Physics of Plasmas_L.C.Woods

Terms corresponding to pressure and temperature appear in the moment equations and yet these properties are essentially continuum concepts that require the existence of local thermodynam

L C Woods

Physics of Plasmas

Author

Prof Dr Leslie C Woods

University of Oxford and Balliol College

Upper right: Heating coronal loops

Credit: M Aschwanden et al (LMSAL).TRACE

NASA

Lower right: A soft X-ray image of the sun

Credit: ESA, NASA

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Preface

This text gives an account of the principal properties of a tenuous gas, hot enough for some

of the molecules to shed electrons and become ionized In general a macroscopic volume of such a gas consists of a mixture of free electrons and the ions and neutrals of several molecular species and is called aplasma If the temperature is high enough, e.g N 10 000 K at a pressure

of 1 Pascal, a hydrogen plasma will be fully ionized, which is the case of most interest in this book If there is also a magnetic field present, the ions and electrons will gyrate about the field lines, producing an anisotropic medium with some very interesting properties Because of the

orbiting motions, it is more difficult for the plasma to flow across the magnetic field lines than

along them and with very strong fields both the plasma and its energy are said to be ‘confined’

by the field, although some leakage across the field lines does occur

Examples of naturally occurring magnetoplasmas are found in the Sun’s corona, the solar wind and comet’s tails; laboratory examples include the plasma created in the fusion research machines known as tokamaks and in the application of what is termed ‘plasma processing’ to

the manufacture of semiconductor devices Although molten metal is not a plasma, it is a con-

ductor of electricity and therefore subject to magnetic forces; its behaviour is described by the equations of magnetohydrodynamics (MHD), which are a limiting case of the magnetoplasma equations Electric currents are used in industry to heat metals to the liquid state, when these metals can be stirred, levitated and pumped with magnetic fields New applications of plasma physics arise from time to time; however, in a short book such as this there is space for little more than the basic principles of the subject

One of the attractions of plasma physics is the range of subjects required for its understand- ing; these include fluid mechanics, electricity and magnetism, kinetic theory and thermody- namics, although for this text relatively little experience in these topics is assumed There are many equations, so some effort has been made to cross-reference them at each stage of devel- oping the theory To help the reader with mathematical points, I have included ‘mathematical notes’ at appropriate stages in the chapters, and I have also have added some appendices cover- ing standard analyses With a subject like plasma theory, subscripts are essential to distinguish between the properties of the several fluid components, so to avoid doubling up on subscripts,

I have followed the common practice of employing the dyadic notation for tensors and where the vector and tensor analysis is complicated, I have filled in the steps involved

Many texts on plasma theory begin with a description of the collisionless motion of in- dividual charged particles known as particle orbit theory Particles at a given time t and at a

point r in physical space are then grouped according to their velocity w and a ‘kinetic’ equa-

tion describing the evolution of the number density of particles at a point P = P(r, w, t) in phase-space is found It is at this stage that particle collisions enter the model via a collision operator @, which removes particles from P or introduces particles into P by collisional scat- tering Finally, integrals of the kinetic equation over velocity space yield the fluid or MHD equations However, these moments representing the conservation of mass, momentum and

VI Preface

energy, are independent of @, the term containing which vanishes in each integration Hence

C could in fact be zero The standard account thus precedes from a microscopic descrip- tion to what purports to be a collisional macroscopic model, without collisions playing any role at all Terms corresponding to pressure and temperature appear in the moment equations and yet these properties are essentially continuum concepts that require the existence of local thermodynamic equilibrium, a state for which particle collisions are essential

To avoid the confusion and occasional errors that the standard approach has introduced into plasma theory, in this text the subject is developed in the reverse order from that described above, that is we start with collision-dominated classical fluid mechanics in Chapter 1, adding the effects of electromagnetic fields in Chapter 2 At this stage we only need sufficient knowl-

edge of particle orbit theory to determine the length and time scales below which a fluid or continuum description is not valid

Chapter 3 presents the theory of small amplitude plasma waves and shock waves, and

finishes with a brief introduction to magneto-ionic theory, required in studying the reflection and scattering of radio waves in the ionosphere Stability of plasmas is treated in Chapter 4,

covering the usual macroscopic instabilities of ideal plasmas, and also an important instability that depends on the electrical resistivity Finally we remove collisions entirely from the model and introduce the Vlasov theory of plasma waves, applying it to Landau damping and the ion-acoustic instability, which has important applications in solar physics Chapter 5 , which

is concerned with transport in magnetoplasmas, starts from the Fokker-Planck equation and gives an account of the theory of electron-ion collision intervals and several other relaxation times of important in the transport of particle energy and momentum

The final chapter collects a miscellany of important topics, including second-order trans- port theory, thermal instabilities, particle orbit theory, magnetic mirrors, partially ionized plas- mas and a brief introduction to some important applications of plasma physics By second- order transport is meant, for example, the transport of heat in the presence of strong fluid shear, when the heat flux vector depends not only on the temperature gradient as in Fourier’s law, but also on the rate of strain of the fluid This proves to be very important in the presence

of magnetic fields and leads to the thermal instabilities next described in the chapter Particle orbits in the presence of magnetic field gradients is a particularly important phenomenon in near-collisionless plasmas, with applications to transport in tokamaks Partially ionized plas- mas add the complexity of a third fluid comprised of the neutral particles, to the model, so

a brief introduction to Saha’s equation for the dependence of the degree of ionization on the temperature and pressure is included The final section briefly describes a few important ap- plications of the theory - fusion research, solar physics, metallurgy, MHD direct generation

of electricity and dusty plasmas

The treatment ispitched at a level suitable for graduate students in mathematics, engineer- ing and physics who need an introductory account of plasma physics It is recommend that

the reader should aim to get a clear physical picture of the mechanisms at each stage before

checking through the analysis Most of the exercises are straightforward extensions of the theory and therefore worthy of attention

L C Woods Oxford, 1st August, 2003

Contents

1.1 Molecular models and fluids 1

1.1.1 Introduction 1

1.1.2 Microscopic particles 2

1.1.3 The mean free path 3

1.1.4 Fluid particles 4

1.2 Macroscopic variables 4

1.2.1 Number density 4

1.2.2 Fluid velocity 5

1.2.3 Temperature 9

1.2.4 Equations of state 11

1.3 Pressure 12

1.3.1 Macroscopic definition of pressure 13

1.3.2 Kinetic definition of pressure 14

1.3.3 Vanishing pressure gradient 15

1.3.4 Local thermodynamic equilibrium 17

1.4 Macroscopic conservation laws 18

1.4.1 Convection and diffusion 18

1.4.2 A general balance equation in physical space 19

1.4.3 Conservation laws for a simple fluid 21

1.4.4 Specific entropy 22

1.5 Introduction to kinetic theory 23

1.5.1 Kinetic entropy 23

1 S.2 Equilibrium distribution function 24

1 S.3 Averages over velocity space 26

1 S.4 Evolution of the phase-space density 28

1 S 5 Boltzmann's distribution law 29

2 Magnetoplasma Dynamics 33 2.1 Electromagnetic fields 33

2.1.1 Maxwell's equations 33

2.1.2 Galilean transformations 35

2.1.3 Polarization 37

2.2 Basic plasma parameters 39

VIII Contents

2.2.1 Plasma neutrality

2.2.2 The cyclotron frequency

2.2.3 Plasma frequency

2.2.4 The Debye length

2.3 Magnetohydrodyamic equations

2.3.1 Ohm'slaw

2.3.2 Conservation laws in MHD

2.3.3 Lagrangian form and entropy production

2.4 Electromagnetic farces

2.4.1 Stress tensor and Poynting vector

2.4.2 Magnetic forces in MHD

2.4.3 The induction equation

2.4.4 Difision of magnetic fields

2.4.5 Conservation of magnetic flux

2.5.1 Steady state equations

2.5 Magnetostatics

2.5.2 The theta pinch

2.5.3 The linear pinch

2.5.4 Axisymmetric toroidal equilibrium

2.5.5 Force-free magnetic fields

2.6 Transition equations across surface layers

2.6.1 Surface intensities

2.6.2 Boundary conditions

2.6.3 Current sheets and surface charge

2.6.4 Fluid equations

3 Waves in Magnetoplasmas 3.1 MHD waves in an unbounded plasma

3.1.1 Introduction

3.1.2 Linearization

3.1.3 The dispersion equation

3.1.4 MHDwaves

3.2 Coupled plasma waves

3.2.1 High frequency waves

3.2.2 Whistlers

3.2.3 Propagation of wave fronts

MHD waves in cylindrical plasmas

3.3.1 The dispersion relation

3.3.2 Fast wave cut-off

3.4 Group velocity

3.4.1 Transmission of energy

3.4.2 Wave packets

3.5 Shock waves

3.5.1 Jump conditions across an MHD shock

3.5.2 Thermodynamic constraint

3.3

39

39

40

41

42

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46

48

48

49

50

53

54

54

54

55

56

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59

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79

80

80

81

82

82

85

Contents IX

3.5.3 Classification of MHD shocks 87

3.5.4 Perpendicular shock waves 89

3.6 Magneto-ionic theory 92

3.6.1 Electrical conductivity 92

3.6.2 The dielectric tensor 93

4 Magnetoplasma Stability 97 4.1 Rayleigh-Taylor and Kelvin-Helmholtz instabilities 98

4.1.1 Linearized equations 98

4.1.2 Surface waves 99

4.1.3 The dispersion equation 100

4.1.4 Special cases 101

4.2 Interchange instabilities 103

4.2.1 Flute instability 103

4.2.2 Thermal stability 105

4.3 Instabilities of a cylindrical plasma 106

4.3.1 The sausage instability 106

4.3.2 The kink instability 107

4.3.3 Stability condition 108

4.4 The energy principle 110

4.4.1 Potential energy 111

4.4.2 Surface term 112

4.4.3 General stability condition 113

4.4.4 Cylindrical plasma with a volume current 115

4.5 Resistive instabilities 115

4.5.1 The tearing mode 116

4.5.2 Differential equation for By 117

4.5.3 Physics of the tearing mode 118

4.6 The two-stream instability 120

4.6.1 Vlasov theory of plasma waves 120

4.6.2 Solution of the dispersion equation 122

4.6.3 Landau damping 123

4.6.4 The ion-acoustic instability 124

4.7 Fibrillation of magnetic fields 126

5 Transport in Magnetoplasmas 131 5.1 Coulomb collisions 131

5.1.1 Particle diffusion in electric microfields 131

5.1.2 Particle orbits 133

5.1.3 The Rutherford scattering cross-section 135

5.2 The Fokker-Planck equation 136

5.2.1 Friction and diffusion coefficients 136

5.2.2 Scattering in velocity space 138

5.2.3 Super-potential fknctions 139

4.3.4 Stability of an unbounded flux tube 109

X Contents

5.3 Lorentzian plasma 141

5.3.1 Collisional loss rate 141

5.3.2 Expansion of the distribution function 142

5.3.3 Electrical conductivity 143

5.3.4 Conductivity in a fully-ionized plasma 144

5.4 Friction and diffusion coefficients 146

5.4.1 First super-potential 146

5.4.2 Second super-potential 147

5.4.3 Limiting cases 148

5.4.4 Relaxation times 149

5.5 Transport of charge and energy 152

5.5.1 Ohm’slaw 152

5.5.2 Resistivity in a magnetoplasma 152

5.5.3 Fourier’s Law 153

5.5.4 Thermal conductivity in a magnetoplasma 154

5.6 Transport of momentum 155

5.6.1 Classical formula for the viscous stress tensor 155

5.6.2 The viscous stress tensor in a magnetic field 157

6 Extensions of Theory 165 6.1 Second-order transport 165

6.1.1 Convection versus conduction 166

6.1.2 The second-order heat flux 167

6.1.3 The viscous stress tensor 170

6.2 Thermal instability 170

6.2.1 Heat flux in a cylindrical magnetoplasma 170

6.2.2 Unstable current profiles 172

6.2.3 Planar geometry 174

6.2.4 Heating the solar corona 175

6.3 Particle orbit theory 176

6.3.1 Rate of change of the peculiar velocity 176

6.3.2 Guiding centre drifts 177

Drifts due to variations in the magnetic field

6.3.4 Gyro-averages 180

6.3.5 The grad B and field curvature drifts 182

6.4 Magnetic mirrors 184

6.4.1 Constants of motion of gyrating particles 184

6.4.2 Magnetically trapped particles 185

6.4.3 Fraction of trapped particles 186

6.5 Partially ionized plasmas 187

6.5.1 Degree of ionization 187

6.5.2 Ratio of the specific heats 188

6.5.3 Resistivity 190

6.6 Applications of plasma physics 191

6.6.1 Tokamak research 192

Lists of physical constants, plasma parameters and frequently used symbols

In SI units, the constants required in plasma theory are:

Permittivity (Free Space)

Permeability (Free Space)

Speed of light (Vacuum)

Protordelectron mass ratio

1.6022 x 10- l9

1 3 8 0 7 ~ 1 0 - ~ ~ 8.8542~10-”

4n x1OP7 2.9979~10’

1.8362~10~

1.1605~10~

6 6 2 6 2 ~ 1 0 - ~ ~ 5.6703~10-’

1 The Equations of Gas Dynamics

1.1 Molecular models and fluids

A plasma is a mixture of positive ions, electrons and neutral particles, electrically neutral

over macroscopic volumes, and usually permeated by macroscopic electrical and magnetic fields In addition to these ‘smoothed’ or averaged electromagnetic fields, which with lab- oratory plasmas are often imposed from outside the plasma volume, there are the localized micro-fields due to the individual particles The trajectories of the charged particles are thus continuously modified by a range of electromagnetic forces, the average fields acting like body forces and the micro-fields like collisional forces The micro-fields are responsible for the transmission of pressure and viscous forces, for the conduction of particle energy, and

for the friction forces between diffusing components of the plasma Some care is needed

in dividing the continuum of electromagnetic forces into their macroscopic and microscopic components, but with this achieved, there is little formal distinction between the theory of the macroscopic behaviour of neutral gases and that of magnetoplasmas

The aim of this book is to describe the various physical processes that underpin plasma theory and the equations representing these processes The distinction between what we shall term the ‘mechanisms’ and the equations based on them - symbolisms - is particularly im- portant in a complex subject like plasma physics Definitions of physical properties can be taken either from the mechanisms or the symbolisms, but one must take care not to mix the two, e.g to adopt a purely mathematical definition of a property and then to assume that this automatically entails the usual physical attributes of that property

Our method is to commence with the macroscopic description of the individual compo- nents of the plasma, that is we shall treat the collections of electrons, ions and neutrals as comprising separate fluids, and their fluid properties developed In the next chapter they will

be combined to make a plasma, an approach with the merit of making a clear distinction between the fluid and electrical properties of a plasma Readers already familiar with fluid mechanics might skip to Chapter 2, although in § 1 .S there is an introduction to kinetic theory that will be required in later chapters

Except for the basic concept of a ‘mean-free-path’, the trajectories of individual particles will be described in a later chapter In this chapter we shall introduce the standard macroscopic variables of gas dynamics, such as pressure, temperature, fluid velocity and entropy, and de- rive the equations relating them Excepting entropy, these physical properties are best defined

Physics of Plasmas

L C Woods

Copyright 0 2004 WILEY-VCH Verlag GmbH & Co KGaA

2 I The Equations of Gas Dynamics

in terms of mechanisms although sometimes synthetic definitions have a place Consider tem- perature for example; either it is defined physically via thermometers and the mechanism of

thermal equilibrium, which requires close physical contact through molecular particle colli- sions, or it may be defined symbolically as a kinetic temperature, which is a property of the

distribution of molecular velocities and collisions are not explicitly involved The danger of employing the second definition is that it is too easy to adopt properties of temperature that re- ally depend on the first definition For example the conduction of heat, which depends on the temperature gradient, is a collisional process in which the gradient of the kinetic temperature would be misplaced without the additional constraint that the medium is collision-dominated, the precise meaning of which will be discussed later in 1.3.4

1.1.2 Microscopic particles

A substance in the gaseous state consists of an assembly of a vast number of microscopic particles that, excepting when they collide with each other, move freely and independently through the region of physical space available to them The nature of the particles depends largely on the temperature of the assembly At low temperatures, but above the critical value

at which liquefaction can occur, they are molecules At higher temperatures the molecules dissociate into atoms, and at still higher temperatures the atoms become ions by shedding some of their electrons The resulting assembly is termed a ‘plasma’ Partially ionized plasmas consist of a mixture of neutral atoms, electrons, and ions, requiring at least three distinct species of microscopic particles to be included in a complete mathematical representation of their collective behaviour

The simplest model of a microscopic particle is a small featureless sphere, possessing a spherically-symmetric force field For neutral particles this field has a very short range, and the particles can be pictured as being almost rigid ‘billiard-balls’, with an effective diameter equal to the range of the force field As they have no structure, these particles have only energy

of translation The gas is usually assumed to be sufficiently tenuous for collisions involving

more than two particles at a time to be ignored, i.e only binary collisions are considered The

model is appropriate for monatomic uncharged molecules

Diatomic and more complex molecuIes do not have symmetric force fields, but for many purposes they are also well represented by the billiard-ball model Their relative orientations

at collisions may be assumed to be randomly distributed, so that averages taken over a large number of encounters will have values independent of orientation, just as with symmetric force fields It is the internal vibratory energies possessed by multi-atomic particles that give rise to the largest discrepancies between the predictions of the simple billiard-ball model and observation, a phenomenon that is easily included in kinetic theory by adding an average internal energy to the translatory energy of each molecule

Kinetic theory is concerned mainly with the connection between the motions and interac- tions of microscopic particles comprising a gas and the transport of macroscopic properties like fluid momentum and energy through that gas The oldest example relating macroscopic properties to microscopic behaviour is provided by the pressure force acting on the walls of

a gas container That it is due to the near-continuous bombardment of the walls by the vast number of neighbouring molecules, is a concept dating back to Boyle and Newton The more subtle relationship between heat and the energy of molecular agitation required more than an-

1 I Molecular models andjuids 3

other century before it was revealed with increasing detail in the works of Waterston, Clausius, and Maxwell’ Clausius’ main contribution to kinetic theory was the concept of the mean free path, which is the average distance travelled by a molecule between successive collisions, and which led to Maxwell’s introduction of the velocity distribution function, to be discussed in

$1.5

The intermolecular force law plays a central role in kinetic theory and classical kinetic theory proceeds on the assumption that this law has been separately established, either empir- ically or from quantum theory, except with charged particles, when the well-known Coulomb force law applies We shall return to this topic in Chapter 5; for the present it is sufficient to understand the concept of the mean-free-path

Two microscopic parameters play a leading role in our account of the collective behaviour

of an assembly of particles These are the mean free path A, which is the average distance moved by a particle between successive encounters with other particles, and the collision interval T , which is the average time taken by a particle to move this distance The reciprocal

of r is known as the ‘collision’ frequency, v = T - ~ The terminology is particularly fitting for ‘hard’ molecules, i.e those with force fields abruptly falling to zero outside a molecular diameter CT, say

An approximate formula for X can be found as follows Suppose there are n molecules per unit volume, and we assume that all are stationary, save one that has a velocity v, relative

to the others In a tenuous or dilute gas, X >> CT and hence mr2vr is a good approximation

to the volume swept in one second by the sphere of influence of the moving particle Those molecules with centres lying within this volume will experience a collision and therefore the collision frequency per molecule is 7-l = m2nvr Replacing v, by the average molecular speed E relative to the centre of mass of all the similar molecules within a macroscopic volume element, and writing X = re, we arrive at the estimate

The accurate formula for X is 2-4 times this value

‘Soft’ molecules have extended force fields that make only slight changes in the momen- tum and energy of most passing molecules, so many such ‘grazing’ collisions are required to accumulate significant changes in these properties for a given test particle However, by mod- ifying to denote an ‘effective’ diameter, we can extend (1.1) to the case of soft molecules Then X becomes the average distance that a sequence of small-angle collisions takes to stop

a test particle moving in a given direction, i.e to give a 90” deflection, and T is the time it takes for this change to happen Even with hard molecules, a small sequence of collisions is required to ‘stop’ a particle Another consequence of this cascade process is that momentum and energy require related but slightly different times to be transported in a specified direction The Coulomb force fields of electrons and ions have ranges extensive enough to influence great numbers of nearby particles, so that purely binary collisions are very rare The billiard- ball model and the associated concept of a mean free path are not strictly relevant, although

‘See The kind of motion we call heat by Stephan Brush, North-Holland Publishing Company, 1976

4 I The Equations of Gas Dynamics

it is usual to describe the distance required for a 90” deflection of a test particle as being a

‘mean free path’ More precisely, X is defined to be the distance over which this particle loses its momentum along its initial direction of motion

A fluid is sometimes described as being a ‘continuum’, that is a substance that has a

continuous rather than a discrete structure, but since nature is particulate, obviously this is

an approximate model, valid only on a length scale so large that the mean free path appears negligible2 Familiar properties of a fluid are density, pressure, temperature and velocity, which we shall discuss in detail shortly However, there are other important properties of fluids which depend on X having a non-zero magnitude and that would vanish in a genuine continuum, e.g fluid viscosity, electrical conductivity and thermal conductivity, all of which are proportional to relevant mean free paths

The mechanism that qualifies an assembly of particles to be described as being afluid is the frequency of collisions between the particles, which in turn depends on the size C of the assembly Evidently if C is smaller than the mean free path A, there will be few collisions and the assembly will lack the continuum properties required of a fluid If L << A, the assembly

is certainly not a fluid and is described as being a ballistic system; in plasmas the word ‘colli- sionless’ is adopted with the same meaning The smallest physical element that can be called a fluid is termed a ‘fluid particle’ and such elements have the usual properties of thermodynamic systems Neighbouring fluid particles interact with each other via collisions, e.g by exerting

a force on each other and by exchanging particle energy, whereas adjacent ballistic elements can not do this It follows that a key dimensionless parameter in fluid mechanics is the ratio

of a microscopic length (or time) to a macroscopic scale length (or time), a ratio known as the Knudsen number, k, We shall return to this basic concept in 31.3.4

a vast number of microscopic particles of the species under consideration This is termed a macroscopic point Let the number of particles in dr be n dr, then the fluctuations in n due

to particles entering or leaving dr merely due to the discreteness of matter, will be negligible

We may therefore introduce the number density n(r, t ) as a continuum variable provided the size dC = (dr)1/3 of the fluid particle is very much larger than the inter-particle distance,

n 1/3 On the other hand, we must not take d! to be too large, otherwise significant variations

*A ‘fluid’ is to be distinguished from a ‘liquid’, which is a special case of a fluid

TheJfuid density is the macroscopic variable

where m is the particle mass, and for the present we are assuming that only one molecular species is present

Let w be the velocity of a typical particle at P, measured relative to a frame L (the 'lab- oratory' frame), and use ( + ) to denote average values taken over the particles in dr The

fluid velocity at P is the macroscopic variable v(rl t ) = (w) It is the velocity of the centre of mass of the particles at P For a more precise account, we need a statistical treatment The basic variable for statistical mechanics was introduced by Maxwell at the end of the eighteen fifties; this is the number density f ( r , w, t ) in six-dimensional phase space, usually

termed the 'velocity distribution function' Thus f(r, w , t) dr dw is the number of molecules

that at time t have positions lying in a volume element dr (= dz dy dz) about the position

r, and velocities (of translation) lying within the velocity-space element dw (= dv, dv, dv,)

about w

The physical number density is

where the integral is over the whole of velocity space It follows that f dw/n is the probability

that a given molecule at the (macroscopic) point (r, t ) has a velocity in the element dw at w

We shall sometimes adopt the notation

where the average on the left-hand side is the macroscopic variable corresponding to the phase-space function 4

6n/n N N - ' l 2 , where N is the total number of particles in the system For a system of volume

condition that 6n/n << 1 requires that 71(6.!?)~ >> 1, as in (1.2)

6 1 The Equations of Gas Dynamics

and similarly the acceleration of the fluid at P(r, t) is

fi = 1 w f(r,w,t)/ndw,

where w is the acceleration of the typical particle relative to L When there are several distinct components present, the i-th component of which is a fluid with a velocity vi and density pi,

the velocity of the fluid as a whole is

and we can therefore interpret v as being the velocity of mass flux An obvious property of

v is its dependence on the choice of the laboratory frame L, and as the equations describing the behaviour of fluids must be independent of this choice, v can not appear alone in these equations A ‘ frame-indifference’ molecular velocity will be introduced shortly

There is another fluid motion of considerable importance in the theory This is the ‘spin’

n of the fluid at P(r, t), the meaning of which is that the fluid circulates around the mathe- matical point M(r, t) with an angular velocity a Below we shall show that it is related to the fluid vorticity 6 = V x v by4

A macroscopic point that coincides momentarily with P(r, t), that has the same velocity, acceleration, rate of change of acceleration and so on, as the fluid, is called a convectedpoint, and its locus is termed a path line If axes are fixed relative to the fluid at this point and

allowed to rotate with the local fluid spin 52, then the point thus augmented, say P,, is a

convectedframe Viewed from this frame, the fluid near P, will appear to be almost stationary, and without spin This is a frame in which the ambient fluid is stationary and therefore it is the appropriate frame in which to specify the local thermodynamic system By referring to a fluid property ‘p ‘at Pc’, we shall mean that value of ‘p as observed in a convected, spinning

macroscopic point at r, t The spin and acceleration of P, are important when time derivatives

of vectors and tensors at P, are required in the theory

Spatial changes in the fluid velocity v(r, t ) influence the transport of properties between adjacent fluid particles To calculate the effects on transport we need the following analysis Suppose that a convected point Pc(r, t) moves with a velocity v(r, t), then a neighbouring convected point Qc(r + R, t) has the fluid velocity

1.2 Macroscopic variables 7

Figure 1.1: Strain of a fluid element

where R is an infinitesimal displacement vector (see Fig 1.1) The combination Vv is a second order tensor known as the velocity gradient tens03 , which can be analyzed into three distinct components, known as pure rate of strain, dilatation and spin (or vorticity) For this purpose we require the following mathematical analysis

~~ ~

Mathematical note 1 The decomposition of second-order tensors

X

In general a second-order tensor A has a symmetric part A", an antisymmetric part A", a trace A, a

vector A", and a deviator A defined by 0

As = i ( A + A ) , A" = a(A-A), A = 1 : A , A" = fl x l : A , A = A " - g l i, (1.11) where 1 is the unit tensor, i.e 1 A = A and A 1 = A, and the tilde denotes the transposed ten- sor Since 1 : 1 = 3 and 1 : A = 1 : A, it follows that the deviator of i has zero trace With double products like ab : A we shall adopt the convention that ab : A = b - A a = A : ab, e.g if

A = Kij, ab : A = K(a j)(b i) Hence

8 I The Equations of Gas Dynamics

Since a tensor A can always be expressed as the sum of three dyads, e.g A = ab + cd + ef, it follows that

A": 6" =A" x 1 :1 x B" = -2A" *B",

and therefore expanding each tensor, we obtain

fl is the angular velocity el Hence the second right-hand term of (1.16) represents a rigid

body motion of the fluid element with an angular velocity 51 Such motion does not strain (i.e deform) the element, and it will not induce a stress, except in materials of unusual micro- structure The term 52 x R can be removed from (1.16) by transforming to the convected

frame P,

Let R be the unit vector along R, then by (1.16) the 'outward' speed of Qc relative to P, is

IRI times gV - v + RR : Gv If R is distributed isotropically, the average of RR taken over

1.2 Macroscopic variables 9

0

all directions radiating from P, is i 1 (cf (1 30)) and as 1 : V v = 0, the average fluid speed

outwards from P, on the sphere IRI = a is iaV - v Thus the third right-hand term in (1.16)

is due to the changing volume of the fluid element; this type of strain is called dilatation

The remaining term in (1.16), representing pure straining motion without volume change,

is called the deviatoric rate of strain and it plays a central role in transport theory The sym-

metric part of the velocity gradient tensor, viz

since no fluid crosses it

The velocity w of a typical particle p can be divided into two components, (i) the fluid

velocity v = (w) and (ii) the peculiar velocity c, peculiar that is to the particular particle

under consideration Thus w = v + c, and by definition ( c ) = 0 The distinction between v

and c is fundamental in kinetic theory In particular notice that c is independent of the choice

of reference frame, which of course is not true of w The fluid particle P, is the basic ther- modynamic system in fluid mechanics and the kinetic theory approach to the thermodynamic

variables defines them as averages over functions of c

The basic thermodynamic variable is temperature; it is usually introduced with the aid

of the concept of thermal equilibrium between two contiguous macroscopic systems, say G and W Such systems are said to be in thermal equilibrium if no net energy transfer occurs between them when they are in physical contact At the microscopic level ‘physical contact’ means that the molecules of G and W are colliding with each other One of the basic laws of thermodynamics - known as the ‘zeroth’ law - states that

Two systems in thermal equilibrium with a third are in thermal equilibrium with each other

The third system can be regarded as being a ‘thermometer’, and the three systems are said to

be at the same temperature

Now choose G to be a gas and W to be the rigid boundary wall confining it Take the line of impact at the collision to be the axis O X , not necessarily perpendicular to the wall (see Fig 1.2) Let the velocity components of a gas particle GI and a wall molecule W, be

u, v, w, and U, V, W just before the collision and u’, d, w’ and U’, V’, W’ just after it, then from momentum and energy conservation, the velocity components parallel to OY and 02

10 1 The Equations of Gas Dynamics

will be unchanged, whereas in the OX direction,

m u + MU = mu‘ + MU’,

imu2 + MU^ = $mut2 + MU'^,

m ( u 2 - u 12 ) = - M ( u ~ - ~ ’ ~ 1

From the second pair of equations it follows that u + u’ = U + U’, whence the relative velocity

u - U is reversed by the collision, as required by perfect elasticity This condition and the momentum equation gives

(m + M)u’ = (m - M ) u + 2MU, (m + M)U’ = -(m - M)U + 2mu, and hence the gain in the wall’s kinetic energy per collision is

2mM (m + M ) 2 iM(U‘2 - U2) = {mu2 - MU^ + ( M - r n ) u ~ } ,

Gas

Figure 1.2: Collision between gas and wall molecules

As the wall is stationary, WI oscillates about a mean position fixed in the wall, and since u

and U are uncorrelated, over a large number of collisions along OX the product uU will have positive and negatives values with equal probability Hence the average of UU is zero, so the net gain of wall energy is proportional to the average of (mu’ - M U 2 ) , or equivalently to the average of (mu2 + muI2 - MU2 - MUr2) We now extend the averaging to all directions of

OX to find that the average gain of energy by the wall is proportional to Q, where

4mM

Q = (m + M ) 2 { ($mc2)G - ( $ M c ~ ) ~ } ;

1.2 Macroscopic variables I 1

here c is the peculiar speed (the fluid velocities are zero) and the subscripts denote gas and wall molecules Thermal equilibrium therefore requires that

(+mc2)&? = (aMC2)w,

in which case the gas and wall are at the same temperature

If a second gas is present, also in thermal equilibrium with the wall, then

the subscripts denoting the first and second gases Hence

and the gases are in thermal equilibrium with each other; this is the zeroth law described earlier We have now established the result:

When two gases at the same temperature are mixed, the average kinetic energy of their molecules is the same

The above suggests that we could define the absolute temperature T ( r , t) at a point Pc(r, t )

as being proportional to the average energy of translation of the particles in P, Hence we take

%kBT = rn(;c2) (IC, = 1.3807 x 10-23JK-1), (1.19) where the constant of proportionality IC, is termed Boltzmann’s constant However, the def- inition is based on the fact that energy can be transported between adjacent systems, so it is important not to forget the role of collisions; (1.19) does not apply in a collisionless gas

1.2.4 Equations of state

At P, a particle has the energy rn(;cz + E ) , where me is the energy due to its internal motions and its intermolecular potential The average particle energy per unit mass will be denoted by u; thus

is a macroscopic variable u(r, t ) , termed the speciJic energy (i.e energy per unit mass of the

medium) From (1.19) and (1.20),

u = iRT + ( E ) ( R = k B / m ) , (1.21) where R is termed the gas constant As ( E ) is found to depend on e and T , a more general formof(1.21)is

a relation known as the caloric equation of state

12 1 The Equations of Gas Dynamics

If the particles have no internal structure, i.e possess energy of translation only, (1.2 1) gives

where c, is the spec$c heat at constant volume In 51.3.3 we shall show that with such

particles the pressure, i.e the force per unit area, acting normal to a convected surface, has an average value of

This basic relation depends on the definition of temperature in (1.19) and is therefore not valid

in a collisionless gas A gas to which (1.23) and (1.24) apply is said to be aperfect gas More generally the pressure is related to e and T by a relation

known as the thermal equation of state

One mole is a mass of gas in grams numerically equal to its molecular weight M ; e.g one mole of 0 2 is 32 grams of oxygen From (1.24) applied to a volume V of gas containing N molecules,

vessel The microscopic picture defines pressure as being due to the change of momentum of

the particles due to collisions either with other particles or with the wall molecules To quote Maxwell (1 860),

1.3 Pressure 13

“Daniel Bernouilli, Herapath Joule, Kronig, Clausius, etc have shown that the relations between pressure, temperature and density in a perfect gas can be explained by supposing the particles to move with uniform velocity in straight lines, striking against the sides of the containing vessel and thus producing pressure It is not necessary to suppose each particle

to travel any great distance in the same straight line: for the effect in pmducingpressure will

be the same if the particles strike against each other; so that the straight line described may

be very short.”

we need to introduce the concept of a pressure tensor

We start with the macroscopic interpretation of pressure To include the action of viscosity,

Figure 1.3: Pressure components

Consider the force acting on one face of a small parallelepiped as illustrated in Fig 1.3

For the face lying orthogonal to the OX axis the force is (px2i + p,,j + p,,k) dy dz, where the first subscript indicates the orientation of the face and the second the direction of the force and i, j, k are unit orthogonal vectors in a Cartesian coordinate system Similarly the force acting on the rectangle orthogonal to the OY axis is (pyxi +pyyj +py,k) dz dx, and so on for the remaining rectangle Thus to define the force acting on the surface of the volume element

we need to define the nine components p,, , p,, , , or equivalently the second order tensor

(1.27)

Notice that the pressure acting on the surface normal to i is now i - p and more generally, if n

is a unit vector normal to a fluid surface, the pressure force acting on this surface is n - p

For the fluid particle to experience a net force, there needs to be a change in the pressure across the width of the parallelepiped Considering the rectangles normal to OX, the force

is i - p dy dz on one surface and -i ‘ ( p dy dz + ( d p / d x ) dx dy dz) on the opposite surface

14 1 The Equations of Gas Dynamics

Therefore the net force in the OX direction is -i - (&)/ax) dr, where d r = d x dy dz is the volume element, and similarly for the two other directions Hence the total force is

dx d y dz

- (i- + j- + k-) - p d r = -V p d r (1.28)

At the microscopic level the pressure force is due to the uneven molecular bombardment of

a fluid element The off-diagonal components like pzy, p,,, pyz, etc are due to the tangential forces applied to the surfaces by the colliding particles and therefore the net force they apply to the element is of a shearing nature, which at a macroscopic level is termed a ‘viscous’ force It

is readily shown that for the simple microscopic particles occurring in a fully-ionized plasma

the pressure tensor is symmetric, i.e p,, = py,, etc In general the normal components,

p,,, pYy, p,, are equal in magnitude, otherwise it would mean that the normal pressure on a surface would depend on the orientation of that surface They are also much larger than the off-diagonal components and it is therefore convenient to separate them by writing

where p is the usual thermodynamic pressure, the tensor 1 defined by the equation is the unit

tensor introduced in Mathematical note 1 (page 7) and 1T is known as the viscosity tensor A

fluid is said to be ‘ideal’ if TT is zero

1.3.2 Kinetic definition of pressure

In a frame P,, consider the flux of momentum across a small (convected) surface n d C where n is the unit normal (see Fig 1.4) The mass flux per particle is mc and its momentum flux is mcc, where c is now a generic velocity, that is in each appearance it ranges over all the possible values accessible to the particles in an infinitesimal volume containing dC Averaging over all these values, we find that the total momentum flux parallel to unit normal

is F = n - ~ ( c c ) , where ( ) denotes the average taken over the full range of values of c

with allowance for the frequency of occurrence of particular values

Figure 1.4: Momentum flux and pressure

1.3 Pressure 15

Divide the momentum flux parallel to the unit normal n into

then F+ is the momentum flux crossing unit area in the positive sense, from fluid @ to @) as

shown in Fig 1.4 This momentum is absorbed in fluid @) by collisions, and by allowing for all the angles of the particles incident on dC, it can be shown that the transfer is completed on

average at a perpendicular distance N g X from the interface, where A is the mean free path Hence on a length-scale for which X is small, we can say that fluid @experiences a ‘surface’ force F+ per unit area due to the particles arriving from 0 Also, to set up the return flux

-F-, @will suffer a further surface force, F- It follows that

F + + F - = n - p (P = e(cc)),

is the force per unit area exerted by fluid @ on fluid @ across their interface From (lS),

e = rnn and w = v + c, it follows that

When X is comparable with the length-scale of interest, C say, the net normal momen- tum flux across n dC, namely n p dC, remains the pressure force transmitted across n dC,

although, of course, the particle collisions through which the force is manifest in the fluid, no

longer all lie in the close neighbourhood of n dC A moderate change of scale does not alter our interpretation of n - p as being a surface collisional force However when the mean free path greatly exceeds C, i.e the system is collisionless, it is no longer a fluid In this case the pressure vanishes and the formula m = e(cc) merely defines the (unchanging) momentum

flux, which is not a force (cf Newton’s first law) It follows from the above that the length scale for the pressure gradient must be much larger than A, i.e

XIVlnpl << 1

Without collisions in the infinitesimal volume element at P,, the force V - p is zero This

is obvious from the physical description given above, and it can also be verified from the kinetic definition of V p, namely

where 6V is a small volume enclosing the macroscopic point P, at which the divergence is required, and dV is its surface Hence from (1.30)

16 1 The Equations of Gas Dynamics

Figure 1.5: Particle flux across a fluid element P,

Consider a collinear stream of particles crossing 6V, entering at the surface element

nl dCl and departing across n2 dC2, as shown in Fig 1.5 Because this stream is defined

in a convected frame, the particles will be relieved of any body force, like gravity etc., that acts on the fluid in the laboratory frame Hence we may assume that the entering and leaving particles have the same velocity c It follows that (cc n dC)1 = -(cc n dC)z Hence the contribution to the surface integral in V - p from this stream is (f2 - f1)cc - n dC, where

the factor 3 is required to avoid double-counting, An increment to V * p is therefore possible

only if f2 # fl, i.e the particles enter and depart in different numbers, which requires that some particles either join or leave the stream en route across the element The only physical mechanism for this process is particle collisions within the fluid element Hence a necessary condition for V - p to be non-zero is the occurrence of collisions; it is not sufficient since gains and losses may balance

Pressure is transmitted through a neutral gas by the impulsive forces occurring at particle

collisions Between collisions neutral particles are free of the influence of pressure gradi- ents The impulsive collisional forces have two distinct components, namely a purely random component that scatters the particles isotropically, and a non-random component proportional

to -V * p Without the latter, particles would not be driven (on average) down the pressure gradient, i.e their collective behaviour would differ from that of the fluid they comprise - an evident contradiction The same considerations apply to charged particles, with the important modification that since abrupt collisions are rare (see §5.1.1), the non-random force is not applied impulsively, but is present as a continuous influence on particle motions

The normal pressure between the fluids is n p - n Let i, j, k denote the unit Cartesian vectors, and take n to be parallel to each in turn The average value of the normal pressures at

some point in the fluid is

which equals the thermodynamic pressure p Hence

1.3 Pressure 17

1.3.4 Local thermodynamic equilibrium

For a given continuum variable cp, the macroscopic scales are defined by

(1.32) the minimizing being over all relevant values of (r, t ) and all orientations at a point in the gas

The Knudsen numbers

k N , f X/C, k m T/T, ICN E maX{kNL, k m } , (1.33) are a measure of how nearly the medium may be regarded as being a continuum In a ‘true’ continuum k, is zero, but in this limiting case, diffusion is completely suppressed by colli- sions, making it impossible to transmit fluid momentum and energy through the gas except by the collective process of convection

A thermodynamic system P, must have the possibility of reaching uniform values for its macroscopic variables in a relaxation time Tth much smaller than the macroscopic time-scale

7 for, as will be explained shortly, only then can precise values be assigned to Pc’s pressure

and temperature In a gas the mechanism that tends to produce equilibrium is the interaction

of particles via collisions Since 7 t h M 7, for these thermodynamic variables we must have

k , << 1 And as the molecular speed, c = X/r, and the speed C / 7 at which macroscopic perturbations propagate are usually comparable, this constraint entails k,, << 1

In continuum mechanics pressure is force per unit area and in a gas the existence of such

a force requires the particles to collide either with each other or with confining walls Away from boundaries, the ambient particles around a point P, play the role of the confining ‘walls’ This implies that our local thermodynamic system, P,, must have a typical dimension that is

at least a mean free path in length In this case adjacent systems, say PCl and Pc2, can interact, with each one exerting a pressure on the other If PCl has a slightly higher pressure than Pc2, a net force will result, with P c 2 experiencing more numerous or more energetic collisions with particles coming from PC1, than PCl does with particles from Pc2 On a continuum description,

the force is said to be due to the component of the pressure gradient directed from P c 2 to P,1, but the underlying mechanism is an imbalance in the particle collisions In a ‘collisionless’ gas, however great the difference between the values of Q (c2) at two neighbouring points, there would be no pressure force in the gas We therefore define the pressure as being the

force transmitted across a unit surface The concept of ‘equilibrium’ is not required in this definition but collisions are essential

Gas temperature appears to be a variable that does not depend on the presence of col-

lisions Its definition in terms of the average kinetic energy of particles (see 51.2.3) is one

of the most famous results of the early kinetic theory But in classical thermodynamics the notion of ‘thermal equilibrium’ plays a central role in the definition of empirical temperature,

a necessary preliminary to the introduction of the absolute temperature Thermal equilibrium between P,1 and P c 2 requires a collisional interchange just as already described for the pres- sure Now suppose that PCl is hotter than P,2 The transfer of energy between these systems involves two stages First the more energetic particles from PCl move through a free path

and then they deposit their excess energy in P c 2 by collisions It is important to distinguish

between mere energyjux, which like radiation, need not be deposited locally and heat flux,

which does require collisions

18 1 The Equations of Gas Dynamics

The precise definition of pressure and temperature is therefore determined by the force and energy transmitted between neighbouring fluid elements When the laws of conservation

of momentum and energy are obtained using these variables, their physical properties are invoked in the formulation Alternatively, these and related variables like the viscous stress tensor TI and the heat flux vector q are defined implicitly by their roles in the conservation

laws Macroscopic equilibrium is an unnecessary restriction in the definition, but we must ensure that the gas (or fluid) has an internal structure such that the symbols p , TI , T, q, etc really do have the properties implied by in their appearance in the conservation laws

The local transport of momentum and energy sets a lower bound on the size of the ther- modynamic system P, If d l is a typical dimension of P,, then it follows from the above discussion that we need X 2 de Similarly, the macroscopic time element dt cannot be much less than T , for this would imply that the pressure and temperature could respond to changes that occur much faster than the collisional mechanism effecting these changes

We can now define our thermodynamic system at P, to be an element in which, for each

of the thermodynamic variables of interest,

k , << 1, k , << 1, X 2 de << C, T 5 dt << 7 (1.34)

It is not possible to be more precise than this

Convection is the transport of a macroscopic property, such as density, momentum, energy, the concentration level of a contaminant, and so on, by the fluid motion Let @(r, t ) be such

an attribute, specified as an amount per unit mass of fluid - known as a ‘specific’ property - then a volume element of mass e dr will possess an amount e@ dr of it As the volume of

fluid crossing a stationary surface n dC in one second is v n dC, it follows that the transport

of @ due to convection occurs at the rate e@v - n dC across this surface

This description can be generalized by introducing a specific property $(r, w, t), whose

value may depend on the velocity w = v + c of the particles involved in its transport Suppose that the average value of 4 taken over particles at a macroscopic point is @, i.e

across n d S is the average of e4w n dS This gives a total flux

then the local transport of

1.4 Macroscopic conservation laws 19

In some circumstances an expression for J+ of the form

(1.38) can be found; K+ is termed the coeficient of dzffusion for a It follows from (1.38) that K+

has the dimensions: (length)2/time, and since it is due to particle transport, we may write

where Q is a constant of order unity Sometimes there exist several ‘routes’ for the diffusion

of a, i.e a number of distinct processes each contribute to J+ If these are independent, the

total flux is obtained by summation, and K+ becomes

(1.40)

the subscript j denoting a particular process

v - n, is dependent on the choice of reference frame

in which the velocity is measured, whereas the diffusion term is not In fact we shall adopt

‘frame-indifference’ as being the essential property that distinguishes diffusion from convec- tion In some circumstances it is not evident from the physics where to draw the line between these two transport processes, and a mathematical definition is useful

The physical assumption involved in (1.37) is that 4 is a property transported by individual particles, whose random motions are independent of the amount of 4 that they carry In

a simplified description, if we take eq5 to be the density of molecules labelled in a particular way, the intensity 4 becomes the fraction of labelled molecules at a given point If 4 is initially uniform, variations in the density Q cannot affect 4, since labelled and unlabelled molecules will have the same propensity to migrate at each point in the fluid Thus J+ cannot depend

on Ve, at least in a linear model of the phenomenon The random molecular motions will disperse any non-uniformity in @, and the first spatial derivative of is the dominant term driving the system towards equilibrium

For example when 4 is the particle energy ( f e z + E ) (see (1.20)), by (1.37) its diffusion vector is

The convection term in (1.36), viz

q = Q ( C ( $ C 2 + E ) ) , (1.41)

which is known as the heatflux if collisions are dominant and the energyflw otherwise The

transport of energy between adjacent fluid elements requires a non-zero, finite value for the

mean free path A

1.4.2

Consider a volume V of the medium, with a surface dV across which 4 is being convected and diffused Let S+(r, t ) be the rate per unit volume at which is being introduced at points within V due to physical or chemical processes, the details of which will be specified later

A general balance equation in physical space

20 I The Equations of Gas Dynamics

The amount of 4 within V is (&) =

dV and its creation within V Hence by (1 -36) we obtain the balance equation per unit volume Its increase is due to its flux across

By Gauss’ divergence theorem, this can be expressed as

which is the required balance equation for @

There may be several gas components present, in which case it is necessary to distinguish the macroscopic variables 0, v, T , p, q, etc by a different subscript for each species Thus for the ‘ith-fluid’, Pi:

(1.43)

The source term is partly due to mass transfer between the fluid components resulting from chemical or physical interactions between them Let hij be the rate at which Pi gains mass from Pj, then hijdj is the rate at which the property 4 is received from Pj by Pi Conversely,

Pi loses 4 at the rate hji4i to Pj The net gain of 4 by Pi due to mass transfers is therefore

There may also be transfers of 4 without mass exchange Let &jej be the rate per unit volume of such a transfer from Pj to Pi, then by conservation of 4 during this transfer,

The total source due to internal and external transfers is therefore

(1.44) where S g is the rate at which Pi receives 4 per unit volume from sources outside the system

1.4 Macroscopic conservation laws 21

1.4.3

(1.42) with (1.20), (1.30), (1.37), (1.41) and (1.44):

ma^^ : 4 = 1, 0 = 1, J4 = 0, S$ = 0 Hence

Conservation laws for a simple fluid

To write down the conservation laws of mass momentum and energy, we use equation

where

d

at

is the rate of change in a frame moving with velocity v relative to the laboratory frame In this

convected frame (I 45), (1.46) and (1.47) become

22 I The Equations of Gas Dynamics

whence from (1.50)

Now (1.52) can be written

which is the first law of thermodynamics for a fluid element

which formula is often written in the form

(1.58) For a monatomic gas (1.23) gives y = 5/3

If the conditions at the convected point are adiabatic, i.e no heat is removed or supplied to the volume element dr at P,, then q is zero Furthermore, if the fluid is also ideal (see 5 1.3 l),

lT is zero, and (1.55) reduces to

Thus s is constant at P, and if this holds throughout the fluid, the motion is said to be isen- tropic For such flows (1.58) reduces to

where K is another constant

Equation (1.52) can be rearranged as

(1.61) (1.62)

1.5 Introduction to kinetic theory 23

is the entropy production rate per unit volume With Q, = s in (1.48), J, = q/T in (1.61) is

identified as being the difision vector for entropy Here q is interpreted as being the energy

flux into the fluid element (see last paragraph of 5 1.4 l), whereas in O, q is the heat flux,

which by Fourier's law (see $5.5.3) is q = -tcVT, where tc is the thermal conductivity I f

the motions of the particles are reversed, it follows from (1.41) that energy flux changes sign and is therefore a reversible phenomenon, whereas heat flux does not Hence (1.54) can be

expressed in the form

(1) (2)

pDu - { -p@Dp-' - TV (q/T)} = T O ,

in which the terms labelled (I) and (2) represent the power supplied reversibly to the macro-

scopic point P, by compression and heating The stability of the process requires the internal energy of P, to increase at a rate not less than that provided by the reversible power supply Hence to secure stability it is necessary to have a non-negative entropy production rate,

S =Iv e s d r ,

within a volume V,

(1.65) satisfies

$1.2.1 (In the convected frame P,, w is replaced by c.) From the additivity of entropy it

follows that a mixture with all components having the same mass, has an entropy density defined by

(1.67)

24 I The Equations of Gas Dynamics

Now suppose that the i-th component is specified to be those particles of a monatomic gas that lie in the volume element dci of velocity space, at a point where the phase space density

is f i = f (r, ci, t) Then the number density of this component is ni = fi dci The spread in velocities of the i-th component is proportional to ldci 1 l I 3 , and hence by (1.19) the associated temperature Ti is proportional to l d ~ i 1 ~ / ~ Thus (1.67) gives

es = -kBc fi In f i dci + kBC ni( $ InTi - In Idci() + @so

The second right-hand term reduces to a constant times Q, and hence with a particular choice for SO (which has no significant consequences), and allowing the components to form a con- tinuum so that the sums can be replaced by integrals, we arrive at the formula

(We could have introduced this equation as being the dejnition of entropy in phase-space, but

it is instructive to see its connection with fluid entropy.) Now we change the interpretation and take (1.68) to define the entropy density of a single-component perfect gas As s is prescribed only to within an additive constant, the term -f in the integrand is sometimes omitted The entropy within a volume V of the fluid is

S = s, QS dr = -kB f (In f - 1) dc dr , (1.69) which changes at the rate given by (1.66) In an isolated system the surface term is zero, in which case S steadily increases with time until when equilibrium is attained, it will be at its maximum value In this state small variations in S due to changes in f will be zero We shall use this principle to determine the equilibrium value o f f

Let fo denote the equilibrium value ofthe distribution function, then at f = fo, the entropy

S of the system is a maximum From (I 62) and (1.63) this requires the gradients to vanish and therefore maximizing S is equivalent to maximizing Q S subject to any constraints applying to the distribution function We shall define fo to be that distribution function which applies to the same values of the number density n and energy density QU as in the non-equilibrium case Hence, as the gradients relax towards zero and f -+ fa, the functions

1.5 Introduction to kinetic theory 25

Since the variation in f is now arbitrary, d(es) is zero only if the integrand vanishes, i.e i f f has the value fo given by

The constants a and /3 are determined by using the constraints in (1.70)

The first constraint requires that

-ca M-m

The average value of c: is

- whence by (1.72), 3 = 1/(2P)

kBT = m/(2P), so that P = m / ( 2 k B T )

The formula in (1.7 1) now reads

Similarly 3 = c2 = 1/(2/3) Thus from (1.73),

(1.74)

It remains to show that the stationary point is actually a maximum; this is left as an exercise

26 1 The Equations of Gas Dynamics

Figure 1.6: Spherical coordinates for the velocity vector

bility that a particle chosen at random has a velocity in the range First suppose that the particles are in an equilibrium velocity distribution, then the proba- c , c f dc is (fo/n) dc = n-1 C3 exp(-v2) dc ,

where

(1.75) and i: denotes unit vector along c Transforming from the Cartesian coordinates (c~, cy, c,)

to the spherical coordinates (c, 8, $), we have

dc = c2 dc sin 8 d8d4 = c2 dc4r d n , (1.76) where 4n dS2 is the element of solid angle subtended at the origin Thus the probability that the relative speed v falls in v, v + dv and the unit vector C lies in dCl is

1.5 Introduction to kinetic theory 27

i% = ii sin2 8 sin2 4 + jj sin2 8 cos2 4 + kk cos2 0

+(ij + ji) sin2 8 sin 4 cos 4

+(jk + kj) sin 8 cos 8 cos q5 + (ki + ik) sin 8 cos 8 sin 4

From these expansions we find that

J 2 = & l i % s i n O d O d 4 = 5 ( i i + j j + k k ) ,

i.e

e d Q = O , e E d R = il,

where 1 is the unit tensor (see page 7)

We shall also need the standard integrals:

is the Maxwellian speed distribution, plotted in Fig 1.7 Its maximum occurs at v = 1, identifying C as the most probable molecular speed

28 1 The Equations of Gas Dynamics

"0 0.4 0.8 1.2 1.6 2.0 2.4 2.8

v = c / c

Figure 1.7: The Maxwellian speed distribution

1.5.4 Evolution of the phase-space density

Consider a volume V of phase space Let dS = ( d S , , dS,) be a surface element of V ,

with dSr referring to physical space, and d S , to velocity space Suppose that a bunch of particles, moving with velocity w and acceleration w pass through dS, then the number flux across the surface element is f w d S , + f w dS, If d S is directed outwards from V, this flux will be a loss from V Let (aflat), d v , be the net rate at which collisions cause particles

to appear (or disappear, if this number is negative) in the 6-D volume element d v = d w dr, at

some point within V Then particle numbers are conserved if

By Gauss' theorem the surface integrals can be written as phase-space volume integrals,

and as the volume V is arbitrary in extent, it follows that

in which the precise nature of the partial derivatives has been made explicit It follows that, provided