ON LIAPUNOV'S DIRECT METHOD: A
UNIFIED APPROACH TO HYDRODYNAMIC
STABILITY THEORY
By
JOHN PAUL SAN GIOVANNI
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1969
Copyright by
John Paul San Giovanni
1970
my wife, Marie,
my sons, John Paul and Thomas
and my parents.
"The mere formulation of a problem is far more often
essential than its solution which may be merely a matter of mathemati
cal or experimental skill. To raise new questions, new possibilities,
to regard old problems from a new angle requires creative imagina
tion and marks the real advances in science."
A. Einstein
"My only purpose in this work is to explain what I succeeded
to do towards solving the problem I formulated and what may serve
as a starting point for further research of a similar character."
In the introduction to "General Theory of Stability" by
A. M. Liapunov
ACKNOWLEDGEMENTS
The author wishes to express his sincere appreciation to the
following persons and organizations:
Dr. X. B. Reed: for his guidance and assistance in this
investigation and the preparation of this dissertation,
Dr. R. W. Fahien: for his guidance throughout the author's
graduate career and assistance in the initial stages of this investi
gation,
Dr. J. Siekmann: for his interest in the author's studies,
Professor R. D. Walker: for his interest,
Marie San Giovanni: for her typing of the rough drafts of
this manuscript; her many sacrifices and encouragement which led to
the realization of this dissertation,
Ford Foundation and the Chemical Engineering Department: for
financial assistance.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ............................................... v
LIST OF TABLES ................................................. x
LIST OF FIGURES ................................................ xi
ABSTRACT. ....................................................... xii
CHAPTERS:
I. INTRODUCTION...................................... I.I1
1.1 The State Space Approach and Stability....... I.1
1.2 Hydrodynamic Stability Theory................. 1.7
1.3 Liapunov's Direct Method ...................... 1.15
1.4 Scope of the Present Investigation........... 1.24
Bibliography....................................... 1.28
A. On Liapunov's Direct Method................... 1.28
B. Hydrodynamic Stability........................ 1.31
II. THE GENERAL STABILITY PROBLEM...................... II.1
II.0 Prolegomena................................... II.1
II.0.a Introduction......................... II.1
II.O.b On the Scope of the Theory........... 11.4
II.1 The Space of States.......................... II.(
Addendum on the Generation of Perturbations
through Environmental Influences.................. 11.17
11.2 Constraints upon Possible States.............. 11.22
II.2.a Internal Constraints.................. 11.23
II.2.b Constraints Based Upon the Nature of
the Medium........................... 11.41
II.2.c External Constraints ................. 11.46
TABLE OF CONTENTS (Continued)
Page
11.3 Perturbation Problems Associated with
Dynamical Processes.......................... 11.48
II.3.a Perturbation of the Dynamical Pro
cess Itself........................ 11.49
II.3.b The Stability Problem.............. 11.53
11.4 The General Stability Problem for a Dyna
mical Process Associated with a Continuous
Medium....................................... 11.58
Bibliography........................................ 11.75
III. POSITIVE CONES AND LIAPUNOV OPERATORS.............. III.1
III.1 Introduction................................ III.1
111.2 Preliminaries: Relevant Definitions,
Properties, and Concepts .................... 111.4
111.3 Liapunov's Stability Theorems and Positive
Cones........................................ 111.12
111.4 Concluding Remarks .......................... 111.24
Bibliography........................................ 111.26
IV. QUADRATIC OPERATORS AND LIAPUNOV OPERATORS......... IV.1
IV.1 Introduction ................................ IV.1
IV.2 Definitions and Preliminaries ............... IV.2
IV.3 The Method of Squares ....................... IV.7
IV.4 The Method of Squares for Simple Bilirear
Operators................................... IV.11
IV.4.a Banach Space with a Positive
Multiplication ..................... IV.13
IV.4.b Hilbert Space ...................... IV.16
IV.4.c Ndimensional Hilbert Space........ IV.21
Bibliography. ....................................... IV.27
V. DIFFERENCE EQUATIONS FOR A CLASS OF BASIC FLOWS.... V.1
V.1 A Class of Basic Flows...................... V.2
TABLE OF CONTENTS (Continued)
V.2 Equations Describing Velocity and
Vorticity of Difference Motions...........
V.3 Form of Governing Equations for Particular
Classes of Fluids.........................
V.3.a Newtonian Fluids .................
V.3.b Stokesian Fluids with Constant
Coefficients.....................
V.3.c Finite Linear Viscoelastic Fluids
V.3.d Simple Fluids ....................
V.4 Equations Governing Difference Fields for
Parallel Flows ............................
V.4.a Newtonian Fluids .................
V.4.b Stokesian Fluids .................
V.4.c Finite Linear Viscoelastic Fluids
V.4.d Simple Fluids....................
Bibliography .....................................
APPENDICES:
A. ON THE PHYSICAL INTERPRETATIONS OF MATHEMATICAL
STABILITY ........................................
B. ON THE CALCULUS IN ABSTRACT SPACES ...............
B.1 Differentiation...........................
B.l.a Some Concepts of Abstract
Differentiation..................
B.l.b Historical Note..................
B.l.c 5Examples of Abstract Differen
tiation ..........................
B.2 Abstract Integration.......................
B.2.a ftiout the Lebesgue Integral
n, Its Generalizations..........
viii
Page
V.6
V.17
V.22
V.24
V.28
V.32
V.35
V.36
V.37
V.38
V.4C
V.42
A.I
B.1
B.3
B.7
B.13
B. 1,
B.22
B.27
TABLE OF CONTENTS (Continued)
B.2.b
B.3 Methods
B.3.a
Daniell's Theory of Integration...
of Solution of Operator Equations..
The Method of Contracting Opera
tors ..............................
B.3.b Implicit Function Technique.......
B.3.c NewtonRaphsonKantorovich Method.
B.3.d Method of Steepest Descent........
B.3.e Method of Weighted Residuals......
Bibliography.................................... ..
BIOGRAPHICAL SKETCH ............................................
Page
B.29
B.32
B.32
B.34
B.36
B.42
B.45
B.48
B.54
LIST OF TABLES
Table Page
1.1 General Balance Equation and the Fundamental
Principles of Physics ... ........ .. .......... ....... II.42
V.I Calculated Quantities for Parallel Flows............ V.4
V.2 Key for Equations..... ....... .. .................. V.12
V.3 Useful Forms of Describing Equations ,................ V.15
V.4 Integral Formulation for Velocity Equations......... V.18
V.5 Integral Formulation for Vorticity Equation.......... V.19
V,6 Integral Formulation for Velocity Variance ............ V20
V.7 Integral Formulation for Vorticity Variance .,....... V.21
A.1 State Spaces, Topologies, and Concepts of Stability... A.2
B.1 Applications of Daniell's Formulation by Shilov and
Gurevich (1966) .... .......... ........ .. ............. B.31
LIST OF FIGURES
Figure Page
I.1 Geometrical Interpretation . .............. .. ... 1.19
I.2 Methods of Hydrodynamic Stability Analysis .......... 1.27
II.1 Plane Couette Motion. ... ................... 11.19
11.2 Pulsed Alteration of Strain Field .................. 11,21
11.3 Nomenclature....... ....... . ... ............... 11.26
I 4 The Classical Tetrahedron. ..... ...... ............ II.32
B,l Newton's Method for Roots of an Algebraic Equation... B.37
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
ON LIAPUNOV'S DIRECT METHOD: A UNIFIED
APPROACH TO HYDRODYNAMIC STABILITY
By
John Paul San Giovanni
March, 1969
Chairman: R. W. Fahien
CoChairman: X. B. Reed, Jr.
Major Department: Chemical Engineering Department
The objectives of this dissertation are twofold:
1) to develop a unified approach to stability problems for
systems described by operator equations of evolution,
2) to generalize the direct method of Liapunov.
Objective (2) offers the possibility of a more discriminating treatment
of objective (1), although here we only initiate a study of such implica
tions. These objectives are accomplished by a utilization of functional
analytical techniques of modern mathematics.
Regarding (1) we concern ourselves primarily with physico
chemical systems modelled as continuous media. From a somewhat novel
formulation of continuum mechanics a versatile model is developed whose
mathematical and physical complexity is regulated by: (i) specifying
the state space, i.e., the number and nature of the state variables, and
(ii) specifying the operator equation of evolution, i.e., the signifi
cant mechanisms for transport, the internal interactions, and the system's
interaction with both distant and contiguous surroundings.
The formalism is illustrated by considering a subclass of physi
cal systems for which the describing equations are the balance of mass
and linear momentum and for which the state is specified by the velocity
field. Relevant stability analysis equations for the entire class
of parallel flows are developed (i) for constitutive operators with
particular mathematical characteristics, and then (ii) for several classes
of ideal materials. Stability equations pertinent to any particular
parallel flow are precipitated from these by the specification of a
coordinate system and the components of the basic velocity field.
Regarding (2), the Liapunov operators in our generalization need
not have the totally ordered positive portion of the real line as their
range, rather their values may be in a positive cone in an abstract
space thereby offering possibilities for more subtle, delicate, and
sophisticated distinctions in the state spaces of complex systems.
As with the classical method the principal difficulty in applications is
finding a suitable Liapunov operator; thus, we have also generalized
a classical technique utilizing the theory of quadratic operators.
xiii
CHAPTER I
INTRODUCTION
I.i. The State Space Approach and Stability
The goal of this dissertation is a unified theory of stabi
lity analysis utilizing Liapunov's direct method. We do not restrict
ourselves to specific physical systems, nor do we attempt to pre
sent a theory which is all inclusive. Rather, we present a theory
which describes a significant class of physicochemical systems. We
require only the weak restriction that the mathematical description
of the system be in the form of an operator equation of evolution,
that is, of the quite general form of a balance equation
Rate of f Net Rate + Rate of
Accumulation of Input3 Generation
Although this includes a panorama of mathematical models ranging
from kinetic theory to continuum mechanics, we will investigate in
*
detail only the models in continuum mechanics. However, if such a
specific formulation is to be at all ambitious in the sense of
describing several coupled phenomena occurring in a physical system,
then we would expect not simply one, but several coupled equations
of this general form. We have achieved just such quantitative
In Chapters III and IV, however, we do not limit ourselves
by this restriction. The results of these chapters are completely
general, subject only to the condition that the describing equation
is in the form of a matrix operator equation of evolution in a
Banach space.
1.2
descriptions within our formulation by merely treating the above
equation as an equation for a state vector; the elements of which,
relative to an appropriate basis, are column matrices, the components
of which are not simply numbers, however, but rather members of
suitable function spaces. The formulation for our unified theory
thereby provides the capability for treating a wide variety of
currently popular mathematical models.
The initial steps in the formulation of a mathematical model
for any specific physical system or class of physical systems are
the selection of an appropriate state space and the selection of an
appropriate operator equation of evolution. Although one often
discusses these elements of a mathematical model as if they were
independent of one another, when in actuality they are not, the
selection of the most suitable state space and of the operator
equation describing the evolution of states within that space may
not be divorced. Indeed, even the mere choice of whether to use
an integral or differential formulation of the general balance
equation makes a qualitative difference in the selection of the
appropriate state space.
The relationship between these two elements of a mathematical
model is considerably deeper. The selection of a state space
involves at its most elemental yet among many other things, the
selection of the minimum number of variables which characterize
the system. This may not be done, however, until the operator
equation of evolution, which specifies all significant mechanisms
1.3
for transport and transformation within the physical system, is
known. Conversely, one may take the point of view that the selection
of the state variables dictates the mechanisms for transport and
transformation which are significant by requiring, simply, that they
be consistent with the choice of the state variables and thus also
the state space. This latter tack, at least without sufficient
physical motivation, can tend to become a vacuous exercise.
The selection of a topology for the state space is also a
step of considerable content, for until a mathematical structure in
the form of a topology is imposed upon the state space, it is an
amorphous collection of elements, with only the possibility of some
algebraic operations defined over the collection. These algebraic
operations are necessarily defined so that the operator equations
of evolution are meaningful, and therefore they usually consist of
the operations of addition of elements of the linear space and of
multiplication of them by scalars conveniently defined so as to
give the set the algebraic structure of a linear space. The topology
is then imposed upon this algebraic structure in a manner such that
the algebraic operations are continuous. In this manner the topology
provides the algebraic structure (the set and the algebraic opera
tions) with some sustenance as is evidenced by the vastness of the
theory of linear topological spaces. In dealing with mathematical
models of physical systems, this topology is often that induced by
an appropriate selection of a metric, that is, a measure of the
distance between any two states. There are an embarrassing variety
of metrics for any metrizable space, 2 where T is the cardinality
of the set, thereby permitting a considerable latitude in theory;
in actual practice, there are considerably fewer choices because
of various mathematical dictates and physical considerations.
The selection of a particular metric among all possible metrics is
based upon the interplay of physical intuition and mathematical
acceptability of the definition on one hand versus the analytical
utility and the physical accuracy of the model on the other hand.
The utility of a particular metric arises primarily because
of the topological properties it may give to the state space as well
as the effect it may have upon the properties of the operators in
the equation of evolution. Depending upon the metric selected, the
metrized state space may be complete, separable, or compact. Further
more, depending upon the metric selected the operators in the equa
tion of evolution may be continuous, compact, or completely continu
ous. This latter fact is of significance not only in the proof of
various existence and uniqueness theorems for the operator equations
of evolution, but also in the application of various exact and
approximate methods of solution for these equations. Thus the act
of selecting a metric is much more important than it sometimes
appears.
The reader is referred to Appendix B or any standard
functional analysis text for these concepts.
1.5
Just as a metric for the measurement of distance between
elements of a state space is important, so also is it often important
to introduce notions of distance between sets of elements of a
state space. In fact, this notion is essential in stability theory
where we are interested in the closeness of trajectories rather
than the closeness of elements in the state space.
One point of view in defining such a metric is to look
upon the elements of the state space as functions of the variables
of physical space and time; the state space is to be taken as a
psuedometric space with the psudeometric being a continuous posi
tivedefinite function of time. This point of view would allow us
to view the trajectories as a whole that is, a single unit as
well as to consider with probably greater facility nonstationary
basic states as well as stationary basic states.
We have chosen, however, to take the more conventional point
of view, namely, to look upon time as a parameter; the state as a
function of the variables of physical space alone; and to investigate
the magnitude of the metric instantaneously. However, when we take
this second point of view we must explicitly state what is meant by
stability and asymptotic stability. In tnis context stability
means that if a trajectory has started within some bounded neighbor
hood of the basic trajectory then at all subsequent instants of
times the trajectory is within this bounded region. Asymptotic
stability, on the other hand, requires in addition to stability, the
condition that all the trajectories beginning in some bounded
1.6
neighborhood will actually converge to the basic trajectory as time
grows without bound. It is referred to as uniform asymptotic sta
bility if the convergence in time is uniform rather that pointwise.
In stating these notions of stability, we have spoken as
if we had existence and uniqueness in a bounded neighborhood of the
basic trajectory. For example, what does the above intuitive notion
of stability mean if it is possible to have multiple trajectories
emanating from the same point in event space? Some of these tra
jectories may be such that they would imply stability by the above
intuitive notion while others would imply instability. Thus, in the
mathematical models for which multiple trajectories may be emanating
from some points of state space, a precise mathematical definition
based upon the above intuitive notion would be equivocal. One
possible way out of this dilemma, similar to the technique used
in control theory, is to further refine this intuitive notion for the
case when multiple trajectories exist. In particular, if, in the
conventional case (single trajectories) a set "S" is said to have the
quality "Q" when the trajectories originating at the points of "S"
possess the property "P," then in the more unconventional case
(multiple trajectories) the set "S" is said to be "strongly N' if
all the trajectories originating at the points of "S" possess the
property "P," and the set "S" is said to be "weakly N" if at each
point of "S" there exists some trajectory which possesses the property
"P." However, in this dissertation, we will consider only the
conventional case since we suppose as a matter of course that the
I.7
physical system is so formulated that the mathematical problem is
wellposed, that is, the operator equations of evolution and the
necessary auxiliary conditions (initial and boundary conditions) are
such that the solution (i) exists, (ii) is unique, and moreover
(iii) depends continuously upon the loundary data. We should empha
size that this differs from the more standard usage. Wellposedness,
as the term is generally used, also includes a fourth condition that
the solution depends continuously upon the initial data as well as
the boundary data. But stability theory is the study of the behavior
of trajectories that initiate in a given neighborhood and hence is
naturally kept distinct from questions of wellposedness in the
sense of (i)(iii). There is a larger sense in which stability
theory could become sterile and wellposedness could incorporate
continuous dependence upon both boundary and initial data, were it
possible to develop a complete physical theory that would explain
how "perturbations" arise. Wellposedness would then be truly the
measure of the realism of a mathematical model. This matter is
pursued no further in this dissertation, although brief sorties
against the origins of physical perturbations are cursorily made.
1.2. Hydrodynamic Stability Theory
The remarks made above about stability apply equally well
to all types of stability problems. In this dissertation we con
centrate almost entirely on stability problems associated with a
special class of mathematical models, involving the flow of continuous
1.8
media. We also attempt a unification theme: the formulation is
intended to be sufficiently general as to unify a wide range of
stability phenomena, and the approach is a unified one based upon
Liapunov's direct method. In order to place this formulation in
sharper focus, at this juncture we provide a brief review of the
structure and composition of what might be termed classical hydro
dynamic stability.
The immediate objectives of classical hydrodynamic stability
are to understand the mechanism of the instability of laminar flows
and to obtain criteria for their occurrence. The more fundamental 
and therefore more ambitious objectives of this theory are to
understand why, how, and under what circumstances turbulence arises
from laminar flow instability. In every system of which we have
knowledge, in fact, the transition to turbulence from a laminar
instability is by means of a sequence of stages which are in some
cases easily observed whereas in others they are almost unobservable.
Coles (1965) has demonstrated experimentally, for instance, that
the transition in a Couette cell may be from the basic laminar flow
to one of several types of laminar flow regimes and the ostensible
transition directly to turbulence occurs only under certain cir
cumstances. Qualitatively similar results have been reported in
investigations of boundary layer instability phenomena and the
transition to turbulence (see, Benney, 1964; Emmons, 1951; Elder,
1960; Greenspan and Benney, 1963; Klebanoff, Tidstrom, and Sargent,
1962; Kovasznay, Kamoda, and Vasudeva, 1962; Miller and Fejer, 1964),
where the sequential transition to turbulence is even more striking,
proceeding from a laminar boundary layer, to TollmeinSchlichting
waves, to layers of concentrated vorticity, to spots of "turbulent
bursts," and finally to a turbulent boundary layer. Similar results,
both experimental and theoretical, have also been reported for
transition from laminar flows in other systems (see, e.g., Gill, 1965;
Howard, 1963; Malkus and Veronis, 1958; Palm, 1960; Palm and Qiann,
1964; Sch'lter, Lortz, and Busse, 1965; Tippleskirch, 1956, 1957;
Veronis, 1965).
The mathematical formulation of the general problem of
classical hydrodynamic stability is obtained by taking a (generally
steady) solution of the relevant describing equations, by super
imposing a perturbation upon the basic solution, and then by sub
stituting the resultant disturbed motion into the relevant des
cribing equations. A set of nonlinear equations of evolution for the
growth of the disturbance results. As expected, the difficulties
in the classical theory of hydrodynamic stability arise almost
exclusively because the basic equations are nonlinear.
The formulation of the describing equations has taken two
distinctive forms in the literature: (i) a partial differential
equation formulation (see, e.g. Lin, 1955) and (ii) an integral
equation formulation (see, e.g., Serrin, 1959). The former
intensifies the difficulties due to the nonlinear nature of the
equations whereas the latter tends to diminish possible mathematical
difficulties due to the nonlinearity. The partial differential
I.10
equation formulation is arrived at by assuming that, for example,
the difference velocity field
D = v B
where v : perturbed velocity field
v : the basic velocity field whose stability is under
investigation
may be expressed as an expansion in a complete system of normal modes,
or eigenfunctions. The substitution of this expansion into the
equations describing the rate of growth of the disturbance leads
to an infinite system of coupled nonlinear ordinary differential
equations for the amplitudes associated with the normal modes. The
wellrecognized source of the nonlinearity coupling these is the
inertial term in Cauchy's First Law, voVv. Thus, if the expansion
in normal modes is substituted for VD, and if the inertial term is
isolated, two infinite series are then multiplied together;
therefore, the ordinary differential equation for the amplitude
associated with the ith mode is coupled with the amplitudes for
all the other modes.*
Difficulties of this sort occur in many familiar mathematical
models in varied disciplines. It is, therefore, reasonable to expect
that approximation techniques have been proposed to deal with them,
It is of some interest to note that Lamb (1945) ascribes
this difficulty to a "mathematical disability."
I.11
and indeed a majority of them have been developed and applied in
hydrodynamic stability theory.
The early theoretical attempts at hydrodynamic stability
theory (e.g., Rayleigh, 1880, 1887, 1895; Thompson, 1887a, 1887b;
Taylor, 1923) attempted to circumvent these difficulties by concern
ing themselves with infinitesimaldisturbances so that the coupling
term
vD D
was assumed to be of negligible importance. Because of this assump
tion, the timedependent part of the disturbance may be taken as an
exponential form, exp(kt). If perturbations at the boundary are
excluded, the boundary conditions on the disturbance are homogeneous,
and one arrives at an eigenvalue problem for the parameter k.
In this linearized theory, therefore, the flow is said to be unstable
if it is possible for k to have a positive real part; otherwise, it
is said to be stable.
If k does have a positive real part, then the amplitude
of the disturbance grows with time until the coupling term may no
longer be neglected. The socalled "nonlinear" theories of hydro
dynamic stability are constructed to account for this coupling term
in some approximate manner. The two most frequently used techniques
The interested reader may consult the books of Lin (1955)
and Chandrasekar (1961) for further examples, discussion, and
references.
1.12
are: (i) the truncated modal evolution, in which only a fixed
number, N, of modes are assumed to be of importance, and (ii) the
normal mode cascade, in which it is assumed that initially there are
only a finite number of "primary modes" and that all higher harmonics
are formed by interactions of these primary modes. The interested
reader will find examples of the truncated modal evolution in the
works of Dolph and Lewis (1958), Lorentz (1962), Meister (1963),
and Veronis (1963). Examples of the technique of normal mode
cascade will be found in the works of Eckhaus (1965), Palm (1960),
Segel (1962, 1965a, 1965b), Stuart (1960a), and Watson (1960).
Both these techniques accomplish the same thing in that they make
the mathematical problem of stability tractable the truncated modal
evolution by reducing the infinite system of equations to a finite
number of equations while the normal mode cascade allows us to solve
an infinite system successively.
Closely related to both the linear and nonlinear theory is
the "method of parametric expansion." This method is a linearized
stability analysis of a steadystate solution of the equations
describing the evolution of the difference state. Malkus and
Veronis (1958) applied this technique to a particular finite ampli
tude solution for the Benard problem. SchUlter, Lortz, and Busse
(1965) extended the analysis of Malkus and Veronis to a larger
class of possible finite amplitude solutions. This latter research
is, of course, a theoretical attempt to explain why one or another
of the possible finite amplitude solutions (mode of convection in
1.13
the B&nard problem) is preferred.
All the methods of classical hydrodynamic stability discussed
to this point are based upon the partial differential equation for
mulation, and they characteristically contain a number of assumptions
about either the magnitude of the disturbance or the coupling of
various normal modes of the disturbance; these assumptions alone
enable them to achieve the primary objectives of hydrodynamic
stability theory. In contrast, the integral equation formulation and
the associated "generalized energy" method require no such assump
tions of this nature to reach these objectives. In particular, the
nonlinear terms may be fully accounted for. The integral equation
formulation is thus a far more natural framework within which to
view arbitrary but physically realistic perturbations. Unhappily, to
treat a wide variety of disturbances requires that one sacrifice the
more detailed results obtained from the partial differential equation
formulation. Because the purview of the integral equation formula
tion is that of global properties, that is, of integrals over the
flow regime, the details of the flow pattern are necessarily lost.
Although this is a weakness of the "generalized energy" method, it
is precisely the global approach, which deals with the overall
effects rather than in details, that allows the method to be applied
to disturbances of any magnitude. For example, if by critical we
mean the onset of instabilities as predicted by linear theory, then
Joseph (1965) has used the integral equation formulation to show the
nonexistence of these subcritical instabilities for arbitrary
1.14
spatially periodic disturbances. To be redundant for emphasis,
the nonlinearity is retained and results far beyond those of linear
hydrodynamic stability theory may be obtained. Moreover, because
the method does not require a discussion of interactions of normal
modes (the physical significance of which may not be all obvious)
the physics described by the equations is readily apparent.
The integral equation formulation and the associated
"generalized energy" method seems to have originated in the work of
Reynolds (see, e.g., Reynolds, 1895) and Orr (see, e.g., Orr, 1907).
Although their basic approach to stability problems has been used
through the years (see, e.g., Hamel, 1911; Serrin, 1959; Synge,
1938a, 1938b; Thomas, 1943; von KArman, 1924), it received only a
small fraction of the use and the attention that the partial differen
tial equation formulation received. Interest in the method, however,
has risen steeply since the publication of Serrin in 1959. The
work on the method since then may be placed into two classifications:
(i) an improvement of the bounds used by Serrin (see, Velte, 1962;
Sorger, 1966), and (ii) treatment of more complex problems (see,
Conrad and Criminale, 1965; Joseph, 1965, 1966). Thus, Conrad
and Criminale treated the case of time dependent basic velocity
fields whereas Joseph treated the Benard problem with the
Boussinesq equations.
In all these various extensions and generalizations of
Serrin's work, the key lies in the positivedefinite property of
1.15
the "generalized energy" and the inequalities which are used to
obtain sufficient conditions for the time derivative of this "energy"
to be negativedefinite. The essential ingredient for stability is,
to summarize, a generalized energy that is positivedefinite yet
with a negativedefinite total time derivative.
The properties of the "generalized energy" used by Serrin
(1959) are precisely those of a Liapunov functional, and, in fact,
Serrin's use of the "generalized energy" to ascertain sufficient
conditions for stability bears a striking resemblence to the use
of Liapunov functionals in Liapunov's direct method. Because
of this similarity and because one of the major contributions of
this dissertation is a generalization of Liapunov's direct method,
it is appropriate that we now discuss the method, its essential
ingredients, its interpretation, its advantages, its limitations,
and recent research work involving it.
1.3. Liapunov's Direct Method
Mathematics is at times the language of science and at other
times it is the queen of science (Bell, 1955). In its latter role
mathematics does well in its imposition of a logical structure upon
mathematical models of physicochemical systems. On other occasions,
however, it fails in both its roles, notably in dealing adequately
with nonlinear problems. As an unhappy lesson of experience, more
over, the more accurate the model desired or the wider the range of
theory sought, the more probable it is that the formulation will be
1.16
nonlinear. And it long ago became clear that solutions to nonlinear
operator equation are generally unobtainable, at least in the style
to which science became accustomed to when it dealt only with linear
problems. Analytical solutions seldom exist and quantitative
methods of analysis, while helpful, in no way suffice as they do
for linear problems. Qualitative methods, on the other hand, may
tell us much about a nonlinear system, they may even tell us all
that we need to know; but even when analytical methods provide us
with some useful information, qualitative methods will always serve
to complement them.
Within the larger framework of qualitative methods in
mathematics, the names of Henri Poincar6 and Alexander Mikhailovich
Liapunov occupy a revered position. They were among the first to
recognize the futility of attempting to obtain explicit solutions
to all differential equations leading the trend toward obtaining
qualitative characteristics of families of solutions of a differen
tial equation. Whereas Poincare's interests in this area centered
primarily upon the existence of periodic solutions and the geo
metrical properties of families of solutions generated by pertur
bations of an operator, Liapunov's primary contribution was the
development of a method for studying the geometrical properties of
a family of solutions of a given equation, relative to a basic
solution that need not be known. This last is the source of the
terminology, Liapunov's direct method.
The direct method is a generic term for a number of theorems
1.17
centered about several precise definitions of stability and certain
real scalarvalued functionals that have come to be called Liapunov
functionals. These functionals have the nature of a generalized
"distance" or "energy:" (i) in much the same manner that a compari
son of the norms of each of two elements of a linear space indicates
which is closer to the origin, even though the elements are in
different "directions'," so also may different points of the state
space be distinguished from one another, relative to the origin, by
different values of their Liapunov functional and (ii) in much the
same way that a redistribution of energy provides a means of con
sidering transient phenomena, so also may local changes in the
values of Liapunov functional indicate that a trajectory in the
state space locally is tending toward or away from the basic tra
jectory (which may be degenerate, in that it may be a basic state).
Loosely speaking, Liapunov functionals provide a partial ordering of
the admissible states of the system, as well as of trajectories in
the state space (Chapter III).
If we consider the Liapunov operators as real scalar
valued functionals, this geometrical interpretation may be easily
illustrated (Elgerd, 1967; Hahn, 1963, 1967; Lasalle and Lefschetz,
1961). From the statements of the relevant theorems, a Liapunov
functional, V<.>, for an autonomous system described by
d_ = f(x), f(O) = 0
stability, for example, if it is a positive
insures asymptotic stability, for example, if it is a positive
1.18
definite functional, and if its temporal variation along admissible
trajectories of the system must be negativedefinite. For ease of
visualization, we suppose that the state of the system may be
represented as a point x in I2R a particular state space generally
referred to as the phase plane. We then erect a perpendicular to
this plane at the origin to indicate the range space of the Liapunov
functional. In this 3 a Liapunov functional may be represented as
an openended, bowlshaped surface as indicated in Figure I.la, and
in particular that is the graph of a singlevalued function. Thus
the projections of the
Vax> = const. (I.1)
loci onto the phase plane generate a system of closed, non
intersecting curves inR 2 (see Figure I.1), and it is an intuitively
obvious and weilknown result of elementary topology that a closed
curve in a plane divides the plane. Mathematically, the criterion
of negativedefinite temporal variations along all admissible
trajectories may be expressed by
dV = d VV < 0
dt dt 
Now the classical geometric interpretation of the gradient is that
of a vector, the direction field of which is everywhere perpendicular
to loci given by equation (I.1), and whose magnitude is a measure
of the spatial variation of V<.>. Geometrically, therefore, the
mathematical inequality noted above may be interpreted as a require
ment that the tangent to the field of trajectories f(x) in the phase
1.19
V
I
_a) x
x2
V(x)3
V(x) 2
V x)1
dx
\ dt
.__A TYPICAL TRAJECTORY
FIGURE 1.1. GEOMETRICAL INTERPRETATION
1.20
plane must always and everywhere have a negative projection upon
the corresponding gradient vector (see Figure I.l.b). Therefore
the system is asymptotically stable if the admissible trajectories
are such that succeeding states along them correspond to lower
values of the Liapunov function. And in this sense, the succeeding
states may be said to be "closer" to the origin.
The framework of this geometrical interpretation of Liapunov's
direct method may also be used to emphasize an important point
about Liapunov's direct method: it provides only sufficient conditions
for stability which is, incongruously, both a strength and a weak
ness, of which we shall have more to say later. If we had plotted
in this phase plane the vector field, f(x), in some neighborhood of
the origin (the basic solution), we could, by the method of iso
clines and visual observation, determine the behavior of trajectories
in that neighborhood of the origin; it would, however, still remain
to find a suitable Liapunov functional if stability were the case
Conversely, if we had a Liapunov functional with the required
properties in some neighborhood of the origin, then we could avoid
che graphical construction of the method isoclines in that neigh
borhood and yet be assured of stability. Now in state spaces of
higher dimension the first alternative is not a viable one, and
if we were also unable to find a suitable Liapunov functional, then
we would be unable to draw any conclusions about the stability of
the origin.
1.21
This sufficient nature of Liapunov's direct method permeates
every part of its structure and is at once both the greatest
strength and the most significant weakness of the method. We say
strength because, as a consequence of this sufficient nature, one
has an analytical tool powerful enough to assure us of stability
in some neighborhood of the origin not only for one equation but for
a class of equations a monumental accomplishment, Moreover, this
may be accomplished without the necessity of the many oversimplifica
tions usually necessary to obtain an explicit solution to just one
number of this class of equations. On the other hand we say that
this sufficient nature is also a weakness of the method because if
we are unable to discover a suitable Liapunov functional, even after
an extensive search, we know no more abouc the stability of the
origin than before we began. Also, even if a suitable Liapunov
functional has n found and an associated neighborhood of stability
determined, this does not say that the portion of state space outside
this r. orhood is a r n tof instability, Thus the major
dificul, in appi Liapunov's direct method to a particular
n Lem or class of tems is in obtaining a Liapunov functional
with the required pr ties. Consequently, it is reasonable to
expect that much of the research on the method is devoted to techniques
tor construct suiLable Liapunov functionals (see, e.g., Brayton
Another not in onsequential advantage of Liapunov's direct
M ehod, it seems to us, is its simple physical and geometrical
interpretation (cf. above)
1.22
and Miranker, 1964; Hahn, 1963, 1967; Ingwerson, 1961; Leighton,
1963; Letov, 1961;Luecke and McGuire, 1967; Krasovskii, 1963;
Peczkowski and Liu, 1967; Schultz and Gibson, 1962; Szego, 1962;
Walker and Clark, 1967; Zukov, 1961). Generally speaking, the proposed
methods may be classified into three categories: (1) those which
assume a certain form for the gradient of the Liapunov functional,
(2) those which assume a certain form for the Liapunov functional or
its time derivative, and (3) those which make use of the similarity
of two systems, for one of which a Liapunov functional is already
known, with the standard application being an extension from a linear
system to a "slightly" nonlinear system.
Zubov's nowfamous method falls under the first classifica
tion (see, e.g., Hahn, 1963; Zubov, 1961). Essentially, it makes
use of the geometric interpretation of a first order, partial differ
ential equation. If the existence of a solution can be proved for
the partial differential equation, then one will have proved the
existence of a Liapunov functional for the system of ordinary
differential equations. Moreover, Zubov's method actually provides
a constructive method for Liapunov's method based upon the solution
of the related first order inhomogeneous partial differential equation.
The method of "variable gradients" (see, e.g., Hahn, 1967; Schultz
and Gibson, 1962) also belongs to this first classification. In
The reader is referred to any standard textbook in the theory
of partial differential equations.
1.23
this method one assumes a form for the gradient of the Liapunov
functional and requires that the functional be singlevalued. This
latter requirement allows us to find the Liapunov functional by a
line integration along any convenient path. The methods of "separa
tion of variables (Letov, 1961), "canonical variables" (see, e.g.,
Brayton and Miranker, 1964; Letov, 1961; Zubov, 1961), and "squares"
(see, e.g., Hahn, 1963, 1967; Letov, 1961; Krasovskii, 1963) all
belong to the second classification. In particular, the method of
squares focuses upon the fact that any positivedefinite form is a
possible Liapunov functional, an essential feature to which we
return in Chapter IV. The most widely used method under the third
classification at least among engineers, is the one using a Liapunov
functional for a linear system to determine the region of asymptotic
stability for a nonlinear system which is somehow close to the linear
system (see, e.g., Krasovskii, 1963, Chapter IV).
These techniques of constructing Liapunov functionals 
and to a lesser extent the theorems themselves had, until 1960,
been applied mainly to stability problems associated with systems
of ordinary differential equations, or in engineering parlance, to
lumped parameter systems. However, since the appearance of Zubov's
monograph (see, Zubov, 1961), there has been an increased interest
in applying the method to mathematical models involving difference
differential equations (see, e.g., EL's gol'c 1964, 1966; Krasovskii,
1963) and to partial differential equations (see, e.g., Brayton and
Miranker, 1964; Hsu, 1967; Mochvan, 1959, 1961; Parks, 1966;
1.24
Wang, 1964, 1965), that is to socalled distributed parameter systems.
1.4. Scope of the Present Investigation
Although our interests lie generally with systems that are
described by the general equation of balance, in this dissertation
we consider, almost exclusively, systems arising from continuum
mechanics. The primary objectives, then, of this dissertation are
to present a unifying formulation of stability problems associated
with continuous media and a unified approach to their analysis by
means of Liapunov's direct method. The methods and concepts which
have been employed in realizing these objectives are those of modern
mathematics, particularly, functional analysis.
The main significance of this unifying formulation and unified
approach is the fresh outlook from which to view not only the very
old problems of hydrodynamic stability but also those of arbitrary
physical and mathematical complexity. This dissertation presents a
systematic way of proceeding from the hypothetical mathematical
model of a physical system, through the corresponding operator
equations of evolution for the difference state, and, finally,
through operator equations whose solution will be a Liapunov opera
tor for that particular physical system. In effect, therefore, we
have divorced the physical problems involved with modelling from the
mathematical problems involved with solving an operator equation for
the Liapunov operator. From another point of view, however, we
have directly related the physical problems to the mathematical
1.25
problems, in that once the mathematical model for a physical system
is selected, the corresponding operator equations for a Liapunov
operator may be immediately obtained by mere substitution in the
relevant equations. Again, from a still more distant vantage point,
we have actually provided but a simple illustration of the power
and versatility that the abstract spirit of modern mathematics may
bring to bear on the complicated problems facing scientists and
engineers of today.
Chapter II is devoted to the preliminaries necessary to
optimally utilize the power latent in Liapunov's direct method.
Thus, Chapter II relates and discusses a state approach to the
models of continuum mechanics and the place of stability problems
within the class of perturbation problems. In tone if not in
accomplishment, this chapter provides a somewhat novel approach to
continuum mechanics. We close the chapter with a natural generaliza
tion of some of the work of Serrin (1959) and Joseph (1965, 1966)
in order to illustrate the advantages of an abstract approach to
stability problems by using the simplest type of Liapunov operator 
a quadratic form to arrive at sufficient conditions for stability,
one of our few concrete results.
In Chapter III, we turn to a consideration of Liapunov's
direct method and develop a generalization of the method. Instead
of restricting ourselves to the real line as the range space for
Liapunov functionals, we consider Liapunov operators with partially
ordered linear topological spaces and, in particular,positive
1.26
cones in these spaces as the range spaces. It is reasonable to expect
that the richer and more delicately structured the domain of the
Liapunov operators is because this is the state space the more
desirable it becomes for us to consider range spaces richer than
the real line if we are to provide a more highly discriminating
ordering of the elements and trajectories in the state space by
means of these operators. In other words, if one considered the
set of all possible Liapunov operators, it would be greatly enriched.
Moreover, it is likely that certain systems may be especially apt
for the application of these Liapunov operators, whereas they may
have hitherto been unamenable to analysis by means of Liapunov
functionals, and because of the sufficient nature of the method they
may have gone unanalyzed by thedirect method of Liapunov. Of the
many theorems that could have been developed from this conceptual
breakthrough, we emphasize only generalizations of the main theorems
on stability, asymptotic stability, instability, unbounded instability,
and also the conditional nature of the new method.
In Chapter IV we employ these theorems and a slight generaliza
tion of the notion of a quadratic form to arrive at operator equations
for the construction of these Liapunov operators, and in Chapter V
we take a very special class of physical systems, namely, those whose
state is specified by the velocity field to illustrate the versa
tility of our formulation and its possible range of application.
For the convenience of the reader in investigating the refer
ences, we present a block diagram of Hydrodynamic Stability Theory in
Figure 1.2.
1.27
BIBLIOGRAPHY
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__ ___1957, Uber DifferentialDifferenzengleichungen,
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1.29
Ingwerson, D. R. 1961, A modified Liapunov method for nonlinear
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Kalman, R. E. and Bertram, J. E. 1960, Control analysis and design
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Krasovskii, N. N. 1959, On the theory of optimal control, Appl. Math.
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___ 1963, Stability of Motion, Stanford Univ. Press,
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certain autonomous nonlinear differential equations,
Contrib. to Diff. Eq., 2, 10.
Letov, A. M. 1961, Stability in Nonlinear Control Systems, Princeton
Univ. Press, Princeton, N.J.
Liapunov, A. M. 1967, General Problem of Stability of Motion,
Academic Press, N.Y.
Luecke, R. H. and McGuire, M. L. 1967, Stability analysis by Liapunov's
direct method, IEC Fundamentals, 6, 432.
Malkin, I. G. 1950, Certain questions in the theory of stability of
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2, 20.
1959, Theorie der Stabilitat einer Bewegung, Verlag R.
Oldenbourg, Muchen.
Movchan, A. A. 1959, The direct method of Liapunov in stability
problems of elastic systems, Appl. Math. Mech., 23, 483.
1961, Stability of processes with respect to two matrics,
Appl. Math. Mech., 24, 988.
Parks, P. C. 1966, A stability criterion for panel flutter via the
second method of Liapunov, AIAA J., 4, 175.
Peczkowski, J. L. and Liu, R. W. 1967, A format method for generating
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Rekasius, Z. V. and Gibson, J. E. 1962, Stability analysis of
1.30
nonlinear control systems by second method of Liapunov,
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Wang, P. K. C. 1964, Control of distributed parameter systems,
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__ 1965, Stability analysis of a simplified flexible
vehicle via Liapunov's direct method, AIAA. J., 3, 1764.
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Zubov, V. I. 1961, The Methods of A.M. Liapunov and Their Application,
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1.31
B. HYDRODYNAMIC STABILITY
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Eckhaus, W. 1965, Studies in Nonlinear Stability Theory, Springer
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1.32
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CHAPTER II
THE GENERAL STABILITY PROBLEM
11.0. Prolegomena
II.O.a. Introduction
Given a physical system, the selection of an appropriate state
space and an operator equation are primarily questions of utility and
of accuracy. As important as such questions are in formulating a
mathematical model of a physical system, we shall defer them until
Sections 11.2 and 11.3. The operator equation of evolution may be
looked upon as an abstract operator that maps a given initial state
s into the sequence of states of the system; a sequence of states
{s/ fIso generated will be referred to as a state space trajectory.
In actuality, however, if we write the operator equation in the form
then the operator S may be viewed as a temporal connection* that
maps the state of the system at one instant of time, s_ into the
state of the system at the "next" instant of time, s t. p This sug
gestive terminology for the right side of the equation of evolution
will often be used in the sequel, but it must be kept in mind that
.*The interpretation given to the words "temporal connection"
is analogous to the use of the words connection coefficient in
differential geometry. The operator S<.> may also be looked upon as
a map from the state space to the space of linear and continuous
operators from the real line to the space state.
11.1
11.2
the temporal connection will be singlevalued only if the boundary
conditions are incorporated into S<.> which means that the equation
must be reinterpreted, say as an integral operator incorporating the
boundary conditions  and if the problem were wellposed to begin with h
so that there are single trajectories emanating from each element of
the state space.
It is often the situation that arbitrary initial states or
arbitrary classes of initial states are to be considered, in which case
we speak of the flow of a set of states from the initial set.
The classical theory of stability is the study of the fate of
a perturbation superimposed upon a given state space trajectory, or it
may be the study of a state space trajectory acted upon by intermittently
or continuously acting perturbations. Or, more generally, one can study
a class of perturbations acting either once and for all, intermittently,
or continuously but superimposed on either a state space trajectory or
upon a flow of state space elements.
If the evolution of the state of a system can be described ac
curately by a finite set of ordinary differential equations, then the
state space is finitedimensional. There are standard definitions of
thedifferent types of stability in finitedimensional state spaces
(see, e.g., Kalman and Bertram, 1960; Hahn, 1963). If the changes of
state of a system must be described by a system of partial differential
equations to achieve the desired accuracy, then the state space is
infinitedimensional. There also are available standard definitions of
stability in infinitedimensional state spaces (see, e.g., Zubov, 1961;
11.3
Hahn, 1963, 1967). In stability theory, as in any mathematical formu
lation of a specific physical system, the proper selection of a specific
metric, or at least a specific topology is of utmost importance. This
is especially true in stability theory for we are interested in the
closeness of both trajectories and states. Conversely, the selection
of a specific metric may limit a theoretical analysis needlessly, No
specific metric or other type of topology is considered explicitly: we
consider abstract spaces as state spaces and an intuitive (essentially
topological) notion of stability and asymptotic stability,*
Not content with this level of generality, we shall consider
(see Sections II.3.a., II.3.b., II.3.c., and 11.2) the formulation in
the context of a general dynamical process  by which we mean the set
of state space elements of a flow, the operator equation of evolution
generating that flow, and the boundary and initial conditions delineat
ing the succession of state space elements. By considering perturba
tions of dynamical processes as well as perturbations of basic states
(please see Section 11.3), classical perturbation theory and stability
theory under perturbations to both boundary conditions and initial
conditions are brought under the same province as classical stability
theory. There is, of course, a certain unity to be gained from this.
It is to this and to the abstract formulation with an "arbitrary" set
*In Appendix A, however, we do consider, for illustrative
purposes,.specific normed linear spaces appropriate to specific
equations drawn from continuum mechanics, and we consider the physi
cal meaning of different norms and their relations to different
types of stability.
II.4
of general balance equations that we refer when we speak of a unified
stability theory.
II.O.b. On the Scope ofthe Theory
Although the fate of a perturbation, loosely speaking, is our
ultimate aim, it is first necessary to attempt to say what is meant .by.
the terms "state space," "perturbation" and "relevant describing
equations."
Thus, with regard to the last, the derivation of a quite general
form for the operator equations of evolution in terms of essentially
three possible representations (integral equations, differential
equations, difference equations and combinations thereof) is discussed.
The two factors which are helpful in making possible the formulation
of a unified approach to stability theory based upon the general
balance equations are: (i) each of the principles of physics  that
is, laws of our method of representation  has the form of this
equation but with a different physical interpretation attached to the
mathematical symbols in each principle, and (ii) the facility of Li
apunov's direct method for dealing with classes of equations as well
as with specific equations. In particular, we state at the onset
that the mathematical terms of the general balance equations are to
be interpreted within the context of continuum mechanics, thereby,
also including mass point mechanics.* The unification of the various
*Hamel (1908) has shown that when the motion of a body about
and relative to the center of mass may be neglected, then the equations
of continuum mechanics reduce to those of mass point mechanics. In his
formulation it is stress rather than force which is of central impor
tance. See also Noll (1959).
11.5
principles of continuum physics in the form of a single general balance
equation emphasizes the now traditional, essentially axiomatic formu
lation of physics. In this way we are able to present a unified treat
ment of the class of stability problems associated with the objects
and phenomena within the range of the physical theories referred to as
continuum physics. Accordingly, subclasses of stability problems
associated with physical systems which involve continuous media,
classes of phenomena, classes of materials, and classes of state
space trajectories may be delineated by the specification of a set of
hypotheses.
It is clear, moreover, that the selection of the relevant
describing equations cannot be divorced from the selection of the
state space, nor can the selection of a state be separated from the
selection of a set of describing equations. The two are but different
aspects of a whole, the description of the processes occurring in a
given physical system. It isnevertheless useful to make the dis
tinction, however useless it may seem, in formulating the problem,
for it is the very essence of solving the problem.
In other words, the stability problem is so formulated that
its physical  and mathematical complexity may be regulated by
specifying the number and nature of the state variables of the sys
tem, by specifying the nature of all significant mechanisms for trans
port  the internal couplings in the system, by specifying the nature
of volume interactions with the surroundings, and by specifying the
nature of the interactions of the system with the continuous sur
11.6
roundings. This formulation is accomplished by simultaneously con
sidering a finite number of equations of the form of the general
balance equation, by considering the basic state whose stability is
being investigated as a solution to this set of equations, and finally
by employing slight revisions of some familiar mathematical relations
in order to obtain a general operator equation of evolution for the
difference between a perturbed state and a basic state.
II.l. The Space of States
Because we have chosen to interpret the terms in the general
operator equation of evolution within the context of continuum mechan
ics, the state variables naturally appear as fields defined over a re
gion of physical space. The values of these fields at a point in the
region of physical space  that is, the physical properties such as
temperature, velocity, stress, density, concentration, polarization
density, etc.  describe, indicate, and provide some measure of the
physical situation at that point. To be more precise, we have
Definition B.l.I: At an instant of time, the local state of a point
Ln the region of physical space,R is said to be known if the values
of an independent set of these physical properties are known.
The members of this independent set are referred to as the
local state variables. In a particular description of some physical
phenomena, the meaning of an "independent" set is intimately linked
to both the nature of the continuous medium occupying the region of
11.7
interest, R of physical space and the physical theory being used to
describe the phenomena. Alternatively, the choice of what constitutes
an independent set of these physical properties.may be used to charac
terize the range of a proposed physical theory (cf. e.g., Toupin's
theory of an elastic dielectric [Toupin, 1956]; Coleman and Noll's
theory of thermostatics [Coleman and Noll, 1958]). We do not take up
the general and profound question of what constitutes an independent
set of state variables, although we do take it up peripherally as we
consider specific (but still somewhat general) situations of the basic
equations that describe them.
Continuum mechanics is constructed, however, so as to deal
directly with gross phenomena occurring in finite portions of physical
space. For this reason it deals with fields whose independent var
iables are the points within the region of interest, R and whose
values are the various physical properties. Thus, the specifications
of a field is actually a specification of a continuum of local proper
ties. In view of this, we define,
Definition 11.1.2: At any instant of time, the global state of a
region of physical space, k is said to be known if the field
yielding the local state at each point of 'R is known.
Now corresponding to local and global states, two types of
state space may be constructed. In particular, the local state at
some point, x in the region, 1 may be of particular interest.
The instantaneous local state at that point, x may be given as
an Ntuple whose elements are the values of an independent set of the
11.8
physical properties. Some of these elements will be real numbers,
some vectors, some dyadics, etc., but for convenience, this Ntuple
may be represented as a point in an appropriate state space formed by
taking (N) direct products of spaces to which each of the components
of the Ntuple belong. In general, the local state at the point,
x will be different at different times. It would, therefore, be
represented by different points in the local state space of the
point, xo of $ The totality of all the local states associated
with a point, xo of #1'as time elapses is a curve in this local
state space. This curve is called the trajectory of local states
associated with the point, x of 1 .
On the other hand, our interest may be in the instantaneous
global state of a region, tR In order to specify the global state,
a continuum of these local states must be specified. The concept of
a field, however, precisely specifies this global state by means of an
Ntuple of fields  one element corresponding to each of the local
state variables. Again, for convenience, one may look upon this
Ntuple of fields as a point in an appropriate state space. This
state space will now be formed, however, by (N) direct products of
appropriate function spaces (or, in general, operator spaces). The
choice of the particular types of function spaces involved in the
(N) direct product operations depends upon the nature of the physical
properties associated with the fields describing the system and in
volved in the Ntuple. Because the local states may vary with time,
the global states of necessity will also vary with time. Different
11.9
global states associated with different times are represented by
different points in this global state space. The totality of all
these points as time is varied continuously is called the trajectory
of global states. Any point on this curve, along with the associated
instant of time, is referred to as an event. An event is thus speci
fied by an (N+l)tuple.
Intimately connected with this concept of "state" is the con
cept of "system." The conceptual division of the universe into sys
tem and surroundings allows a discussion of process of primary
concern without discussing all processes in the entire universe, a
problem of trivially insurmountable proportions. Although this
division of the universe is arbitrary, a system may only be isolated
for study if the influence of the rest of the universe  the sur
roundings  may be adequately controlled or described or both. The
intimate relation between the state of a system and the surroundings
of a system thus occupies a central position in the construction of
any meaningful theory.
Another distinction which will be found convenient because
we intend to deal with stability problems is that of environment of
a system, by which we mean all alterable (i.e., controllable) ex
cernal influences on the system. Suppose, for example, that a
particular physical system has been designated for study. The sur
roundings are immediately fixed by this and an appropriate selection
of the state variables. However, the environment is not automatically
fixed. For example, one might only be interested in and able to
II.10
control the effect of alterations of only a specific type of external
influence (e.g., an external electric field or a condition at the
physical boundary), in which case only that external influence would
constitute the environment. The environment is thus a controllable
subset of the surroundings of a given system. Needless to say, fields
that are parts of the environment in one class of systems may be parts
of the system or nonenvironmental parts of the surroundings in other
classes of systems.
The choice, then, of what constitutes the environment is
directly related to the question "stability with respect to what?"
because it may specify some of the ways in which perturbations may
be generated. Thus, as observers we are necessarily a part of the
surroundings and, presumably, are the modus operandi behind alter
ations of the controllable external influences that can perturb the
state of a system. In classical stability theory, no allowance is
made for perturbing the system to its new state nor of permitting
the system any "inertia" in that direction. One simply assumes that
perturbations are achieved instantaneously and then the system ceases
co be acted upon by the external influences that provided the original
(step) change of state. The tacit assumption, of course, is that by
considering all possible perturbations (at least of a given class),
one accounts for many of these effects. More generally, intermittent
or continuous perturbations provide a more reasonable description
of these effects. (A further discussion of perturbations arising
from environmental influences is presented in an addendum at the end
II.11
of this section). There are, however, uncontrollable external in
fluences that can perturb that state. Worse, there are molecular
sources providing for random fluctuations of the continuum (or
macroscopic) variables, and these sources of perturbations lie outside
the framework of the commonlyused theories. By the same token,
there has been little study of the influence of macroscopic fields
upon the microscopic processes, for example, upon the theory of
fluctuations. Some usual choices for the environment are: (1) cer
tain classes of boundary conditions (the specification of fields or
of an operator equation for the fields on a mathematical surface or
surfaces bounding the region and representing the interaction of
the system with its contiguous surroundings); (2) certain classes
of body force fields (the specification of the interaction of system
and external influences acting throughout the regionP ),
A variety of terms such as physical property, global state,
event, trajectory, system, surroundings, and environment, have been
introduced. They have been discussed separately, and the relation
ships between them have not been emphasized. To provide a degree of
unity another concept must be introduced, that of a dynamical process.
The dynamical process (DP) consists of all the information that is
needed to transform the present state into the next state in a manner
which conforms with physical reality.* Consequently, the particular
*This is the ultimate criterion of any physical theory, and
although it is universally recognized, it should nevertheless not
go unstated.
1112
elements in a dynamical process depends upon the previous selection
of (i) the state variables (ii) system and, (iii) the surroundings,
including the environment.
Although the choice of specific quantities for these three
elements of a dynamical process should be made simultaneously, let us
discuss separately how the "next" state is affected by them, A
choice of the state variables (i.e. an independent set of fields)
limits the class of possible "next" states in three ways. First, it
indicates the types of physical phenomena which are included in the
study (the range of a physical theory). Second, it suggests which
of the principles of physics will supply an independent set of
governing equations. Thus, if the state variables for a particular
problem have been correctly chosen, the relevant principles of physics
are precisely those sufficient to describe the evolution in time of
these state variables.* The third effect precipitated by a choice
of the state variables is a restriction it places upon the choice of
an environment.**
*As a simple illustration, the models of isothermal and
aonisothermal flow of a singlecomponent newtonian fluid may be
considered. In the model of isothermal flow the NavierStokes
equations are taken to be the describing equations. On the other
hand, in the nonisothermal model these equations must be considered
simultaneously with an equation governing the temperature field.
**Again, the model of isothermal flow of an incompressible
pure newtonian fluid may be used as an illustration. If the den
sity and velocity fields are selected as the state variables, then
it will not be consistent to choose for the environment  i.e.,
the alterable external influences  anything that may significantly
alter the temperature field.
11.13
Because the selection of a system and its environment are
intimately related, the question of specifying the state variables
as distinct from the environmental influences is quite delicate. It
traditionally hinges more upon theoretical and experimental (or both)
tractability than upon a bona fide effort at dealing with coupled
fields. For example, suppose we have a newtonian fluid flowing, g
subject to an electromagnetic, as well as a gravitational, field.
The describing equations are the NavierStokes equations (with Max
well stresses) and Maxwell equations for a flowing (continuous)
medium. The relevant boundary conditions plus the (applied) body
force fields constitute the surroundings, and the problem is then
formulated. Unfortunately, we are not yet in a position to solve
nonlinear partial differential equations, much less highly coupled,
nonlinear sets of such equations. The uncoupling device is the
following: assume electromagnetic fields that result as solutions
of Maxwell's equations for certain similar geometries, etc., and use
them as the entries in the NavierStokes equations. In the latter
situation, then, electromagnetic fields are taken to be part of the
environment, whereas in the former they are part of the system and
are state variables. It is clear, then, that were we able to solve
the full, coupled, set of equations, the state space trajectories
would provide us a full description, but because we are mathemati
cally inept, we must resort to a specification of certain of the
state variables once and for all in order to find the approximate
temporal and spatial variations of the remaining fields. Clearly,
11.14
the selection of a "smaller" state space will strongly prejudice the
accessible next states* of the system, simply by excluding the full
range of values for those "state variables" that we have been forced
to specify as part of the environment.
The way in which the environment influences the next state
may be loosely described as follows. In the absence of environmental
influences and presuming uniqueness (uniqueness is assumed throughout
the discussion; the argument may be generalized to nonunique situations),
there will be a single trajectory passing through a given state, with
the "next" state being thus defined. For a given nonzero value of
the environment, a different trajectory will in general pass through
that state, and as the environment is varied, still other trajectories
will result. With each change of trajectory, there will be, of course,
a change of "next" states.
The class of possible "next" states is, of course, also af
fected by the past history of the system simply because we consider
a state space trajectory as beginning somewhere, at some initial
event. In general, if the system passed through different initial
events in the past, they will occupy different current events at
present and therefore have different future events. For certain
classes of equations of evolution, however, the past history of a
system assumes an even more significant role, in that a complete
*To reiterate, the colloquialism "next" state is used solely
for descriptive purposes. Time is taken throughout to be a continu
ous, not a quantified, independent variable,
II.15
description requires not simply an initial event, but rather an initial
history.* The equations of evolution appropriate for the model of a
material with a memory, for example, requires an initial history.
In closing, we should note that if, rather than proceeding
from a set of hypotheses about the state variables of a system, the
particular form of the principles of physics, and the environment, we
had started from a particular physical situation and attempted to de
termine the elements of that particular dynamical process, then the
following steps would have been taken:
(1) A consideration of the class of physical systems and the
range of the physical theory necessary to describe the events that
can take place in the system.
(2) A system which is consistent with (1).
(3) An environment which is consistent with both (1) and (2).
(4) and based upon (1) (3), decide upon the minimum a
mount of information needed to predict subsequent events in a manner
such that at least for a certain range of the variables (i.e&, level
of operation of the system) this prediction (i) exists, (ii) is
unique, and (iii) depends continuously upon prescribed data.**
The decision mentioned in (1) amounts to a specification of
the type of phenomena under consideration and hence the relevant form
*An example of this situation is a situation in which the
relevant describing equations are differential equations with retarded
arguments. The interested reader is referred to El'sgol' (1964, 1966)
for a more detailed discussion.
**These three conditions are classically referred to as Hada
mard's conditions for a wellposed problem. The interested reader may
wish to consult a text in partial differential equations (e.g. Garabed
ian, 1964; Courant and Hilbert, 1952).
11.16
of the principles of physics involved as well as those principles to
be used. Thus, from (1) and (2) onemay find which and how many of
the principles of physics lead to an independent set of equations of
evolution (as well as the conditions imposed by the surroundings).
The part of thesurroundings which will be  or may be  considered
alterable.isspecified in (3).
If the above steps are used to determine a dynamical process
for phenomena associated with continuousmedia, the following con
siderations result:
.() the principles of physics provide the governing equa
tions and their specific form.
(2) the state variablesprovide, together with (1), the
knowledge about an independent set of above equations.
(3) the past history and the initial state provide initial
datafor equationsof evolution.
(4) the surroundings and environment provide the boundary
conditions and body force terms for the equations.
Alternatively, the above four points may be viewed as a
.larification .of points concerning the necessary elements of a set of
hypothesesthat delineate a class of stability problems associated
with a particular dynamical process or a class of dynamical processes
from a still larger class of problems.
11.17
Addendum on the Generation of Perturbations
through Environmental Influences
The fate of perturbations superimposed upon solutions of rele
vant describing equations is indeed the main topic of stability analysis,
However, another important question which naturally arises is: "how
was the basic flow actually perturbed?" We, as observers of the sys
tem, are part of the surroundings. Consequently, in order to exert
our influences upon the internal fields in the system  in the form of
perturbations  we must be able to affect the interaction of the sys
tem with its surroundings. In this chapter interactions were divided
into two classes: (i) volume or body interactions and influences,
and (ii) surface or contact interactions and influences. They are,
therefore, the means by which we may perturb the existing internal
fields.
As may be inferred from the above, this question "how" pro
vides one link between the physics and the mathematics of hydrodynamic
stability theory. The relationship and the inherent difficulties may
be therefore illustrated most easily by the examination of a concrete
example. Therefore, let us focus attention upon the idealization re
ferred to as plane Couette flow. The corresponding physical situation
is a fluid contained between two infinite parallel plates separated
by a small gap(Figure II.1). The basic velocity field is generated
by translating the upper plate at a constant velocityU, parallel
to itself,
11.18
Suppose the region over which are stability question is posed
is that contained within the dotted lines on Figure II.1, The boun
dary of this region may be decomposed into the six parts which are
also illustrated on the figure. Two of these surfaces, namely W
and 31?4, are at the interface between the fluid medium and the solid
boundary. Another difference between Y and v)4 and the other sur
faces is that they are exposed to us the experimentalist, at least if
he. so designs the equipment. If we, as experimenters, decide to per
turb the internal field or fields through contact influences it must
be done on the surfaces R3 or 41 In particular, suppose that we
decide upon this approach and instantaneously pulse the velocity of
the upper plate from U1 to 1U and then back to MU .I Because
real actions always take a finite time interval to perform, we have
already an "almost" type of experiment, which we will generally
idealize as a pair of step changes, thereby introducing a certain
error, or difference, into our mathematical model of the experiment.
There are several other important features of this "almost"
experiment. The act of pulsing the velocity of the upper plate alters
che strain field along This, in turn, alters the stress field
along which in turn affects the velocity field throughout
Consequently, the velocity field within R has been perturbed by
altering conditions  the strain field or the stress field  at one
of the exposed surfaces. In this case it was the strain field along
) which was directly altered, whereas a pressure pulse in the case
of plane Poiseuille flow is an example of an alteration of the stress
field.
11.19
11.20
Difficulties, however, still remain in the details of our
"almost" experiment. Essentially they have arisen because our system
is: (i) an open system, and (ii) because we have direct control over
only two of the surfaces, namely, A 3 and I4" The difficulty is
"what are the conditions along )R and 1 ?" (Figure V.2). Al
thought the velocity field is specified directly on 1?3and DIR ,
all that is known about the velocity field on R and 2after the
pulse is that it must satisfy the describing equations at each of
their points. Because the basic velocity field does satisfy the
describing equations, the special case might be thought of where the
velocity field along 1?I and) is the basic velocity field. For
plane Couette motion, this can only be a "thought" experiment which
may be imagined while for some other flows, such as plane Poiseuille
flow, it might actually be obtained in practice if '1 and were
chosen carefully.
Only plane Couette flow has been discussed, and similar diffi
culties arise in analyzing other members of the class of parallel flows,
For still more complicated flows the situation is still more compli
cated, for as one attempts to develop a tractable theoretical model
that is useful in analyzing a given experiment, more and more dis
crepancies can arise. Thus, as we investigate the stability of even
this simplest class of basic flows, it is well to keep in mind that
the stability investigation is above all a "Gedanken" experiment.
11.21
(a)+t
(b) t=t0
(c) t=to+e
 I I 
(d) trt1>to
FIGURE 11.2. PULSED ALTERATION OF STRAIN FIELD
II.22
11.2. Constraints upon Possible States
Of all the states, state space or trajectories, in a given
state space, only a limited number yield identities upon insertion
into the describing equations; that is, only a small subset of states
or trajectories are solutions of the operator equations. A state
that does satisfy the basic equations is termed an admissible state
or an admissible trajectory, as it is a solution of the steady or in
steady equations. The equations of evolution may thus be said to
constrain the system, being in fact sometimes described as governing
equations. From a consideration of the general form of a dynamical
process associated with a continuous medium, two different classes
of constraints may be distinguished, namely, internal and external
constraints, of which the basic equations are of the former class
and the body forces couples,and, the boundary conditions imposed on
the system (reflecting the influence of the surroundings) are of the
latter class. We hasten to add three points: first, that the
division is artificial because the body forces and couples are present
in the basic equations; second, because the boundary conditions are
also in an integral formulation of the basic equations, and third,
that our use of the terminology of internal and external constraints
as distinct follows that of, for example, Noll (1959). Our usage
of the qualifiers internal and external is, however, clear, and the
notion of the term constraint is equally transparent, for constraints
follow the class of admissible states and they serve to select the
next state or states, depending upon whether or not uniqueness prevails.
11.23
II.2.a. Internal Constraints
Within the class of internal constraints, there are also two
major subdivisions: those arising from the fundamental principles of
physics, by which we shall always mean classical physics, and those
arising because of the nature of the material within the region P ,
of interest. In the first subdivision there are essentially six of
these fundamental principles of physics:
(1) Balance of Total Linear Momentum
(2) Balance of Total Angular Momentum
(3) Balance of Total Energy
(4) Balance of Total Mass
(5) Balance of Total Electric Charge
(6) Balance of Magnetic Flux
These principles  or statements  when expressed in a mathematical
form yield the operator equations of evolution (in more standard but
inverted terminology, the governing equations) which must be satis
fied by the admissible states. The mathematical representations of
these principles may all be placed within the framework of the general
equation of balance, i.e.,
Rate of Accumulation Net influx of the specified
of a specified quantity quantity through the surface(
in a given control volume) bounding the control volume
+ Rate of Generation of the
specified quantity within (II.1)
the control volume
Conversely, these principles provide the specific items to appear in
the braces. In itself the equation of balance is nothing more than a
11.24
bit of common sense; it is thus only a skeleton  the riesh is
supplied by the principles of physics.* To be embarrassingly speci
fic, the principles are, in this context, what the flux of the
specified quantity is and how the specified quantity may be generated
within the volume. To reiterate, the mathematical representation of
the principles of physics, in the form of equations of balance, re
stricts the class of all possible next states and delineates the
class of admissible states.
Among the various possible ways of interpreting this general
equation of balance we have selected the viewpoint provided by con
tinnum mechanics.** However, rather than the usual approach taken
therein of dealing with body manifolds*** (see, e.g., Noll, 1959;
Truesdell, 1960), we choose to focus our attention upon a certain
region of physical space  the control volume, b. Furthermore,
specific representations of the principles of physics are viewed as
plausible postulates, rather than hard and fast laws,
*In speaking of this equation, Truesdell (1960) says, "This
statement is sufficiently general in itself; it doesn't say anything
ac all."
**In that we will use its defining concept (to again borrow
the words of C. Truesdell) of a generalization of the "stress hy
pothesis of Cauchy"  or in our terminology, "the equipollence
hypothesis."
***In this way we have avoided any questions dealing with the
"actual" structure of the body  that is, about the connections be
tween "material points" within the medium  which are necessary
hurdles to overcome in defining the mathematical operations inte
gration over a set of material points. We feel that the approach
we have taken is a more practical and a more convenient  and
further, one that is still consistent with the fundamental goal of
continuum mechanics  that is; to explain, represent, and predict
gross phenomena in a macroscopic portion of physical space.
11.25
Thus, for purposes of description at the outset, we are inter
ested in phenomena occurring in some compact region, b, of physical space.
We adopt the viewpoint that the phenomena occurring fall within the
range of physical theories referred to as continuum mechanics (so that
the local physical properties may be specified by their associated phy
sical fields), and we assume that what we call an equipollence hypothesis
may be used to characterize internal interactions* within the region of
interest, b.
Let us denote by q) a "typical"** physical field (of the per unit
()
volume nature), which is used to describe the phenomena of interest. Let
a continuous medium,*** 33 occupy the spatial regionPBCE (physical
space) at time t, and letdu denote the surface in ET coinciding with
the elements of the material surface O3 bounding the body 13 at time t
(see Figure 11.3). Further, suppose that the influence of the universe
exterior to ID which we will assume is also occupied by a continuous
medium, upon the rate of change of the field 7T)may be characterized
specifying: (i) a surface influence, I of the exterior to 3 that
.ts upon 12 by acting only upon (and being defined upon)3 B and
ii) a body influence that acts through 13 (being defined per unit volume),
*This is not to say that one must consider the media which are the
subject of continuum mechanics to have no molecular structure. Rather one
avoids the explicit use of any molecular structure they may have, taking
iL into account, loosely speaking, only implicitly through its macroscopic
effects. To this end, molecular theories may and do play a qualitative
role in continuum physics.
**The quotes are to remind the reader that the principles of phy
sics are unified by eq. (II.1).
***By a continuous medium we mean any matter which, as far as the
phenomena being considered areconcerned, falls under the range of the phy
sical theory called continuum mechanics (cf., our previous remarks on the
range of continuum mechanics above).
11.26
, i i=h(, t; n2);h3b
\
I
/
FIGURE 11.3. NOMENCLATURE
n2
11.27
Because at time t, 3 occupies the region T) we may view J ) as
being defined on)CEBbyl ( ), and +t )as defined in
at the instant t. Further, suppose that at time t an observer
selects an arbitrary control volume, b as indicated in Figure
11.3. This control volume, bounded by the mathematical surface b ,
need not be contained entirely within, nor need it contain entirely
within itself, the spatial region > ; but, again, the matter within
is to be viewed as a continuous medium. Thus, the influence of the
surroundings upon bB may be characterized by: (i) a volume in
fluence per unit volume,f[r and (ii) a surface influence,tLb
defined over the surface 46B.
This surface influence, L, unlike the surface influence,
I ', which is imposed by some external agent  reflects the in
ternal interactions between the various material points.* In the
special case of the balance of linear momentum this hypothesis is
usually referred to as the stress hypothesis. In its general con
text, we refer to it as the equipollence hypothesis, for it asserts
hat the interaction of the material points external to a4(D
a~ time t upon those within (c ) R )3 at time t is
equipollent  as far as its contribution to the temporal rate of
*It is of interest to note that Poincare in his writings on
the philosophy of science argued against the use of the stress hy
pothesis for exactly this reason. He felt that the concepts of physics
should all be real  that is, observable  and not owe their exis
tence to a hypothesis which may not be directly tested. The reader
is referred to Synge (1960, pp.45) and Truesdell and Toupin (1960,
p.229) references contained therein for further discussion of this
"operational" philosophy of science and its ramifications.
1128
change of '() in 6(B is concerned  to a field ib( )defined on the
surfaceoOE)B. Further, .it is assumed that this field is dependent*
only upon the position on the surface, time, and tangent plane to the
surface at that point, that is,
A
where FXIEand is the normal to this surface pointing away
from bjs.
By the use of the equipollence hypothesis again, this time
to characterize the internal interaction of material points in ced
at time t upon material points of tlIn ,** we obtain the field
_ (,)defined on the surface Db s Again,V 1 is assumed to
depend only upon position on the surface, time, and the tangent
plane to the surface at that point, that is,
A
where X _bl and is the normal to this surface
*We have followed the conventional presentation (e.g., Noll,
1959) by enumerating the dependence of hbupon a single geometric pro
perty of the surface, the normal. This is often accurate assumption
at physical boundaries, but the recent work of Scriven, Eliassen, and
Murphy (Eliassen, 1963; Murphy, 1965; Murphy and Scriven, 1968) demon
strates that other geometric properties enter if the boundaries of
the control volume are of high curvature. These points must therefore
be kept in mind when selecting a control volume.
**It should be mentioned that because of the conventions we
have chosen, any internal interaction of matter exterior to o 
other than material points in 1 at time t  upon material in 6b(1
ar time t must be accounted for in the volume interaction term. This,
however, is consistent with the usual convention in electromagnetic
theory (see, e.g., Elsasser, 1956; Sommerfeld, 1952; Toupin, 1956).
11.29
pointing into P&6 For convenience, we define the following fields,
t) I~ T5(7)
(TT~ ) ) (~L&YT M6 V
0 ^.; ^s
((J)
so that equation (II.1) when applied to the region, b  subject to the
above decompositions of influences and interactions  becomes, at time t,
x +b
br}3 1 (11.2)
In words, the above equation and proceeding discussion may be collected
in the form
Rate of Accumulation)
of the quantity C
I within the control
volume b
Net influx of ()
= due to internal 4+
)interactions sub
ject to equipollenc,
hypothesis
Net increase of )
& e) to the
surface sources
due to surround
ing actions uponN
the body 3
Rate of Generationl
+ of within b
due to volume in
Ifluences
Here, x and 3 have been used to indicate dummy integration variable
along the various surfaces (see, Figure 11.1) and
: element of volume in b,
S : element of surface area on b ,
Bj : element of surface area onD .
11.30
The term
deserves further comment, for, although an analogous term commonly
appears in the balances taken in electrostatics, it is scarcely seen
in the balances taken in other fields of continuum mechanics. This
term plays a versatile role in the balance equation, depending upon
the physical situation at hand; at times it may represent the contri
bution from a flux which is presented at the boundary of B (possibly
by the dictation of some external agent), represent the contribution
from a surface source, or both of these. As an example of its rep
resentation as a surface source term we may imagine a mass transfer
problem in which a chemical reaction is occurring only at the sur
face of a catalyst particle and we have selected our control volume
to include some of the catalyst as well as some of the surrounding
fluid mixture. In this type of physical situation the term
J ( ) '
would represent the rate of generation of material within the con
trol volume .due to chemical reaction at the surface of the catalyst
particles. On the other hand, physical situations where it repre
sents the contribution from a prescribed flux field are most easily
found in heat transfer problem, e.g., a solid whose surface is com
pletely covered by heating wires. It is of importance to note that
this term does not account 'for all the flux of heat across the
11,31
surface of the solid  just that which is externally prescribed,
If the volume and surface integrals of eq. (11.2) are placed on
different sides of the equation and  is brought within the in
dt
tegral, there results
b 1u (11.3)
We now invoke a general form of the classical tetrahedron
argument* whereby it is demonstrated that the internal interaction
at time C between a material point at x>l and the material points
in U9  as given by (x C'  is the value of a linear oper
ation on the vector VT The argument goes as follows. Suppose
that the control volume is selected such that lr}%)r. G." the null
set  that is, b is either completely within or completely external
to B. Further, suppose that the control volume, b, is selected to
be a small tetrahedron of volume, Ub( and bounded by a surface of
area, A.0 which does not contain any points of )B (see, Figure
11.4). Then the mean value theorem for volume integrals is invoked,
~''i:h leads to the equation
At this juncture, the classical derivation divides this re
sult by'A;c6 and takes the limit of both sides as AL>O ; thereby,
*The reader unfamiliar with the following development is re
ferred to the discussion of the stress tensor as given by Truesdell
and Toupin (1960); Eringen (1962),and Aris (1962).
11.32
THE CLASSICAL TETRAHEDRON
FIGURE II.4.
II.33
obtaining the classical result
where Lim = 0Q has been used
By following the classical development, this surface integral may be
decomposed into four contributions,
(11.5)
th
3
where x = dummy integration variable indicating points on i
L
coordinate surface
= element of surface area .on side of tetrahedron which
is perpendicular to ith direction
= unit vector in i coordinate direction.
In the limiting situation as the tetrahedron shrinks to a point (see,
e.g., Cesari, 1956) it is assumed that
P 5 10,1
11.34
or
where Kt denotes the normal to the slanted surface of the tetrahedron
v4
and is an element of surface area on this face. If one substitutes
these relations into eq.(II.5), the following equation results
i Yh(T) T) tJ 0eVjo^ jjt
At this point the classical derivation draws the implication
from this equation that
where x*denotes some point within the infinitesmal volume. Now the
quantity, ) is defined by
which allows us to rewrite eq. (11.6) as
This seemingly innocent relation is actually one of the most pro
found relations of continuum mechanics. In essence it implies that
it is not necessaryto specify the value \ 5 for each of all
the possible mathematical surfaces that may be constructed through
X rather all these possible values may be characterized by the
specification of a quantity, which is independent of the
11.35
mathematical surface  thus, loosely speaking, this relation
separates the mathematical geometry (as embodied ingY\) from the
physics (as embodied in ).
Because the e 5 occurring in the definition of (T> de
pend, at time T, only upon x the following identifications* may
be made:
and
L
By combining these identifications with eq. (11.7), one is led to
at each instant t. If eq. (11.9) is substituted into eq. (11.8),
one obtains the representation
Ihis representation may be viewed in either of two ways.** It may
be looked upon as defining a linear operator (p C .) 7 at the place
x at time which operates on n to yield the local influence\ i C ,
*The reader looking for analogies is referred to similar
identifications made in developing the concept of a stress tensor
(see, e.g., Truesdell and Toupin, 1960; Eringen, 1962; Aris, 1962).
**The reader is reminded of the distinctions made in finite
dimensional and abstract spaces between the differential, the
derivative, and the two possible interpretations of the derivative
(please see Appendix B, section II.l.a).
11.36
which is a 2.formin E3 inthemathematicaltheory of differential
forms (see, e.g., Buck, 1956; Guggenheimer, 1963; Spivak, 1965;
Willmore, 1959). An alternative interpretation is obtained by viewing
it as a definition of a field C(.TC'V  that is, a global quan
tity  at time t associated with the quantity y*) whose values at a
point XtB are the linear operators yielding local values of the
associated quantity, 1 ]. In the classical context of Cauchy's law,
where T .') assumes the role of andtC!*) assumes the role of
Vi ( )T' eq. (II.10) is referred to as the fundamental theorem
of Cauchy which asserts the existence of a stress tensor field,
T7 ) which characterizes the local stress vectors, ( xTv).
In its more general context we refer to it as the generalized funda
mental theorem of Cauchy,
Within this general context, the primary utility of the gen
eralized fundamental theorem of Cauchy  and therefore, also the
classical theorem  becomes strikingly apparent. Because YC ) is a
2form in E3 the analogs of the classical integral theorems of vector
analysis may be applied to it. The classical divergence theorem may
be applied to the surface integral of ) with an impact upon con
tinuum mechanics at the deepest level. Before proceeding with the
development, however, it is well to list the five key points which
led to eq.(II.10); (i) the form of the general balance equation,
(ii) the equipollence hypothesis, (iii) the assumption that the value
of r 7at a point on )b depended only upon the point and the tangent
plane to b at that point, (iv) the fact that control volume b
11.37
b contained no points of the surface B where there was an imposed
surface traction, and (v) the assumption that the passage to the
limit in the generalized tetrahedron argument is a valid procedure.
These points must be kept in mind as we now make use of the repre
sentation of n as a 2form in E
(7)
These points must be kept in mind as we now apply the volume
to surface integral theorem* (see, e.g., Bergman and Schiffer, 1953;
Buck, 1956; Kellogg, 1929; Sommerfeld, 1952) by selecting a control
volume such that b == and by substituting eq. (II.10) into
eg. (II.3)to arrive at
Then applying the volume to surface integral theorem under suitable
conditions on 3 and 30 (see, e.g., Kellogg, 1929)  for example,
providing ) is a continuously differentiable field and b is a
regular surface**  eq. (II,11) may be expressed solely in terms of
volume integrals, that is
m f) ^,t( (T ()
On the other hand, if we impose weaker conditions on the field ( ,
namely, that it may be discontinuous on a set of volume measure zero,
*This theorem is alternatively referred to as Divergence
Theorem, Gauss Divergence Theorem, Ostrogadskii's Theorem, Green's
Theorem, and various permutations of these names. Not wishing to
enter the controversy we refer to it simply as the volume to surface
integral theorem.
**Kellogg (1929) defines a regular surface as one which may be
decompressed into a finite number of surfaces which have a continuous
normal.
11.38
then a suitably revised formof the volume to surface integral theorem
(see, e.g., Bergman and Schiffer, 1953, p. 363; Truesdell and Toupin,
1960, p. 526) leads to the equation,
where ( is the k+ region of volume measure zero on which dp is
discontinuous; I is the jump in the field D across ( is
(t 1 (M (kj. ) T is
the outward normal to positive when pointing from )0 to G in
relation to the convention chosen for I) It should be noted
that eq. (11.12) is a special case of eq. (11.13) when the field is
continuous throughout b. We have, in deriving eq. (11.13), allowed
the fields to be discontinuous on some surface or surfaces. Although
it may be argued that these discontinuous fields are not in the true
spirit of continuum mechanics, they are frequently convenient and
sometimes necessary idealizations; indeed, they are necessary if one
wishes to analyze phenomena such as shock waves within the realm of
continuum mechanics. Furthermore, just as surfaces of discontinuity
have sometimes proved convenient, so also have the idealizations of
lines and even points of discontinuity.* We do, however, limit our
development only to cases of surfaces of discontinuity.
*Although these idealizations are abundantly used in the
theory of electrostatics, they are also used in many other
disciplines. In mass:transfer, for example, the model of diffusion
from a point source to a moving stream is just such an idealization.
Often these idealizations have experimental significance; the analysis
of the example cited is used to determine "eddy diffusivities (see,
Sherwood and Pigford, 1952, pg. 42).
11.39
The .quantity' maybedecomposedinto twoparts: a convective
kT)
contribution due to a transport of the physical properties associated
with material points as the material points themselves move across the
surface b into b and a diffusive contribution due tothe transport,
across o into 6 of the values of the physical properties associated
with material points whichneednot cross b that is,
The classicalandcurrently modern theories all suppose that the dif
fusive contributions are due only to physical properties associated
with material points in a small neighborhood of b .* The plausibility
of this classical supposition finds support from the molecular view
point under a wide range of circumstances because of the effectively
short distances over which intermolecular forces act. The extent (in
terms of distances between interacting material points) and the
strength (in terms of how strong the interaction is between the
material points) of the diffusive contribution depends upon the na
ture of the medium which, in turn, is mathematically described by
making a constitutive assumption. Thus, ) generally takes on
a different form for each material or class of materials. On the
other hand,dC depends only upon the density 0, kinematical quan
city, the velocity of the material point instantaneously occupying
*This, in fact, was one of the reasons we choose to charac
terize the internal interactions of Be o upon bG)B as being
volume forces, e.g. B is a dielectric and surroundings is some non
polarizable medium.
11.40
the place x, and the particular physical property in the multilinear
combination,
If this relation  namely that Ptxk)v(x,1) is the linear operator
on 4J whose value is the convective contribution mentioned above 
J)
is substituted into eqs. (11.12) and (11.13), one obtains
t[V ^,( ^0)*^ , (11.14)
and
j kk. "oo ( 1 1 1 5 )
Because these equations have been developed for an arbitrary
control volume, b, the familiar assumptions and arguments (see, e.g.,
Truesdell and Toupin, 1960; and similar arguments in the calculus of
variations used to obtain the EulerLagrange equations, e.g., Courant
and Hilbert, 1952) used to derive local relations (differential
equations) from global relations (integral equations) may be applied
:o obtain the local relations
and
4Q (11.17)
11.41
Each of the fundamental principlesof physics may be placed
within the framework developed in eqs. (11.14) to (11.17) by different
interpretations of the quantities p 4 They are
cataloged in Table (11.1). For a given physical system, the number
of equations of this form which constitute an independent set of
equations of evolution depends upon the range of the physical theory,
and both the range and nature of activity within the associated dy
namical process. Thus, an explicit expression p by an enumera
tion of its independent variables, is directly related to what
constitutes this independent set  that is, the nature of the
medium has an effect in determining the state variables.
II.2.b. Constraints Based Upon the Nature of the Madium
An explicit expression for the constraints based upon the
nature of the medium  that is, a constitutive assumption  serves
a twofold purpose, namely, it describes the significant mechanisms
of diffusive transport, and it relates the constructs, to
he physical observables of a system. Moreover, the selection of
explicit expressions for has three major effects:
(i) It quantifies our physical intuition* about the inter
actions (or couplings) between various fields;
*The interested reader is referred to Birkhoff (1960) for
discussion of the role played by intuition in science, in general,
and in hydrodynamics, in particular.
I 4 C 0
0 0 U1
0 4 N
0 10 0
M 0
b0 4
0 q *H U
+ H *H 0
N O4
"o
0 w0
0
mn
So
41
4J I
0 0
*H
, ,
41
w
0 u
S. 3t
w 0
o W
I aj 0 1
, p
0) b
>W
I
r4
. 0
11.42
1 w
i *'i
0
0
r1 *
4J
0 u
,4
0
a4
>
r4 4
0
4s
> 0
4 *H
4J 
AC
'N
e
a *He
: 4J rI
0 *Hr 0
c t >
.r U
W m
C 0
r
CiL CiL4
0
0) 3
o W
(V W
w n
W
01
o 0
, o
0 0
I W
0 >{ 0 4Ji
6 0 (3
0 4J C
w4
II.43
0
o I
*OH
0 IW C
) ,*4 4n M4 w 4 14
H Wu aw) a) ( rH t C 0r
4 H I 4 r. I= i r C a) 4 C 4x 3 0
CO a O ** o 4 u o o
o) WU1 Y tfh ( 4' (0. W n 0i >i C s C^ 3>''4141o
SW ** o m i' 0 0 %
>l ~ ~ !q l (l  J 41 I, t" 01% l) ^ (
4)O 0 T fl. o W T0 4 ) > o To 44o
4 C ZU rVI aW F= 4 0 41 W4i 4HC(
C m w r 00 44. 00 C 4
0 H u i i 0 0 et coWT4
0
4 W 0 I ** i4
,AWC 0 Ha)04 I W0
L 4 U C 3 (U *.4 H 2
W *H 1 l r CW t 
'1 n > 4 tos o > 4
cu ii ni p w co _z co 4 a)> U En pcs ^W
(U W U0 U= *r c
U *H 0 w& X w w *0 *H a) H 44
W'4 u 41
 ) 4 P 4 4 4
ow > w 0 >
M H W r 3 uo 41 0 W
r) w ci C a) rC ,.
oO Z) w Q 0 O r* 0 Q) (0
1 4CZ VC 0 A Q 0o *0
'4 I 1 W
P ^ro 1 Hj3 ct4 nE l
Q) W3441 X 00
0 ~ 0 W 0 Q **OCH
u Hl) C C) 00 rA 41
0 *Uo 0 5 Q0
W0 :3 L r 4 0 W ca
Cr 000 0
0 Lo 43 >1 p
u Q
) 0 i 0 CO( UC
I1 : ) 1 1; El 44 C a
> H4) $ U 401
O w 4 C.) W (1) 4W O (1)
(ao *C (1)
p( 0 0 )
M n V jo i lz. o1 4 >,e & tn*
314 W.Hz4
41Q (U 4 (1)^ <
41 4 H 4 0
0 n
U *w ci ) ** 0 ** 0
,C > c1 /^ i c 41 0,
4J <1 L 41CO
(4 1 H 1M
0 0 c 0
P14J
0) 4 4
u .r 0 P C 0 .4
C 4 00 C w to4
M & 0 W> w U CU
CL 4 co <^ 00 0o 4
0 0 OO
0 rl = 0 4
441 A41) E(4w u1qw0 0U
11.44
(ii) it restrictsattention toa specific ideal material or
class of ideal materials;
(iii) it introduces into the formulation various phenomeno
logical coefficients as parameters describing the "level of operation"
of a dynamical process within a given class of ideal materials.
Effects (i) and (ii) are complementary in that an explicit
expression for ) will exclude possible couplings between a set
of fields q within the constitutive assumptions. To be sure,
the velocity field is linearly coupled to all other fields by its
appearance in the convective terms of the field equations of the
dynamical process, so that it occupies regal position within the
class of all physical fields, as does the density field. The three
effects may be illustrated by considering the constitutive assumption
for an incompressible newtonian fluid, namely,
where T is the total stress dyadic, r is the viscosity coefficient,
and b) is the deformation rate dyadic. This expression conveys the
idea that the material depends upon the deformation rate and does so
in a linear manner; therefore, we have focused attention upon a
certain class of ideal materials  ideal because a specific real
material may behave as indicated under a certain set of circumstances,
and yet behave differently under;a different set of circumstances.
Furthermore, if one places the restriction upon that it is a
constant, then this class of ideal materials does not have the
possibility.of any explicitcoupling between the other physical
fields, e.g., temperature, composition  except, of course,
through the coupling of..the velocityfield with all others in the re
maining equations of balance. This parameter, t. may be used as an
example to demonstrate the third effect, inthat a specific value for
it characterizes the strength of response to a certain stimulus 
within the class of incompressible newtonian fluids. In some de
tail, if we considered two different members of this class, A and B,
with their corresponding viscosity coefficients such that
then they will have different values for the stress field for the
same deformation rate field and vice versa; thus, the stress field
may be viewed as the stimulus or the response and the formation
rate as the converse. If the stress is viewed as the stimulus, al
though the mechanism for the response; as characterized by both the
form of ( and its arguments, is the same for fluids A and B, it may
be said the A reacts.morestrongly than B. In a sense, then, a value
of f. thephenomenological coefficient of this ideal material 
determines a "level of operation" within the class of incompressible
newtonian fluids.
This "level of operation" is often expressed in a di
mensionless form, as again may be illustrated by considering the
example of incompressible newtonian fluid. In particular, if this
constitutive assumptionis substituted into the balance of linear
II.46
momentum, which then becomestheNavierStokes equation,
which may be made dimensionless by taking
L L
where u. i ,7Tj p C are the dimensionless variables, thus
giving rise to the dimensionless parameter, the Reynolds number.
This parameter, a number, reflects "level of operation" of a material
within the class of ideal materials mentioned above. It may be used to
do much more than that. Indeed the Reynolds number may serve to in
dicate the "level of operation" of a system, an ability obtained by
including characteristic geometrical kinematical, as well as the
physical properties, of the material. The reader wishing to pursue
the subject of similarity and modelling in greater detail is referred
to the delightful little book by Birkhoff (1960) and references
therein.
II.2.c. External Constraints
The interaction of the system with its surroundings comprises
the external constraints. External constraints may be of three basic
types: (i) those due to configuration or kinematical conditions (e.g.,
conditions on the location or movement of the bounding surfaces);
(ii) those due to the interaction with or influence from contiguous
surroundings, in the form of boundary conditions; (iii) the influence
of distant surroundings upon the system, in the form of body forces
acting throughout the system. The last have already been accounted
for, having appeared as source terms in the general balance equations,
and whereas boundary conditions may sometimes describe a genuine coup
ling between the system and its (contiguous) surroundings and some
times not, body forces always influence the system, yet are themselves
uninfluenced by the system and its behavior. The division into in
ternal and external constraints is, to be sure, somewhat artificial,
and depending upon whether the equations are formulated and con
sidered in integral or differential form, it may seem more or less
artificial, depending on one's taste. Thus, as a set of partial
differential equations, the field equations require appropriate
boundary conditions that appear separately, yet the body forces occur
in the equations themselves. Conversely, in integral form the basic
equations incorporate both external and internal constraints, both
body forces and boundary conditions on an equal footing. As a final
point with regard to all three kinds of external constraints that is
perhaps worth recalling, they may each be subdivided into those con
ditions that are susceptible to control ("pure influence") and those
that are not (an interaction, or coupling).
The boundary conditions may be made dimensionless in the same
manneras the partial differential field equations. But whereas the
dimensionless numbers that appeared there measured, or were at least
indicative of, the level of operation of the system, the dimensionless
parameters that arise from nondimensionalizing the boundary conditions
describe the level of communication between system and surroundings.
11.3. Perturbation Problems Associated with Dynamical Processes
Because stability theory is the topic of this dissertation
and because stability problems are concerned with the fate of per
turbations, it is appropriate that the position occupied by stability*
problems within the class of perturbation problems be considered.
Moreover, the vantage point constructed in Section II.1  that is,
the concepts of a state space and a dynamical process  will allow
us to distinguish the classes of perturbation problems in a straight
forward manner. Once the class of stability problems is clearly
distinguished from other types of perturbation problems, we will
concentrate upon a specific class of dynamical processes (in Section
11.4), those for which the equations of evolution are of the form
of the general balance equation of Section 11.2.
In this classification of perturbation problems associated
with dynamical processes, it is advantageous to view the dynamical
process as an operation, (DP), which transforms the present state of
a system into a succeeding state, both states, of course, belonging
to the state space associated with the system. Among all the ele
ments of this state space, a particular set of elements may be dis
tinguished, those that are transformed into themselves by the
dynamical process  that is, they are the set of fixed points of
the operation (DP). Any element which belongs to this set is referred
*By stability problems we mean not the classical problems of
stability with respect to boundary perturbations, or with respect to
perturbations in the body force field; rather we mean the stability
of a basic state of a particular dynamical process (see, e.g., Lin,
1955).
II.49
to as a basic state of that dynamical process. One type of pertur
bation problem is concerned with perturbations of the basic states
of a specific dynamical process, while the other is concerned with
perturbations of the operation  the dynamical process (DP) 
itself. In the first type of perturbation problem, which we have
taken as our prototype stability problem, is usually centered upon
a particular basic state without regard for the method in which
this basic state will be perturbed. On the other hand, in the
second type of perturbation problem  that is, perturbation of the
dynamical process  the method of perturbation is of primary concern.
II.3.a. Perturbation of the Dynamical Process Itself
Although the main topic of this dissertation is with the
perturbation of a basic state of a dynamical process, we would like
to first briefly discuss the class of problems which involve the
perturbation of the dynamical process itself. This class of per
turbation problems, which includes perturbation of boundary con
ditions, perturbations of the shape of the physical boundaries,
perturbations of the external influence fields, and perturbations of
the operator in the constitutive assumption, is currently a topic of
interest in both mathematics and engineering (see, e.g., Van Dyke,
1964; Seeger and Temple, 1965).
In its broadest sense, the question raised in this type of
perturbation problem is: if one is given or can easily find the
trace of a particular dynamical process, (DP), as t is varied in
I
11.50
event space, then how maythe trace of some other dynamical process,
(DP )  which is close in some sense to (DP )4  be expressed in
terms of a modification of that of (DP ). Perhaps the most straight
forward modification one could imagine is a series expansion in terms
of a scalar perturbation parameter. Indeed, this is the technique
proposed by Poincare in 1892  when such a modification was not so
obvious. Essentially Poincar4's method consists of expanding the
dependent variables (e.g., the state variables) of the relevant
governing equations in a power series in the perturbation parameter;
substituting the series into the relevant equations; equating terms
of similar powers in the perturbation parameter (powers of a scalar
real variable are linearly independent);then solving the resulting
system of equations successively. The equation corresponding to
the zero power in the perturbation parameter is the governing equation
for (DP ). It is found that for a sizeable class of perturbation
problems  referred to as regular perturbation problems  this tech
nique provides a valid representation of the trace of (DP ) based
upon that of (DP ) throughout physical space.
However, there exists an important class of dynamical pro
cesses in mathematical physics for which this technique is not ade
quate. For example, if the governing equations are differential
equations and the scalar parameter appears as a coefficient of the
highest order derivative, then Poincare's technique does not yield a
valid representation throughout all of physical space. The most fre
quently used techniques of handling this type of perturbation problem
11.51
for differential equations are: (i) Prandtl's method of inner and
outer expansions, and (ii) Lighthill's method (see, e.g., Van Dyke,
1964). In Prandtl's method, series expansions are obtained that are
for valid (approximate) representations of different (hypothetical)
dynamical.process occurring indifferent regions of physical space,
and it is assumed that there is a region of physical space in which
both expansions are valid and can be matched. The result, if suc
cessful, is a uniformly valid approximation to the trace of the
actual dynamical process. Prandtl's original development was based
solely on physical intuition and certain orderly arguments in the
NavierStokes equation. It was half a century before any degree of
mathematical vigor could be given to Prandtl's arguments, despite
the pioneering efforts of Lagerstrom, Kaplun, and Cole (see, e.g.,
Lagerstrom and Cole, 1955; Kaplun and Lagerstrom, 1957; Kaplun,
1954), many open questions remain. In Lighthill's technique, on the
other hand, the dependent as well as the independent variables
(e.g., positions in physical space) are expanded in terms of a per
turbation series in a third set of variables. The choice of ex
plicit relations for this third set of variables is subject to a set
of guidelines that lead, under certain conditions, to a uniformly
valid representation of (DP) These guidelines provide a systematic
scheme for finding a solution by a change of variable  as evidenced
by the expansion of the independent variables as well as the de
pendent variables.
11.52
However, the validity of using the various techniques of
handling perturbation problems of this type rests upon the answer
to a more basic question: "Is the problem correctly set (or well
posed)?" This question is a major consideration in determining the
elements of a dynamical process as well as the validity of any of the
above perturbation solution techniques. It is concerned with the
validity of a mathematical representation and thus is inherently
based upon the belief that if the mathematical representation of
some stable (in the physical sense) physical phenomena is suffici
ently well formulated, then the presence of small errors in the
prescribed data should result in small changes in the solutions.
On the other hand, if the physical situation was physically stable,
yet the mathematical model did not behave in this manner, then one
would conclude that the model was badly formulated. Consequently,
if we wish to relate physical instability to the instability of a
dynamical process, it is first necessary that the dynamical process
provide a valid description of the physical situation. By valid, we
mean in the sense that the dynamical process provides a mathematical
formulation for which a solution (i) exists, (ii) is unique, and
(iii) depends continuously upon prescribed data. In order that we
may attach some physical significance to any stability analysis of
a dynamical process, we must suppose that it is a valid represen
tation of the physical situation.
