Title: On Liapunov's direct method
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Title: On Liapunov's direct method a unified approach to hydrodynamic stability theory
Physical Description: 1 v. (various pagings) : ill. ; 29 cm.
Language: English
Creator: San Giovanni, John Paul, 1941-
Publication Date: 1969
Copyright Date: 1969
 Subjects
Subject: Lyapunov stability   ( lcsh )
Stability   ( lcsh )
Chemical Engineering thesis, Ph.D   ( lcsh )
Dissertations, Academic -- UF -- Chemical Engineering   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Thesis: Thesis (Ph.D.)--University of Florida, 1969.
Bibliography: Includes bibliographical references.
Additional Physical Form: Also available on World Wide Web
Statement of Responsibility: by John Paul San Giovanni.
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Bibliographic ID: UF00097778
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 002201817
oclc - 36517811
notis - ALE1738

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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 N-dimensional 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 Newton-Raphson-Kantorovich 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
Co-Chairman: X. B. Reed, Jr.
Major Department: Chemical Engineering Department


The objectives of this dissertation are two-fold:

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 physico-chemical 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

psuedo-metric space with the psudeo-metric being a continuous posi-

tive-definite 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

well-posed, 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. Well-posedness,

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 well-posedness in the

sense of (i)-(iii). There is a larger sense in which stability

theory could become sterile and well-posedness 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. Well-posedness 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 Tollmein-Schlichting

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

well-recognized 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 time-dependent 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 so-called "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 steady-state 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 positive-definite 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 negative-definite. The essential ingredient for stability is,

to summarize, a generalized energy that is positive-definite yet

with a negative-definite 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 physico-chemical 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 scalar-valued 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 negative-definite. 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 open-ended, bowl-shaped surface as indicated in Figure I.la, and

in particular that is the graph of a single-valued 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 weil-known result of elementary topology that a closed

curve in a plane divides the plane. Mathematically, the criterion

of negative-definite 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

dif-icul, 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 now-famous 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 single-valued. 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 positive-definite 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 so-called 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








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1.28




1.29


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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 single-valued 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 well-posed 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 finite-dimensional. There are standard definitions of

the-different types of stability in finite-dimensional 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

infinite-dimensional. There also are available standard definitions of

stability in infinite-dimensional 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 of-the 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 is-nevertheless 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 N-tuple 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 N-tuple

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 N-tuple 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

N-tuple of fields -- one element corresponding to each of the local

state variables. Again, for convenience, one may look upon this

N-tuple 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 N-tuple. 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 commonly-used 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 single-component newtonian fluid may be
considered. In the model of isothermal flow the Navier-Stokes
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 Navier-Stokes 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 Navier-Stokes 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 non-zero 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 well-posed 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) one-may 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 the-surroundings which will be -- or may be -- considered

alterable.is-specified in (3).

If the above steps are used to determine a dynamical process

for phenomena associated with continuous-media, the following con-

siderations result:

-.() the principles of physics provide the governing equa-

tions and their specific form.

(2) the state variables-provide, together with (1), the

knowledge about an independent set of above equations.

(3)- the past history and the initial state provide initial

data-for equations-of 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

hypotheses-that 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 2-after 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) tr-t1>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.4-5) 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 U-9 -- 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
v-4
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^ j-jt

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 necessary-to 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.-form-in E3 in-the-mathematical-theory 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 Xt-B 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, ( xT--v).

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

2-form 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 2-form 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 form-of 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' maybe-decomposed-into two-parts: 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 to-the transport,

across o into 6 of the values of the physical properties associated

with material points which-need-not cross b that is,





The classical-and-currently 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 Euler-Lagrange 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 principles-of 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 two-fold 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 U-1
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











,- ,-
4-1


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



r-1 *
4J

0 u



,-4
0
a4-











>


r4 -4



0
4-s




> 0





4 *H
4J -

AC


'-N

e-


a *He
: 4J r-I
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) ( r-H 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 4-1 I--, t" 01% l) ^ (

4-)O 0 T fl. o W -T0 4- ) > o To 44o
4- C ZU rVI aW F= 4- 0 4-1 W4-i 4HC(
C m w r- 00 44. 00 C 4-
0 H u i -i 0 0 et coWT-4
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) r-C ,.
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 00-0 0
0 Lo 43 >1 -p
u Q





) 0 i 0 CO( UC
I-1 : ) -1 1; El 44 C a




> H4) $ U 401
O w 4 C.) W (1) 4-W O (1)



(ao *C (1)
p( 0 0 )
M n V jo i lz. o1 4- >,e & tn*




31-4 W.Hz-4
4-1Q (U 4 (1)-|^ <





4-1 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- r-l = 0 -4
441 A41) E(4w u1qw0 0U






11.44


(ii) it restricts-attention to-a 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 explicit-coupling between the other physical

fields, e.g., temperature, composition -- except, of course,

through the coupling of..the velocity-field with all others in the re-

maining equations of balance. This parameter, t. may be used as an

example to demonstrate the third effect, in-that 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.more-strongly than B. In a sense, then, a value

of f.-- the-phenomenological 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 assumption-is substituted into the balance of linear






II.46


momentum, which then becomes-the-Navier-Stokes 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

manner-as 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 may-the 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 in-different 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

Navier-Stokes 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.




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