy namely, ρ

d → = 0 e + div Jq + ← τ k : grad u − Jc · E dt

(2.24)

where as introduced above, = E + u × B E Eqs. (2.17a)-(2.17c), (2.20) and (2.24) are the sought result for the conser→ τ k and Jq have to be determined vation equations. Clearly the unknowns Ji , ← by seeking solutions to Eq. (2.2), a task to be dealt with later.

2.2

The H Theorem and Local Equilibrium

Before discussing these important properties of the Boltzmann equation we need to specify clearly the domain of its applicability. In the absence of a magnetic field Eq. (2.2) is valid in the so called kinetic regime characterized by time t ∼ τ the mean free time where τ tc the duration of a collision time. However, in the presence of a magnetic field we have two characteristic frequencies competing in the mixture, the collision frequency ωc ∼ 1/τ and . For electrons ωe ∼ 1.76 × 1011 B whereas the Larmor frequencies ωi = |e|B mi

2.2. The H Theorem and Local Equilibrium

19

e for ions ωi = ωe m . If the field is weak enough ωi τ is of the order of 1 for mi both cases implying that the field does not interfere in the collisional regime of the mixture. We shall limit ourselves to this case. When ωi τ 1 radical modifications have to be made to the whole approach to the problem and we shall not discuss it here at all (see however Ref. [7]). Once this is clarified we proceed with our discussion. If we multiply Eq. (2.2) by ln fi integrate over d vi and sum over i the left hand side vanishes since the only extra term, · ∂fi ln fi d ( vi × B) vi vanishes after integration by parts. Therefore, using ∂ vi the same procedure for the right hand side as in the single component case remembering Eq. (2.2) and Klein’s inequality one obtains that for H≡ vi , (2.25) fi ln fi d

i

∂H(r, t) ≤0 (2.26) ∂t for all binary collisions and their corresponding inverses (i, j

i , j ). Remember that in Eq. (2.5), H ≡ H(r, t) is still function of r and t. So the irreversibility criteria imposed by Eq. (2.26) is still valid in the weak field approximation and moreover, the quantity usually associated with the entropy production σ(r, t) is always positive definite for all exact solutions to Eq. (2.4) σ = −k vi (2.27) ln fi J(fi fj )d i,j

This result will be used later on. We also notice that the solution to the homogenous Boltzmann equation, namely, (0) (0)

(0) (0)

J(fi fi ) + J(fi fj ) = 0 for i, j = a, b is a local Maxwellian distribution function. This arises from the well known argument stating that ∂H = 0 for every binary collision. By the standard ∂t argument of kinetic theory this implies that

(0) fi

= ni (r, t)

mi 2πkT (r, t)

provided we define

32

e−

m(vi − u( r ,t))2 2kT ( r ,t)

(2.28)

ni (r, t) =

(0)

fi d vi

(2.29a)

20

2 The Problem

ρu =

ρi

(0)

vi fi vi d

(2.29b)

i

1 3 ρi c2i ρe(r, t) = nkT = 2 2 i

(2.29c)

Nevertheless Eq. (2.28) is still not a solution to the full Boltzmann equation since it is necessary that

Fi ∂ ∂ ei ∂ ∂ (0) · + vi · + · + ( vi × B) (2.30) ln fi = 0 ∂t ∂r mi ∂ vi mi ∂ vi is satisfied for i = a, b. The procedure is, once more, the standard one [1]-[2]. We write (0) (2.31) ln fi = ν(r, t) + k(r, t) · vi − h(r, t)vi2

32 mi β mi β 2 mi β u , k = βmiu; h = ; A = ni where ν = ln A − with 2 2 2π β = (kT )−1 . · vi = 0 we Substitution of (2.31) into (2.30) and noticing that ( vi × B) get that,

∂h ∂h ∂k ∂ν − vi2vi · + −vi2 + vi · vi · + ∂t ∂r ∂t ∂r

2h ∂k ∂ν ei + Fi · k = 0 Fi + k · ( + − vi · vi × B) ∂t ∂r mi mi mi which must hold for all values of v . The coefficients of order vi3 and vi2 do so by the standard procedure h = h(t) and k = r ∂h + r × not depend on B ∂t Ω(t) + k0 (t). For conservative forces (including F = −e grad φ) the linear coefficient in vi yields

2hei ∂k ei + grad ν + φi − k×B =0 ∂t mi mi yields in turn and φi is the electrical potential. Scalar multiplication by B, that

k ∂ 2he i · B + grad ν + φi =0 ∂t mi

2.2. The H Theorem and Local Equilibrium

21

= 0 and ignoring the possible but unlikely occurrence that B is which for B perpendicular to the term in parenthesis, ∂k + grad ∂t

2hei ν+ φi mi

=0

∂ is a constant vector and, once more by the This implies ∂t rotk = 0 or Ω argument for a one component system, and non-pathological external forces,

m 32 mi vi2 i eq + φi (r) f i = ni exp −β for i = a, b (2.32) 2πkT 2

where the potential energy is φi = φext + ei φ. Thus equilibrium is achieved and characterized by the Maxwell distribution function Eq. (2.32).

Bibliography [1] S. Chapman and T. G. Cowling; The Mathematical Theory of NonUniform Gases; Cambridge University Press, Cambridge (1970), 3rd ed. [2] G. W. Ford and G. E. Uhlenbeck; Lectures in Statistical Mechanics; American Mathematical Society, Providence, R. I. (1963). [3] J. R. Dorfman and H. van Beijren; The Kinetic Theory of Gases, in Statistical Mechanics, Pt. B.; Bruce J. Berne, ed. Plenum Press, New York (1977). [4] J. H. Ferziger and H. G. Kaper; Mathematical Theory of Transport Processes in Gases; North-Holland Publ. Co., Amsterdam (1972). [5] C. Cercignani; The Boltzmann Equation and its Applications; SpringerVerlag, New York (1988). [6] L. Garc´ıa-Col´ın and P. Goldstein; La F´ısica de los Procesos Irreversibles; Vols. 1 and 2, El Colegio Nacional, Mexico D. F. (2003) (in Spanish). [7] R. Balescu; Transport Processes in Plasma Vol. I, Classical Transport; North-Holland Publ. Co., Amsterdam (1988). [8] W. Marshall; The Kinetic Theory of an Ionized Gas; U.K.A.E.A. Research Group, Atomic Energy Research Establishment. Harwell U.K. parts I, II, and III (1960).

23

Chapter 3 Solution of the Boltzmann Equation In this section we want to discuss the solution to Eq. (2.2). The first problem where we encounter concerns the correct interpretation of the drift term vi ×B vi is the velocity of a particle of species i. Since vi = ci + u and | ci | >> |u| being the velocity associated with the thermal agitation of the molecules we in the drift term follow Chapman and Cowling’s suggestion in keeping u × B and bring ci × B to the collisional contribution. Using the definition of the , Eq. (2.2) may be rewritten as follows, “effective” electric field E ∂f 1 ∂fi ∂fi ei i · ∂fi + F i + ei E · + vi · + = − ( ci × B) J (fi fj ) ∂t ∂r mi ∂ vi mi ∂vi for i, j = a, b (3.1) is a conservative force whence the solu = 0, Fi + ei E Notice that when B tion to Eq. (3.1) is trivial extension of the solution obtained for an inert mixture in the presence of a conservative force. What will complicate the solution to Eq. (3.1) is the presence of the first term in the r.h.s. As usual, we now assume that in the weak field approximation the distribution functions fi (r, vi , t) can be taken as functionals of the locally conserved variables namely fi (r, vi |ni (r, t), u(r, t), e(r, t))1 and further, they may be expanded in power series of Knudsen’s parameter around the local equilibrium distrib(0) ution function fi defined in Eq. (2.28). Omitting unnecessary arguments, (0)

(1)

(2)

fi = fi (1 + ϕi + 2 ϕi + · · · ) for i = a, b 1

(3.2)

Notice that ui is not taken as a local variable.

L.S. Garc´ıa-Col´ın, L. Dagdug, The Kinetic Theory of a Dilute Ionized Plasma, Springer Series on Atomic, Optical, and Plasma Physics 53 © Springer Science + Business Media B.V. 2009

25

26

3 Solution of the Boltzmann Equation

the well known Hilbert-Chapman-Enskog approximation. We remind the reader that the parameter is a measure of the unitary change of a local variable in a mean free path. Notice that since (0)

∂fi (0) mi ci = −fi ∂ vi kT · ci = 0, f (0) is still a solution to the homogeneous part and that ( ci × B) i of Eq. (3.1) as stated in page 19. Substitution of Eq. (3.2) in Eq. (3.1), collecting terms of order and for clarity taking i = a, Eq. (3.1) reduces to (0) (0) ∂f (0) ∂fa 1 ∂fa a F a + ea E · + va · + = ∂t ∂r ma ∂ va

−

ea (1) (0) (1) (1) · ∂ fa(0) ϕ(1) C(ϕ + f ( ca × B) ) + C(ϕ + ϕ ) a a a a b ma ∂ va (1)

(1)

(3.3)

(1)

where C(ϕa ) and C(ϕa + ϕb ) are the linearized collision kernels for collisions among particles of species a and species a, b, respectively. Their explicit form will be written later.2 Clearly there is an identical equation for species b simply obtained by exchanging the indices a and b in (3.3). As the next step we must evaluate explicitly the left hand side of Eq. (3.3) since by the functional assumption, (0)

(0)

(0)

(0)

∂fa ∂fa ∂ni ∂fa ∂u ∂fa ∂T = + · + ∂t ∂ni ∂t ∂u ∂t ∂T ∂t where the local temperature T (r, t) is introduced as usual through the equation 3 (3.4) ρe(r, t) = nkT , n = na + nb 2 The time derivatives of the local variables are to be computed to first order in the gradients namely, through the Euler equations. Clearly, a similar (0) expression holds for ∂f∂ar which is unnecessary to write down explicitly. Also, (0)

(0)

fa ∂fa = ∂na na 2

(0)

(0)

(0)

ma fa ∂fa ∂fa = ca fa(0) ; = ; ∂u kT ∂T T

See Eqs. (31), (32) in Ref. [1].

ma c2a 3 − 2kT 2

(3.5)


27

Euler’s equations are readily obtained as follows. Firstly, from Eq. (2.17c) (the local variable is u!) noticing that Jieq = 0 since ci eq = 0 and div ci ∼ 2 we have

∂na = −div nau (3.6) ∂t 0 where subscript 0 is to emphasize the lowest (first) order in the gradients u) u + u div u = ρ d we do not ignore the nonterm. In Eq. (2.19) since ∂(ρ ∂t dt ← → linear term u · grad u; also τ = pI and Jc ∼ since Ja is a first order in term. Whence,

∂u + Jc × B + = −ρu · grad u − grad p + QE ni Fi (3.7) ρ ∂t 0 i Finally, since from Eq. (2.17b) ρ dtd e = de + ρe div u using Eqs. (3.4), and dt de ∼ 2 (3.6) it is readily seen that dt + ρe div u = 32 nk dT and since Jc · E dt is a first order in the gradients term, div Jq ∼ 2 , one finally because E arrives at the result,

∂T 2p div u (3.8) = − u · grad T + ∂t 0 3nk Eqs. (3.6)-(3.8) are the Euler equations of magnetohydrodynamics. (0)

Using Eqs. (3.6)-(3.8), the corresponding equation for ∂f∂ar and writing the explicit form for Jc in Eq. (3.7), after some tedious algebra one arrives at the following expression namely,

→ ma c2a 5 ma ←− na 0 (0) fa ca ca : grad u + − grad ln T · ca + ca · dab = kT 2kT 2 n

b ∂ ϕ(1) m e a a a (0) (1) · · ca + ca × B − f (0) ej −fa(0) d cj fj ϕj cj × B ma ∂v ρkT a j=a (1) (1) + C ϕ fa(0) C ϕ(1) + ϕ a a b

(3.9)

← → and an identical equation for species b. In Eq. (3.9) A 0 denotes the symmetric traceless part of any tensor A. Here,

28


na na nb (mb − ma ) grad p + dab = −dba = grad n nρ p Fb ρa ρb Fa na nb (mb ea − ma eb ) · E − − − pρ ma mb pρ

(3.10)

is the well known diffusive force accounting for all external forces and gradients in the system, others than gradT , which induces in the system the dependent term in E , is responcorresponding currents. In particular the E sible for all the electrical effects in the mixture arising from the presence of the electric field. Notice further the fact that the solution to Eq. (3.9) is now severely complicated by the presence of the first two terms in its r.h.s. and which really act as “collisional” contributions to ϕ(1) . containing B i The solution to Eq. (3.9) may be constructed by the standard procedure. = 0, Eq. (3.9) has no unique solutions since the We recall that when B homogeneous equation has five particular solutions, the collisional invariants. Hence there are solutions composed by a solution to the inhomogeneous term plus an arbitrary linear combination of the solutions to the homogeneous equation. The coefficients appearing therein are determined through the subsidiary conditions namely, ⎧ ⎫ m i ⎪ ⎪ ⎨ ⎬ (0) (n) m c i i d ci = 0 n ≥ 1 (3.11) f i ϕi ⎪ ⎩ 1 mi c2 ⎪ ⎭ i i 2 arising form the fact that we have arbitrarily chosen to determine the local (0) variables only through fi . The argument is still valid for Eq. (3.9) except that we now notice that the homogeneous equation only has ma and 12 ma c2a as particular solutions since, as the reader may readily verify the first two terms on the right hand a a side do not vanish for the collisional invariant maca . Recall that ∂ϕ = ∂ϕ ∂ va ∂ ca since u is constant in v -space, so that, ← → (1) i ( ϕi = Bi ( ci , T, B · · · ) : grad u + A ci , T, B · · · ) · grad ln T + i ( ci , T, B · · · ) · dij + α1 + α2 mi c2i D

(3.12)

← → are to be determined. Since these coefficients are conwhere B , A and D as we shall see below the resulting intestructed from the vectors ci and B grals, when (3.12) is substituted into the first and third condition in (3.11),


29

are odd in ci and therefore vanish. Thus the contributions from α1 + α2 mi c2i can be shown to vanish following the same arguments as in the non magnetic case and proven that α1 = α2 = 0. Thus, ← → (1) i · grad ln T + D i · dij (3.13) ϕi = Bi : grad u + A is the most general solution to (3.9) where the coefficients are the most ← → D which can be constructed from the only general tensor B , and vectors A, These are (non-skew vectors) base vectors available ci and the vector B. and ( ×B ci , ci × B ci × B)

(3.14) × B] ×B = −B 2 ci × B. since [ ci × B ← → ← → The term B : grad u where B is now the most general tensor that may be constructed from the set (3.14) and gives rise to viscomagnetic effects will be ignored in the subsequent discussion. This tantamounts to taking a shear free mixture for which gradu = 0. For the remaining two terms we have that i = AI (ci , B, · · · ) + AIII (ci , B, · · · )( ×B A ci + AII (ci , B, · · · ) ci × B ci × B) (3.15a) ci + DII (ci , B, · · · ) ci × B + DIII (ci , B, · · · )( ci × B) × B Di = DI (ci , B, · · · ) (3.15b) for i = a, b. The quantities AI , DI , · · · etc. are functions of all the scalars T ( that can be formed with these vectors namely ci = | ci |, B = |B|, ri , t), · · · etc. so from here on we shall omit writing their arguments. Even in this simplified case, what follows is much more cumbersome to = 0 case. When Eqs. (3.15a), (3.15b) are substituted deal with than in the B back in to Eq. (3.13) and this in turn plugged into the expressions for Jq and Ji we will get all direct effects like heat conduction and mutual diffusion , plus the cross effects plus the electromagnetic ones arising from the force E such as the Dufour and Soret effects and others. The program is to see if one ← → ← → ← → can construct a matrix L relating all of them and such that L = L † , the = 0 this Onsager reciprocity relations. For the much simpler case when B is readily accomplished by a direct extension of our recent work in ordinary mixtures (Ref. [1]) and the reader is referred to the original source. When = 0 this is still a challenge and an attempt to accomplish it, will be done B elsewhere.

30


We start by substituting Eq. coefficients of grad ln T and dab , equations for the vector functions fa(0) ma −fa(0) ρkT

ma c2a 5 − 2kT 2

ej

(3.13) back into (3.9) and equating the respectively. This leads to two integral i which read as: Ai and D

ca = −fa(0)

(0) j × B d cj f j A jc

ea · ∂ Aa ( ca × B) ma ∂ ca

a ) + C(A a +A b) · ca + fa(0) C(A

j

(3.16a) and

ea na · ∂ Da ca fa(0) = −fa(0) ( ca × B) n ma ∂ ca

ma (0) (0) −fa(0) ej c + f ) + C( D + D ) d cj f j D × B · c C( D j j a a a b a ρkT j

(3.16b) and two identical equations for species b. The vectors Aa and Da are themselves given by Eqs. (3.15a) and (3.15b) so that still involved manipulations have to be carried out to simplify these equations. Due to the similarity of the terms involved in both equations, it is only necessary to give the details ×B = B( cj · B) − B 2 cj we rewrite for a single case. Noticing that ( cj × B) Eq. (3.15a) as a = (A(1) − B 2 A(3) )ca + ca × BA (2) + B( ca · B)A (3) A a a a a (1) (1) (2) (3) and where Aa ≡ AI , etc. and Aa , Aa , Aa are scalar functions of |ci |, |B| although we shall arbitrarily neglect here the dependence on the latter |ci · B| (1) (3) term. Also, Aa − B 2 Aa is a coefficient that is a function of the scalars B 2 , (1) etc. so that we shall simply relabel it as Aa . Thus

a = A(1)ca + (ca × B)A (2) ca · B)A (3) A a a + B( a

(3.15’)

As shown in Appendix A when this function is substituted in (3.16a) the first two terms on the right side yield −fa(0)

ea 2 (2) · ∂ Aa = −fa(0) ea ( (1) ca × B)A ( ca × B) − [B c − B( c · B)]A a a a a ma ∂ ca ma


31

and

−ca ·

ej

(0) cj d cj f j A j (

× B)

(2) (1) − [B 2ca − B( ca · B)]G = (ca × B)G B B

j

where 1 = ej 2 j

(1) GB

1 = ej 2 j

(2) GB

(0)

(1)

(0)

(2)

d cj fj Aj [c2j −

1 2] ( cj · B) 2 B

(3.17a)

1 2] ( cj · B) B2

(3.17b)

and d cj fj Aj [c2j −

Therefore, Eq. (3.16a) yields,

ma c2a 5 (0) (0) ea (1) 2 (2) ( ca × B)Aa − [B ca − B( − ca · B)Aa ] fa ca = −fa 2kT 2 ma ma (0) (2) (1) − [B 2 ca − B( ca · B)]G ca × B)G fa ( + B B ρkT (1) (1) +fa(0) C( ) + C( c A + c A ) ca A(1) a a b b a (2) (2) (2) +fa(0) C ( ca × B)A + C ( c × B)A + ( c × B)A a b a a b +fa(0)

(3)

C( ca A(3) ca A(3) b Ab ) a ) + C( a +c

B ·B

(3.18) (i)

and an identical equation for species b. The corresponding equations for Da (i) and Db where i = 1, 2, 3 have exactly the same structure except that the inhomogeneous term is the one appearing in Eq. (3.16b). Moreover, it is and ( B that now clear from the three independent vectors ca , ca × B ca · B) we have three independent equations namely,

ma c2a 5 ea B 2 (2) ma B 2 (2) (0) − G ca Aa ca − fa(0) fa ca = fa(0) 2kT 2 ma ρkT b (1) (0) (1) (1) +fa ca Aa + cb Ab ) (3.19a) C( ca Aa ) + C(

32


ea (0) ma (1) (1) ( ca × B)G ( ca × B)A a + fa b ma ρkT (2) (2) A(2) + C ( c +fa(0) C ca × B × B)A + ( c × B)A a b a a b 0 = −fa(0)

(3.19b)

B, and taking out the factor B ea (2) (3) (0) ma (0) (3) (3) C( c c ca A(2) + f G + f A ) + C( c A + c A ) a a a b a a a a a b b ma ρkT (3.19c) This is a set of three linear coupled integral equations for the three functions (i) (1) (1) Aa characterizing ϕa . An analogous set exist for ϕb and the same reason(i) (i) ing leads to the equations for Da and Db coefficients. Eq. (3.19b) may be so that it reads, also simplified by factoring out ×B ea (1) (2) (0) ma (0) (2) (2) C( c c + f G + f A ) + C( c A + c A ) 0 = −fa(0) ca A(1) a a a b a a a a a b b ma ρkT (3.19b ) We may reduce this system to only two equations by the following trick. We multiply Eq. (3.19c) by B 2 and add it to Eq. (3.19a). The first two terms cancell out yielding

ma c2a 5 (0) − ca Ra ) + C( ca Ra + cb Rb )} (3.20a) fa ca = fa(0) {C( 2kT 2 0 = −fa(0)

(1)

(3)

where Ra = Aa + B 2 Aa . But this equation is identical to that obtained for an ordinary mixture so that the form the function Ri (i = a, b) is already known. We need not worry any more about its solution. On the other hand if we multiply Eq. (3.19b) by iB and add the result to Eq. (3.19a) we get that3

ma c2a 5 ma (0) − (iB) ca G fa ca = fa(0) 2kT 2 ρkT −fa(0)

ea ca (iB)Aa + fa(0) {C( ca Aa ) + C( ca Aa + cb Ab )} ma

(3.20b)

where (1)

(2)

Ai = Ai + iBAi (1) (2) G = GB + iBGB 3

B 2 = iB(−iB).

(3.21)


33

Eq. (3.20b) has a completely different structure as that of standard linear integral equations encountered in kinetic theory due to the appearance of the iB terms in the right hand side. Its solution remains as the main challenge of this section. Notice should be made of the fact that an identical procedure leads to the equations for the D coefficients. The final result is that na (0) 2 (3) 2 (3) ca (D(1) fa ca = fa(0) C( ca (D(1) a + B Da )) + C( a + B Da ))+ n (1) (3) cb (Db + B 2 Db )) (3.22a) and

ma ea na (0) fa ca = fa(0) (iB) ca K − fa(0) (iB) ca Di + n ρkT ma ca Da ) + C( ca Da + cb Db )} fa(0) {C(

where

(1)

(2)

Di = Di + iBDi (1) (2) K = KB + iBKB

and

1 1 (0) (1) 2 2 = ej d cj fj Dj cj − 2 (cj · B) 2 j B 1 1 (0) (2) 2 2 = ej d cj fj Dj cj − 2 (cj · B) 2 j B

(1) KB

(2)

KB

(3.22b) (3.23)

(3.24a) (3.24b)

The question is now how can we solve Eqs. (3.20b) and (3.22b) which as mentioned, are of a new type in kinetic theory. Nevertheless, before we do so (1) we must see if ϕi has no further restrictions imposed on it by the subsidiary conditions, Eqs. (3.11). Omitting the gradu term, substitution of Eq. (3.13) into Eq. (3.11) yields, ⎧ ⎫ 1 ⎪ ⎪ ⎨ ⎬ (0) c i mi f i Ai d ci · grad ln T + ⎪ ⎩ 1 c2 ⎪ ⎭ i 2 i ⎧ ⎫ 1 ⎪ ⎪ ⎨ ⎬ (0) c i mi f i Di d ci · dij = 0 1 ⎪ ⎪ 2 ⎩ c ⎭ i 2 i

34


Since the two terms are similar, we fix our attention in the first one. Using i we have for the non-vanishing integrals that, the expression for A (0) (1) (0) (2) × grad ln T )+ mi ci · grad ln T + fi Ai ci ci d ci · ( B fi Ai ci ci d i

(0) (3) B ci B( fi Ai ci ci d

· grad ln T )

=0

←−→ Since ci ci = ca 0 ca + 13 c2i I and noticing that all integrals with the symmetric traceless part vanish, we are left with three conditions, (0) (1) mi fi Ai c2i d ci · grad ln T = 0 i

(0) (2) × grad ln T ) = 0 ci I · ( B fi Ai c2i d

mi

i

mi

(0)

(3)

ci B 2 grad ln T = 0 fi Ai c2i d

i

ˆ Only then B( is taken, say along the z-axis, B = kB. B·grad provided B T) = 2 B grad T . This assumption is unnecessary but shall be kept for didactical ˆ be a unit vector along B = (Bx , By , Bz ). Multiply the reasons. In fact, let h ˆ so that h ˆ · grad T is the parallel component of grad T first condition by h ˆ ˆ = B · h) along h. The same operation with the third condition yields B( 2 ˆ B (h · grad T )h so adding both we get (0) (1) (3) ˆ h ˆ · grad T ) = 0 mi fi (Ai + B 2 Ai )c2i d ci h( i

from which the first of Eqs. (3.25) follows at once. Now multiply the first Eq. ˆ and add to the second one multiplied by i. Since I · h ˆ =h ˆ vectorially by h we get (0) (1) (2) ˆ × grad T ) = 0 mi fi (Ai + iBAi )c2i d ci ( h i

ˆ = ˆı cos θ + from which the second of Eqs. (3.25) follows at once. In general h ˆj cos φ sin θ + kˆ sin φ sin θ in a cartesian coordinate system, thus emphasizing is in any arbitrary direction. Nevertheless by choosing that in general B


35

many algebraic operations are simplified but this has to be z-axis along B kept in mind specially when discussing magnetohydrodynamics. (1) (3) (1) (2) So for Ri = Ai + B 2 Ai and for Ai = Ai + iBAi we finally get that the subsidiary conditions are: (0) 2 fi Ri ci d ci = 0 i mi (3.25) (0) 2 fi Ai ci d ci = 0 i mi (j)

and similar relations for the Di functions. Eqs (3.25) follow also if as menˆ ˆ so that B( = kB B · grad T ) = B 2 ∂T k. tioned, B ∂z The last step in this methodology is to propose convenient series expansions for the unknown functions in terms of a complete set of ortho-normal functions. For this purpose we select, as is now familiar in kinetic theory, (m) the Sonine (Laguerre) polynomials Sp . Recall briefly that [2], Sp(m)

=

p (−x)r (m + p)p−r r=0

where (m + p)q =

r!(p − r)! q

(m + p + s)

s=0

so we get that for the first terms, (0) Sm (x) = 1 ;

and

(1) Sm (x) = −x + m + 1 ;

x2 1 (2) Sm (x) = (m + 1)(m + 2) − (m + 2)x + 2 2! ∞ Γ(m + p + 1) (p) (q) δpq e−x Sm (x)Sm (x)xm dx = p! 0

(3.26)

are a few properties of these polynomials. We now propose that for species a, b, ∞ j = a, b (i) (i) (m) (aj )(m) S 3 (c2j ) (3.27) Aj = i = 1, 2, 3 2 m=0

(i) (aj )(m)

where the coefficients are functions of the scalars B 2 , n, T , etc. A (i) similar expression holds also for Dj which we shall not write down.

36


In the case of the functions Ai and Di introduced in Eqs. (3.21) and (3.23), we shall write (i) Aj

=

(1) Aj

+

(2) iBAj

=

∞

(m)

(m)

αj S 3 (c2j )

(3.28)

2

m=0

with (m)

αj

(1)

(2)

= (aj )(m) + iB(aj )(m) (m)

and similar equations for Di and dj respectively. Eqs. (3.27) and (3.28) are very useful to considerably reduce both, the structure of the integral equations (3.20b) and (3.22b), as well as the subsidiary conditions expressed by Eqs. (3.25). In fact introducing the dimensionless velocity mi ωi = ci (3.29) 2kT m 32 2 i (0) e−ωi and using the orthogonality condition noticing that fi = ni 2πkT for the Sonine polynomials, Eq. (3.26) one readily finds that Eqs. (3.25) reduce simply to b (1)(0) (3)(0) =0 (3.30a) ni a(i) + B 2 ai i=a b

(0)

ni a(i) = 0

(3.30b)

i=a (m)

thus implying that all coefficients a(i) for m > 0 are not restricted. By the same argument, it is readily seen that, ej nj (1) (1)(0) G(B) = kT a(j) (3.31a) mj j and (2)

G(B) =

ej nj j

and using Eq. (3.28), G=

mj

ej nj j

mj

(2)(0)

kT a(j)

(3.31b)

(0)

(3.32)

kT a(j)


Analogously, K=

37

ej nj j

mj

(0)

kT d(j)

(3.33)

With these results we can now begin to tackle the two immediate remaining questions namely, the evaluation of the physical fluxes Ja and Jq involving diffusive phenomena, heat conduction and corresponding cross effects. According to Eq. (2.15) the calculation of the conduction current requires evaluation of Ja which also involves, through dab , mechanical and thermal diffusion. Therefore chapter 4 will be devoted to the computation of the physical fluxes (or currents).

Bibliography [1] P. Goldstein and L. Garc´ıa-Col´ın, J. Non-equilib. Thermodyn. 30, 173 (2005). [2] See references in Chap. 1. Specially Ref. [8] part III.

39

Chapter 4 Calculation of the Currents 4.1

Diffusion Effects

We begin with calculating explicitly the expression for the mass current namely, Ja = ma ca fa(0) ϕ(1) ca a d which according to Eq. (3.13), omitting grad u terms yields Ja = ma

ca fa(0)

Aa · grad ln T + Da · dab dca

a and D a as well, have the form expressed in Eq. (3.15’). The next where A step is obvious. We introduce Eq. (3.15’) into Ja , use (3.27) and notice that the three integrals appearing in the resulting expression are fa(0)

∞ m=0

(m)

a(m)(i) S 3 caca dca = 2

na kT a(0)(i) I ma

when use is made of the fact that the contribution arising from the symmetric ←−→ traceless part ca 0 ca vanishes and we resort to Eq. (3.26). Here I is the unit tensor. L.S. Garc´ıa-Col´ın, L. Dagdug, The Kinetic Theory of a Dilute Ionized Plasma, Springer Series on Atomic, Optical, and Plasma Physics 53 © Springer Science + Business Media B.V. 2009

41

42

4 Calculation of the Currents

Using this result and Eq. (3.13) in the definition of Ja we get that grad ln T + a(2)(0) grad ln T × B+ Ja = na kT a(1)(0) a a (2)(0) (3)(0) ab d(1)(0) + dab × B ab Bd · grad ln T Ba (3)(0) + d d + B · d B a a a a (4.1) with an analogous expression for Jb (= −Ja ) remembering that dab = −dba . Eq. (4.1) is a very eloquent result: to know Ja we do not need the full information contained in the infinite series (3.27) but only six coefficients, three for each of the thermodynamic forces. This will considerably simplify the solutions to the integral equations derived in the previous section. Notice also how the magnetic field strongly influences the diffusive, thermal and mechanical effects giving rise to the mass flux Ja . This is readily appreciated examining Eq. (3.10) which even in the absence of an external field has two contributions, the first arising from the standard diffusive first two terms which includes and a second one arising from the term proportional to E the effect of the magnetic field. In order to understand the full character of Eq. (4.1) let us examine the structure of its terms, one by one in a three dimensional space assuming that the direction of the magnetic field is taken along the z-axis. Thus, ˆ = Bk B ˆ being a unitary vector along z-axis. Now we rearrange Eq. (4.1) convek niently. The third term is simply B 2 ∂T which may be combined with the ∂z (1)(0)

(3)(0)

+ aa B 2 ∂T . corresponding contribution in the first term to yield aa ∂z × grad T = B ˆj∂T − ˆı∂T , ˆj and ˆı being unit vectors along the Also, B ∂x ∂y y and x axis, respectively. With a similar procedure for vector dab , we are readily lead to the following expression, Ja = na k(a(1)(0) + a(3)(0) B 2 )(grad T ) + na ka(1)(0) (grad T )⊥ + a a a (1)(0)

(2)(0) 2 (3)(0) da + B da na kaa B(grad T )s + na kT + dab

ˆ (dab )⊥ + na kT d(2)(0) B(dab × k) (4.2) na kT d(1)(0) a a ˆ ∂T , (grad T ) ≡ ˆı∂T +ˆj∂T and (grad T ) ≡ where, for simplicity (grad T ) ≡ k ⊥ s ∂z ∂x ∂y ∂T ∂T ˆ ˆ ˆ −ˆı ∂y + ˆj ∂x (see Fig. (4.1)). Also dab · k k ≡ dab k is the total diffusion

4.1. Diffusion Effects

43

Figure 4.1: The different vectors which result from terms B×grad T and B·grad T = B ∂T ∂z . The fluxes can be parallel to B, perpendicular to B (x-y plane) and perpendicular to this plane and to the z-axis.

along the B direction and dab

⊥

= ˆı (da )x +ˆj (da )y . The term kˆ × dab merits

a closer look to be discussed later. Thus we have in general matter (and charge) flowing in the direction parallel to the magnetic field, in the (x, y) plane in a direction perpendicular to the field and in a third direction, the (−x, y) plane which is both perpendicular to the field and the (x, y) plane. = 0 the first three terms simply yield an expression for the ordinary When B thermal diffusion term, the Soret effect. Ja = na ka(1)(0) grad T a Soret

also in the absence of an electric field. To exploit Eq. (4.2) let us study the flow of electrical charge according to Eq. (2.15). For this purpose we write na nb (0) (mb ea − ma eb )E dab = d ab − pρ

(4.3)

(0) where d ab is the standard non electromagnetic part of dab (cf. Eq. (3.10)). = 0, Since u × B z

ˆ E = E + u × B = (Ex + uy B)ˆı + (Ey − ux B)ˆj + Ez k

44


where ˆ×E = (Ex + uy B)ˆj − (Ey − ux B)ˆı k so calling n2a nb kT (mb ea − ma eb ) ≡ θ pρ

(4.4)

and re-arranging terms, the electrical conduction current for species a is given by ˆ − d(1)(0) θ {(Ex + uy B)ˆı + (Ey − ux B)ˆj} − + B 2 d(3)(0) )Ez k Ja(c) = −θ(d(1)(0) a a a d(2)(0) θB {(Ex + uy B)ˆj − (Ey − ux B)ˆı} a

(4.5)

Using Eqs. (2.15), (4.4) and (4.5) we see that we have three electrical conductivities namely, σa =

n2a nb (ma + mb )2 2 e kT (d(1)(0) + d(3)(0) B2) a a ma mb pρ

(4.6a)

corresponding to the current flow parallel to the magnetic field if Ez = 0; a current in the (x,y) plane whose conductivity is, σa⊥ =

n2a nb (ma + mb )2 2 e kT d(1)(0) a ma mb pρ

(4.6b)

and the one corresponding to the current perpendicular both to the (x, y) plane and the direction of B, namely σas =

n2a nb (ma + mb )2 2 e kT d(2)(0) a ma mb pρ

(4.6c)

Further, If B = 0, σa = σa⊥ and the current flows along the direction of E. the respecif the electric field is fixed in a certain direction with respect to B tive flows can be easily derived from the general expression, Eq. (4.5). We further insist that to explicitly compute the conductivities, only three quan(i)(0) tities are needed, the coefficients da where i = 1, 2, 3 (since Ja = −Jb ). (0) The several cross effects which arise from Eq. (4.2) when dab is taken into account will be left out for a later discussion (see Chapter 6).

4.2. Flow of Heat

4.2

45

Flow of Heat

To study this quantity we use the same definition as in the no magnetic field case. The standard definition for a mixture, according to classical irreversible thermodynamics is [3, 7],

Jb 1 1 5 Ja Jq= Jq − + (4.7) kT kT 2 ma mb so that using Eqs. (2.12) and (2.23) we get that 1 Jq= kT i=a b

mi c2i 5 − 2kT 2

ci ci fi d

and substituting the explicit form for fi , Eq. (3.13), 1 Jq= kT i=a b

mi c2i 5 − 2kT 2

(0)

ci fi

a · dij d a · grad ln T ± D A ci

where the ± sign appears since dij = −dji . If we now make use of. Eqs. a we readily see that all six integrals that appear in a and A (3.15a,b) for D the resulting expression are of the form ∞ b b mi c2i 5 1 2 (0) (j)(m) (m) 2 ni (j)(1) 5 − ci f i ai S 3 (ci )dci = − kT a , 2 2kT 2 3 2 mi i m=0 i=a i=a after Eq. (3.26) has been used and integrations performed using Eq. (3.27). Thus we arrive at the result that b ni (1)(1) 5 (2)(1) + J q = − k 2 T ai grad T + ai grad T × B 2 m i i=a (3)(1) 2 ai B

5 − (kT )2 2

( grad T )

b ni (1)(1) (2)(1) (3)(1) B × dab + di (±) di B 2 dab dab + di m i i=a (4.8)

46


where ± means that a minus sign appears when i = b to account for the property that dab = −dba . Eq. (4.8) contains all possible contributions to the flow of heat namely those coming from gradT modified by the presence of the magnetic field together with all the cross effect provided by the purely (0) , diffusive term dab and the electromagnetic contributions coming from θE = 0 the ordinary thermal θ being defined in Eq. (4.4). Notice that if B conductivity is simply given by ni (1)(1) 5 a (4.9) κ = k2T 2 mi i i whose explicit values will be discussed later. Using the same notation as in ˆ Eq. (4.8) may be written as follows, = B k, the previous case with B 5 J q = − k 2 T 2

b ni (1)(1) (3)(1) (1)(1) (ai + B 2 ai )( grad T ) + ai ( grad T )⊥ mi i=a

(2)(1)

+ai

(1)(1)

(3)(1)

+ B 2 di (1)(1) (2)(1) (dab )⊥ + di (dab × B)] di

B( gradT )s + (±)T [(di

)(dab ) + (4.10)

Eq. (4.10) shows there are three different thermal conductivities. One corresponding to flow in the direction of the field b 5 2 ni (1)(1) (1)(3) ai + B 2 ai κ = k T 2 mi i=a

(4.11)

A second one corresponds to the heat flux in the xy plane, thus the thermal current flowing along the direction of the magnetic field and the one flowing = 0, there is a unique along the (x, y) plane are not the same. Only when B current along the direction of grad T , b 5 2 ni (1)(1) κ⊥ = k T a 2 mi i i=a

(4.12)

so that κ = κ⊥ . The third one is a thermal current flowing in the (−x, y) plane, perpendicular both to the z axis and the (x, y) plane with a thermal conductivity given by b 5 2 ni (2)(1) κs = k T a B (4.13) 2 mi i i=a

4.2. Flow of Heat

47

which is the so-called Righi-Leduc effect (see Ref. [1] p. 337 and Refs. [2], [3]). (0) The Dufour effect arising from the term dab as well as the electromagnetic contribution to the heat current can be easily extracted from the second line in Eq. (4.8) but we shall leave those details for a discussion in Chapter 6. (i)(1) What is altogether important to stress is that only the six coefficients aj (i)(1) and dj where i = 1, 2, 3 and j = a, b are needed to explicitly compute any transport property associated with the heat transport. With all this information in our hands it is now convenient to tackle the most difficult part of this work, namely the solution to the linear integral equations (3.20b) and (3.22b) which as said before, are not the standard type of equations appearing in the simple one component system in kinetic theory.

Bibliography [1] S. Chapman and T. G. Cowling, loc. cit. Chap. 1. [2] D. Miller, Chem. Rev. 60, 15 (1960). [3] R. Haase, Thermodynamics of Irreversible Processes; Addison-Wesley Publ. Co., Reading, Mass (1969). [4] R. Balescu, Transport Processes in Plasma Vol. I, Classical Transport; North Holland Publ. Co., Amsterdam (1988). [5] L. Garc´ıa-Col´ın, A. Sandoval-Villalbazo and A. L. Garc´ıa-Perciante, Physics of Plasmas 14, 012305 (2007); ibid 14, 089901 (2007). [6] L. S. Garc´ıa-Col´ın, A. L. Garc´ıa-Preciante, A. Sandoval-Villalbazo, J. Non-equilib. Thermodyn. 32, 379 (2007). [7] S. R. de Groot and P. Mazur, Non-equilibrium Thermodynamics, Dover Publications, Mineola N. Y. (1984).

49

Chapter 5 Solution of the Integral Equations As pointed out before, Eqs. (3.20a) and (3.22a) are of the standard type of linear integral equations encountered in the kinetic theory of neutral systems so we need not worry about their solutions at all. We shall give a summary of their main properties in Appendix C. Here we wish to deal with Eqs. (3.20b) and (3.22b) whose structure becomes quite complicated due to the presence of the three magnetic field dependent terms appearing in their right hand side. Since they essentially differ only in structure by their inhomogeneous term, let us fix our attention to one of them namely, Eq. (3.20b) whose solution will yield the values of the “a” coefficients required to compute the three thermal conductivities. We shall seek a solution to this equation using a variational method which is apparently due to Davison [1]. We take Ii (i = a, b) as a trial function for A and construct a functional D(Ii ) by multiplying the full equation (3.20b) by Iici integrating over ci and summing over i. Thus, b mi c2i 5 (0) (0) ei − D(Ii ) = (iB)ci dci Iici · −fi ci − fi 2kT 2 mi i=a b (0) mi (0) (iB)Gci + +fi dcici · fi (C(ci Ii )) + C (ci Ii + cj Ij ) (5.1a) ρkT i=a Now, from the definition of A = A(1) + iBA(2) and Eqs. (3.17a-b) the third term reads as

b 1 m 1 i (0) (0) 2 (iB) dci Iicici fi dci fj Ii c2j − 2 (ci · B) kT 2 B i=a j L.S. Garc´ıa-Col´ın, L. Dagdug, The Kinetic Theory of a Dilute Ionized Plasma, Springer Series on Atomic, Optical, and Plasma Physics 53 © Springer Science + Business Media B.V. 2009

51

52

5 Solution of the Integral Equations

but the curly bracket no longer depends on ci so this expression is equal to iB 1 (0) mi dci Ii c2i fi = 0 kT 3 j since Ii must obey the subsidiary conditions, Eq. (3.25). Thus, b mi c2i 5 iBei (0) (0) − D(Ii ) = fi Iici + dci Iici · −fi ci − 2kT 2 m i i=a

b

(0) dci fi

(C(ci Ii )) + C (ci Ii + cj Ij ) ci

(5.1b)

i=a

Now, we want to seek a solution which satisfies the extremal condition δD(Ii ) = 0 consistent with Eqs. (3.25). To perform the variation we use the expression for the linearized kernel C(ci Ii ) and C (ci Ii + cj Ij ) so the last two terms in the above expression may be written as (0) (0) · · · dci dc j dc i fi fj gij σ(cicj → c i c j )Ii · ci · i,j

(c i I i ) + (c j I j ) − (ci Ii ) − (cj Ij )

the convention of notation being the same as in Eq. (2.4). In fact this (0) (1) expression arises simply from Eq. (2.4) substituting fi by fi (1 + ϕi ) (1) and keeping terms linear in ϕi . Now, we perform the two transformations leading to the H theorem, exchange first i with j and next exchange ci with c i and cj with c j using the fact that σ satisfies Eq. (2.5). This yields for Eq. (5.1b), ei mi c2i 5 (0) (0) 2 − D(Ii ) = − dci ci dci fi I2i c2i + fi Ii − iB 2kT 2 mi i i 1 4 i,j

···

2 (0) (0) dci dc j dc i fi fj gij σ (c i I i ) + (c j I j ) − (ci Ii ) − (cj Ij )

where [ ]2 implies [ ] · [ ].


53

Now we carry out the variation of Ii and get that ei mi c2i 5 (0) (0) 2 − δD(Ii ) = − dci ci dci fi I2i c2i δIi + fi δIi − 2iB 2kT 2 m i i i 1 (0) (0) · · · dci dc j dc i fi fj gij σ 2 i,j (c i δI i ) + (c j δI j ) − (ci δIi ) − (cj δIj ) = 0 Next we perform all the H-theorem operations again over this last terms so that the last bracket appears only as ci δIi so that mi c2i 5 ei (0) (0) − δD(Ii ) = − dci ci δIi fi ci − 2iB fi c2i Ii + 2kT 2 mi i (0) (0) )=0 dc i dc j fi fj gij σci · j

for all of Ii . Hence the resulting equation for Ii is mi c2i 5 ei (0) (0) (0) − fi = −2iBci fi ci Ii + 2fi [C (ci Ii ) + 2kT 2 mi (0)

C (ci Ii + cj Ij )] + 2αfi ci

(5.2)

where α is an undetermined parameter that does not depend on i and requires that Ii satisfies Eq. (3.25). But we have shown in p. 51 that indeed this holds true since in the original Eq. (3.20b) the term with the G function vanishes so that we can assert that Eq. (5.2) is equivalent to Eq. (3.20b). This implies that Ii , a solution to Eq. (5.2) which yields a stationary condition for D(Ii ) is the appropriate solution to Eq. (3.20b). Notice further that if we take Ii = Ai + O(δ 2 ) where δ is small, Di (I) = D(Ai ) + O(δ 2 ) so that a not so good trial function gives already a reasonably good value for D(Ii ) which is what we are seeking for. Further, this now entitles us to fully use Eq. (3.27) and propose as a solution to Eq. (5.2), the Sonine expansion Ii = A i =

M m=0

(m)

(m)

ai S 3 (ci ) 2

(5.3)

54


so that to apply the variational procedure we must evaluate each term in Eq. (5.1a); the limit M in the sum, taken to label the order of approximations. Now M ni (1) (0) mi c2 5 15 (m) (m) i 2 − ai S 3 dci = kT a fi ci 2 2kT 2 2 mi i m=0 i i when Eq. (3.26) is used. The second term yields: iBei (0) dci c2i I2i fi = mi i M M ei ∞ (m) (m) (m ) (m ) (0) dci c2i ai S 3 (ci ) ai S 3 (ci )fi iB 2 2 mi 0 m=0 i m =0

which after a transformation to the dimensionless velocity ωi , setting m = m by the orthonormality property of Sonine polynomials, using Eq. (3.26), and carrying out the integration over ωi yields, ei ni (0) (1) 2 2 2 (0) iBei 2 6(ai ) + 15(ai ) = iBkT dci ci Ii fi mi m2i i i The third term in (5.1a) vanishes due to the subsidiary conditions so that we are left with the expression for the linearized collision kernel. However, its adequate handling requires of some results concerning the properties of collision integrals discussed at length in the Appendix B. In fact with the same approximation as used for the second term, namely M = 0, 1 the full expression for the linearized collision kernels, reads 5 (0) (0) (0) (1) 2 I≡ · · · fi fj dv i dv j σ(Ω)dΩgij ai ci − ai ci (ωi − ) 2 i,j

5 5 (0) (1) (0) (1) ai c i − ai c i (ωi2 − ) + aj c j − aj c j (ωj2 − ) 2 2

5 5 (0) (1) (0) (1) −ai ci + ai ci (ωi2 − ) − aj cj + aj cj (ωj2 − ) 2 2 Exchanging subscripts i and j and subsequently setting ci → c i , cj → c j , the same transformation used in proving the H-theorem, and using Eqs. (B.1) and (B.2) of appendix B it follows at once that 1 I=− ni nj [Gij , Hij ] (5.4a) 2 i,j


55

where (0)

(1)

(0)

(1)

Gij = ai ci − ai ci (ωi2 − 52 ) + aj cj − aj cj (ωj2 − 52 )

(5.4b)

Hij = Gij If we now define

(0)

(1)

Ki = ai ci − ai ci (ωi2 − 52 ) (5.5) Lj =

(0) aj cj

−

(1) ai cj (ωj2

−

5 ), 2

taking Mi = Ki , Mj = Lj and using Eq. (B.4) together with Eq. (5.5) after some tedious algebra we find that 5 5 2 (1) 2 2 2 I = 2na (aa ) ca (ωa − ), ca (ωa − ) 2 2 aa 5 5 2 (1) 2 2 2 +2nb (ab ) cb (ωb − ), cb (ωb − ) 2 2 bb 5 (0) 2 (0) (1) 2 +na nb (aa ) [ca , ca ]ab − 2aa aa ca , ca (ωa − ) 2 ab 5 5 2 +(a(1) [ca (ωa2 − ), ca (ωa2 − ) a ) 2 2 ab 5 (0) 2 (0) (1) 2 +(ab ) [cb , cb ]ab − 2aa ab cb , cb (ωb − ) 2 ab 5 5 (1) +(ab )2 cb (ωb2 − ), cb (ωa2 − ) + 2 2 ab 5 5 (0) (1) (0) (0) (0) (1) 2 2 +2aa ab [ca , cb ]ab −2aa ab cb , cb (ωb − ) −2ab aa [cb , ca (ωa − ) + 2 ab 2 ab 5 5 (1) (5.6) +2a(1) ca (ωa2 − ), cb (ωb2 − ) a ab 2 2 ab where the twelve collision integrals appearing here are evaluated in Appendix D. In that appendix we see that all of them, when properly written and evaluated using the dimensionless velocity ωi are related to a single collision integral [ωa , ωa ]ab ≡ ϕ, in the limit when ma >> mb . Writing I in terms of 1 ϕ, introducing the appropriate factor ( 2kT ) 2 for each ci in the collision term, mi and collecting all three terms composing Eq. (5.1a) we finally find that

56


15 D(Ii ) = − kT 2

ei ni (0) na (1) nb (1) (1) 2 2 2(ai ) + 5(ai ) a + a + 3iBkT ma a mb b m2i i

1 3 (0) (1) 13 (1) 2 M1 (0) 2 2 (a(0) aa aa + (a ) + (a ) a ) − ma ma 4ma a mb b M1 (0) (0) 3M12 (0) (1) 3 15M1 (1) 2 (0) (1) (ab ) − 2 −√ ab ab + a a − a a ma mb 2 ma mb a b mb a b 3 M1 (0) (1) 27M12 (1) (1) +3 a a − √ a a ma mb b a 2 ma mb a b √ √2 2M1 (1) 2 2 (1) 2 2 (5.7) +2kT ϕ (a ) na + (ab ) nb ma a mb +na nb ϕkT

which by the computational condition δD(I) = 0 will lead to a set of algebraic (0) (0) (1) (1) equations for the unknown coefficients aa , ab , aa and ab . Here, M1 = ma , ϕ is known and its value is given in Appendix C. We still have to ma +mb introduce the subsidiary condition given in Eq. (3.30b) namely, (0)

na a(0) a + nb ab = 0 Nevertheless before taking the variation of Eq. (5.7) it is convenient to write it in a slightly different way. To do so we introduce ea = −e, eb = e, the characteristic frequencies ωi = |emi |iB , and use Eq. (3.30b) to get that, D(Ii ) 15 =− ϕkT 2ϕ

na (1) nb (1) a + a ma a mb b

nb 5 nb 3i na 5 na (0) 2 (1) 2 (0) 2 (1) 2 ωb (ab ) + ωa (aa ) + ωb (ab ) − ωa (aa ) + − ma mb ϕ 2 ma 2 mb 1 3 (0) (1) 13 (1) 2 M1 (0) 2 2 +na nb (a(0) (aa )(aa ) + (a ) + (a ) a ) + ma ma 4ma a mb b √ M1 3 15M1 (0) 2 3 M1 (0) (1) (0) (0) (1) (0) (ab ) − 2 −√ a a + (a )(ab ) + √ a a ma mb a b 2 ma mb a ma mb b a

6i + ϕ

27M12 (1) (1) √ n2a (1) 2 − √ a a + 2 (aa ) + 2 ma mb a b ma 3

√

2M1 n2b (1) 2 (a ) mb mb b

(5.8)


57

We now take the variation of Eq. (5.8) collect terms in the independent (0) (1) (1) variation of δaa , δaa and δab and set each coefficient equal to zero which leads to a set of three equations with three unknowns, namely, (0)

(1)

(1)

A1 aa − A2 aa − A3 ab = 0 (0)

(1)

(1)

−B1 aa + B2 aa − 92 M12 ab = 5τ (0)

(1)

(1)

−C1 aa − 92 M1 aa + C2 ab =

(5.9)

5τ M1

when we set na = nb = n2 , a fully ionized plasma. Further, nτ = ϕ−1 is taken as the mean free collision time. The coefficients in Eq. (5.9) are given by, A1 = 2(1 + M12 − 2M1 − 6iωa τ ) A2 = −3(1 + M1 ) √ A3 = 32 M12 ( M1 − 1) B1 = −(1 + M1 ) B2 =

13 16

+

√ 4 3

(5.10)

2 − 5iωa τ

C1 = −1 + 12 M1 C2 = 5 +

4 3

2 M1

+ 5iωa τ

For the simple case B = 0 the resulting set is trivial to solve, at least approx1 )n , n ≥ 1 ∼ 0 in all additions, imately, by setting all powers of M1n ∼ ( 1836 namely 1 + M1n 1. The result is that (0)

aa = 2.94τ (1)

aa = 1.96τ (1)

ab =

(5.11)

0.7265 τ M1

as can be easily checked with a desk computer. The solution for B = 0 is outlined in Appendix E.

58


To compute the dis the procedure is completely similar. The form for the D(Ii ) function only changes in the inhomogeneous term which is now (0) given by nni fi ci . When this is multiplied by −τci and integrated over the velocities after setting M (m) (m) di S 3 (ci ) Ii = 2

m=0

and sum is carried over the two species we get that −n

b ni i=a

n

fi(0)ci · ci

M

(m)

(m)

di S 3 (ci )dci = 3kT 2

m=0

(0)

nb n2a n(1 − M1 ) ma na

(1)

The resulting algebraic set of equations for di , di , i = a, b is identical to the one given by Eq. (5.9) except that the inhomogeneous term changes from ⎛

⎞ 0 ⎝ 15 τ ⎠ 2 15 τ 2M1

⎛ to

⎝

3 (1 2

⎞ − M1 )τ ⎠ 0 0

(5.12)

Once more the solution in the simple case B = 0 is readily obtained to read (0)

da = 1.191τ (1)

da = 0.294τ (1)

db = 0.0234τ For the case B = 0 the solution is also outlined in Appendix E.

(5.13)

Bibliography [1] P. C. Clemmow and J. P. Dougherty; Electrodynamics of Particles and Plasmas, Addison-Wesley, Pub-Co., Reading, Mass (1990). [2] B. B. Robinson and I. B. Bernstein; Ann Phys, 110 (1962). [3] J. O. Hirschfelder, C. F. Curtiss and R. B. Byrd; The Molecular Theory of Liquids and Gases, John Wiley & Sons, New York (1964), 2nd ed. [4] W. Marshall; The Kinetic Theory of an Ionized Gas; U.K.A.E.A. Research Group, Atomic Energy Research Establishment. Harwell U.K. parts I, II, and III (1960).

59

Chapter 6 The Transport Coefficients In this section we wish to give a detailed account of all the transport coefficients related to the vectorial fluxes discussed in the previous chapters. These are the mass flux Ja (= −Jb ), the corresponding charge flux or electrical current Jc , closely related to Ja , and Jq the heat flux. In every case = B kˆ so that the magnetic field is chosen as the direction of the z-axis, B will follow from the for any vector, its different components respect to B decomposition illustrated in Fig. (4.1). We shall begin our discussion with the heat flux Jq which, according to Eq. (4.10) is given by 5 J q = − k 2 T 2

ni (1) (1)(1) (2)(1) αi (gradT ) + ai (gradT )⊥ + ai B(gradT )s + mi i

(1) (1)(1) (2)(1) (dij )⊥ + di (B × dij ) (±1)ab T δi (dij ) + di

(6.1)

where in the second term the minus sign has to be taken into account since (1) (1)(1) (3)(1) (1) (1)(1) dij = −dji . All the coefficients αi = ai + B 2 ai and δi = di + 2 (3)(1) B di are the coefficients given in Eqs. (5.11) and (5.13) and Appendix E. Also, the mean free time has been defined as the inverse of the collision integral [w 1, w 1 ]12 ≡ ϕ in terms of which all collision integrals may be expressed. Thus, 3√ 3 4(2π) 2 me 20 (kT ) 2 1 (6.2) nτ = = ϕ e4 ψ Here 0 =8.554 × 10−12 F/m, me is the electron mass and ψ is the shielding L.S. Garc´ıa-Col´ın, L. Dagdug, The Kinetic Theory of a Dilute Ionized Plasma, Springer Series on Atomic, Optical, and Plasma Physics 53 © Springer Science + Business Media B.V. 2009

61

62

6 The Transport Coefficients

or logarithmic function and is defined as 2

16kT d0 0 ψ = ln 1 + e2 where d0 is Debye’s length given by, d0 =

0 kT ne2

12 (6.3)

It follows then from Eq. (6.1) that there are three components for the which is unaffected by the field, conventional heat current, one parallel to B one in the (x, y) plane perpendicular to B and the third one in the (−x, y) plane whose direction is perpendicular to both (Jq ) and (Jq )⊥ . This is the current known in the literature as the Righi-Leduc effect, the thermal analog of the well known Hall’s effect in electromagnetism. But we also have three other contributions to the heat current arising from the diffusive dij . In a completely ionized gas, ni = n/2, grad(ni /n) = 0. In this case the ordinary or “Fickian” contribution to the heat current does not exist but there will be two other contributions arising from the pressure gradient present in the diffusive force, and another one which includes Thomson’s well . These known thermoelectric effect arising from the term proportional to E cross terms have been rarely dealt with in the literature. Let us first analyze the three conventional heat currents. Clearly from Eq. (6.1), using the fully me 1 ionized plasma condition and noting that M1 = m = 1836 < 1, we have that p (J q ) = −κ (gradT ) where

5 nk 2 T 2.01τ 4 me

(6.4a)

5 nk 2 T τ (38.7 + 2270x2 + 161x4 ) 4 1 m e

(6.4b)

5 nk 2 T τ (206x + 2644x3 ) 4 1 m e

(6.4c)

κ = and, in a similar fashion, κ⊥ =

κs =

6 The Transport Coefficients

63

Figure 6.1: The thermal conductivities as function of x for n = 1021 cm−3 and T = 107 K. The full line is κ , the dotted line is κ⊥ and the dashed line is κs .

where 1 = 19 + 2078x2 + 2650x4 + 9x6

(6.4d)

eB . The three conductivities are plotted in Fig. and x = ωe τ , where ωe = m e 5 6.1 for 0 < x ). Indeed according to classical irreversible thermodynamics [8] this term must be included to get Jq as done in section 4.2 of this book. Marshall on the other hand deals with it a la old fashioned form of Chapman and Cowling leading to complicated expressions for the thermal conductivities. Nevertheless the θ and κ have been here compared and the results shown in Fig. (7.2), see Ref. [10]. It is important to emphasize that the expression here used as definition of the heat flux allows a clear separation between thermodynamic fluxes and forces as has become standard since the classical work of de Groot and Mazur [8]. Thus we may assert that what we achieve in this book is a rather comprehensive derivation of the transport processes of a dilute inert plasma within the tenets of irreversible thermodynamics.

Bibliography [1] See Ref. [6] in Chap 5. [2] R. Balescu; Transport Processes in Plasmas; Vol. I. Classical Transport; North-Holland Publ. Co., Amsterdam (1988). [3] H. Grad; Principles of the Kinetic Theory of Gases; in Handbuch der Physik, S. Fl¨ ugge, Ed., Vol. 12; Springer-Verlag, Berlin (1958). [4] L. S. Garc´ıa-Col´ın and R. M. Velasco; J. Nonequilib. Thermodyn 18, 157 (1993) and Ref. cited therein. [5] R. M. Velasco and L. S. Garc´ıa-Col´ın; J. Stat. Phys., 69, 217 (1992). [6] R. M. Velasco, F. J. Uribe and L. S. Garc´ıa-Col´ın; Phys Rev. E, 66, 32103 (2002). [7] L. S. Garc´ıa-Col´ın, R. M. Velasco and F. J. Uribe; J. Nonequilib. Thermodyn. 29, 257 (2004). [8] S. R. de Groot and P. Mazur; Non-equilibrium Thermodynamics; Dover Publications, Inc., Mineola, N.Y. (1984). [9] See Ref. [5] Chap. 4. [10] L. S. Garc´ıa-Col´ın, A. L. Garc´ıa-Perciante and A. Sandoval-Villalbazo; Phys. Plasmas 14, 012305 (2007); ibid 14, 089901 (2007).

79

Part II Tensorial Transport Processes

Chapter 8 Viscomagnetism 8.1

The Integral Equation

← → In this section we wish to consider the contribution of the term B : gradu to the transport processes occurring in a magnetized dilute plasma. For this purpose we must seek the most general solution to Eq. (3.9) which reads as: ← → → (1) i · grad ln T + ← ϕi = B i : grad u + A D i · dij

(3.13)

← → where on account of the inhomogeneous term in that equation, B i has to be the most general symmetric traceless tensor which may be constructed and (ci × B) ×B = B( ci · B). Clearly all scalars from the vectors ci , ci × B, 2 , n, T, etc. appearing in such a tensor will be function of c2i , B 2 , (ci · B) When Eq. (3.13) is substituted back into Eq. (3.9) we get that → → mi ←− ei · ∂ ← ci 0 ci = (ci × B) B i− kT mi ∂ci mi ρkT

← → (0) ej dcj fj cj × B B j

← → ← → ← → · ci + C( B i ) + C( B i + B j ) (8.1)

j

for i, j = a, b Besides the unit tensor I only six symmetric traceless tensors may be constructed from the three independent vectors and by inspection they readily L.S. Garc´ıa-Col´ın, L. Dagdug, The Kinetic Theory of a Dilute Ionized Plasma, Springer Series on Atomic, Optical, and Plasma Physics 53 © Springer Science + Business Media B.V. 2009

83

84

8 Viscomagnetism

follow, (0) ← → τ i =I

←−→ (1) ← → τ i = ci 0 ci → (← τ i )αβ = (2)

1 2

β + ciβ (ci × B) α ciα (ci × B)

(3) → α (ci × B) β− (← τ i )αβ = (ci × B)

→ (← τ i )αβ = (4)

1 2

→ (← τ i )αβ = (5)

1 3

· ci )2 δαβ B 2 c2i − (B

(8.2)

β + ciβ B α (ci · B) − 1 (ci · B) 2 δαβ ciα B 3 1 2

α (ci × B) β +B β (ci × B) α (ci · B) B

(6) → β (ci · B) αB 2 − 1 B 2 (ci · B) 2 δαβ (← τ i )αβ = B 3

We now propose following the same technique as in the vectorial processes, that 6 ← → (n) →(n) τ j Γi ← for i = a, b (8.3) Bi= n=0 (n) → where Γi is a function of all scalars in the field. Notice that for all n, ← τ i is an even function of the velocities ci so upon substitution of Eq. (8.3) into Eq. (8.1) the integral (0) →(n) = 0 for all n τ i (cj × B) dcj fj ←

since its integrand is an odd function of cj , whence Eq. (8.1) reduces to 6 → mi ←− ei (n) →(n) · ∂ τ i + collision terms (ci × B) Γ ← ci 0 ci = kT mi ∂ci n=0 i

(8.4)

Moreover, (n) → · ∂ Γ(n) ← · τ i = (ci × B) (ci × B) i ∂ci

(n)

∂Γi ∂ci

(n) ← → τ i

· Γ(n) + (ci × B) i

→ ∂← τ i ∂ci

(n)

8.1. The Integral Equation (n)

for all n. But

∂Γi ∂ci

85 (n)

=

ci ∂Γi |ci | ∂ci

(n)

since Γi is a scalar function of ci whence, · ci = 0. Thus we need to evaluate the first term always vanishes, (ci × B) (n) → only the action of the operator gradci ≡ ∂∂ci on each tensor ← τ i . Clearly, (0) → ∂ ← τ i = 0 and the rest of the terms are given by, ∂ci

· (ci × B)

(1) (2) → → · ∂ ← τ i τ i = 2← (ci × B) ∂ci

(8.5a)

∂ ← (2) (1) (3) (4) → → → → τ τ i +← = −B 2 ← τ i +← τ i ∂ci i

(8.5b)

(3) (2) (5) → → → · ∂ ← (ci × B) τ i = −2B 2 ← τ i + 2← τ i ∂ci

(8.5c)

(4) (5) → → · ∂ ← τ (ci × B) =← τ i ∂ci i

(8.5d)

· (ci × B)

∂ ← (5) (4) (6) → → → = −B 2 ← τ i τ τ i +← ∂ci i

(6) → · ∂ ← (ci × B) τ i =0 ∂ci

(8.5e)

(8.5f)

Eqs. (8.5a), (8.5d) and (8.5f) follow almost by inspection but Eqs. (8.5b), (8.5c) and (8.5e) require some lengthy and abnoxious algebra. Probably the easiest way of proving them is by components αβ and after some hard work the equations prove to be correct. These equations show that the space spanned by the seven tensors of Eq. (8.2) form a closed space, acting by the operator gradci does not create new tensors. Nevertheless Eqs. (8.5) are rather inappropriate to manipulate Eq. (8.4) since the inhomogeneous ←−→ term contains ci 0 ci which does not appear in the base defined by Eqs. (8.2). Therefore we must perform a transformation to another base in which only ←−→ ←−→ the symmetric traceless tensors ci 0 ci and B 0 B appear as well as the constant tensors I and ijk the Levi-Civita tensor. Eight symmetric tensors may be

86

8 Viscomagnetism

constructed from these basic ones, namely, ← →(0) ( Q i )i,αβ = δαβ ←−→ ← →(1) Q i,αβ = δαγ δβλ (ci 0 ci )γλ ←−→ ← →(2) Q i,αβ = 12 (δαγ βλϕ + δβγ αλϕ )Bϕ (ci 0 ci )γλ ←−→ ←−→ ← →(3) Q i,αβ = αγϕ βλψ (B 0 B)ϕψ (ci 0 ci )γλ ←−→ ←−→ ← →(4) Q i,αβ = δαβ (B 0 B)γλ (ci 0 ci )γλ

(8.6)

←−→ ← →(5) Q i,αβ = c2i (B 0 B)αβ

←−→ ←−→ ←−→ ← →(6) Q i,αβ = 12 δαγ (B 0 B)βλ + δβγ (B 0 B)αλ (ci 0 ci )γλ ←−→ ←−→ ←−→ ← →(7) Q i,αβ = 12 ( βγϕ (B 0 B)αϕ + αγϕ (B 0 B)βϕ )Bλ (ci 0 ci )γλ ←−→ ←−→ ←−→ ← →(8) Q i,αβ = (B 0 B)αβ (B 0 B)αγ (ci 0 ci )γλ The construction of these tensors follow from certain rules on isotropic cartesian tensors too involved to explain here. Also, in Appendix F we outline the ← → → proof that the tensors ← τ given in Eq. (8.2) are related to the Q tensors as follows, ← →(0) ← → τi (0) = Q i ← →(1) ← → τi (1) = Q i ← →(2) ← → τi (2) = Q i ← →(7) ← → τi (5) = Q i (8.7) ← →(3) ← →(4) ← →(1) ← →(5) ← → τi (3) = Q i + 13 Q i − 13 B 2 Q i − 13 Q i ← →(6) ← →(4) ← →(1) ← →(5) ← → τi (4) = Q i − 13 Q i − 13 B 2 Q i + 13 Q i ← →(6) ← →(5) ← → τi (4) = Q i + 13 B 2 Q i

8.1. The Integral Equation

87

The first and last lines follow at once, but the two middle expressions require ← →(3) ← →(6) → → τi (4) = Q i + Q i and a little bit of work. Notice however that, ← τi (3) + ← ← →(3) ← →(6) ← →(4) →(5) ← →(1) ← → → τi (4) = Q i − Q i + 23 ( Q i − B 2 Q i − Q i ). With these that ← τi (3) − ← relations and Eqs. (8.5a-8.5f), it is readily seen that · (ci × B) · (ci × B) · (ci × B)

→(2) ∂ ← Qi ∂ci

· (ci × B) · (ci × B)

→(4) ∂ ← Qi ∂ci

→(1) ∂ ← Qi ∂ci

=0

← →(2) =2Qi

→(3) ← →(6) ← →(1) ← = −B 2 Q i + Q i + Q i

→(3) ∂ ← Qi ∂ci

← →(7) ← →(2) = − 43 B 2 Q i + 2 Q i

· = (ci × B)

· (ci × B)

→(0) ∂ ← Qi ∂ci

→(6) ∂ ← Qi ∂ci

→(5) ∂ ← Qi ∂ci

· = (ci × B)

→(8) ∂ ← Qi ∂ci

(8.8) =0

← →(7) ← →(2) = Q i − 23 B 2 Q i

← →(8) ← →(6) ← →(4) ← →(1) = Q i − B 2 Q i + 13 (B 2 Q i − B 4 Q i )

(n) = 0 for n = 4, 5 and 8. · (ci × B) where use is made the fact that ∂∂ci Qi If instead of Eq. (8.3) we now propose that · (ci × B)

→(7) ∂ ← Qi ∂ci

← → n← →(n) B = Υi Q i 8

(8.9)

n=0

then it is easy to show, equating Eqs. (8.3) and (8.9) and using Eqs. (8.7), that one may obtain the scalars Γni in terms of the Υni ’s. The result is that, Γ0i = Υ0i Υ1i = Γ1i − 13 B 2 Γ3i + 13 B 4 Γ4i Γ2i = Υ2i Γ3i = Υ3i Υ4i = 13 (Γ3i − Γ4i ) Υ5i = 13 (Γ4i − Γ3i + B 2 Γ6i )

(8.10)

88

8 Viscomagnetism

Υ6i = Γ4i Υ7i = Γ5i Υ8i = Γ6i (n) → · ∂ acts upon ← Furthermore, notice that when (ci × B) τ i it generates the ∂ci 2 . Thus, the scalar same tensor multiplied by c2i and/or B 2 never by (ci · B) functions that appear in Eqs. (8.3) or in Eq. (8.9) will not depend on this last term. This means that we may safely assume that the Υni ’s will not depend on this scalar. We are now in a position to deal with the integral given by Eq. (8.4) since ←−→ ← → all tensors Q ni now contain the symmetric traceless tensor ci 0 ci . Substitution of Eq. (8.9) into Eq. (8.4) and collecting terms we get a set of nine integral ←−→ equations for the Υni functions namely, C(Υ0i , ci 0 ci ) = 0 which implies Υ0i = 0 and,

←−→

mi c 0 ci kT i

0=

ei mi

=

ei mi

←−→ ←−→ B 2 Υ2i + 13 B 4 Υ7i ci 0 ci + C(Υ1i ci 0 ci )

2

←− ←−→ → 4 2 3 2 6 1 2 0 0 B Υ + B Υ − 2Υ c + C(Υ ci ci ) c i i i i i i 3 3 ←−→ ←−→ 0 = − meii Υ2i ci 0 ci + C(Υ3i ci 0 ci ) ←−→ ←−→ 0 = − 13 meii B 2 Υ7i ci 0 ci + C(Υ4i ci 0 ci ) ←→ 0 = C(Υ5i , c0i ci )

0=

ei mi

(8.11) ⇒

Υ5i = 0

←−→ ←−→ (B 2 Υ7i − Υ2i ) ci 0 ci + C(Υ6i ci 0 ci )

←−→ ←−→ 0 = − meii (2Υ3i + Υ6i ) ci 0 ci + C(Υ7i ci 0 ci ) ←−→ −→ (8) ← 0 = − meii Υ7i ci 0 ci + C(Υi ci 0 ci ) Eqs. (8.11) is a set of eight linear coupled differential equations for the Υni ←−→ functions (Υ0i = 0) where C(Υni ci 0 ci ) is an abbreviation for the full linearized collision term containing all interactions between the two species i = a, b (see


89

Eq. (8.1)). Besides knowing already that also Υ5i = 0 as pointed out above, we now proceed to find out if the subsidiary conditions, namely ⎧ ⎫ 1 ⎪ ⎪ b ⎨ ⎬ (0) (1) c i d ci = 0 m i f i ϕi ⎪ ⎩ 1 c2 ⎪ ⎭ i=a 2 i impose further conditions on the Υni ’s. Since for our purposes ← → (1) ϕi = Bi : grad u ← → and Bi is given by Eq. (8.9) we get that ⎧ ⎫ 1 ⎪ ⎪ 8 b ⎨ ⎬← (n) →n (0) c i mi f i Υi ci = 0 Q i d ⎪ ⎩ 1 c2 ⎪ ⎭ n=0 i=a 2 i ←−→ ← → (0) Now, Υi = 0 and for the remaining terms all Q ni are proportional to ci 0 ci ←−→ ← → except Q 5i = B 0 Bc2i . So except for this tensor all other integrals are odd ←−→ functions of ci and those who are not vanish since ci 0 ci is a traceless tensor that it reduces to ⎧ ⎫ 1 ⎪ ⎪ b ⎨ ⎬ ←−→ (0) (5) c i B 0 Bc2i d mi f i Υi ci = 0 1 ⎪ ⎪ 2 ⎩ ⎭ i=a c 2 i (5)

This implies that to satisfy the vanishing of the integral even in ci , Υi = 0 as already seen, which in turn implies that the set in Eq. (8.11) is reduced indeed to seven integral equations. Using this result and inverting Eqs. (8.10) we find that (2)

(5)

Υ2i + 13 B 2 Υ7i = Γi + 13 Γi 2 2 6 B Υi 3

(3)

(1)

+ 43 B 2 Υ3i − 2Υ1i = 2(B 2 Γi − Γi ) (8.12) B

2

(7) Υi

−

Υ2i

=

(5) B 2 Γi (3)

−

(2) Γi

(6)

2Υ3i − Υ4i = 2Γi + Γi

90

8 Viscomagnetism

Eqs. (8.12) simply imply that solving for the Υ functions will allow the computation of the early Γi functions introduced in Eq. (8.3) which, as we shall see, are more convenient when determining the coupling of the stress tensor with the magnetic field. Writing the integral equations in terms of the Γni functions we easily find that, → − → mi ←− ei 2 (2) 1 2 (5) 1 2 (4) (1) (3) ← 0 0 ci ci = B (Γi + B Γi )+C Γi + B (Γi − Γi )ci ci (8.13a) kT mi 3 3 −→ −→ 2ei 2 (3) (1) ← (2) ← (B Γi − Γi )ci 0 ci + C(Γi ci 0 ci ) mi → −→ ei (2) ←− (3) ← Γi ci 0 ci + C(Γi ci 0 ci ) 0= mi → −→ 1 (3) −ei 2 (5) ←− (4) ← (B Γi )ci 0 ci + C( (Γi − Γi )ci 0 ci ) 0= 3mi 3 −→ −→ ei (5) (2) ← (4) ← (B 2 Γi − Γi )ci 0 ci + C(Γi ci 0 ci ) 0= mi −→ −→ −ei (3) (4) ← (5) ← (2Γi + Γi )ci 0 ci + C(Γi ci 0 ci ) 0= mi → −→ −ei (5) ←− (6) ← Γi ci 0 ci + C(Γi ci 0 ci ) 0= mi 0=

(8.13b) (8.13c) (8.13d) (8.13e) (8.13f) (8.13g)

Moreover, using some straightforward transformations we may still reduce this set to three linear coupled integral equations. Indeed, multiplying Eq. (8.13g) by 13 B 2 , remembering that C is a linear operator and comparing the resulting equation with Eq. (8.13d) we immediately see that (6)

Γi =

1 (3) (4) (Γi − Γi ) 2 B

(8.14a)

Using this result once more in Eq. (8.13g), yields 2 → − → B ei 4 (5) ←− (3) (4) ← 0 0 (Γ − Γi )ci ci B Γi c i c i = C 3mi 3 i which when used in (8.13a) reduces this equation to a simpler one, → −→ → mi ←− ei 2 (2) ←− (1) ← ci 0 ci = B Γi ci 0 ci + C(Γi ci 0 ci ) kT mi

(8.14b)


91

Substitution of Eq. (8.13e) into (8.13d) yields that 0=−

→ −→ ei (2) ←− (3) ← Γi ci 0 ci + C(Γi ci 0 ci ) mi

(8.14c)

Multiplying this equation by B 2 and add to Eq. (8.14b) to get ←−→ → mi ←− ci 0 ci = C(Li ci 0 ci ) kT

(8.15a)

where (1)

(3)

Li = Γi + B 2 Γi

(8.15b)

Eqs. (8.15a,8.15b) are the first of the three sought equations. Notice now its immense simplicity. Now look at Eqs. (8.14b), (8.13b) and (8.14c). We rewrite (8.14b) as, → −→ → −iBei (2) ←− mi ←− (1) ← (iΓi B ci 0 ci ) + C(Γi ci 0 ci ) ci 0 ci = kT mi Now multiply Eq. (8.14c) by −B 2 = (iB)(iB) and add to this one so that,

→ −2iBei (2) ←− → −→ mi ←− (2) (3) ← ci 0 ci = (iΓi B ci 0 ci ) + C (iΓi B − B 2 Γi )ci 0 ci kT mi Multiplying Eq. (8.13b) by iB and adding the result to this one, yields, → −2iBei (1) −→ mi ←− (2) (3) ← ci 0 ci = (Γi + iΓi B − B 2 Γi )ci 0 ci + kT mi

←− → (1) (2) 2 (3) 0 C (Γi + iΓi B − B Γi )ci ci Define: (1)

(2)

(3)

Gi = Γi + iΓi B − B 2 Γi to get,

→ ←−→ → −2iBei ←− mi ←− Gi ci 0 ci + C(Gi ci 0 ci ) ci 0 ci = kT mi

(8.16a) (8.16b)

Eq. (8.16b) is the second desired integral equation. Notice that from its solution, (1) (3) Re Gi = Γi − B 2 Γi (8.17) (2) Im Gi = BΓi

92

8 Viscomagnetism

From Eqs. (8.15b) and (8.17), (1)

Γi = (Li )B=0 = (Re Gi )B = 0 represents a consistency condition for the solutions. To obtain the last equation we rewrite Eq. (8.13f), after multiplication by B 2 , as follows, → ei B 2 (4) −→ −2ei B 2 (3) ←− (5) ← Γi c i 0 c i = Γi − C(B 2 Γi ci 0 ci ) mi mi If we now substitute this equation into Eq. (8.13b) multiplied by iB, we get that

−→ −→ −iBei (1) (4) ← (2) (5) ← 0= (2Γi + B 2 Γi )ci 0 ci + C (iB 2 Γi + B 2 Γi )ci 0 ci mi Multiply Eq. (8.13e) by B 2 and add the result to this equation to get 0=

−→ −iBei (1) (2) (4) (5) ← (2Γi − iBΓi + B 2 Γi + iB 3 Γi )ci 0 ci + mi

−→ (4) (2) (5) ← C (B 2 Γi + iBΓi + iB 3 Γi )ci 0 ci

Now we multiply Eq. (8.14b) by Eq. (8.2) and add it to this equation to get finally that → ←−→ → −iBei ←− 2mi ←− ci 0 ci = Pi ci 0 ci + C(Pi ci 0 ci ) (8.18a) kT mi where (1) (4) (2) (5) (8.18b) Pi = 2Γi + B 2 Γi + iB(Γi + B 2 Γi ) Eqs. (8.18a) and (8.18b) are the third desired equations. Notice that, (1)

(4)

Re Pi = 2Γi + B 2 Γi (2)

(5)

Im Pi = B(Γi + B 2 Γi ) (6)

Γi =

(3) 1 (Γi B2

(8.19)

(4)

− Γi )

which together with Eqs. (8.15b) and (8.17) determine completely the six (n) (0) unknown functions Γi (since Γi = 0). Furthermore, examination of the

8.2. The Stress Tensor

93

integral equations (8.15a), (8.16b) and (8.18a) clearly indicates that we only need to solve the generic equation (8.16b). Indeed, Eq. (8.18a) reads, → → −iBei ←− mi ←− 1 ←−→ Pi ci 0 ci + C( Pi ci 0 ci ) ci 0 ci = kT 2mi 2

(8.20)

which is identical to (8.16b) if in equation (8.20) 12 Pi is identified with Gi and B is changed to 2B. On the other hand Eq. (8.15a) follows from (8.20) setting B = 0 and Li ≡ Gi = 12 Pi . Thus we have reduced the problem to the solution of a single linear integral equation, equation (8.21) which is readily solved by the same variational method described in Chapter 4. Nevertheless before embarking in this calculation it is very convenient to first study the general structure of the stress tensor. As we shall see, this will simplify the task of solving Eq. (8.21), → ←−→ → −2iBei ←− mi ←− Gi ci 0 ci + C(Gi ci 0 ci ) ci 0 ci = kT mi

8.2

(8.21)

The Stress Tensor

As we pointed out in Eq. (2.18), the stress tensor for the plasma is defined as b ← → m c τ = f c c d i

i i i

i

i=a

i and D i are all Substitution of Eq. (3.13) into this definition, noticing that A linear functions of ci times even scalar functions of this variable, consistently with Curie s principle the terms in grad T and dij vanish and we are thus left with b ← → (0) ← → m c B : grad u (8.22) τ = f c c d i

i

i i

i

i

i=a

← → Using now the proposed expansion for B i given in Eq. (8.3) we obtain that b 6 (0) → ← → mi d Γni ← (8.23) τ ni : grad u τ = ci fi ci ci i=a

n=1

→ → τ − pI, p being the hydrostatic pressure. Eq. (8.23) indicates where ← τ =← that the evaluation of each term in the integral has to be performed individu→ ally for each of the tensors ← τ ni listed in Eq. (8.2). Now, the scalar functions

94

8 Viscomagnetism

Γni which are unknown, are now expanded in a complete set of orthonormal functions the Sonine polynomials of order 5/2. Thus, Γni

=

∞

(p)(n)

αi

(p)

S 5 (c2i )

(8.24)

2

p=0 (p)(n)

where the αi will be functions of the scalars n, T , B, c2i and so on. Substitution of Eq. (8.24) in Eq. (8.23) apparently complicates things, since ← → τ =

b

mi

6

(0) d ci fi ci ci

n=1

i=a

∞

(p)(n)

αi

→ S 5 (c2i )← τ ni : grad u (p)

(8.25)

2

p=0

→ From the structure of the tensors ← τ ni we see that all of them are bilinear functions of ci whence they have the form of | ci | times some angular depen2 dence. The same happens with the dyad ← c→ c i i ≡ ci × angular dependence. Thus the integral over c (magnitude) is the same for all terms in Eq. (8.25) and has the form

∞ 0

(0) (p) c6 fi S 5 d ci 2

m 32 2kT 72 ∞ 2 (p) (0) i = ni wi6 e−wi S 5 S 5 dwi 2 2 2πkT mi 0

when using dimensionless velocities (c.f. Eq. (3.29)). But by the properties of the Sonine polynomials we finally get that

∞

(0) (p) c6 fi S 5 d ci

0

2

= ni

2kT mi

2

15 δp,0 16π

so that Eq. (8.25) now reads, ← → τ =

b 6 ni (0)(n) ← →n 2 15 (2kT ) αi u dΩ(← c→ i ci )Ω ( τ i )Ω : grad m 16π i n=1 i=a

(8.26)

where dΩ = sin θdθdϕ and the subscript Ω means taking the angular part only of the involved quantities. Moreover the infinite number of terms involved in (n) Eq. (8.24) for the Γi ’s reduce to only six coefficients for each species since (n) (0)(n) Γi = α i . This considerably simplifies all calculations. Now the rest of the calculations are a simple matter of algebra.


95

Let us define, ← →n 15 (0)(n) α I = 4π i

2π

π

dϕ 0

→ sin θdθ(cici )Ω (← τ ni )Ω : grad u

0

In general → → → ← → x x x τ ni )xx ∂u + (← τ ni )xy ∂u + (← τ ni )xz ∂u τ ni : grad u = (← ∂x ∂y ∂z → → → y y y +(← τ ni )xy ∂u + (← τ ni )yy ∂u + (← τ ni )zy ∂u ∂x ∂y ∂z → → → z z z +(← τ ni )xz ∂u + (← τ ni )yz ∂u + (← τ ni )zz ∂u ∂x ∂y ∂z → → and (← τ ni )kl = (← τ ni )lk since it is a symmetric tensor. ←−→ → Taking n = 1, ← τ 1i = ci 0 ci = ci ci + 13 I

x ∂uz 2 1 2 ∂ux ∂uy ← → ∂ux n + ∂x τ i : grad u = cx − 3 c ∂x + cx cy ∂y + ∂x + cx cz ∂u ∂z

z 2 1 2 ∂uy y z + c2z − 13 c2 ∂u + ∂u + cy − 3 c ∂y + cy cz ∂u ∂z ∂y ∂z Now, ← → ( I (1) )xx = ×

cos2 ϕ sin2 − 13

15 (0)(n) α 4π i

∂ux ∂x

2π 0

dϕ

π 0

sin θdθ cos2 ϕ sin2 θ

y 2 1 ∂uz + sin2 ϕ sin θ2 − 13 ∂u + cos θ − ∂y 3 ∂z

using spherical coordinates. Using further the auxiliary table of integrals in Appendix G we see that, ← →(1) ∂ux 1 (0)(1) 2 − div u + ( I )xx = αi 3 ∂x By an identical procedure,

← →(1) ∂uy 1 (0)(1) 2 − div u + ( I )yy = αi 3 ∂y ← →(1) ∂uz 1 (0)(1) 2 − div u + ( I )zz = αi 3 ∂z

Now, ← → ( I 1 )xy =

15 (0)(1) α 4π i

2π 0

dϕ

π

2

2

∂ux ∂y

∂uy ∂x

2

0

dθ sin θ cos ϕ sin ϕ sin θ

=

(0)(1) αi

+

∂ux ∂y

+

∂uy ∂x

96

8 Viscomagnetism

Also, ← → ( I 1 )xz =

15 (0)(1) α 4π i

2π 0

π

dϕ

0

dθ sin θ cos2 ϕ sin2 θ cos2 θ (0)(1)

= αi

∂ux ∂z

+

∂uz ∂x

∂ux

+

∂z

∂uz ∂x

→ and similarly for the yz component. Thus, ← τ − pI for n = 1 is determined by these six quantities. → From the structure of ← τ (2) in Eq. (8.2) we see that 2 ∂ux ∂uy → 1← 1 ∂ux (2) 2 τ : grad u = cix ciy ∂x − 2 cix − ciy + ∂x B ∂y + 12 ciz ciy

∂ux ∂z

+

∂uz ∂x

+

y ciy cix ∂u ∂y

−

1 c c 2 iz ix

∂uy ∂z

+

∂uz ∂y

← → ← → ← → ← → The only non vanishing terms in ( I (2) ) are ( I (2) )xx , ( I (2) )yy , ( I (2) )zz , ← → ← → ← → ( I (2) )xy , ( I (2) )zy and ( I (2) )zx . ← → ( I 2 )xx =

15 (0)(2) α B 4π i 2 2

2π 0

dϕ

π

dθ sin θ 12 cos2 ϕ sin 2 θ

0

x (cos ϕ sin θ − sin2 ϕ cos2 θ) ∂u + ∂y

(0)(2) y x B ∂u + ∂u = −αi ∂y ∂x

← → ( I 2 )yy =

2π 15 (0)(2) α B 4π i 0 2 2

dϕ

π 0 2

2 dθ sin θ( 12 sin2 ϕ sin θ)

x (cos ϕ sin θ − sin ϕ cos2 θ) ∂u + ∂y

(0)(2) y x B ∂u + ∂u = αi ∂x ∂y

← → ( I 2 )zz

← → ( I 2 )xy =

15 (0)(2) α B 4π i 2 2

=

∂uy ∂x

2π

dϕ

π

∂uy ∂x

dθ sin θ

0 0

∂uy x + (− 12 )(cos ϕ sin θ − sin2 ϕ cos2 θ) ∂u ∂y ∂x 2π 2 15 4 (0)(2) 2 B 0 dϕ(cos ϕ − sin ϕ) = 0 = − 8π 15 αi

15 (0)(2) α B 8π i

2π 0

dϕ

π

2

2

2

2

dθ sin θ cos ϕ sin θ sin ϕ sin θ

(0)(2) ∂uy x B ∂u − = 12 αi ∂y ∂x

0

∂ux ∂y

2π π ← → 2 15 (0)(2) 1 2 2 ( I 2 )xz = − 8π αi B 0 dϕ 0 dθ sin θ 2 cos θ sin θ cos ϕ (0)(2)

= − 12 αi

∂uy ∂z

+

∂uz ∂y

−

∂uy ∂x


97

→ Take n = 3. From the structure of ← τ 3 we see immediately that

(3) ∂uy ← → 2 ∂ux x τ i : grad u = 13 2c2iy − c2ix B 2 ∂u − c c B + ix iy 2 2∂x∂uy 1 2 ∂y2 ∂x 1 2 z + 3 2cix − ciy B ∂y − 3 2cix − ciy B 2 ∂u ∂z Using the fact that c2ix + c2iy = c2 − c2iz and carrying out the integrals over the angles it is now straightforward to see that

← →3 (0)(3) 2 1 5 ∂uy ∂uz ( I )xx = αi B − 3 div u + 3 ∂y − ∂z

← → (3)(0) 2 ∂uy ∂uz x B 5 ∂u − − ( I 3 )yy = 13 αi ∂x ∂y ∂z ← → (3)(0) 2 1 z B 3 div u − ∂u ( I 3 )zz = 3αi ∂z

← →3 (3)(0) 2 ∂ux ∂uy B + ( I )xy = −αi ∂y ∂x all other components are zero. For n = 4, (4) → 1 ← τ i B2

x : grad u = − 13 c2iz div u + c2iz ∂u + 1c c ∂z 2 ix iz

+ 12 c2iy c2iz

∂uy ∂z

+

∂ux ∂z

+

∂uz ∂x

∂uz ∂y

Once more after integrating over the angles we get that ← → (0)(4) 2 1 z B − 3 div u + ∂u ( I 4 )xx = αi ∂z ← → (0)(4) 2 ∂ux z B ∂z + ∂u ( I 4 )xz = 12 αi ∂x

← → (0)(4) 2 ∂uy ∂uz B + ( I 4 )yz = 12 αi ∂z ∂y ← → ← →4 (0)(4) 2 z = 13 ( I 4 )zz B − 13 div u + ∂u ( I )yy = αi ∂z For n = 5 we have that x ← → τ (5) : grad u = 12 B 3 ciz ciy ∂u + ∂z so that

∂uz ∂x

− 12 B 2 ciz cix

← → (0)(5) 3 ∂ux z B ∂z + ∂u ( I 5 )yz = 12 αi ∂x

← →5 1 (0)(5) 3 ∂uy ∂uz B + ∂y ( I )xz = − 2 αi ∂z

∂uy ∂z

+

∂uz ∂y

98

8 Viscomagnetism

and all other terms are zero by parity considerations. Finally, for n = 6 we have that,

z ∂uy ← → 1 4 2 ∂ux (6) : grad u = − 3 B ciz ∂x + ∂y + 23 B 4 c2iz ∂u τ ∂z z = B 4 c2iz − 13 div u + ∂u ∂z so that,

← → (0)(6) 4 1 z B − 3 div u + ∂u ( I 6 )xx = αi ∂z ← →6 (0)(6) 4 z B − 13 div u + ∂u ( I )yy = αi ∂z ← →6 (0)(6) 4 z B − 13 div u + ∂u ( I )zz = 3αi ∂z

Now we collect terms in Eq. (8.26), → (← τ − pI) =

b 6 ← →(n) ni 2 (kT ) I m i n=1 i=a

and we proceed in each case by components. Thus from the results in all previous pages, → → (1) ← (2) ← → ni 2 2α (kT ) − 2α B S xy S (← τ − pI)xx = bi=a m xx i i i ← → (3) +αi B 2 (2 S yy

← → ← → ← → (4) (6) − S zz ) + αi B 2 S zz + αi B 4 S zz

← → where we have defined the tensor S as follows, ← → 1 ∂uα ∂uβ 1 S αβ = + − div u δαβ 2 ∂xβ ∂xα 3 → (← τ − pI)yy =

b

ni 2 i=a mi (kT )

(8.26b)

→ → (1) ← (2) ← 2αi S yy + 2αi B S xy

← → ← → ← → (4) (6) − S zz ) + αi B 2 S zz + αi B 4 S zz → → (1) ← (3) 2 ← → ni 2 S S zz 2α (← τ − pI)zz = bi=a m (kT ) − 3α B zz i i i ← → ← → (4) (6) +3αi B 2 S zz + 3αi B 4 S zz ← → (3) +αi B 2 (2 S xx

(8.26a)

(8.26c)

(8.26d)


← → τ xy =

99

b → ← → ← → ← → ni (1) ← (2) (3) (kT )2 2αi S xy + αi B( S xx − S yy ) + 2αi B 2 (2 S xy mi i=a (8.27a) b ni → → (1) ← (2) ← ← → (kT )2 2αi S xz − αi B S xz τ xz = mi (8.27b) i=a → → (4) 2 ← (5) 3 ← +αi B S xz − αi B S yz

← → τ yz =

b → ← → ni (1) ← (4) (kT )2 2αi S yz + αi B 2 S yz mi i=a → ← → (4) ← (5) +2αi B S xz − αi B 3 S xz

(8.27c)

These are the six component of the stress tensor in the presence of a magnetic field. The superscript (0) in the α’s has been omitted since it is (n) (n) no longer necessary and clearly, αi = Γi in Eq. (8.24) since only the first term in the summation survives. So there are twelve coefficients αi to be determined, six for each species which will be obtained by solving Eq. (2.21). (1) Notice also that when B = 0 only αi survives which corresponds to the shear viscosity of a simple dilute mixture. Indeed, η=

b i=a

2(kT )2

ni (1) α mi i

(8.28)

a well known result. There is no bulk viscosity for a dilute mixture, one needs the existence of collisional transfer. Also it is important to notice that since (6) (4) (3) − B1 αi = αi − αi the last three terms in the bracket in the zz component cancel out so that → (← τ − pI)zz =

b → ni (1) ← (kT )2 2αi S zz mi i=a

(8.26d)

the stress tensor is not modified along the direction of the field.

8.3

The Integral Equation

To solve Eq. (8.21) we shall proceed in exactly the same way as we did with Eq. (5.7) free from the restrictions imposed by the subsidiary conditions as

100

8 Viscomagnetism

we mentioned in Section 8.1. This allows for the direct of ←− →constructions (0) 0 Davison’s function for which purpose we multiply by Ii ci ci fi , sum over i and integrate over ci . This yields b ←−→ ←−→ ←−→ ←−→ mi ei (0) dci fi Ii − ci 0 ci : ci 0 ci − 2i BIi ci 0 ci : ci 0 ci + D(Ii ) = kT mi i=a ←−→ ←−→ ←−→ C(Ii ci 0 ci ) + C(Ii ci 0 ci + Ij cj 0 cj )

(8.29)

We now propose that Ii ≡ G i =

M

(q)

(q)

gi S 5 (c2i )

(8.30)

2

q=o

←−→ ←−→ and noticing that ci 0 ci : ci 0 ci = 23 c4i as may be easily verified we may calculate the first two integrals namely, −

(0) 2 4 ci

dci fi

3

M

(q)

(q)

gi S 5 = −10 2

q=o

ni (0) (kT )2 gi m2i

whereas ei 2i B mi

(0) 2 4 ci dci fi

3

M q=o

(q)

(p)

(q)

2

2

gi S 5 S 5 = −20ωi

ni (0) (kT )2 (gi )2 2 mi

since the integral is only different from zero if p = q. Here, ωi = meii B as usual. Substituting these results in Eq. (8.29) and writing the collision kernel in full, we get that

(0) 2 (0) 2 nb (0) eb nb 1 na (0) ea na 2 − 10i(kT ) D(I ) = −5kT g + g B (g ) + (g ) a a 3 i b 2 ma mb b m3 m a

1 2

i,j

···

b

(0) ←−→ (0) (0) fi fj dvi dcj σ(Ω)dΩgji gi ci 0 ci

−→ (0) ←−→ (0) ← (0) ←−→ gi c i 0 c i + gi c i 0 c i + gj c j 0 c j

(8.31) where have retained only the first term in the series (8.30). Using exactly the same arguments of part I in dealing with the collision integrals, identifying


101

the function Gij of Eq. (5.4a), we find that to a first approximation the third term in (8.31) reads as 2 na (0) 2 ←−0−→ ←−0−→ n2b (0) 2 ←−0−→ ←−0−→ 2 4(kT ) (g ) w , w w + (g ) wb wb , wb wb + w a a a a m2a a m2b b aa bb

←−−→ ←−−→ ←−0−→ ←−0−→ n a nb (0) wa wa , wa wa (ga(0) )2 + wb 0 wb , wb 0 wb (gb )2 + 2ma mb ab ab

←−−→ ←−−→ (0) (0) 2ga gb wa 0 wa , wb 0 wb ab

Thus our trial function written in full, Eq. (8.31) is

(0) nb (0) 1 na (0) ea na 2 − 10i(kT ) D(I ) = −5kT g + g B (ga )2 + i 2 ma a mb b m3 a

4(kT )

2

(0) n2a (g )2 m2a a

na nb

1 m2a

←−−→ ←−−→ wa 0 wa , wa 0 wa

+ aa

n2b (0) (gb )2 m2b

(0) eb nb (gb )2 m3b

+

←−−→ ←−−→ wb 0 wb , wb 0 wb + bb

←−−→ ←−−→

←−−→ ←−−→ (0) (0) wa 0 wa , wa 0 wa (ga )2 + m12 wb 0 wb , wb 0 wb (gb )2 + b ab ab

←−−→ ←−−→ (0) (0) 1 2ga gb wa 0 wa , wb 0 wb ma mb ab

(8.32) Using the properties of the collision integrals given in Appendix D one finds that the five collision integrals appearing in Eq. (4.4) are related to the collision integral ϕ defined in Eq. (D.23) (see table in Appendix D) so that Eq. (4.4) when divided by kT yields finally that na (0) nb (0) D(Ii ) ea na (0) 2 eb nb (0) 2 = −5 g + g (g ) + 3 (gb ) + − 10ikT B 2kT ma a mb b m3a a mb √ n2a (0) 2 √ n2b (0) 2 4kT ϕ 2 2 (ga ) + 2M1 2 (gb ) + n ma mb 10 (0) 8 na nb 2 (0) 2 (0) (0) g ) + g − M1 g a g b (8.33) 2 m2a a 3m2b b 3ma mb In Appendix H the main steps outlining the variational procedure to (0) (0) compute ga and gb are given as well as the results for all the required coefficients to compute all the quantities involved in the pressure tensor.

102

8 Viscomagnetism (n)

(n)

Emphasis should be placed upon the fact that since αi = Γi (first approximation), the sought values for the Γ’s are given through Eqs. (8.15b), (8.16a) and (8.18b) since Li , Gi and Pi are all related to the solution of Eq. (0) (8.21) so that to first order Gi ≡ gi and from these follow all the values for the Γ’s, or in other words, as we shall see, the values of the five independent viscosities.

8.4

Comparison with Thermodynamics

Eqs. (8.6) and (8.7) have to be shaped into a form allowing for direct comparison with the form predicted, from very general symmetry arguments, for the stress tensor in the presence of a magnetic field. This form is derived in the well known monograph of de Groot & Mazur (see Ref. [5] appendix C) and we shall reproduce it here for the sake of completeness. In our language, along the z-axis it takes the following remembering that we have taken B form: Sxx Syy Szz Sxz Syz Sxy ← → ( τ − pI)xx −η η − η3 0 0 0 η4 ← → ( τ − pI)yy −η + η3 −η 0 0 0 η4 → (← τ − pI)zz 0 0 −η 0 0 0 ← → τ xy η4 −η4 0 0 0 η3 ← → τ xz 0 0 0 η2 −η5 0 ← → τ yz 0 0 0 η5 η2 0 recalling that Sαβ

1 = 2

∂uα ∂uβ + ∂xβ ∂xα

1 div uδαβ 3

(8.34a)

(8.34b)

and that the bulk viscosity contributions are not present in the dilute gas approximation. To derive this tensor we notice first in Eq. (8.26a) that the combination (3) (4) [−B 2 (αi − αi ) + B 4 αi6 ] vanishes on account of Eq. (8.19). Exactly the same cancellation occurs in Eqs. (8.26c, d) so that in its final form the stress tensor reads, → (← τ − pI)xx = (kT )2

b → → → ni (1) ← (2) ← (3) 2 ← 2 Γi S xx − Γi B S xy + Γi B S yy (8.35a) mi i=a

8.4. Comparison with Thermodynamics

→ (← τ − pI)yy = (kT )2

103

b → ← → → ni (1) ← (2) ← (3) 2 Γi S yy − Γi B S xy + Γi B 2 S xx (8.35b) mi i=a

→ (← τ − pI)zz = (kT )2

b → ni (1) ← 2Γi S zz mi i=a

(8.35c)

b ← → ← → ← → ni (1) (3) (2) 2 (Γi + Γi B 2 ) S xy + Γi B 2 ( S xx − S yy ) mi i=a (8.35d) b ni ← → ← → (1) (4) (2) (5) → 2(Γi + Γi B 2 ) S xz − B(Γi + Γi B 2 )( S yz ) (← τ )xz = (kT )2 mi i=a (8.35e) and finally

→ (← τ )xy = (kT )2

b ← → ← → ni (1) (4) (2) (5) 2(Γi + Γi B 2 ) S yz − B(Γi + Γi B 2 )( S xz ) mi i=a (8.34f) ← → → One can appreciate the fact that since S is a symmetric tensor so is ← τ i. b ni 2 Further, let us define quantities, all multiplied by i=a 2 mi (kT ) namely

→ (← τ )yz = (kT )2

η = 2(kT )2 η4 = 2(kT )2

b

b

η − η2 = 2(kT )2 η − η3 = 2(kT )2 η5 = 2(kT )2

b

ni (1) i=a mi Γi

(2) ni i=a mi BΓi

b

ni 2 (4) i=a mi B Γi

(8.36)

b

ni 2 (3) i=a mi B Γi

(2) ni i=a mi B(Γi

(5)

+ B 2 Γi )

This allows to write the system of Eq. (8.35a-8.34f) in the following way ← → → (← τ − pI)xx = v ηSxx − η4 S xy + (η − η3 )Syy ← → ← → ← → → (← τ − pI)yy = −η S yy + η4 S xy + (−η + η3 ) S xx

(8.37a) (8.37b)

104

8 Viscomagnetism

← → → (← τ − pI)zz = −η S zz ← → ← → ← → → (← τ )xy = η3 S xy + η4 ( S xx − S yy ) ← → ← → → (← τ )xz = η2 S xz − η5 S yz ← → ← → → (← τ )yz = η2 S yz + η5 S xz

(8.37c) (8.37d) (8.37e) (8.37f)

From Eqs. (H.1)–(H.4) we immediately see that η1 , η2 and η5 are symmetrical → −B (even function of |B|) whereas η4 and η5 under the transformation B such as required by Onsager’s reciprocity theorem. are odd functions of |B| Also η = η3 when B = 0 so all the results correspond to the viscosity of a nonmagnetized mixture of dilute gases. With a careful translation of symbols Eqs. (8.36) correspond to the result predicted in LIT as given by Eq. (8.34a). To understand the behavior of these five viscosities we simply extract (0) from Eqs. (H.1) to (H.4) their real and imaginary parts of gi where i = a, b to get (0)

Rega =

ma τ (6.38 Δ1 kT

+ 0.16x2 ) (8.38)

(0) Imga

and

=

(0)

τ − Δm1akT (24.5x

3

+ 0.64x )

Regb =

mb τ (0.2386 Δ1 kT

(0) Imgb

τ − Δm1akT (0.037x

+ 4.091x2 ) (8.39)

=

− 0.6x ) 3

and Δ1 = 6.1 + 104.77ωa2 τ 2 + 2.56ωa4 τ 4

(8.40)

(0)

The corresponding values for pi follow readily from the scaling mentioned in p. 99 (c.f. eq. H.2). For instance, for η we have η=

n(kT )2 (1) Γa ma

for a fully ionized plasma, na = nb = 12 n. Using Eq. (H.4) and the value for τ which we rewrite here for convenience, 3√ 24π 2 me 20 3 1 (kT ) 2 τ= = 4 ϕ ne ψ

8.4. Comparison with Thermodynamics

105

Figure 8.1: The five viscosities as functions of x for n = 1021 cm−3 , T = 107 K. Notice the somewhat unexpected result showing that in the presence of a magnetic field additional four viscosities are not negligible when compared with represented by the full line.

we find that

3√ π 2 me 20 5 η = 72.85 (kT ) 2 4 eψ

(8.41)

where ψ is the logarithmic function defined in Eq. (6.3). This result is in agreement with that obtained both by Balescu and by Braginski. The behavior of the five viscosities is better appreciated in Fig. (8.1), their calculation resulting from simple addition and subtraction of complex numbers. Their 5 relevant characteristic is that they all show the dominating (kT ) 2 depen = 0 the values of the viscosities is dence with temperature. Further, for B by no means negligible as compared with η for values of ωe τ field dependent < 10. This result may have strong implications in several situations in astrophysics including the problem of angular momentum transfer in accretion disks in binary systems.

Chapter 9 Magnetohydrodynamics The subject to be discussed in this chapter is one of the most important and relatively old aspects of fluid dynamics. When a charged fluid is set in motion the charge and current densities generate an electromagnetic field which through Maxwell’s equations couples with the fluid’s own equations of motion giving rise to what is now called magnetohydrodynamics. As we already pointed out in Chap. 2 the resulting macroscopic equations do not constitute a closed set of equations. Additional information must be supplied through the so-called constitutive equations, relating the different currents or fluxes present to the state variables chosen to describe the fluid. For the particular case of a dilute inert plasma this information was extracted in Chaps. 4–6 from the microscopic model furnished by Boltzmann’s kinetic equation when the magnetic field is either homogeneous or weakly inhomogeneous. When the equations are substituted into the conservation equations obtained in Chap. 2 a rather complicated set of non linear partial differential equations will result, the set being complete. Since the calculations have been carried out keeping terms which are linear in the gradients, the thermodynamic forces, we may refer to this set of equations as the Navier-Stokes-FourierMaxwell equations of magnetohydrodynamics. Let us choose for the moment, to describe the non-equilibrium states o a binary plasma, the state variables ρi (r, t) (i = a, b) the local mass densities of each species, u(r, t) the local barycentric velocity and e(r, t), the total local energy density. Recalling the results from Chap. 2, the equations governing their time dependence are: ∂ρi + div(ρiu) = −div Ji ∂t L.S. Garc´ıa-Col´ın, L. Dagdug, The Kinetic Theory of a Dilute Ionized Plasma, Springer Series on Atomic, Optical, and Plasma Physics 53 © Springer Science + Business Media B.V. 2009

(9.1) 107

108

9 Magnetohydrodynamics

the mass conservation equation, ∂ → + (u × B)) + Jc × B (ρu) + div (← τ k + ρuu) = Q(E ∂t

(9.2)

the momentum conservation equation; and 1∂ ∂(ρe) 1 → = 0 (9.3) (ρu2 ) + + div (Jq + u · ← τ k + ρue + ρuu2 ) − JT · E 2 ∂t ∂t 2 the energy equations. In these equations the mass current Ji and the charge conduction current Jc are related through, ma + mb e Jc = Ja eJa ∼ ma mb ma

(9.4)

the last equality holding when mb >> ma . Recall that Ja + Jb = 0. The heat current Jq is defined as 1 Jq = ρi ci c2i (9.5) 2 i The total charge current is, ma + mb JT = ni ei vi = Qu + Jc = eJa + Qu ma mb i since vi = u(r, t) + ci and Q is the total charge. The momentum current or stress tensor is defined as ← → ρi ci ci τ k=

(9.6)

(9.7)

i

r, t) and B( r, t) are obtained from Maxwell’s equations and the local fields E( which in the MKS system are, = 1Q div E 0 ∂B =0 + rot E ∂t

=0 div B ∂E = μ0 JT − + rot B ∂t

(9.8)

We insist on the fact that Eqs. (9.1-9.3) constitute a set of six equations with 21 unknowns. There are six independent variables (ρi , u, e) and the


109

remaining fifteen constitutive equations, namely, Eqs. (9.5-9.7). The fields and B depend also on the two sources Q and JT so that, in principle, the E whole time evolution problem of the fluid’s state for well specified boundary conditions, is uniquely determined. This constitutes the basic theoretical framework of magnetohydrodynamics at the Navier-Stokes level. The remaining steps required to handle these results are, nevertheless, quite subtle. Examination of the equations that we have derived for the currents in the previous chapters clearly indicates that their direct substitution into the conservation equations would result in a set of highly non linear partial differential equations so complicated that they are practically unmanageable to deal within a concrete application. But, on the other hand, it is precisely at this stage where in many cases, workers in the field have proceeded in a very arbitrary way. Let us deal with Eq. (9.2). Let H be the characteristic hydrodynamical length, the length of a gradient namely, −1 H = M ax

|grad M (r)| M (r)

for any local variable M (r). Let τH be a hydrodynamical time, τH−1 = vs /H where vs is the velocity of sound. Then, from Maxwell’s equations, the ratio E| −1 E2 0 |Ediv vs2 H ∼ 0 μ0 −1 vs 2 ∼ 2 1 c H B |B||JT × B|

μ0

is negligible using the fact that |E| ∼ |τB| . So for a non-relativistic plasma QE H H = JT × B. If ∂ E > ma , that if B where θ = ne2 /4ma . Therefore, with the results of Appendix E, Eq. (9.12) yields, Jc = −σ|| grad φ − τ|| grad T (9.13) where σ|| =

ne2 τ × 1.191 4ma

(9.14a)

nkeτ × 2.94 4ma

(9.14b)

is the electrical conductivity and τ|| =


111

is Thomson’s thermoelectric coefficient. The first term in Eq. (9.13) is simply the well known Ohm’s coefficient. Here arises the first major objection against the approximation leading to Eq. (9.9). These affects are there lost! = 0. Eq. (9.9) ignores the possible electrical effects on the plasma even if B = 0, Eq. (9.2) becomes much more complicated. However, for a If B (0) fully ionized plasma and very small pressure gradients dab may be neglected. → Even though, its electromagnetic component together with div ← τ k gives a rather awkward expression. The x-component of the equation reads, ρ

dux ∂τxx ∂τxy ∂τxz x + + + = (JT × B) dt ∂x ∂y ∂z = Quy B + uy B(Jc )x

Here, na ke (1)(0) ∂T ∂T − a(2)(0) )+ (aa B a ma ∂y ∂x na kT e (−θd(1)(0) (Ey + ux B) − θd(2)(0) (Ex − uy B)) a a ma

(Jc )y =

which finally yields, dux ∂τxx ∂τxy ∂τxz + + + = Bux ρ dt ∂x ∂y ∂z

na kT e Quy − ma

(1)(0)

aa T

(2)(0)

∂T aa − ∂y T

(Ey + ux B) + d(2)(0) (Ex − uy B) −θ d(1)(0) a a

∂T ∂x

(9.15)

This is not a very inspiring result. The as and ds coefficients are given in Appendix E and are complicated expressions which depend on B, n and T . In the l.h.s the derivatives of the stress tensor are also nasty. We must remember that τxx = p + ηSxx + η4 Sxy + (η − η3 )Syy τxy = η3 Sxy + η4 (Sxx − Syy ) τxz = η2 Sxz − η5 Syz ← → and S is defined in Eq. (8.26b). Thus the five independent viscosities are involved in the momentum current possibly overshadowing the NavierStokes term ηSxx . It is difficult to say anything in general as to their relative

112


importance. Even for an isothermal plasma and vanishing small or inexistent electric fields Eq. (9.15), is impressive, ∂ na × k × T × e → (ρux ) + (div ← τ )x = B θ(−uy d(1)(0) + ux d(2)(0) ) a a ∂t ma In contrast however, Eq. (9.9) seems to provide a simpler version to the × rot B) x . But two equation of motion since the r.h.s becomes (1/μ0 )(B problems arise. The first one is that the connection with LIT is lost, the is related to JT ? second one which requires a deeper thought is how then B We shall come back to this question later. Nevertheless, Eq. (9.9) and Eq. (9.15) contain a term, the divergence of the stress tensor which is barely mentioned, if mentioned at all, in the vast literature on the subject of magnetohydrodynamics. In spite of its impressiveness, it is still linear and gives rise to what we could call the extension of the Navier-Newton viscous effects present in an ordinary real fluid. Indeed → if one carries out the operations indicated in the term (div ← τ )x using the explicit form for the τij components and collects terms in a convenient way, the following result is obtained namely, 1 ∂ 2 ux ∂ 2 ux ∂ 2 ux η ∂ 2 ux 2 ∂ ← → div u + − (div τ )x = η 2 + η3 2 + η2 2 + 2 ∂x2 3 ∂x 2 ∂x ∂y ∂z 2 ∂ ux 1 ∂ ∂ 2 uy ∂ 2 uz 1 + η2 − div u + η3 + (η − η3 ) 2 ∂x∂y ∂x∂z ∂x∂y 3 ∂x 2 2 ∂ ux ∂ uy ∂ 2 uy ∂ 2 uz 1 η4 − + − η5 (9.16a) ∂x∂y ∂y 2 2 ∂z 2 ∂z∂y This equation and the two similar ones for the y and z components respectively, show that the Navier-Newton symmetry characteristic of nonmagnetized fluids is completely lost. When B = 0, η = η2 = η3 , η4 = η5 = 0 and the first three terms reduce to

1 1 ∂ 2 η ∇ ux + (div u) 2 3 ∂x which is the usual Navier-Newton term if we redefine the viscosity to absorb the 1/2 factor. This is a remarkable result. The presence of a magnetic field is reflected in the “viscous modes” of the plasma through the viscosities η2 , η3 , η4 and η5 . Further, all terms in Eq. (9.16a) are linear in the velocities


113

so there is no priori reason to ignore them except in those (B, n, T ) regimes where such viscosities are unimportant. This is clearly indicated in Fig. 8.1. Just for completeness since they have never been written anywhere, the two other components are: ∂ 2 uy ∂ 2 uy 1 ∂ 2 uy η ∂ 2 uy 2 ∂ ← → div u + (div τ )y = − η3 2 + η 2 + η2 2 + 2 ∂x ∂y ∂z 2 ∂y 2 3 ∂y 2 ∂ ux 1 ∂ 1 ∂ 2 ux ∂ 2 uz + η2 − div u + η3 + (η − η3 ) 2 ∂x∂y ∂z∂y ∂x∂y 3 ∂y 2 2 ∂ ux 2 ∂ 2 uy ∂ uy 1 ∂ 2 ux ∂ 2 uz 1 η4 −− − + − η5 (9.16b) ∂x2 3 ∂x∂y 2 ∂y 2 2 ∂z 2 ∂z∂y and

→ (div ← τ )z =

∂ 1 ∂ 2 uz ∂ 2 uz div u + η2 2 + η 2 ∂z 2 ∂x ∂y 2 2 2 ∂ ux ∂ uy η ∂ uz − + η5 + 2 2 ∂z ∂x∂z ∂x∂z 1 1 η2 − η 2 6

(9.16c)

Even in the direction of the magnetic field, the z-axis in our case, the two dimensional Laplacian of the velocity component uz is affected by the field through η2 . To our knowledge this effect has never been accounted for in elementary magnetohydrodynamics. Recall from Chap. 8 that η2 is a complicated expression involving the parameter x = ωτ and its values are important for x up to 5. We insist here on the fact that Eqs. (9.16a-c) are all linear in the velocities so that they deserve a much closer attention to find their influence in transport phenomena involving non-relativistic ionized dilute plasmas. A similar situation arises from the energy equation. Indeed a straightforward application of the local equilibrium assumption transforms Eq. (9.11) into the temperature representation. In fact, since e = e(ρ, T ) and noticing ∂e that ∂ρ = 0 for an ideal system we get that T

ρ

3 dT dT de = ρCv = nk dt dt 2 dt

so Eq. (9.11) reads, 5k 3 dT → nk + div Jq + ← div (Jc T ) = −Jc · E τ k : (grad u)s + 2 dt 2e

(9.17)

114


Using the expression for Jq obtained in Chap. 4, Eq. (4.8) we readily find that its temperature dependent part contributes to div Jq by

−κ⊥ ∇2(2) T − κ

∂ 2T ∂z 2

and by ∂2p ∂z 2 where ∇2(2) ≡ ∂x∂ 2 + ∂y∂ 2 , the κ’s and D’s are the parallel and perpendicular expressions for the thermal conductivity and diffusion coefficients, respectively. The contribution arising from dij is readily obtained and reads

∂Ex ∂Ey ∂uy B ∂ux B ∂Ez (e) + + B⊥ + div dij = B⊥ + B + ∂x ∂y ∂z ∂x ∂y ∂ ∂ Bs − (Ey + ux B) + − (Ex + uy B) ∂x ∂y −D⊥ ∇2(2) p − D

where the B’s are the Benedicks transport coefficients defined in Chap. 8. Their relative importance in magnetohydrodynamics problem has never been mentioned, even less, assessed. So it is not altogether clear that they are = 0, and therefore B = B⊥ we have that negligible. Even if B (e) div dij = B ∇2 φ

a linear contribution in the electrical potential. In astrophysical systems = 0 and the first two terms disappear. If further, B is homogeneous often E

∂u ∂u ∂u ∂u y x x y (e) div dij = BB⊥ + + − BBs ∂x ∂y ∂x ∂y so its influence depends linearly on the magnitude of the velocity gradients. The contributions arising from the stress tensor are relatively easy to deal with if one is going to neglect non-linear terms. Indeed in the product ← → ∂ui τ k : (grad u)s all contributions are of the type τij ∂x , i, j = x, y, z and j thus nonlinear. The only term remaining linear in u is −pdiv u so that this simplifies the temperature equation considerably. Now, the term 5k 5k 5k div (T Jc ) = Jc · grad T + T div Jc 2e 2e 2e


115

The first term is clearly a non-linear one in the thermodynamic forces involving grad T · grad T and grad T · dij terms, so we neglect it. Also, since div (grad T )s = 0 the temperature contribution to the T div Jc yields,

2 na k (1) ∂ T (0)(1) 2 T αa + aa ∇ T − ma ∂z 2 For a fully ionized plasma, ignoring pressure diffusion effects, we must eval(e) uate the terms arising from dij . Indeed

(e) ∂E ⊥ ∂ (s) ∂ − σa − σa div Jc = − σa (Ex + uy B) + ∂z ∂x ∂y

⊥ ∂ (s) ∂ + σa − σa (Ey − ux B) ∂x ∂y (s) = 0, σa(s) = 0, σa = σa⊥ where σa , σa⊥ and σa are defined in Chap. 4. When B and div Jc(e) = −σa div E

Finally, = − σ Ez2 − σ⊥ Jc · E

2

E − Ez2 + B (ux Ey − uy Ex ) −

σa(s) B [(uy Ey − ux Ex )] if B = 0 = − σ E 2 the well known Ohmic dissipation term. Inserting all these results in Eq. (9.17) the temperature equation, we obtain that, ∂ 2T ∂2p 3 dT nk − κ⊥ ∇2(2) T − κ 2 − D⊥ ∇2(2) p − D 2 + 2 dt ∂z ∂z

∂Ex ∂Ey ∂uy B ∂ux B ∂Ez + + B⊥ + B⊥ + B + ∂x ∂y ∂z ∂x ∂y ∂ ∂ (Ex + uy B) Bs − (Ey + ux B) + ∂x ∂y

− σ Ez2 − σ⊥ E 2 − Ez2 + B (ux Ey − uy Ex ) −

2 na k (s) (1) ∂ T (0)(1) 2 T αa + aa ∇(2) T σa B [(uy Ey − ux Ex )] − ma ∂z 2 + p div u = 0

(9.18)

116


Equations (9.1), (9.15), and (9.18) are the linear approximation to the full set of the equations of magnetohydrodynamics to first order in the gradients for a fully ionized hydrogen-like plasma (mb >> ma ). Even with the approximations introduced they are still quite unmanageable and also far from clear how to devise a suitable systematic way to obtain appropriate approximations. Thus, as already expressed by Balescu [4], for any specific problem one will have to find the way of reducing this set into one which is simpler to operate with. It is pertinent at this point to discuss a different approach to the interpretation of the energy equation similar to that leading to Eq. (9.9) for the may be substituted by momentum equation. Indeed, in Eq. (9.17) Jc · E −1 1, JT = μ0 rot B and assuming that ∂t E so that JT · E 1 JT · E = rot B E + u × B μ0 Now we introduce a second assumption, namely that the constitutive equation for Jc given by ˆ Jc = σ Ez k + σ⊥ (ˆıEx + ˆjEy ) + σs E + uB (9.19a) ignoring pressure diffusion and thermoelectric effects as inferred from Eq. (9.12). This equation is invariably substituted by + uB JT = σ E (9.19b) which is difficult to understand. Indeed σ⊥ < σ is overrated in the first two terms by assuming equality and σs completely misunderstood in evaluating is neglected and σs = σ . Nevertheless, from this the third term, where E result + u × B = 1 (rot B) E μ0 σ and therefore, = JT · E

1 μ20 σ

2 (rot B)

(9.19)

The temperature equation is completely modified since also =0 div Jc = div JT = div (rot B)

(9.20)


117

whence 3 dT ∂2T 1 2 nk − κ⊥ ∇2(2) T − κ 2 + p div u = − 2 (rot B) 2 dt ∂z μ0 σ

(9.21)

ignoring pressure diffusion. Setting aside the fact that these results are hard to believe less to understand, they are in complete contradiction with the tenets of LIT. Jc , the conduction current is the outcome of the presence of all thermodynamic forces present in the system. This means that the direct electromagnetic effects represented in the second term of Eq. (9.12) are accompanied by (0) the cross effects arising from diffusive forces dij and temperature gradients. Some may predominate over the others but clearly having all acting in a way such that div Jc = 0 as required by Eq. (9.20) is simply untenable. Indeed even in the absence of a magnetic field, div Jc = −σ ∇2 φ − τ11 ∇2 T and grad T acted in a rather pecuwhich could only be zero if grad φ = −E liar way. We heartfully believe that this approach to magnetohydrodynamics is wrong and that the correct form for the temperature equation is that given by Eq. (9.17). In order to get a better assessment of the results here derived, it is important at this stage to seek for their relationship with the Euler equations, also named ideal or non-resistive equations, of magnetohydrodynamics. They were derived in Chap. 3 but we repeat them here keeping the approximation is negligible. The continuity equation is the same, the other two that Q|E| are ∂u ρ = −ρu · grad u − grad p + JT × B (3.7) ∂t and

2p ∂T = − u · grad T + div u (3.8) ∂t 3nk were, for the time being, we restrain from using the questionable result as + Jc × B and this serting that JT = μ−1 u×B 0 rot B. Further, JT × B = Q second term follows readily from Eq. (9.12) indicating clearly the source terms still present at the Euler level. Yet, in this canonical form they are never used in the literature.

118


On the other hand, let us go back to Eq. (3.7) and keep all terms. Using the first of Maxwell’ equations ρ

∂u E + JT × B = −ρu · grad u − grad p + 0 (div E) ∂t

Using the fourth of Maxwell’ equations, 1 JT × B = ∂t E + rot B × B μ0 where ∂t ≡ ∂/∂t, the step being completely legitimate. Now, using once more Maxwell’ equations 1 ∂ 1 1 1 ∂t E + rot B × B = E × B − B × rot E + rot B × B μ0 μ0 ∂t μ0 μ0 But,

· grad B B − B ×B = div B rot B E div E = − E · grad E + div E E

and Therefore,

div E + JT × B · grad E + 0 div E = − 0 E E − 0 E 1 − 1 Bgrad − 1 div (B B) B ∂t E × B μ0 μ0 μ0 Defining

and

= 1 E ×B G μ0 ← → τ e=

0 2 1 1 2 E + B B |E| + |B| I − 0 E 2 2μ0 μ0

(9.22)

(9.23)

the electromagnetic stress tensor, and introducing these expressions in Eq. (3.7) we get, ρ

∂t G ∂u → = −ρu · grad u − div (pI) − + div ← τ e=0 ∂t ∂t

(9.24)


119

Eq. (9.24) is the complete Euler equation for an ionized plasma. It it is completely shows that even without explicitly writing the term JT × B equivalent to Eq. (3.7). If we call M ≡ ρu the mechanical momentum, Eq. (9.24) reads after integration over an arbitrary volume V , ∂ → → +G dV + (← M τ +← τ e ) · dσ (9.25) ∂t V S → where ← τ = pI, no dissipative terms are present. In words: The rate of increase (or decrease) of the momentum of particles and fields equals the rate at which momentum is following out of (or into) an arbitrary volume V . As for the energy equation (3.8) it is self explanatory so that together with (3.7) and the continuity equation may be regarded as the set of Euler equations for a dilute plasma derived from the microscopic model furnished by the Boltzmann equation. For completeness, it should be mentioned that the conservation theorem expressed in Eq. (9.25) follows readily from the full momentum equation, → Eq. (9.25) provided the mechanical stress tensor ← τ is now considered as → → the full tensor ← τ = pI + ← τ k . The interested reader may easily verify this statement. We would like to finish this chapter on magnetohydrodynamics by discussing what most authors in the subject accept as the equations of “resistive magnetohydrodynamics” emphasizing the complete lack of support they have from the Boltzmann microscopic model of a plasma. We start from Eq. (9.12) which may be regarded as the canonical form for the constitutive equations for the conduction current to first order in the gradients. If in this equation we keep only the electromagnetic terms arising from the electric component of dij , which means ignoring thermoelectricity, pressure diffusion and so on, we get Eq. (9.19a), where the three electrical conductivities are thoroughly discussed in Chap. 6. This equation, as mentioned earlier, is arbitrarily substituted by Eq. (9.19b). In Eq. (9.19b) σ⊥ is notoriously overrated setting it is ignored and vanishes; the term σs E equal to σ which is only true when B worst σs is set equal to σ . All these assumptions are completely wrong since σs < σ⊥ < σ for all values of B = 0. Nevertheless they allow writing that 1 E = JT − u × B σ 1, an assumption which destroys the symmetry behind Assuming that ∂t E the argument leading to Eq. (9.25), one may set JT = μ−1 0 rot B therefore

120


leading to + u × B = E

1 rot B μ0 σ

(9.26)

Even worst, from the first of Maxwell’s equations and Eq. (9.26) Q=−

1 div u × B 0

the two sources of the fields turn to depend only on the fields themselves. This is preposterous. In fact from Maxwell’s equations, ∂B 1 = −rot E = rot u × B − (rot ) rot B ∂t μ0 σ whence = grad (div B) − ∇2 B But rot rot B ∂B + 1 ∇2 B = rot u × B ∂t μ0 σ

(9.27)

implying that the magnetic field is autonomous but still has to comply with = μ0 JT !! the condition that rot B Moreover, the equation of motion which in the Euler’s regime is given by Eq. (3.7), it is now transformed into ρ

∂u 1 × rot B = −grad p − B ∂t μ0

(9.28)

→ thus ignoring all dissipative effects coming from div ← τ k. Finally the equation for the pressure is obtained from Eq. (9.18) neglecting all thermoelectric and Benedick’s effects, κ,⊥ = 0, B,⊥ = 0; all diffusive terms setting σ⊥ B(ux Ey − uy Ex )=0, σs = 0 and letting σ = σ⊥ so that 3 dT nk + p div u = σ E 2 2 dt Using the equation of state, Dalton’s law p = nkT so that

if mb >> ma

dn dT p dn p dT dp = kT + nk = + dt dt dt n dt T dt

(9.29)


121

we obtain that, so nk

dp p dρ dT = − dt dT ρ dt

With the continuity equation this result is transformed into, nk

dp dT = + pdiv u dt dt

so that equation (9.29) reads, 2 1 dp 5 · rot B + pdiv u = rot B dt 3 3 μ0 σ2

(9.30)

Eqs. (9.27), (9.28) and (9.30) are referred to in the literature as the equations for “resistive magnetohydrodynamics”. We believe that there is not much to add about them. Within the framework provided by kinetic theory they constitute a rather arbitrary model whose implications can be seriously considered suspect. They are completely unjustified. It is clear that according to this work, the correct equations for dissipative magnetohydrodynamics, to first order in the gradients are provided by Eqs. (9.1), (9.15) and (9.18). Their application requires we insist, a very careful examination of the problem to be dealt with before any kind of simplifications may be imposed on them. This is the great and deep value of kinetic theory.

Bibliography [1] Plasma Physics by S. Chandrasekhar. The University of Chicago Press, Chicago (1960). [2] J. L. Delcroix; Introduction to the Theory of Ionized Gases; Interscience Publishers Inc., New York (1960). [3] R. M. Kulsrud; Plasma Physics for Astrophysics; Princeton Univ. Press, Princeton, N. J. (2005), Chap. 3 and references there in. [4] R. Balescu; Transport Processes in Plasmas; Vol. 1. Classical Transport; North-Holland Publishing Co., Amsterdam (1988). [5] L. Spitzer Jr.; The Physics of Fully Ionized Gases; Wiley-Interscience, New York (1962).

123

Appendix A Calculation of M We start by substituting Eq. (3.15’) in the first two terms of Eq. (3.16a) to get ∂ (0) ea (1) (2) (3) · a + B( ca · B)A a −fa (ca × B) ca Aa + (ca × B)A ma ∂ca m a (0) ca ≡ M ej dcj fj Aj (cj × B) −fa(0) ρkT j and examine term by term. (1)

· ∂ (ca A(1) (1) · ca ∂Aa a): (ca × B) ca × B)A ca × B) a ) = ( a + ( ∂ca ∂ca The second term vanishes so we keep only the first one

∂ ∂ (2) · a = (ca × B) · A(2) b): (ca × B) (ca × B)A (ca × B) a + ∂ca ∂ca

(2) ∂A a · (ca × B) (ca × B) ∂ca The second term vanishes since using the vector identity,

(2)

∂Aa ∂ca

∼ ca and the first term we compute

· grad(ca × B) + 2(ca × B) × rot(ca × B) grad (ca × B) · (ca × B) = 2(ca × B) L.S. Garc´ıa-Col´ın, L. Dagdug, The Kinetic Theory of a Dilute Ionized Plasma, Springer Series on Atomic, Optical, and Plasma Physics 53 © Springer Science + Business Media B.V. 2009

125

126

Appendix A. Calculation of M

where grad and rot are taken with respect to ca . Using well known vector identities, we get · grad(ca × B) = grad c2a B 2 − (ca · B) 2 − 2(ca × B)

× ca · gradB − B grad ca + ca div B −B div ca 2(ca × B) grad ca = −B, since B is constant and all second term vanishes, except −B 2 2 2 ×B = (ca · B) B − B ca . But grad(c B − (ca · B) 2) = in ca space and (ca × B) a B so adding the two terms we get that 2B 2ca (ca · B) · grad (ca × B) = 2(B 2ca − B( ca · B)) (ca × B) · ca = 0. c): The third term vanishes since (ca × B) Thus the first term in M reads, −fa(0)

ea (0) 2ea 2 (1) (ca × B)A − f c − B( c · B)) A(2) (B a a a a a ma ma

(A.1)

We first simplify the second term in M to read ma (0) ej dcj fj Ajcj · (ca × B) = −fa(0) ρkT j ma (0) (1) (2) + B( cj · B)A (3) (B×ca )· ej dcj fj cj Aj cj + (cj × B)A fa(0) j j ρkT j Again, proceed term by term, 1 (0) (1) (0) (1) ej dcj fj Aj cjcj = dcj fj Aj c2j I 3 j j The second term is (0) (2) (0) (2) cj = ej dcj fj Aj (cj × B) ej dcj fj Aj cjcj × B j

j

=

1 ej 3 j

(0)

(2)

dcj fj Aj c2j I

×B

(A.2)

Appendix A. Calculation of M

127

However

= (ca × B) ×B (ca × B) · I × B

and this term yields

B B ca − (cj · B) 2

1 3

ej

(0)

(2)

dcj fj Aj c2j

(A.3)

j

·B =0 The third term vanishes since (ca × B) Using the identity 1 2 (0) (0) 2 2 cj − 2 (ca · B) fj R(c)dc = dcj fj R(c)c2j B 3 which is valid for any arbitrary function of c, R(c), we finally find combining Eqs. (A.2-A.3) that ea (1) (0) 2ea 2 B ca − B( ( ca × B)Aa − fa ca · B) A(2) M= a + ma ma (1) (2) (0) ma (0) ma 2 ( ca × B)GB + fa B ca − B( ca · B) GB fa ρkT ρkT −fa(0)

(1)

(2)

which are the results quoted in the text. GB and GB are defined in the text in page 31.

Appendix B Linearized Boltzmann Collision Kernels In what follows we shall discuss several important properties related to the nature and structure of the linearized Boltzmann collision kernels. Let Gij = Gi (ωi , ωj ) and Hij = Hij (ωi , ω j ) be any two tensors functions of the velocities ω i and ω j . Define 1 (0) (0) · · · Gij : (Hij − Hij )fi fj gij σ(Ω)dΩdvi dvj (B.1) [Gij , Hij ]ij ≡ ni nj where the subscript in the bracket denotes an integration over the variables vi and vj and the differential cross section σ(Ω)dΩ = bdbd where is the inclination of the orbit and b the impact parameter. Setting vi → v i , vj → v j (0) (0) (0) (0) noticing that gij = gij , fi (vi )fj (vj ) = fi (v i )fj (v j ) and changing signs in (B.1) we get that 1 (0) (0) [Gij , Hij ]ij ≡ − · · · (Gij −Gij ) : (Hij −Hij )fi fj gij σ(Ω)dΩdvi dvj 2ni nj (B.2) The bracket is symmetrical with respect to an exchange of the indexes i and j in Gij and Hij and also between brackets. Hence the following set of equations holds: [Gij , Hij ]ij = [Hij , Gij ]ij = [Gij , Hij ]ji = [Hij , Gij ]ji

(B.3)

It is also clear from its definition that [ ]ij is a linear operator. Suppose that Gij = Ki + Lj Hij = Mi + Nj L.S. Garc´ıa-Col´ın, L. Dagdug, The Kinetic Theory of a Dilute Ionized Plasma, Springer Series on Atomic, Optical, and Plasma Physics 53 © Springer Science + Business Media B.V. 2009

129

130

Appendix B. Linearized Boltzmann Collision Kernels

Ki and Li depend only on ωi and Lj , Nj depend only on ω j , then by inspection, [Ki + Lj , Mi + Nj ]ij = [Ki , Mi + Nj ]ij + [Lj , Mi + Nj ]ij = [Ki , Mi ]ij + [Ki , Nj ]ij + [Lj , Mi ]ij + [Lj , Nj ]ij

(B.4)

In Eq. (B.4), Ki ≡ Ki (ωi ), Lj ≡ Lj (ωj ) and so on, and [ ]ij indicates that the integral is evaluated for collisions between molecules of species i and j. Let us now consider two sets of tensor functions Ki and Li and define ni nj [Ki + Kj ; Li + Lj ]ij (B.5) {K; L} = ij

From the properties of the square bracket it follows immediately that {K, L} = {L, K}

(B.6a)

{K; L + M } = {K, L} + {K, M }

(B.6b)

and Since {K; K} represents integrals whose integrands are non-negative, it follows that {K, K} ≥ 0 (B.7) and further, for obvious reasons, the equality sign holds if K is a linear combination of the collisional invariants. For a binary mixture, expanding (B.5) we get that {K, L} = n2a [Ka + Ka , La + La ]aa + 2na nb [Ka + Kb , La + Lb ]ab + n2b [Kb + Kb , Lb + Lb ]bb which upon expansion and appropriate collection of terms yields {K, L} = 2n2a [Ka , La ]aa + 2n2b [Kb , Lb ]bb + 2na nb ([Ka , La ]ab + [Ka , Lb ]ab + [Kb , La ]ab + [Kb , Lb ]ab )

(B.8)

Eqs. (B.7) and (B.8) are of importance in the solution to the integral equations.

Bibliography [1] J. O. Hirschfelder, C. F. Curtiss and R. B. Byrd; The Molecular Theory of Liquids and Gases; John Wiley & Sons, New York (1964), 2nd printing.

131

Appendix C = 0 The Case when B There are at least two reasons to take the time and space to consider this case in some detail. Firstly and above all, the fact that the full thermodynamic theory can be derived including the explicit form of all transport for the mixture, and a rigorous proof showing that the transport matrix is symmetric in full agreement with Onsager’s reciprocity theorem. Secondly, the explicit expressions for the thermal and electrical conductivities can be compared with those derived earlier by Spitzer [1]. Moreover the cross coefficients for the Soret and Dufour effects are also readily obtained which to the author’s knowledge have never been appropriately assessed in the case of a ionized gas. Their values could be significant in some astrophysical systems such as cooling flows and planetary nebulae. Many of the results to be given here arise simply from those in the main = 0. Others require some additional attention which will text just setting B be offered in detail. The conservation equations are given by Eq. (2.17c) which remains unchanged. The momentum equation is ∂ (ρu) + div (τ k + ρuu) = QE ∂t which may be readily derived [c.f. Eq. (2.9)] and from Eq. (2.19) d e ρ + div Jq + τk : grad u = Jc · E dt ρ

(C.1)

(C.2)

Nothing else changes with respect to the H-theorem, the validity of Eq. (2.26) giving the equilibrium distribution function nor the fact that the solution to the linearized homogeneous term of the Boltzmann equation J(f 0 f 0 ) = 0 L.S. Garc´ıa-Col´ın, L. Dagdug, The Kinetic Theory of a Dilute Ionized Plasma, Springer Series on Atomic, Optical, and Plasma Physics 53 © Springer Science + Business Media B.V. 2009

133

= 0 Appendix C. The Case when B

134

is the local Maxwell-Boltzmann distribution function. The first substantial = 0 case is Eq. (2.29a) for the first order in gradients difference with the B solution to the BE, namely, (0) (0) (0) Fa ∂fa ∂fa ∂fi (1) (1) + va · + · = fa(0) C(ϕ(1) + C(ϕ + ϕ a a b ∂t ∂r ma ∂ va (e) Since both Fa (e) and ea E are conservative forces where Fa = Fa + ea E. it may be readily identified with the linearized version of the Boltzmann equation in the traditional case, F conservative. Evaluation of the left hand side using the explicit form for f (0) and Eqs. (3.5)-(3.8) in the text where leads to the result that the last two terms in (3.7) are replaced by QE

fa(0)

→ ma ←− ca 0 ca : grad u + kT fa(0)

where

ma c2a 5 − 2kT 2

C(ϕ(1) a )

+

na grad ln T · ca + ca · dab = n

(0) C(ϕ(1) a ϕb )

=

(C.3)

na na nb (ma − mb ) dab = −dba = grad + grad ln p n nρ −

ρa ρb e n a nb (Fa − Fbe ) − (mb ea − ma eb )E pρ pρ

(C.4)

Just as in the magnetic field free case all the theorems showing that Eq. (C.3) has no unique solution but an infinite number composed by a solution to the inhomogeneous part plus as arbitrary linear combination to the solution of the homogeneous part in these case being mi , mici , and 12 mi c2i where (i = a, b), hold true. The latter solution is uniquely determined by the subsidiary conditions so one gets that (1) −ϕi = Ca 0 Ca Bi : grad u + Ai ci · grad ln T + Di dij · ci

(C.5)

where Ai , Bi and Di are scalar functions of ci , ni , T , etc. Ai and Di are still subjected to the subsidiary condition that (0) Ai mi f i ci = 0 (C.6) c2i d Di i


135

These functions depend on the particular interaction potential between the ions and the electrons in the gas and are solutions to the equations mi c2i 5 (0) − − (C.6a) f (0) ci = {C(Aici ) + C(Aici + Ajcj )} fi 2kT 2 a and

ni (0) (0) ci fi = fi {C(Dici ) + C(Dici + Dici + Djcj )} (C.6b) n The solution for Bi associated with the tensor grad u will not couple with the vectorial fluxes dab and grad T so we shall simply ignore it in what follows. For isotropic systems there are no visco-electric effects. The minus sign in Eq. (C.5) has been included for convenience. Ignoring Bi , −

(1)

ϕi = −Aici ·

1 grad T − Di · ci dij T

(C.5 )

can now allow us to compute the different fluxes in the mixture. This we shall do in detail to clearly exhibit how does Onsager’s reciprocity theorem holds true. There are essentially three currents, the heat flow, the mass flow (0) (e) and the electrical flow, the forces being grad T , dij and dij where n a na n b (0) + (ma − mb ) grad ln p dab = grad n nρ

(C.7a)

na n b (e) (mb ea − ma eb )E dab = pρ

(C.7b)

(e) (0) for simplicity other external forces, Fi = 0. Therefore, dij must be re(e) lated to conventional diffusion processes and dij with electrical phenomena together with their respective cross effects. For this purpose we shall follow closely the treatment contained in references [2] and [3]. Onsager’s symmetry arises then in a very simple way. We recall that Ji = mi ci fi (ci )d ci = mi ni ci = ρi ci

Ji = 0

(Ja = −Jb )

i

Also

ea eb Ja + Jb Jc = na ea ca + nb eb cb = ma mb

(C.8a)


136

We now use the fact that the results of ref. [1] and [2] for a binary mixture do not depend on the explicit form of the diffusive force dab . In particular, as shown in Chap. 2, 1 1 mb − ma J = Ja + Jb = Ja ma mb ma mb 5 J q = Jq − kT J 2 as in the inert mixture and in addition we have the electric current ea eb eb m a + ea m b Je = Ja + Jb = Ja ma mb ma mb

(C.8b) (C.8c)

(C.8d)

On the other hand, as shown in Ref. [4], one may write that, Ja = −Laq grad ln T − Lab dab

(C.9)

and Laq , Lab obey the OR theorem. It is also clear that by (C.8b) and (C.8d) the expressions for J and Je will also fulfill such theorem so indeed we have the three linear relations, (0) (0) J = −Laa dab − Lae dab − Laq grad T (0) (0) J q = −Lqa dab − Lqe dab − Lqq grad T (0) (0) Jc = −Lea d − Leq grad T − Lee d ab

(C.10)

ab

(0) = −grad φ. and by its own structure, dab ∝ −grad φ since E The reader may now see that we may write the Onsager matrix in a more canonical form, namely

⎞⎛ ⎛ ⎞ ⎛ (0) ⎞ J Laa Lae Laq −d ab ⎝Jc ⎠ = ⎝Lea Lee Leq ⎠ ⎝ −d (e) ⎠ ab Lqa Lqe Lqq − grad T Jq

(C.11)

Although Eqs. (C.10) are those which follow from microscopic reversibility and are therefore the holders of symmetry, in Eq. (C.11) it is no longer obvious that indeed one can make Lae = Lea , etc. but only their ratio can be shown to be constant. Nevertheless in their comparison with experiment the relations following from Eq. (C.11) as well as (C.8b-d) will be used. This


137

point has to be kept in mind. (See Ref. [5] Chap. 9). We shall come back to this point later on. We now proceed with the evaluation of Ja (= −Jb ) and Jq in order to evaluate the corresponding transport coefficients. For this purpose we shall assume that (p) (p) Ai (ci , . . .) = ∞ p=0 ai S 3 (ci ) 2

Di (ci , . . .) =

(C.12)

∞

(p) (q) q=0 di S 3 (ci ) 2

(m)

S 3 (ci ) are Sonine polynomials whose properties are summarized in p. 35 of 2

the main text. First of all the subsidiary conditions (C.6) now read, ∞ (p) ai (0) (p) ci = 0 fi c2i S 3 (ci )d (p) 2 di p=0 Using the dimensionless velocity, ωi = ci

mi 2kT

and

m 3/2 2 i = ni exp−ωi 2πkT dropping irrelevant constants we get that (p) ai ni δp,0 = 0 (p) d i i (0) fi

which yields (0)

(0)

(0) na d a

(0) nb d b

na aa + nb ab = 0 (C.13) +

=0

the resulting conditions to be imposed on the expansions in (C.12). Using Eq. (C.5 ) and (C.12) in the expression for Ja we find that (0)

∂ ln T aa Ja = −na kT − nna kT d(0) a dab ∂r

(C.14a)


138

Introducing this last equation into Eq. (C.8a) and Eq. (C.8b) we can calculate Jc and the mass flow, namely, m m + m ∂ ln T + m a b a b (0) − nna kT Jc = −na kT eaa ed(0) a dab (C.14b) ma mb ∂r ma mb mb − ma (0) ∂ ln T (0) J= − nna kT da dab −na kT aa (C.14c) ma mb ∂r Finally, using these two last equations and Eq. (C.13) in Eq. (C.8c) we have that, (1) (1) m a a + m 5 n n a a b a b 2 b J q = (kT ) + grad ln T − nna (kT )2 d(0) a dab 2 ma mb ma mb (C.14d) (0) (e) Since in these equations dab = dab + dab and E = −grad φ we have the transport coefficients matrix in which the diffusive force dab reduces to grad na if p is constant. Insertion of the explicit form for dab in Eqs. (C.14b-d) leads precisely to the general set of equations (C.10). Notice should be made of the fact that Onsager’s symmetry cannot be expected any longer. The proof of the (1) ORT requires the full form of φi of which only four coefficients survive (0) (0) (1) for the explicit calculation of the transport coefficients namely, aa , da , aa (1) and ab . These will be obtained by solving the integral equations which arise when (C.5 ) is inserted back in the linearized Boltzmann equation, Eq. (C.3). Indeed, if this substitution is performed, we get that ma c2a 5 (0) (0) − − fa ca = {C(Aaca ) + C(Aaca + Abcb )} fa 2kT 2 (C.15) na (0) (0) − fa ca = {C(Daca ) + C(Daca + Dbcb )} fa n and two identical equations for species b. These equations, far simpler than those arising in the B = 0 case will be solved by a rather straightforward variational method using the results of Appendix B. The argument goes as follows: Eqs. (C.15) are of the form, b

(0) (0) Ri (v ) = dv i gσ(Ω)dΩfi (v )fj (v1 ) Ti (v i ) + Tj (v j ) − Ti (vi ) + Tj (vj ) and

j=a

(C.16)


139

where Ri (v ) is known, Ti is unknown and v i and v j are the velocities of the particles after collision, and for collisions among of the same species (v , v 1 → v , v1 ). Also, C(Aaca ) = dv a gσ(Ω)dΩ c a Aa (c a ) + c a1 Aa (c a1 ) − ca Aa (ca ) (0) (0) −ca1 Aa (ca1 )} fa (ca )fa (c a )

C(Aaca + Abcb ) =

dv a gσ(Ω)dΩ c a Aa (c a ) + c b Ab (c b ) − ca Aa (ca ) (0)

(0)

−cb Ab (cb )} fa (ca )fa (cb ) and in each case g =| vi − vj |, the relative velocity of the colliding particles. Let now ti (v ) be a trial function proposed as solution to Eq. (C.15). Integrating over vi after multiplication by ti (v ), one gets (0) (0) dvi Ri (v ) : ti (v ) = ij dvivj gσ(Ω)dΩfi ( i

vi )fj (vj ) ti (v ) : ti (v i ) + tj (v j ) − ti (vi ) + tj (vj ) This equation according to the definition of collisional brackets and Eq. (B.8) yield that 1 dvi Ri (v ) : ti (v ) = − {ti , ti } = 2 i −n2a [ta , ta ]aa − na nb [ta + tb , ta + tb ]ab − n2b [tb , tb ]bb Clearly, from Eq. (C.16) i

(C.17)

1 dvi Ri (v ) : ti (v ) = − {ti , Ti } 2

so if ti (v ) in fact satisfies the integral equation (C.15) we would have that {ti , ti } = {ti , Ti } By Eq. (B.7), {ti − Ti ; ti − Ti } = {ti , ti } − 2 {ti , Ti } + {Ti , Ti } ≥ 0 then −2

i

dvi Ri (v ) : ti (v ) = {ti , ti } ≤ {Ti , Ti }

(C.18)


140

Eq. (C.18) is the basis of the variational principle. The proposed solution ti (v ) must be such that it maximizes the collision integral {Ti , Ti }. We start with the second of Eqs. (C.15) which after multiplication by (0) fa on both sides and summation over the two species, reads ∞ 2 (0) p ni dcicici fi (ci ) d(i) dci ti : Ri = − −2 p S 3 (ωi ) 2 n i p=0 i after substitution of the trial function ti = ci

∞ p=0

(p)

di S p3 (ci ) 2

Introducing the dimensionless velocity ω i and carrying the integrals recalling that ∞ 3√ 4 dωω 4 exp−ω S p3 S 03 = πδp,0 2 2 8 0 and introducing the subsidiary condition given by Eq. (C.13) we finally arrive at the result that na (na mb − nb ma ) (0) −2 da (C.19) dci ti : Ri = −6kT n ma mb i The inequality (C.18) thus yields that na (na mb − nb ma ) (0) da = ca d(p) ca d(p) −6kT a S 32 (ωa ); a S 32 (ωa ) n ma mb p p If we now use Eq. (B.8) for the binary mixture, to first approximation, (p) noticing that [ca , ca ]aa = [cb , cb ]bb = 0 we have that (da = 0, p > 0) so that n a na m b − n b m a (0) (0)2 −6kT = 2n [ωa , ωa ]aa + 2d(0) ωa , ωb ]ab + d(0) b da a a db [ n ma mb

(0)2 db [ωb , ωb ]bb (0)

(0)

Once more, since db = −(na /nb )da we have (na mb − nb ma ) (0) na n2a (0)2 −3kT d a = nb d a [ωa , ω b ]ab + 2 [ωb , ωb ]bb [ωa , ωa ]aa − n(ma mb ) nb nb


141 (0)

so that the trivial solution is da = 0 and the sought one is given by d(0) a

3kT =− n

na m b − nb m a ma mb

na 1−2 nb

ma n2a + mb n2b

−1

1 (C.20a) [ωa , ωb ]ab

As we see everything may be expressed in terms of a single collision integral using the results of Appendix D, namely ma [ωa , ωa ]ab (C.20) [ωa , ωb ]ab = − mb and b ]ab = [ωb , ω

ma [ωa , ωa ]ab mb

where √ 2 4kT d 2πe4 1 1 ln 1 + [ωa , ωa ]ab = √ (4π0 )2 (kT ) 32 ma e2

and d=

kT 0 e2 n

(C.20b)

(C.20c)

is Debye’s length. Eqs (C.20a-b-c) are the final result of this calculation. Improvement on the values for da0 can be obtained by considering more terms in the evaluation of { } but we leave that to the reader if and when he considers it necessary. For the first of the equations (C.15) we have that −2

i

dci ti : Ri = −2

∞ i mi c 2 i

(p) ai ni

p=0

mi 32 4π 2πkT

mi c2i 5 − 2kT 2

exp− 2kT c4i dci S 3 (ωi ) (p) 2

where the trial function used is the one in the previous case with different coefficients. Introducing the velocity ωi , recalling that ∞ 15 √ 2 (1) (p) S 3 (ω)S 3 (ω)ω 4 exp−ω dω = πδp,1 2 2 16 0


142

we wind up with the result that na (1) nb (1) a + a 15kT = {t, t} ma a mb b The calculation of {t, t} is a rather cumbersome procedure. One resorts to Eq. (B.3) setting (a)

ca ) Ka = La = a0 ca + a(1) a S 3 ( 2

(b)

(1)

Kb = Lb = a0 cb + ab S 3 (cb ) 2

and makes use of all the values for the corresponding collision integrals which are given in Appendix D. After a lengthy but straight forward algebraic procedure one finds that {ti , ti } τ is given precisely by the terms arising from the collisional integrals which are written in the right hand side of Eq. (5.8) after the ω-dependent terms namely the second and third rows are ignored. We find unnecessary to repeat that equation here. The remaining steps of the variational procedure lead precisely to Eqs. (5.9) and (5.10) when ωa = ωb = 0. As expected the results in this case are those quoted in Eqs. (5.11). The procedure to obtain the di ’s is completely analogous.

Bibliography [1] L. Spitzer; The Physics of Fully Ionized Gases, Wiley-Interscience, New York (1962). [2] S. Chapman and T. G. Cowling; loc. cit. Chap. 1. [3] G. W. Ford and G. E. Uhlenbeck; loc. cit. Chap. 1. [4] P. Goldstein and L. Garc´ıa-Col´ın; J. Non-Equilib. Thermodyn. 30, 173 (2005). [5] S. R. de Groot and P. Mazur; Non-Equilib. Thermodynamics, Dover Publications Inc., Mineola, N.Y. (1984), Chap. XI. [6] J. O. Hirschfelder, C. F. Curtiss and R. B. Byrd; The Molecular Theory of Liquids and Gases, John Wiley & Sons, New York (1964). 2nd ed.

143

Appendix D The Collision Integrals In order to determine the coefficients of viscosity, thermal conduction, and diffusion of a gas, it is necessary first to evaluate the collision integrals. In this Appendix we shall consider collisions and the evaluation of the various collision integrals in detail and how the cross section can be simply expressed for Coulomb interactions. Suppose we have two particles colliding, the first of mass m1 , charge e1 , velocity v1 and the second mass m2 , charge e2 and velocity v2 . For Coulomb forces between the particles the equation of motion is given by d2r ke e1 e2 μ 2 = r dt r3 where μ=

(D.1)

m1 m2 m1 + m2

is the reduced mass, and ke = 1/4πε0 where ε0 the permitivity of vacuum (ε0 = 8.85 × 10−12 C2 /Jm). The geometry of the collision is shown in Figure D.1. Initially particle 1 has a relative velocity (D.2) g = v1 − v2 and asymptotic distance of approach b. We suppose that its relative position makes an angle β with the direction −g , so initially β = 0. Finally β = π − χ where χ is the scattering angle we wish to find as a function of g and b. The differential cross section for scattering into unit solid angle shall be L.S. Garc´ıa-Col´ın, L. Dagdug, The Kinetic Theory of a Dilute Ionized Plasma, Springer Series on Atomic, Optical, and Plasma Physics 53 © Springer Science + Business Media B.V. 2009

145

146

Appendix D. The Collision Integrals

Figure D.1: Geometry of a collision.

replaced by the well known Rutherford formula, namely, σ(χ, ε) =

2 ke e1 e2 4 1 cosec χ 2μg 2 2

(D.3)

The angles of this expression are depicted in Figure D.1. From Figure D.1 we see that, χ = π − 2θ μbg 2 cot 12 χ = ke e1 e2

(D.4)

is convenient to make use of the unit vector kˆ drawn as show in Figure D.1. Clearly kˆ · g = g cos θ = −kˆ · g (D.5) where g = v 2 − v 1 = c 2 − c 1

(D.6)

and it is the relative velocity after the collision. Also ˆ g = g − 2(g · k) = g − 2g cos θkˆ

(D.7)


147

It is also convenient to introduce the dimensionless numbers M1 =

m1 m1 + m2

m2 M2 = m1 + m2

(D.8)

The center of gravity velocity is, relative to the drift velocity u = M1c1 + M2c2 G

(D.9)

and from (D.8) and (D.9) the following equations can be derived + M2g c1 = G + M1g c2 = G (D.10) c 1

+ M2g =G

+ M1g c 2 = G Also, c 1 = c1 + 2g M2 cos θkˆ c 2 = c2 + 2g M1 cos θkˆ

(D.11)

Now define new variables x and y by x = g

μ 2kT

y = G

(D.12) m1 + m2 2kT

148


Then

1

1

w 1 = M12 y + M22 x 1

1

w 2 = M22 y − M12 x (D.13) 1 2

w 1 = w1 − 2M2 x cos θkˆ 1

w 2 = w2 − 2M12 x cos θkˆ

where

mi 2kT Since the Jacobian of the transformation is, √ m1 m1 ∂(x, y ) = J= ∂(c1 , c2 ) 2kT w i =

(D.14)

(D.15)

and from Eq. (D.4) in terms of this new variables, the collision integral can be written as, 1 2kT 2 1 dxdy dεbdbx exp(−x2 − y 2 ) [G1 , H2 ; K1 L2 ]12 = − 3 μ π 1 ) + H2 ( w 2 )] : [K1 (w 1 ) + L2 (w 2 ) − K1 (w 1 ) + L2 (w 2 )] [G1 (w

(D.16)

Using this formula all the integrals for collisions between an electron and an ion can be worked out. Then the integrals for collisions between like particles can be obtained by setting the masses equal. We shall give the details of just one calculation, namely of [w 1, w 1 ]12 . From Eq. (D.16) this is 1 2kT 2 1 1 ]12 = − 1 · (w 1 − w 1) · · · dxdy dεbdbx exp(−x2 − y 2 )w [w 1, w μ π3 (D.17) Now from Eq. (D.11) 1 1 1 1 − w 1 ) = − M12 y + M22 x 2M22 x cos θ · kˆ (D.18) w 1 · (w Since the y integral is odd, it averages to zero on integrating over y , and finally 1 2kT 2 2π 3 2 1 ]12 = 2M2 (D.19) dxx exp(−x ) bdb cos2 θ [w 1, w 3 μ π2


149

Using Eq. (D.4) this last integral is d 0

bdb cos2 θ =

d

0

1+ 1 = 2

ke e1 e2 2kT

2

bdb 2kT x2 ke e1 e2

1 ln 1 + d2 x4

2

b2 2 2

(D.20)

2kT x ke e1 e2

This expression diverges if d goes to infinity but the divergence is only logarithmic and is therefore very slow. Whence, it does not matter very much what choice we make for d within reasonable limits. Because the value we should use for d is not fixed precisely and because the answer is insensitive anyway we might just as well replace x2 where it appears inside the logarithm by its average value which is 2. Hence Eq. (D.20) becomes 2 1 ke e1 e2 1 ψ 2 2kT x4 where ψ is the logarithm factor

ψ = ln 1 +

4kT d ke e1 e2

2 (D.21)

So Eq. (D.19) becomes 1 2 2kT 2 ke e1 e2 2ψ ∞ 1 √ 0 dx exp(−x2 ) 1 ]12 = M2 [w 1, w μ 2kT x π √ (ke e1 e2 ) ψ 2π 3 (kT ) 2 2

=

m2 m1 (m1 + m2 )

12

(D.22)

If m2 it is much grater than m1 [w 1, w 1 ]12

√

(ke e1 e2 )2 ψ 2π √ 3 = ϕ m1 (kT ) 2

Interchanging the masses in Eq. (D.23) gives 12 √ (ke e1 e2 )2 ψ m1 m1 2 ]12 2π = [w 1, w 1 ]12 [w 2, w 3 m2 (m1 + m2 ) m1 (kT ) 2

(D.23)

(D.24)

150


Now we shall calculate [w 1, w 2 ]12 as follows. By definition [w 1, w 2 ]12 = −

2kT μ

12

1 π3

1 · (w 2 − w 2) dxdy dεbdbx exp(−x2 − y 2 )w (D.25)

and from Eq. (D.13) 2 = − w 2 − w

m1 m2

12

(w 1 − w 1)

(D.26)

Comparing Eqs. (D.25) and (D.26) with Eq. (D.22) we therefore see that [w 1, w 2 ]12 = −

m1 m2

12 [w 1, w 1 ]12

(D.27)

From Eq. (D.22) and Eq. (D.27) 12 12 √ (ke e1 e2 )2 ψ m1 m2 1, w 2 ]12 = 1 − 2π [w 1 , 0; w 3 m2 m1 (m1 + m2 ) (kT ) 2 (D.28) We can now get an expression for [w 1, w 1 ]1 by setting m1 = m2 in Eq. (D.28). This gives [w 1, w 1 ]11 = 0 (D.29) Similarly [w 2, w 2 ]22 = 0

(D.30)

In a similar way all other integrals we need can be evaluated and the results are given in the following list. [w 1, w 1 ]11 = 0 [w 2, w 2 ]22 = 0 1 ]12 = ϕ [w 1, w 1

[w 1, w 2 ]12 = −M12 ϕ 2 ]12 = M1 ϕ [w 2, w 3 1 S 13 (w 12 )]12 = ϕ [w 1, w 2 2


151

3 [w 2, w 2 S 13 (w 22 )]12 = M12 ϕ 2 2 3 3 [w 1, w 2 S 13 (w 22 )]12 = − M12 ϕ 2 2 3 1 [w 1 S 13 (w 12 ), w 2 ]12 = − M12 ϕ 2 2 1 2 1 S 3 (w 1 )]1 = 0 [w 2, w 2

1

2 S 3 (w 22 )]2 = 0 [w 2, w 2

13 ϕ 2 2 4 15 1 [w 2 S 13 (w 22 ), w 2 S 13 (w 22 )]12 = M12 ϕ 2 2 2 3 27 2 16 1 2 1 2 [w 1 S 3 (w 1 ), w 2 S 3 (w 2 )]12 = − M1 ϕ 1 + δ 2 2 4 27 √ [w 1 S 13 (w 12 ), w 1 S 13 (w 12 )]1 = 2ϕ(1 − δ) 2 2 [w 2 S 13 (w 22 ), w 2 S 13 (w 22 )]2 = 2M1 ϕ(1 − δ) [w 1 S 13 (w 12 ), w 1 S 13 (w 12 )]12 =

2

where

2

2 4kT d 1 ke e1 e2 δ= 2 ψ 4kT d 1+ ke e1 e2

which, for all except very extreme conditions is 1/ψ and very small compared to unity. Hence under almost all conditions δ can be set equal to zero in this list. It remains to examine the validity of Eq. (D.1) which assumed that while the particles were interacting all forces other than their Coulomb interaction could be ignored. This will be valid provided the Debye distance, d, is smaller than the gyromagnetic radius, namely, provided 12 kT mv d= > ma and neglecting obvious small terms we find that the determinant for this system of equations is, setting x = ωa τ Δ = 47 + 1.6x2 + 9.842x × 103 −


159

160

Appendix H

Trivially then

10τ ma 258.4 − 40ix kT Δ 10τ mb 9.66 + 40ix = M1 kT Δ

ga(0) =

(H.1)

(0)

(H.2)

gb

When Gi is substituted by Pi /2 and B by 2B we obtain the solution to Eq. (8.20), 20ma τ 258.4 − 40ix (0) pa = (H.3) kT Δ 20τ ma 9.66 + 40ix (0) mb (H.4) pb = M1 kT Δ where ωa = 2ωa and Δ = Δ when ωa = 2ωa . Finally, the solution to Eqs. (8.15a) is obtained when B = 0, ωa = ωb = 0 (0)

la = 1.046 mkTa τ (H.5) (0) lb

=

τ 0.039 Mm1bkT

and τ , the mean free collision time is τ = 1/ϕn. Omitting the unnecessary superscript naught, the relation between these results and the Γi ’s is now summarized as follows, li = Γ1i + B 2 Γ3i Repi = 2Γ1i + B 2 Γ4i Impi = B(Γ2i + B 2 Γ5i )

(H.6)

Regi = Γ1i + B 2 Γ3i Imgi = Γ2i so that B 2 Γ5i = Impi − Imgi , whereas the shear viscosity η is determined by li when B = 0, in this case all other coefficients vanish. This completes the solution to the problem.

Appendix I List of Marshall’s Equations and Notation I.1

Equations n1 n2 (m2 − m1 ) ρ1 ρ2 n1 ∇p − (X1 − X2 )− −d2 = d1 = ∇( ) + n pnρ pρ

n 1 n2 (e1 m2 − e2 m1 )E (M3.14) pρ ni ei I,0 ni ei II,0 1 1 × ∇ log T } 1 (2kT ) 12 j = −{∇ log T } (2kT ) 2 √ ai − {H √ ai m 2 2 mi i i i H · ∇ log T } 1 (2kT ) 12 −H{ 2

ni ei III,0 ni ei I,0 1 1 − {ndi } (2kT ) 2 √ ai √ ei mi 2 mi i i

ni ei II,0 ni ei III,0 H · nd1 )} 1 (2kT ) 12 √ ei − {H( √ ei m mi 2 i i i (M3.61) I II ⊥ III ⊥ I II ⊥ III j = σ D + σ D + σ h × D + ϕ {∇T } + ϕ {∇T } + ϕ h × {∇T }⊥ (M3.62) m2 − m1 m1 m2 − + 1 u × H ∇p − (X1 − X2 ) D=E c n(e1 m2 − e2 m1 ) e(m1 + m2 ) × ndi } 1 (2kT ) 12 {H 2

−

n1 pρ ∇( ) n1 n2 (e1 m2 − e2 m1 ) n


(M3.63) 161

162

Appendix I. List of Marshall’s Equations and Notation

σI =

ne2 τ 1.931 2m1

σ II =

ω 2 τ 2 + 1.802 ne2 τ 2m1 ω 4 τ 4 + 6.282ω 2 τ 2 + 0.933

σ III =

−ωτ (ω 2 τ 2 + 4.382) ne2 τ 2m1 ω 4 τ 4 + 6.282ω 2 τ 2 + 0.933

where ω=−

(M7.8)

eH cm1

and

√ 3 m1 (kT ) 2 3 τ=√ (M7.10) ne4 ψ 2π + ξ II D ⊥ + ξ IIIh × D ⊥ q = −θI {∇T } + θII {∇T }⊥ + θIIIh × {∇T }⊥ + ξ I D (M7.16) nτ k2 T I θ = m1 3.59 θII =

nτ k2 T

0.458ω 2 τ 2 +3.01

m1

ω 4 τ 4 +6.282ω 2 τ 2 +0.933

+

4.458 ω 2 τ 2 +12.716

θIII =

I.2

nτ k2 T m1

1.25

(M7.19)

ωτ ω 2 τ 2 +12.716

−

ωτ (ω 2 τ 2 +6.2) ω 4 τ 4 +6.282ω 2 τ 2 +0.933

Notation

= E + 1 u × H E c c : Velocity of light. em i : coefficients of the expansions. : The magnetic field. H h : A unit vector in the direction of H. i, j : Subscripts labeling electrons and ions, 1 for electrons and 2 for ions.

I.2. Notation

163

j : The conduction current. k : Boltzmann s constant. mi : The mass of particle i. ni : The number density of particles i. n = n1 + n2 : The total number density. q : The heat flux vector. T : Temperature. u : The drift velocity. Xi : any non-electromagnetic force per unit mass which acts on particles i. ξ n : Coefficients giving the contribution to the heat flux from the gener alized electric field D. ϕn : Thermal diffusion coefficients. ψ : Logarithmic cut-off factor. θn : Thermal conduction coefficients. σ n : Coefficients of electric conductivity. ρi : Density of particles i. ρ : Total density. τ : A collision time for electrons. ω: The gyromagnetic frequency for electrons.

Index diffusion flux, 15 diffusive force, 28, 77 distribution function, 13 Donder, T. de, 6 Dorn, 69, 77 Dufour effect, 8, 29, 47, 63, 69, 133

Balescu, 73, 75, 76, 105, 116 barycentric velocity, 15 Benedicks effect, 8, 65, 66, 114, 120 Boltzmann, 9, 18, 75, 129, 134 Boltzmann equation, 9, 13, 18, 19, 20, 107, 119, 133 Boltzmann’s constant, 6, 163 Braginski, 63, 75, 105, 133 chaotic velocity, 15 Chapman, 25, 78 characteristic hydrodynamical length, 109 charge current, 15, 65, 108 CGS, 76 CIT, 5, 8, 10 Clausius, 6, 7 collision integrals, 145, 159 conservation equations, 18 Coulomb, 14, 145, 151 Cowling, 25, 78 cross section, 14 Curie’s principle, 93 Dalton’s law, 120 Davison, 51 Davison’s function, 100 Debye distance, 151 Debye-H¨ uckel, 14 Debye’s length, 62, 141 diffusion effects, 41

electrical conductivity, 44, 67, 110 entropy flux, 6 Euler, 26, 27, 119, 120 Fickian diffusion, 62, 67, 69 flow of heat, 45 Fourier’s equation, 63 Grad, 75 Groot and Mazur, 78, 102 gyromagnetic radius, 151 H Theorem, 18 Hall’s effect, 62 heat flow vector, 7 heat flux, 61 Hilbert-Chapman-Enskog, 26, 75 hydrodynamic velocity, 15 internal energy density, 17 Kelvin, 9 Klein’s inequality, 19 Knudsen, 75

165

166

Index

Landau-Fokker-Planck, 9, 75 Larmor, 18, 159 LEA, 5, 6, 7 Levi-Civita tensor, 85 LIT, 104, 112 local energy density, 6 local entropy, 6 local particles density, 6, 14 logarithmic function, 62, 105

Onsager matrix, 136 Onsagers’ reciprocity theorem, 9, 104, 133, 135 ORT, 9

Marshall, 76, 77, 78 mass conservation, 16 mass conservation equation, 108 mass current, 41, 108 mass flux, 8 Maxwell-Enskog, 16 Maxwell distribution, 19, 21 Maxwell’s equations, 14, 74, 107, 108, 109, 118, 120, 134 Meixner, 6 momentum, 6 momentum current, 108 momentum flow, 7

single particle distribution functions, 13 Sonine (Laguerre) polynomials, 35, 94, 137 Soret effect, 29, 43, 133 Spitzer-Braginski calculations, 63, 75, 133 stress tensor, 93 subsidiary conditions, 28

Navier-Newton, 112 Navier-Stokes-Fourier, 7, 107, 109 numerical charge density, 15 Ohm’s coefficient, 111, 115 Onsager, 9, 10, 29

particle density, 6 Righi-Leduc effect, 47, 62, 67, 68, 76 Rutherford, 146

thermal conductivity, 46, 64 thermoelectric coefficient, 67 Thomson coefficients, 67, 68, 69, 77, 111 Thomsons’ thermoelectric effect, 8, 62, 111 total mass density, 13

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