Tài liệu THE MATHEMATICAL THEORY OF NON-UNIFORM GASES pptx

THE

MATHEMATICAL THEORY

OF

NON-UNIFORM GASES

AN ACCOUNT OF THE KINETIC THEORY

OF VISCOSITY, THERMAL CONDUCT ION AND

DIFFUSION IN GASES

SYDNEY CHAPMAN, F.R.S.

Geophysical Institute, College, Alaska

National Center for Atmospheric Research, Boulder, Colorado

A N D

T. G. COWLING, F.R.S.

Professor of Applied Mathematics

Leeds University

THIRD EDITION

PREPARED IN CO-OPERATION WITH

D. BURNET T

CAMBRIDGE

UNIVERSITY PRESS

Published by the Press Syndicate of the University of Cambridge

The Pitt Building, Trumpington Street, Cambridge CB2 IRP

40 West 20th Street, New York, NY 10011-4211 USA

10 Stamford Road, Oakleigh, Melbourne 3166, Australia

Copyright Cambridge University Press 1939, 1932

© Cambridge University Press 1970

Introduction © Cambridge University Press 1990

First published 1939

Second edition 1932

Third edition 1970

Reissued as a paperback with a Foreword by Carlo CerclgnanI

in the Cambridge Mathematical Library Series 1990

Reprinted 1993

ISBN 0 321 40844 X paperback

Transferred to digital printing 1999

CONTENTS

Foreword

Preface

Note regarding references

Chapter and section

List of diagrams

List of symbols

Introduction

Chapters I-I Q

Historical Summary

Name index

Subject index

titles

References to numerical data for particular gases

(simple and mixed)

page vii

xiii

xiv

XV

XX

xxi

I

10-406

407

4 "

415

4*3

M

FOREWORD

The atomic theory of matter asserts that material bodies are made up of small

particles. This theory was founded in ancient times by Democritus and

expressed in poetic form by Lucretius. This view was challenged by the

opposite theory, according to which matter is a continuous expanse. As

quantitative science developed, the study of nature brought to light many

properties of bodies which appear to depend on the magnitude and motions

of their ultimate constituents, and the question of the existence of these tiny,

invisible, and immutable particles became conspicuous among scientific

enquiries.

As early as 1738 Daniel Bernoulli advanced the idea that gases are formed

of elastic molecules rushing hither and thither at large speeds, colliding and

rebounding according to the laws of elementary mechanics. The new idea,

with respect to the Greek philosophers, was that the mechanical effect of the

impact of these moving molecules, when they strike against a solid, is what

is commonly called the pressure of the gas. In fact, if we were guided solely

by the atomic hypothesis, we might suppose that pressure would be produced

by the repulsions of the molecules. Although Bernoulli's scheme was able to

account for the elementary properties of gases (compressibility, tendency to

expand, rise of temperature in a compression and fall in an expansion, trend

toward uniformity), no definite opinion could be formed until it was investi￾gated quantitatively. The actual development of the kinetic theory of gases

was, accordingly, accomplished much later, in the nineteenth century.

Although the rules generating the dynamics of systems made up of molecules

are easy to describe, the phenomena associated with this dynamics are not so

simple, especially because of the large number of particles: there are about

2X7X IO'9

molecules in a cubic centimeter of a gas at atmospheric pressure

and a temperature of 0 °C.

Taking into account the enormous number of particles to be considered, it

would of course be a perfectly hopeless task to attempt to describe the state

of the gas by specifying the so-called microscopic state, i.e. the position and

velocity of every individual particle, and we must have recourse to statistics.

This is possible because in practice all that our observation can detect is

changes in the macroscopic state of the gas, described by quantities such as

density, velocity, temperature, stresses, heat flow, which are related to the

suitable averages of quantities depending on the microscopic state.

J. P. Joule appears to have been the first to estimate the average velocity

of a molecule of hydrogen. Only with R. Clausius, however, the kinetic theory

of gases entered a mature stage, with the introduction of the concept of mean

free-path (1858). In the same year, on the basis of this concept, J. C. Maxwell

developed a preliminary theory of transport processes and gave an heuristic

derivation of the velocity distribution function that bears his name. However,

[vii]

viii FOREWORD

he almost immediately realized that the mean free-path method was inadequate

as a foundation for kinetic theory and in 1866 developed a much more accurate

method, based on the transfer equations, and discovered the particularly simple

properties of a model, according to which the molecules interact at distance

with a force inversely proportional to the fifth power of the distance (nowadays

these are commonly called Maxwellian molecules). In the same paper he gave

a better justification of his formula for the velocity distribution function for

a gas in equilibrium.

With his transfer equations, Maxwell had come very close to an evolution

equation for the distribution, but this step must be credited to L. Boltzmann.

The equation under consideration is usually called the Boltzmann equation

and sometimes the Maxwell-Boltzmann equation (to acknowledge the impor￾tant role played by Maxwell in its discovery).

In the same paper, where he gives an heuristic derivation of his equation,

Boltzmann deduced an important consequence from it, which later came to

be known as the //-theorem. This theorem attempts to explain the irreversibil￾ity of natural processes in a gas, by showing how molecular collisions tend to

increase entropy. The theory was attacked by several physicists and

mathematicians in the 1890s, because it appeared to produce paradoxical

results. However, a few years after Boltzmann's suicide in 1906, the existence

of atoms was definitely established by experiments such as those on Brownian

motion and the Boltzmann equation became a practical tool for investigating

the properties of dilute gases.

In 1912 the great mathematician David Hilbert indicated how to obtain

approximate solutions of the Boltzmann equation by a series expansion in a

parameter, inversely proportional to the gas density. The paper is also repro￾duced as Chapter XXII of his treatise entitled Grundzige einer allgemeinen

Theorie der linearen Integralgleichungen. The reasons for this are clearly stated

in the preface of the book ('Neu hinzugefugt habe ich zum Schluss ein Kapitel

iiber kinetische Gastheorie. [...] erblicke ich in der Gastheorie die glazendste

Anwendung der die Auflosung der Integralgleichungen betreffenden

Theoreme').

In 1917, S. Chapman and D. Enskog simultaneously and independently

obtained approximate solutions of the Boltzmann equation, valid for a

sufficiently dense gas. The results were identical as far as practical applications

were concerned, but the methods differed widely in spirit and detail. Enskog

presented a systematic technique generalizing Hilbert's idea, while Chapman

simply extended a method previously indicated by Maxwell to obtain transport

coefficients. Enskog's method was adopted by S. Chapman and T. G. Cowling

when writing The Mathematical Theory of Non-uniform Gases and thus came

to be known as the Chapman-Enskog method.

This is a reissue of the third edition of that book, which was the standard

reference on kinetic theory for many years. In fact after the work of Chapman

and Enskog, and their natural developments described in this book, no essential

FOREWORD ix

progress in solving the Boltzmann equation came for many years. Rather the

ideas of kinetic theory found their way into other fields, such as radiative

transfer, the theory of ionized gases, the theory of neutron transport and the

study of quantum effects in gases. Some of these developments can be found

in Chapters 17 and 18.

In order to appreciate the opportunity afforded by this reissue, we must

enter into a detailed description of what was the kinetic theory of gases at the

time of the first edition and how it has developed. In this way, it will be clear

that the subsequent developments have not diminished the importance of the

present treatise.

The fundamental task of statistical mechanics is to deduce the macroscopic

observable properties of a substance from a knowledge of the forces of

interaction and the internal structure of its molecules. For the equilibrium

states this problem can be considered to have been solved in principle; in fact

the method of Gibbs ensembles provides a starting point for both qualitative

understanding and quantitative approximations to equilibrium behaviour. The

study of nonequilibrium states is, of course, much more difficult; here the

simultaneous consideration of matter in all its phases - gas, liquid and solid

- cannot yet be attempted and we have to use different kinetic theories, some

more reliable than others, to deal with the great variety of nonequilibrium

phenomena occurring in different systems.

A notable exception is provided by the case of gases, particularly monatomic

gases, for which Boltzmann's equation holds. For gases, in fact, it is possible

to obtain results that are still not available for general systems, i.e. the

description of the thermomechanical properties of gases in the pressure and

temperature ranges for which the description suggested by continuum

mechanics also holds. This is the object of the approximations associated with

the names Maxwell, Hilbert, Chapman, Enskog and Burnett, as well as of the

systematic treatment presented in this volume. In these approaches, out of all

the distribution functions / which could be assigned to given values of the

velocity, density and temperature, a single one is chosen. The precise method

by which this is done is rather subtle and is described in Chapters 7 and 8.

There exists, of course, an exact set of equations which the basic continuum

variables, i.e. density, bulk velocity (as opposed to molecular velocity) and

temperature, satisfy, i.e., the full conservation equations. They are a con￾sequence of the Boltzmann equation but do not form a closed system, because

of the appearance of additional variables, i.e. stresses and heat flow. The same

situation occurs, of course, in ordinary continuum mechanics, where the system

is closed by adding further relations known as 'constitutive equations'. In the

method described in this book, one starts by assuming a special form for /

depending only on the basic variables (and their gradients); then the explicit

form of f is determined and, as a consequence, the stresses and heat flow are

evaluated in terms of the basic variables, thereby closing the system of

conservation equations. There are various degrees of approximation possible

X FOREWORD

within this scheme, yielding the Euler equations, the Navier-Stokes equations,

the Burnett equations, etc. Of course, to any degree of approximation, these

solutions approximate to only one part of the manifold of solutions of the

Boltzmann equation; but this part turns out to be the one needed to describe

the behaviour of the gas at ordinary temperatures and pressures. A byproduct

of the calculations is the possibility of evaluating the transport coefficients

(viscosity, heat conductivity, diffusivity,...) in terms of the molecular param￾eters. The calculations are by no means simple and are presented in detail in

Chapters 9 and 10. These results are also compared with experiment (Chapters

12, 13 and 14).

In 1949, H. Grad wrote a paper which became widely known because it

contained a systematic method of solving the Boltzmann equation by expanding

the solution into a series of orthogonal polynomials. Although the solutions

which could be obtained by means of Grad's 13-moment equations (see section

15.6) were more general than the 'normal solutions' which could be obtained

by the Chapman-Enskog method, they failed to be sufficiently general to

cover the new applications of the Boltzmann equation to the study of upper

atmosphere flight. In the late 1950s and in the 1960s, under the impact of the

problems related to space research, the main interest was in the direction of

finding approximate solutions of the Boltzmann equation in regions having a

thickness of the order of a mean free-path. These new solutions were, of

course, beyond the reach of the methods described in this book. In fact, at

the time when the book was written, the next step was to go beyond the

Navier-Stokes level in the Chapman-Enskog expansion. This leads to the

so-called Burnett equations briefly described in Chapter 15 of this book. These

equations, generally speaking, are not so good in describing departures from

the Navier-Stokes model, because their corrections are usually of the same

order of magnitude as the difference between the normal solutions and the

solutions of interest in practical problems. However, as pointed out by several

Russian authors in the early 1970s, there are certain flows, driven by tem￾perature gradients, where the Burnett terms are of importance. For this reason

as well for his historical interest, the chapter on the Burnett equations still

retains some importance.

Let us now briefly comment on the chapters of the book, which have not

been mentioned so far in this foreword. Chapters 1-6 are of an introductory

nature; they describe the heavy apparatus that anybody dealing with the kinetic

theory of gases must know, as well as the results which can be obtained by

simpler, but less accurate tools. Chapter 11 describes a classical model for

polyatomic gases, the rough sphere molecule; this model, although not so

accurate when compared with experiments, retains an important role from a

conceptual point of view, because it offers a simple example of what one should

expect from a model describing a polyatomic molecule. Chapter 16 describes

the kinetic theory of dense gases; although much has been done in this field,

the discussion by Chapman and Cowling is still useful today.

FOREWORD zl

Where is kinetic theory going today? The main recent developments are in

the direction of developing a rigorous mathematical theory: existence and

uniqueness of the solutions to initial and boundary value problems and their

asymptotic trends, but also rigorous justification of the approximate methods

of solution. Among these is the method described in this book. It is unfair,

however, to criticise, in the light of the standards and achievement of today,

the approach described in this book, as is sometimes done. In addition to still

being a good description of an important part of the kinetic theory of gases,

this book has played the important role of transmitting the solved and unsolved

problems of kinetic theory to generations of students and scholars. Thus it is

not only useful, but also historically important.

Carlo Cercignani

Milano

EXTRAC T FRO M

PREFAC E TO FIRST EDITIO N

In this book an account is given of the mathematical theory of gaseous

viscosity, thermal conduction, and diffusion. This subject is complete in

itself, and possesses its own technique; hence no apology is needed for

separating it from related subjects such as statistical mechanics.

The accurate theory originated with Maxwell and Boltzmann, who

established the fundamental equations of the subject. The general solution

of these equations was first given more than forty years later, when within

about a year (1916-1917) Chapman and Enskog independently obtained

solutions by methods differing widely in spirit and detail, but giving iden￾tical results. Although Chapman's treatment of the general theory was

fully effective, its development was intuitive rather than systematic and

deductive; the work of Enskog showed more regard for mathematical

form and elegance. His treatment is the one chosen for presentation here,

but with some differences, including the relatively minor one of vector and

tensor notation.* A more important change is the use of expansions of

Sonine polynomials, following Burnett (1935). We have also attempted

to expound the theory more simply than is done in Enskog's dissertation,

where the argument is sometimes difficult to follow.

The later chapters describe more recent work, on dense gases, on the

quantum theory of collisions (so far as it affects the theory of the transport

phenomena in gases), and on the theory of conduction and diffusion in

ionized gases, in the presence of electric and magnetic fields.

Although most of the book is addressed to the mathematician and

theoretical physicist, an effort has been made to serve the needs of labora￾tory workers in chemistry and physics by collecting and stating, as clearly

as possible, the chief formulae derived from the theory, and discussing

them in relation to the best available data.

S.C.

1939 T. G. C.

* The notation used in this book for three-dimensional Cartesian tensors was devised

jointly by E. A. Milne and S. Chapman in 1916, and has since been used by them in many

branches of applied mathematics.

(»•]

PREFAC E TO THIR D EDITIO N

Until now, this book has appeared in substantially its 1939 form, apart

from certain corrections and the addition, in 1952, of a series of notes

indicating advances made in the intervening years. A more radical revision

has been made in the present edition.

Chapter 11 has been wholly rewritten, and discusses general molecular

models with internal energy. The discussion is primarily classical, but in

a form readily adaptable to a quantum generalization. This generalization

is made in Chapter 17, which also discusses (in rather more detail than

before) quantum effects on the transport properties of hydrogen and

helium at low temperatures. The theory is applied to additional molecular

models in Chapter 10, and these are compared with experiment in Chapters

12-14; the discussion in these chapters aims for the maximum simplicity

consistent with reasonable accuracy. Chapter 16 now includes a short

summary of the BBGKY theory of a dense gas, with comments on its diffi￾culties. A new Chapter 18 discusses mixtures of several gases. Chapter 19

(the old Chapter 18) discusses phenomena in ionized gases, on which an

enormous amount of work has been done in recent years. This chapter has

been much extended, even though attention is confined to aspects related

to the transport phenomena. Finally, in Chapter 6 and elsewhere, a more

detailed account is given of approximate theories, especially those that

illuminate some feature of the general theory.

To accommodate the new material, some cuts have been necessary.

These include the earlier approximate discussion of the electron-gas in

a metal, and the Appendices A and B. The Historical Summary, and the

discussion of the Lorentz approximation have been curtailed. The discussion

of certain other topics has been modified, especially in the light of the work

of Kihara, of Waldmann, of Grad and of Hirschfelder, Curtiss and Bird.

A few minor changes of notation have been made; these are set out at the

end of the list of symbols on pp. xxv and xxvi.

The third edition has been prepared throughout with the co-operation

of Professor D. Burnett. We are deeply indebted to him for numerous

valuable improvements, and for his continuous attention to details that

might otherwise have been overlooked. He has given unstinted assistance

over a long period.

Our thanks are due to many others for their interest and encouragement

Special mention should be made of Professors Waldmann and Mason for

their helpful interest. Our thanks are also due, as earlier, to the officials

of the Cambridge University Press for their willing and expert help both

before and during the printing of this edition.

s. c.

1969 T. G. C.

{"< ' 1

NOTE REGARDING REFERENCES

The chapter-sections are numbered decimally.

The equations in each section are numbered consecutively and

are preceded by the section number, (3.41. 1), (3.41. a)

References to equations are also preceded by the section

number and where a series of numbers occur they are elided

(3.41,2,3,...) or (3.41,1-16).

References to periodicals give first (in italic type) the name of the

periodical, next (in Clarendon type) the volume-number, then

the number of the page or pages referred to, and finally the

date in parenthesis.

[Xiv]

CHAPTER AN D SECTION TITLES

Introduction

1. The molecular hypothesis (i)—2. The kinetic theory of heat (i)—3. The three states of

matter (i)—4. The theory of gases (2)—5. Statistical mechanics (3)—6. The interpretation

of kinetic-theory results (6)—7. The interpretation of same macroscopic concepts (7)—

8. Quantum theory (8).

Chapter 1. Vectors and tensors

1.1. Vectors (10)—1.11. Sums and products of vectors (11)—1.2. Functions of position

(12)—1.21. Volume elements and spherical surface elements (13)—1.3. Dyadica and

tensors (14)—1.31. Products of vectors or tensors with tensors (16)—1.32. Theorems on

dyadics (17)—1.33. Dyadics involving differential operators (18).

Some results on Integration

1.4. Integrals involving exponentials (19)—1.41. Transformation of multiple integrals

(20)—1.411. Jacobians (20)—1.42. Integrals involving vectors or tensors (21)—1.421. An

integral theorem (22).

1.5. Skew tensors (23).

Chapter 2. Properties of a gas: definitions and theorems

2.1. Velocities, and functions of velocity (25)—2.2. Density and mean motion (a6)—

2.21. The distribution of molecular velocities (27)—2.22. Mean values of functions of the

molecular velocities (28)—23. Flow of molecular properties (29)—2.31. Pressure and the

pressure tensor (32)—2.32. The hydrostatic pressure (34)—2.33. Intermolecular forces

and the pressure (35)—2.34. Molecular velocities: numerical values (36)—2.4. Heat (36)—

2.41. Temperature (37)-2.42. The equation of state (38)-2.43. Specific heats (39)—

2.431. The kinetic-theory temperature and thermodynamic tempcrature(4i)—2.44. Specific

heats: numerical values (42)—2.45. Conduction of heat (43)—2.5. Gas-mixtures (44).

Chapter 3. The equations of Boltzmann and Maxwell

3.1. Bottzmann's equation derived (46)-3.11. The equation of change of molecular pro￾perties (47)—3.12. £*/expressed in terms of the peculiar velocity (48)—3.13. Transforma￾tion of )<j>9fdc (48)—3.2. Molecular properties conserved after encounter; summational

invariants (49)—3.21. Special forms of the equation of change of molecular properties

(50)—3.3. Molecular encounters (5a)—3.4. The dynamics of a binary encounter (53)—

3.41. Equations of momentum and of energy for an encounter (53)—3.42. The geometry of

an encounter (55)—3.43. The apse-line and the change of relative velocity (55)—3.44.

Special types of interaction (57)—3.5. The statistics of molecular encounters (58)—3.51.

An expression for A(5 (60)—3.52. The calculation of ijtit (61)—3.53. Alternative expres￾sions for nA^; proof of equality (64)—3.54. Transformations of some integrala (64)—

3.6. The limiting range of molecular influence (65).

[xv]