Table Of ContentDr. of T'jchnical Sciences
Professor in the Bauman Institute of Technology, Moscow, U.S.S.R.
Translation edited by
AAAURICE HOLT
Associate Professor of Applied Mathematics
Brown University, Rhode Island, U.S.A.
Literal Translation by
J. GEORGE ADASHKO
P E R G A M ON PRESS
LONDON OXFORD PARIS NEW YORK
1960
I960
Pergamon Press Ltd,
Library of Congress Card No. 59-13721
AUTHOR'S POREWORD TO THE ENGLISH EDITION
The science and methods of the mechanics of continuous media
are assuming an ever-increasing importance in modern physics.
This volume deals essentially with a relatively recent
branch of this imOortant field of physics - the unsteady
motion of a medium.
In addition to its purely technical application in the study
of rapidly occurring processes, particularly detonation and
explosion, the theory of unsteady motion has many applications
of a general scientific nature in astrophysics, cosmogony and
cosmology, and the theory of multiple particle production.
Finally, a knowledge of the general premises of this theory
is almost indispensable in the still younger science of
magnetohydrodynaraics, which makes use not only of the same
methods of solving nonlinear partial differential equations,
but frequently also of the general procedures of formulating
various Oroblems.
It is my sincere wish that English and American research
workers in physics and mechanics will find this book useful,
in spite of the availability of the expellent monograOh
Supersonic Plow and Shock Waves (New York, 1948) by
R". Courant and K.O. Friedrichs.
The Author
26 January 1959
Moscow
VII.
Editor's Foreword
The study of Unsteady Gas Dynamics, or the macroscopic
treatment of the unsteady motion of gases, dates back to the
middle of the nineteenth century when many scientists were
interested in the theory of the Interior Ballistics of Guns,
In the following fifty years the subject was developed exten
sively by Riemann, Hugoniot, Rankine, Mikhelson and others.
In the first part of the present century, apart from notable
contributions by Becker to Shock Wave Theory, by Jouguet to
the theory of Detonation and by Love and Pidduck to Interior
Ballistics, interest in the subject was dormant. The second
world war stimulated new activity in this field on a large
scale, as a result of the advent of high speed aircraft,
supersonic wind tunnels and nuclear weapons. Scientists in
all the principal combatant countries made many contributions
to the subject, both during and after the war. The state of
knowledge at the end of the war, based on classical theory and
wartime contributions made in England and the United States,
is described in the famous book "Supersonic Flow and Shock
Waves" by Courant and Friedrichs. A German account of the
subject at this time, less well publicized, was given in a
series of notes entitled "Non-Stationary Problems of Gas
Dynamics" by Robert Sauer. Other works, at a more elementary
level and only partly concerned with unsteady effects, are
*¸laments of Gas Dynamics" by Liepmann and Roshko, and "Gas
Dynamics" by Oswatitsch.
The present volume contains a restatement of Classical
Theory and also contains, for the first time, a complete
account of contributions to Gas Dynamics by Russian scientists.
These are both important and numerous and account for the much
greater length of the book compared with previous works. The
size of the book also reflects the many new applications of
the subject which have been found in recent years, notably to
problems of explosions, fission or fusion reactors and the
motion of space vehicles.
The book is completely self contained and no previous know
ledge of Gas Dynamics is required. It can be used as a text
book in a course for Graduate Students and as a work of refer
ence for the many research scientists concerned with present
day problems in Gas Dynamics.
IX.
The first two chapters contain the basic Thermodynamic
Principles and Equations of Motion used in Qas Dynamics. In
the third chapter the author describes his own and other con
tributions to the theory of Self Similar Motions. Chapters
IV and V concern the theory of one dimensional gas motions
which can be treated \yy Classical Analysis and include many of
the author's contributions to wave interaction problems.
Chapter VI gives the theory of shock waves and Chapter VII the
theory of detonation waves* Chapter VIII is devoted to a
number of one dimensional problems involving detonation or de
flagration waves. Chapter IX is an extension of Chapter V to
include shock waves while Chapter X is concerned mainly with
spherical and cylindrical explosions. In Chapter XI explosions
in water and other dense media are treated. Chapter XII dis
cusses Lagrange's Ballistic Problem. The last three chapters
show the application of Gas Dynamics to astrophysical problems.
Chapters XIII and XIV introduce gravitational effects, while
relativity is taken into account in the final chapter.
The support of the Office of Naval Research and Air Force
Office of Scientific Research, in meeting the cost of literal
translation by J. George Adashko, is gratefully acknowledged.
I also wish to thank Miss Nancy Bowers and Mrs. Ezoura Fonseca
for their careful production of the final manuscript and the
staff of Pergamon Press in preparing such a long text for the
press.
Above all, I wish to thank K. P.Stanyukovich very warmly
for his cooperation in preparing the translation. His
continual interest in the project and his help in supplying
extra copies of the Russian text, with corrections inserted,
have been of very great value.
Maurice Holt
Division of Applied Mathematics
Brown University
Providence 12, Rhode Island
September 30, 1959
X.
Foreword
This hook treats problems in the theorir of unsteady motion
of any medium, with particular attention to the unsteady
motion of gases. Since unsteady motion is related in many
cases to equilibrium or to stationary motion, some attention
is paid also to certain aspects of the stationary motion of
media. (By stationary motion we mean steady motion. Ed.)
We shall study both isentropic motion, i.e., motion occur
ring at constant entropy over the entire range of motion under
consideration, as well as non-isentropic motion, i.e., motion
in which shock waves play an important role.
We shall consider first the mathematical and thermodyn.amic
tools of gas dynamics, with considerable attention being paid
to the investigation of the so-called self-similar motions.
These tools will be used to solve various problems of funda
mental or technical importance in unsteady gas dynamics.
It is impossible to consider v/ithin the framework of this
monograph all the problems that can be solved with the aid of
the theories developed. We restrict ourselves to considera
tion of the most important problems.
In addition to the classical problems of gas dynamics, we
consider many new problems: the theory of detonation of con
densed explosive media, the theory of escape of compressed
gases and liquids, and the theory of self-similar motion of
a gas (diverging and converging waves). Much attention is
paid to the study of the motion of gas in a gravitational
field as related to problems in astrophysics and cosmology.
In the treatment of these problems we employ certain new
methods for the approximate integration of the equations of
non-isentropic gas motion.
The last chapter treats certain problems of notion of gas
at very high velocities; here we succeed in deriving results
that apparently can find application in the theory of
multiple particle formation as well as in cosmology.
This book is intended for readers specializing in the field
of gas dynamics. The reader should be acquainted with the
theory of differential equations of mathematical physics,
with vector and tensor analysis, and with the elements of
thermodynamics. Many recently solved problems of fundamental
significance are referred to by author and date; references
to basic literature by both Russian and foreign authors are
listed at the end of the book.
In conclusion we feel obliged to express our indebtedness
to F. A. Baum, A. S. Kompaneets, and G. I. Pokrovskl for a
critical review of the manuscript.
XI.
Introduction
Gas dynamics covers the flow of all compressible media,
3-ncluding liquids and solids (the latter under conditions of
high pressure). The study of such flow involves not only the
methods of mechanics, but also methods of other branches of
physics, particularly thermodynamics.
If the parameters characterizing the state and the motion of
the medium are invariant with time in the region where the
motion of the medium is studied, this motion is said to be
stationary© If these parameters change with time, the motion
is called unsteady. Unsteady motion is of very great interest
not only in the solution of many applied technical problems,
such as the investigation of the motion of explosion products
and of the medium in which the explosion occurs, or the in
vestigation of fluctuations of gas inside various engines, but
also in the study of the fundamental problems of modern physics
and cosmology. In fact, various processes occurring in the
\miverse, such as the formation of stars or the gigantic
eruptions that originate in the sun and in stars are caused by
imsteady motion of tremendous masses of gas.
In classical gas dynamics, the medium, the motion of which
is being studied, is considered to be a continuum, i.e., it is
assumed that each elementary volume of the space contains a
continuum of particles of the medium. This makes it possible
to assume that parameters of the medium vary continuously in
space and in time. In fact, ordinary media, not too highly
rarefied, consisting of particles (molecules), do in practice
have this property, for the distance between molecules is
small and each elementary volume contains a large number of
them. In the case of a very highly rarefied medium, for
example the medium filling interstellar space, the medium
cannot be idealized as a continuum: such a medium must be
treated on a discontinuous basis, i.e., it must be assumed to
consist of discrete particles. However, the methods of gas
dynamics make it possible to consider even the motion of such
an ultra-rarefied medium.
The basic equations of gas dynamics are derived from three
fundamental laws of nature, namely the conservation of mass,
momentum, and energy. To study the motion of a medium, we
need to determine the three components of velocity, the
density, and the pressure of the medium as functions of three
spatial coordinates and time. The components of the vector
momentum equation give the three equations of motion, the law
of conservation of mass and the law of conservation of energy
give two more equations; thus, to determine the five taiknovm
functions we have a system of five equations, which evidently,
will be partial differential equations of the first order.
The equations of gas dynamics can be written in two forms:
in one we can determine the quantities characterizing the
motion and the state of the Tuedium at a given point In space
XII.
and at a given instant of time, in the other we follow the
fate of individual particles of the medium. The first form of
the equations is called the Eulerian form, and the second the
Lagrangian form«
Knowing the equation of state of the medium, we can readily
determine its temperature at each point of space and each
Instant of time. If the motion is adiabatic, i.e.. no heat is
added or removed from the medium, the energy equation assumes
the simplest form, since the entropy of each particle remains
constant. The energy equation becomes identical with the
equation of state, in which the variables are the pressure,
density, and entropy. In the case of isentropic motion, this
equation becomes the isentropic equation.
The use of mathematical methods of gas dynamics is to study
this system of non-linear partial differential equations of
the first order. Exact analytic solutions are obtained in the
case of one-dimensional motion.
The main use of thermodynamic methods in gas dynamics is to
connect by means of thermodynamic equations the basic param
eters of the gas, namely the density, pressure, temperature,
entropy, and also the heat content and the velocity of souna.
For an ideal gas, this problem becomes trivial.
In the case of dense media, the problem becomes somewhat
more complicated, since the equation of state is more compli
cated than in an ideal gas, but suitable approximations to the
equation of state and to the isentropic equation make it
possible to solve completely many problems in the motion of
solid media.
The development of gas dynamics is the result of work by
many scientists of many countries. Russian scientists have
made a considerable contribution to gas dynamics.
XIII.
Chapter I
Ma^tft^matlcal an4 Thermodynamic Methods of Gas Dynamics
1. Fnflcta^BQntal Thermoctynamlc Relationship? In Gas Dynamicg
Thermodynamic relations and concepts play a substantial
part in the study of gas dynamics. The system of equatlcms
describing the motion and properties of the medium Includes
the equation of state of the medium, which Is one of the
fundamental thermodynamic equations; It Is thus appropriate
to start this chapter with a discussion of the fundamental
thermodynamic relationships that will be used later.
If the number of particles is assumed constant, four inde
pendent variables are usually used in thermodynamics to
characterize the state of the medium; the pressure p, the
specific Tolume y. the temperature T. and the entropy S.
Correspondingly, four Independent differential equations
define four fundamental thermodynamic functions E. 1. F, and
Φ, where Ε is the Internal energy of the medium, i its heat
COTitent (enthalpy), F the free energy, and Φ the so-called
thermodynamic potential. If the equation of state is known,
any two variables chosen from the dependent and independent
variables determine fully the state of the medium, and hence
the remaining variables. Here the concepts of dependent and
lndepend«it variables are, naturally, arbitrary. Any
variable can be chosen to be either dependent or independent.
In gas dynamics we usually employ two functions — the in
ternal energy Ε and the heat content 1. The independent
variables p, v, T, and S, as well as the potentials S, I9 F,
and Φ are connected by the following differential equations
(thermodynamic identities):
dE= TdS — pdw di — TdS-\-Ydp\
(1; 1)
dP = — SdT—pdv; άΦ —— S dT-\-y dp;
All these functions are defined per unit mass of the medium.
The first equation in (1.1) is a thermodynamic identity
stating directly the law of conservation of energy. The
remaining equations in (1.1) are the consequences of this
thermodynamic identity and of the following equations, which
define the heat coitent, the free «lergy, and the thermo
dynamic potential:
i = E-\^py; F = E-TS', Φ = ί—Τ8. (1*. 2)
For the Identities defining E, 1, F, and Φ we have respec
tively:
Unsteady Motion of Continuous Media
(i;3)
The subscripts denote that the corresponding quantities are
assumed constant. Using the above relationships, ve can
readily change from one pair of independent variables to
another. This system of equations is a complete one; any
newly defined potential will depend on the previous ones 9
while the potentials E, i, F, and Φ are independent in the
sense that they can be expressed in terms of the variables p,
V, T, and S.
If the number of particles is variable, another term μαΝ,
where μ is the chemical potential and Ν is the number of
particles, is added to the right sides of the thermodynamic
identities. For example,
=rd5—/></v + |*dM (1; 4)
where
while
Differentiating this expression with respect to N, we get
Hence
It follows from this that the chemical potential of a medium,
consisting of identical particles, is the thermodynamic
potential referred to a single molecule (particle).
In the absence of heat sources or heat sinks, the motion of
the medium is adiabatic; furthermore, if there are no dissi-
pative phenomena in the medium, the entropy of any given
particle is constant: dS/dt = 0. If the number of particles
is constant, we have
