Table Of ContentFLOW EQUATIONS FOR COMPOSITE GASES
/. M. BURGERS
INSTITUTE FOR FLUID DYNAMICS
AND APPLIED MATHEMATICS
UNIVERSITY OF MARYLAND
COLLEGE PARK, MARYLAND
ACADEMIC PRESS New York and London 1969
APPLIED MATHEMATICS
AND MECHANICS
An International Series of Monographs
EDITORS
F. N. FRENKIEL G. TEMPLE
Applied Mathematics Laboratory Mathematical Institute
David Taylor Model Basin Oxford University
Washington, D. C. Oxford, England
Volume 1. K. OSWATITSCH : Gas Dynamics, English version by G. Kuerti
(1956)
Volume 2. G. BIRKHOFF and Ε. H. ZARANTONELLO: Jet, Wakes, and
Cavities (1957)
Volume 3. R. VON MISES: Mathematical Theory of Compressible Fluid
Flow, Revised and completed by Hilda Geiringer and G. S. S.
Ludford (1958)
Volume 4. F. L. ALT: Electronic Digital Computers—Their Use in Science
and Engineering (1958)
Volume 5A. WALLACE D. HAYES and RONALD F. PROBSTEIN : Hypersonic
Flow Theory, second edition, Volume I, Inviscid Flows (1966)
Volume 6. L. M. BREKHOVSKIKH : Waves in Layered Media, Translated
from the Russian by D. Lieberman (1960)
Volume 7. S. FRED SINGER (ed.) : Torques and Attitude Sensing in Earth
Satellites (1964)
Volume 8. MILTON VAN DYKE : Perturbation Methods in Fluid Mechanics
(1964)
Volume 9. ANGELO MIELE (ed.) : Theory of Optimum Aerodynamic Shapes
(1965)
Volume 10. ROBERT BETCHOV and WILLIAM O. CRIMINALE, JR. : Stability
of Parallel Flows (1967)
Volume 11. J. M. BURGERS: Flow Equations for Composite Gases (1969)
Preface
The following collection of flow equations for composite gases, that is,
gases having more than one species of particle, is the outcome of work
done and seminars given at the Institute for Fluid Dynamics and Applied
Mathematics of the University of Maryland mainly during the years
1956-1965. The preSwntation is directed to the practical use of those equations
which can be derived from the Boltzmann equation by the construction of
its moments. It is not meant as an investigation into deeper reaching problems
of statistical mechanics. An attempt has been made to arrive at a unified
treatment, and it is hoped that the collection may be helpful to workers in
the field of gas dynamics as material for comparison with expressions
obtained by other procedures, or as results to be used as such.
Institute for Fluid Dynamics J.M.B.
and Applied Mathematics
University of Maryland
ν
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Acknowledgments
The work presented here has been supported by the United States Air Force
through the Air Force Office of Scientific Research under Contracts
AF-18(600)993, AF-49(638)401, F-44620-67-C-0023, and Grants AFOSR-
141-63, AFOSR-141-64, AFOSR-141-65, AFOSR-141-66. The constant
interest and advice of Mr. Milton Rogers, Chief of the Mechanics Division
of AFOSR, has been very encouraging and helpful, and has always been a
great pleasure. Interest and advice, moreover, has been received from my
colleagues at the Institute for Fluid Dynamics and Applied Mathematics,
of whom I would mention in particular Professor M. H. Martin, Director
until the end of 1967, and Professors E. L. Resler (now at Cornell University),
S. I. Pai, D. A. Tidman, J. R. Weske, H. Lashinsky, and J. R. Dorfman.
Many discussions have taken place with them and with other colleagues at
the Institute, as well as with friends, colleagues, and correspondents with
whom I have come in contact, often through the meetings and visits which
are characteristic of present day scientific life. Dr. Ζ. I. Slawsky has always
shown a wide interest in the work; early discussions with Dr. C. M. Tchen
have initiated some of the ideas applied in Sections 27 and 28; Professor
B. S. Tanenbaum has supplied a list of errata for an earlier report covering
a part of the work; also Professor T. I. Li has given attention to a provisional
partial text; and there have been inspiring contacts on numerous further
occasions. For all of this I am extremely grateful.
I very much appreciate the extensive help in the typing of notes and
manuscripts which I have received over the years from the secretaries of the
Institute's office. In particular I thank Mrs. Lana Einschlag, who retyped
the manuscript with its many equations during the summer of 1967, and
Mrs. Irene Huvos, who took care of numerous alterations coming later.
vii
viii ACKNOWLEDGMENTS
To the Academic Press, Inc. goes my gratitude for the publication of this
work, and to Dr. F. N. Frenkiel and Professor C. Temple for the invitation
to include it in their Series on Applied Mathematics and Mechanics.
I am much indebted to the Addison-Wesley Publishing Company,
Reading, Massachusetts; to Wiley (Interscience), New York; to Dr. Jerome
Fox of the Polytechnic Institute of Brooklyn, New York; to the Pergamon
Press, New York; and to the Regents of the University of Wisconsin,
Madison, Wisconsin, all of whom have graciously given permission for
making use of material which had appeared in books published by them,
mentioned at the end of the list of annotated references.
Observations on notation3"
Vectorial quantities, when not marked by coordinate subscripts, are
printed in boldface type.
Boldface has also been used in contracting the notation for a volume ele
ment dx dx dx into dx; similarly, άξ άξ άξ (element of velocity space)
x 2 3 χ 2 ζ
has been contracted into d\.
In integrals extended over the entire (ordinary) space or (and) over the
entire velocity space, the limits (—00 and +00 for all variables) have not
been indicated. Usually a single integral sign has been written referring to dx,
with another one referring to d\. Limits are indicated explicitly only when
integrals over the velocity space have been expressed in polar coordinates,
and in some related cases.
The Laplacian (occurring on pp. 133, 218, 220) is written
_ d2 d2 d2
a
dx2 dx2 dx2
3
In the equations written in component notation, the summation rule must
be applied with respect to coordinate subscripts which occur twice (p. 12).
Summation with respect to the subscripts s or t for particle species is indi
cated explicitly.
The Kronecker symbol <5 is equal to + 1 if the two subscripts are equal;
l7
it is equal to zero when the two subscripts are different (p. 13).
The alternating tensor 5 is equal to zero when two (or three) of the sub
hij
scripts are equal ; it has the value + 1 when the three subscripts form an even
permutation of 1, 2, 3; and it is equal to — 1 when the three subscripts form
an uneven permutation (p. 13).
* See also Section 2.
xiii
xiv OBSERVATIONS ON NOTATION
The rate of volume expansion and the rates of deformation for the field of
flow of the particles of species s are indicated by (p. 14):
du du du
sh sj si
£s = τ—\ \z)ij = -τ- + ζ ioe.
s u s
dx όχι
dXj
h
Time differentiation following the mean flow of species s is indicated by
(p. 14):
Dt dt Usi dx, '
Abbreviations for the effective electric field in the case of motion through
a magnetic field:
for the flow of particle species s (p. 21),
Ê = E + ô uBjlc;
sh h hij si
for th e mea n mas s flow of th e gas (p . 82),
Ê = E + ô uBjlc.
h h hiJ i
Thes ystem ofu nitsw hich liesa tt h e basiso ft h ee quation s ist h e centimeter-
gram-second system, althoug h all equation s ar e formulate d in such a way
that the y ca n be applie d with an y othe r consistent system of units .
With regar d t o electri c an d magneti c quantities , Gaussia n unit s hav e bee n
used, tha t is , electrostati c unit s for th e electric charg e an d for th e electric
field strength , wit h magneti c unit s for th e magneti c field strength . A s men
tioned o n p . 18,e ffects of dielectric o r diamagneti c polarizatio n i n th e atom s
or molecules have bee n neglected i n view of th e small densities of th e gases
under consideration . Henc e th e dielectric coefficient an d th e magnetic sus
ceptibility bot h ar e assume d t o bee qua l t o unity (in electrostati c unit s for th e
dielectric coefficient; i n magneti c unit s for th e magneti c susceptibility). N o
distinction is mad e betwee n E , th e electric field strength , an d a n electric dis
placement vector ; no r is a distinctio n made between magneti c field strengt h
and magneti cf luxB . Th e use of bot h electric an d magneti c unit s necessitates
the insertio n of th e vacuu m light velocity c= 3 χ 1010 cm/sec (more exactly,
2.998 χ 1010 cm/sec) in those equations where the two types of units occur
simultaneously. The factor 4π has been retained, in the way this is done
in the Gaussian system. It has been thought that the use of this system leads
to the most convenient description, in view of the circumstance that e.g., the
electronic charge in practically all tables is given in electrostatic units.
For conversion to the M-K-S system, the reader is referred to handbooks.
Introduction
The purpose of this volume is to present flow equations for composite gases
which can be used for problems in which collisional effects are prepon
derant, as well as sets which apply to cases where collisional effects are of
little importance. Since the standard treatment of collisional effects with the
aid of the Maxwell-Boltzmann collision integral is restricted to particles
acting upon one another with spherically symmetric force systems, it has
been considered of interest also to investigate what can be obtained with the
aid of the Bhatnagar-Gross-Krook approximation.
To provide the reader with some lines of guidance through the text, the
main topics treated are summarized here.
All flow equations to be considered are obtained by calculating moments
of various orders of the Boltzmann equation for the distribution function of
the particle velocities. As written in Eqs. (1.1) or (1.2), the left-hand side of
the Boltzmann equation has terms which refer to the changes of the distribu
tion function resulting from nonuniformity in space and from the action of
field forces, while its right-hand side carries an expression for the effects of
collisions. In calculating the moments of the Boltzmann equation there is an
advantage in treating the left-hand side and the right-hand side separately.
Calculations concerning the left-hand side are given in Chapter 1. The results
of that chapter can be immediately applied to flow problems in which the
effects of collisions can be neglected entirely (Section 5). Calculations refer
ring to the right-hand side are given in Chapter 2 on the basis of the standard
form of the collision integral. Chapters 3,4, and 5 are concerned with applica
tions of the results.
The application of the Bhatnagar-Gross-Krook approximation is con
sidered in Chapter 6, while Chapter 7 presents an extension of the Bhatnagar-
-Gross-Krook scheme to collisions in which chemical reactions or related
phenomena occur.
1
2 INTRODUCTION
Finally, Chapter 8 brings a brief discussion of the effect of radiation
phenomena upon the moment equations.
In the construction of the equations, primary attention has been given to
the so-called multifluid picture, in which each species of particles has its own
flow velocity. Pressure, temperature, and heat flow have then been defined
for each separate constituent with reference to its own flow velocity (see the
definitions in Section 2). Contrasted with this is the single-fluid picture, in
which a mean mass flow velocity is introduced for the gas as a whole, while
the differences between the flow velocities of the constituent species and the
mean mass flow velocity are treated as diffusion velocities (definitions in
Sections 2,7, and 17). Care has been taken not to confuse the two points of
view and to indicate which picture is used in the cases worked out.
The reader who is looking for a point of comparison with conventional
forms of the flow equations can start from the equations of Section 17 in
Chapter 3. These equations, in which the collisional effects are of decisive
importance, are written in terms of the single-fluid picture, with diffusion
velocities for the separate constituents of the gas. They are based upon the
13-moment description and use the fully linear approximation with regard to
diffusion velocities, differences in temperature between the constituents,
velocity gradients, and temperature gradients.
Equations (17.9), (17.14), (17.20), respectively, are the continuity, momen
tum, and energy equations for the mean mass flow of the gas as a whole.
Equations for the separate constituents of the gas are:
continuity equation (17.8) or (17.8a);
momentum equation (17.12), rewritten in (17.15) and (17.17) as the
equation governing the diffusion velocities;
energy equation (17.18) or (17.18a);
equation for the deviator components of the pressure tensor (17.22);
equation for the heat flow components (17.24).
Since the right-hand sides of these equations have been obtained from the
Maxwell-Boltzmann collision integral for particles acting upon one another
with spherically symmetric force systems, no account can be taken here of
exchange of internal energy of the particles with their translational energy.
Hence all particles behave as point molecules; the ratio of the specific heats
is y =•§•; and there is no "bulk viscosity." (Exchange of internal energy is
considered in Section 50, Chapter 6, on the basis of the Bhatnagar-Gross-
-Krook approximation.)
The left-hand sides of the equations have been given in more detail than
