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- https://ufdc.ufl.edu/UF00097778/00001
## Material Information- Title:
- On Liapunov's direct method a unified approach to hydrodynamic stability theory
- Creator:
- San Giovanni, John Paul, 1941- (
*Dissertant*) Reed, X. B. (*Thesis advisor*) Fahien, R. W. (*Thesis advisor*) Siekmann, J. (*Reviewer*) Walker, R. D. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1969
- Copyright Date:
- 1969
- Language:
- English
- Physical Description:
- 1 v. (various pagings) : ill. ; 29 cm.
## Subjects- Subjects / Keywords:
- Abstract spaces ( jstor )
Differentials ( jstor ) Equations ( jstor ) Mathematical integrals ( jstor ) Mathematical variables ( jstor ) Mathematics ( jstor ) Topological theorems ( jstor ) Trajectories ( jstor ) Velocity ( jstor ) Velocity distribution ( jstor ) Chemical Engineering thesis, Ph.D ( lcsh ) Dissertations, Academic -- UF -- Chemical Engineering ( lcsh ) Lyapunov stability ( lcsh ) Stability ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- 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 implications. These objectives are accomplished by a utilization of functional analytical techniques of modern mathematics. Regarding (1) we concern ourselves primarily with physicochemical systems modeled 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 significant 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 physical 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.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 1969.
- Bibliography:
- Includes bibliographical references.
- General Note:
- Vita.
- General Note:
- Typescript.
- Statement of Responsibility:
- by John Paul San Giovanni.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- 002201817 ( AlephBibNum )
36517811 ( OCLC ) ALE1738 ( NOTIS )
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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 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 BIBLIOGRAPHY A. ON LIAPUNOV'S DIRECT METHOD Antosiewicz, H. A. 1959, A survey of Liapunov's second method, in Contr. to Nonlinear Oscillations, S. Lefschetz (Ed.), IV, 141. Antosiewicz, H. and Davis, P. 1954, Some implications of Liapunov's conditions of stability, J. Rat. Mech. Anal., 3, 447. Aris, R. 1965, Dynamic Programming, Coll. Lectures in Pure and Applied Science, no. 8, Socony Mobil Oil Co., Dallas, Tex. Brayton, R. K. and Miranker, W. L. 1964, A stability theory for nonlinear mixed initial value problems, Arch. Rat. Mech. Anal., 17, 358. Cartwright, M. L. 1956, The stability of solution of certain equa- tions of fourth order, Quar. J. Mech. Appl. Math., 9, 185. Chetaev, N. G. 1961, The Stability of Motion, Pergamon Press, London. Cunningham, W. J. 1958, Introduction to Nonlinear Analysis, McGraw- Hill Book Co., N.Y. Elgerd, 0. I. 1967, Control Systems Theory, McGraw-Hill Book Co., N.Y. El'sgol'c, L. E. 1964, Qualitative Methods in Mathematical Analysis, American Mathematical Society, Providence, R.I. Ergen, W. K., Lipkin, H. J. and Nohel, J. A. 1957, Applications of Liapunov's second method in reactor dynamics, J. Math. Phys., 36, 36. Hahn, W. 1955, Uber StabilitHt bei nichlinearen Systemen, ZAMM 35, 459. __ ___1957, Uber Differential-Differenzengleichungen, Math. Ann., 133, 251. 1963, Theory and Applications of Liapunov's Direct Method, Prentice-Hall, Inc., Englewood Cliffs, N.J. ___ 1967, Stability of Motion, Springer Verlag, Inc., N.Y. Hsu, C. 1967, Stability analysis of reactor systems via Liapunov's second method, J. Basic Eng., Trans. ASME, ser. D, preprint no. 66-WA/Aut-F. 1.28 1.29 Ingwerson, D. R. 1961, A modified Liapunov method for nonlinear stability problems, IRE Trans. on Automatic Control, May, 199. Kalman, R. E. and Bertram, J. E. 1960, Control analysis and design via the second method of Liapunov, J. Basic Eng., Trans. ASME, ser. D, 82, 371. Krasovskii, N. N. 1959, On the theory of optimal control, Appl. Math. Mech., 23, 624. ___ 1963, Stability of Motion, Stanford Univ. Press, Stanford, Calif. Lasalle, J. P. and Lefschetz, S. 1961, Stability by Liapunov's Direct Method with Applications, Academic Press, N.Y. Leighton, W. 1963, On the construction of Liapunov functions for certain autonomous nonlinear differential equations, Contrib. to Diff. Eq., 2, 10. Letov, A. M. 1961, Stability in Nonlinear Control Systems, Princeton Univ. Press, Princeton, N.J. Liapunov, A. M. 1967, General Problem of Stability of Motion, Academic Press, N.Y. Luecke, R. H. and McGuire, M. L. 1967, Stability analysis by Liapunov's direct method, IEC Fundamentals, 6, 432. Malkin, I. G. 1950, Certain questions in the theory of stability of motion in the sense of Liapunov, Amer. Math. Soc. Translations, 2, 20. 1959, Theorie der Stabilitat einer Bewegung, Verlag R. Oldenbourg, Muchen. Movchan, A. A. 1959, The direct method of Liapunov in stability problems of elastic systems, Appl. Math. Mech., 23, 483. 1961, Stability of processes with respect to two matrics, Appl. Math. Mech., 24, 988. Parks, P. C. 1966, A stability criterion for panel flutter via the second method of Liapunov, AIAA J., 4, 175. Peczkowski, J. L. and Liu, R. W. 1967, A format method for generating Liapunov functions, J. Basic Eng., Trans. ASME, ser. D, preprint no. 66-WA/Aut-17. Rekasius, Z. V. and Gibson, J. E. 1962, Stability analysis of 1.30 nonlinear control systems by second method of Liapunov, IRE Trans. on Automatic Control, 7, 3. Schultz, D. G. and Gibson, J. E. 1962, The variable gradient method for generating Liapunov functions, Trans. of AIEE, 81_, 203. Slobodkin, A. M. 1962, On the stability of equilibrium of con- servative systems with an infinite number of degrees of freedom, Appl. Math. Mech., 26, 356. Szego, G. P. 1962, A contribution to Liapunov's second method: nonlinear autonomous systems, J. Basic Eng., Trans. ASME, ser. D, 84, 571. Tauzsky, 0. 1961, A remark on a theorem of Liapunov, J. Math. Anal. Appl., 2, 105. Walker, J. A. and Clark, L. G. 1967, An integral method for Liapunov function generation for nonautonomous systems, J. Appl. Mech., Trans. ASME, ser. E, 87, 569. Wang, P. K. C. 1964, Control of distributed parameter systems, Advances in Control Systems, 1, 75. __ 1965, Stability analysis of a simplified flexible vehicle via Liapunov's direct method, AIAA. J., 3, 1764. Warden, R. B., Aris, R. and Amundson, N. 1964, An analysis of chemical reactor stability and control XIII, Ch.E.Sci., 19, 149. Zubov, V. I. 1961, The Methods of A.M. Liapunov and Their Application, AEC-tr-4439, Department of Commerce, Washington, D.C. 1.31 B. HYDRODYNAMIC STABILITY Bell, E. T. 1955, Mathematics: The Queen of Science, Dover Pub., Inc., N.Y. Benard, H. 1901, Les tourbillons cellulaires dans une nappe liquid transportant de la chaleur par convection en regime permanent, Annales de Chemie et de Physique, 23, 62. 1927, Sur les tourbillons et la theorie de Rayleigh, Comp. Rend., 185, 1109. Benney, D. J. 1964, Finite amplitude effects in an unstable laminar boundary layer, Phys. Fluids, 7, 319. Chandrasekar, S. 1961, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford. Coles, D. 1965, Transition in circular Couette flow, J. Fluid Mech., 21, 385. Conrad, P. W. and Criminale, N. 0. 1965, The stability of time- dependent laminar flows: parallel flows, ZAMP, 16, 233. Dolph, C. L. and Lewis, D. C. 1958, On the application of infinite systems of ordinary differential equations to perturbations of plane Poiseville flow, Quar. Appl. Math) 16, 97. Eckhaus, W. 1965, Studies in Nonlinear Stability Theory, Springer Verlag, N.Y. Elder, J.1960, An experimental investigation of turbulent spots and breakdown to turbulence, J. Fluid Mech., 9, 235. Emmons, H. W. 1951, The laminar-turbulent transition in a boundary layer, J. Aero. Sci., 18, 490. Gill, A. E. 1965, A mechanism for instability of plane Couette flow and of Poiseuille flow in a pipe, J. Fluid Mech., 21, 503. Greenspan, H. P. and Benney, D. J. 1963, On shear-layer instability, breakdown, and transition, J. Fluid Mech., 15, 133. Hamel, G. 1911, Zum Turbulenzproblem, Nachr. Ges. Wiss. Gottingen, 261. Howard, L. N. 1963, Heat transport by turbulent convection, J. Fluid Mech., 17, 405. 1.32 Joseph, D. D. 1965, On the stability of the Boussinesq equations, Arch. Rat. Mech. Anal., 9, 59. Joseph, D. D. 1966, Subcritical convective instability, J. Fluid Mech., 26, 753. Klebanoff, P. S., Tidstrom, K. D. and Sargent, L. M. 1962, The three-dimensional nature of bdy. layer instability, J. Fluid Mech., 12, 1. Kovasznay, L., Kamoda, H., and Vasudeva, B. 1962, Detailed flow field in transition, Proc. 1962 Heat. Transf. Fluid Mech. Conf., Stanford U. Press, Stanford, Calif. Lamb, H. 1945, Hydrodynamics, sixth edition, Dover Publications, N.Y. Lin, C. C. 1955, Theory of Hydrodynamic Stability, Cambridge Univ. Press, Cambridge. Lorentz, E. N. 1962, Simplified dynamic equations applied to rotating- basin experiments, J. Atmos. Sci., 19, 39. Malkus, W. and Veronis, G. 1958, Finite amplitude cellular convection, J. Fluid Mech., 4, 225. Meister, B. 1963, Die Anfangswertaufgabe fur die storungs differen- tialgleichungen des Taylorschen Stabilitatproblems, Arch. Rat. Mech. Anal., 14, 81. Miller, J. A. and Fejer, A. A. 1964, Transition phenomena in oscil- lating boundary layer flows, J. Fluid Mech., 18, 438. Orr, W 1907, The stability or instability of motions of a liquid. Part II: a viscous liquid, Proc. Roy. Irish Acad. (A), 27, 69. Palm, E. 1960, On the tendency towards hexagonal cells in steady convection, J. Fluid Mech., 8, 183. Palm, E. and Qiann, H. 1964, Contribution to the theory of cellular thermal convection, J. Fluid Mech., 19, 353. Rayleigh, Lord 1880, On the stability, or instability, of certain fluid motions, Scientific Papers, 1, 474, Cambridge Univ. Press, Cambridge. 1887, On the stability, or instability, of certain fluid motion. II, Scientific Papers, 3, 2. 1.33 1895, On the stability, or instability, of certain fluid motions: III, Scientific Papers, 4, 203, Cambridge Univ. Press, Cambridge. Reynolds, 0. 1895, On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Phil. Trans. Roy. Soc. London (A), 186, 123. Sani, R. 1963, Ph.D. Dissertation, Univ. of Minn., Minneapolis, Minn. Schulter, A., Lortz, D. and Busse, F. 1965, On the stability of finite amplitude convection, J. Fluid Mech., 23, 129. Segal, L. A. 1962, The nonlinear interaction of two disturbances in the thermal convection problem, J. Fluid Mech., 14, 97. 1965a, The structure of nonlinear cellular solutions to the Boussinesq equations, J. Fluid Mech., 21, 345. 1965b, The nonlinear interaction of a finite number of disturbances to a layer of fluid heated from below, J. Fluid Mech., 21, 359. 1966, Nonlinear hydrodynamic stability theory and its application to thermal convection and curved flows, in Non-Equilibrium Thermodynamics, Variational Techniques, and Stability, Univ. of Chicago Press, Chicago, Ill. Serrin, J. 1959, On the stability of viscous fluid motions, Arch. Rat. Mech. Anal., 3, 1. Sorger, P. 1966, Uber ein Variationsproblem aus der nichtlinearen Stabilitatstheorie zaher, inkompressibler Stromungen, ZAMP, 17, 201. Stuart, J. T. 1960a, On the nonlinear mechanics of wave disturbances in parallel flows I, J. Fluid Mech., 18, 841. , 1960b, Hydrodynamic Stability, Appl. Mech. Rev., 523. __ 1965, On the cellular patterns in thermal convection, J. Fluid Mech., 18, 841. 1965, Hydrodynamic stability, Appl. Mech. Rev., 18, 523. Synge, J. L. 1938a, Hydrodynamic stability, Semi-centennial Publications of Amer. Math. Soc., 2 (Addresses), 227. 1938b, The stability of plane Poiseuille motion, Proc. Fifth Inter. Cong. Appl. Math., 326, Cambridge, USA. 1.34 Taylor, G. I. 1923, Stability of a viscous liquid contained between rotating cylinders, Phil. Trans. A, 223, 289. Thomas, T. Y. 1943, On the uniform convergence of the solutions of the Navier-Stokes equations, Proc. Nat. Acad. Sci. USA, 29, 243. Thompson, J. J. 1887a, Rectilinear motion of a viscous fluid between parallel planes, Mathematical and Physical Papers, 4, 321, Cambridge Univ. Press, Cambridge. 1887b, Broad river flowing down an inclined plane bed, Mathematical and Physical Papers, 4, 330, Cambridge Univ. Press, Cambridge. Tippleskirch, H. 1956, Uber Konvektionzellen, imbesondere im flussigen Schwefel, Beitr. Phys. frei Atmos., 29, 37. Tippleskirch, H. 1957, Uber die Benard-Stromung in Aerosolen, Beitr. Phys. frei Atmos., 29, 219. Velte, W. 1962, Uber ein Stabilitatskriterium der Hydrodynamik, Arch. Rat. Mech. Anal., 9, 9. Veronis, G. 1963, An analysis of wind-driven ocean circulation with a limited number of Fourier components, J. Atmos. Sci., 20, 277. Veronis, G. 1965, On finite amplitude instability in thermohaline convection, J. Marine Res., 23, 1. von Karman, Theodore 1924, Uber die Stabilitat der Laminarstromung und die Theorie der Turbulenz, Proc. First Inter. Cong. Appl. Mech., 97, Delft. Watson, J. 1960, On the nonlinear mechanics of wave disturbances in stable and unstable flows II, J. Fluid Mech., 9, 371. 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 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. |