The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction, and diffusion in gases

448 Pages·1995·16.17 MB·English

by  Sydney Chapman

THE
MATHEMATICAL THEORY
OF
NON-UNIFORM GASES
AN ACCOUNT OF THE KINETIC THEORY
OF VISCOSITY, THERMAL CONDUCTION AND
DIFFUSION IN GASES
SYDNEY CHAPMAN, F.R.S.
Geophysical Institute, College, Alaska
National Center for Atmospheric Research, Boulder, Colorado
AND
T. G. COWLING, F.R.S.
Professor of Applied Mathematics
Leeds University
THIRD EDITION
PREPARED IN CO-OPERATION WITH
D. BURNETT
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 page vii
Preface xiii
Note regarding references xiv
Chapter and section titles XV
List of diagrams XX
List of symbols xxi
Introduction I
Chapters I-IQ 10-406
Historical Summary 407
Name index 4"
Subject index 415
References to numerical data for particular gases
(simple and mixed) 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.