Introduction to Mathematical Fluid Dynamics

by Richard E. Meyer

Overview

Fluid dynamics, the behavior of liquids and gases, is a field of broad impact — in physics, engineering, oceanography, and meteorology for example — yet full understanding demands fluency in higher mathematics, the only language fluid dynamics speaks. Dr. Richard Meyer's work is indeed introductory, while written for advanced undergraduate and graduate students in applied mathematics, engineering, and the physical sciences. A knowledge of calculus and vector analysis is presupposed.
The author develops basic concepts from a semi-axiomatic foundation, noting that "for mathematics students such a treatment helps to dispel the all too common impression that the whole subject is built on a quicksand of assorted intuitions." Contents include:
Kinematics: Lagrangian and Eulerian descriptions, Circulation and Vorticity.
Momentum Principle and Ideal Fluid: Conservation examples, Euler equations, D'Alembert's and Kelvin's theorems.
Newtonian Fluid: Constitutive and Kinetic theories, exact solutions.
Fluids of Small Viscosity: Singular Perturbation, Boundary Layers.
Some Aspects of Rotating Fluids: Rossby number, Ekman layer, Taylor-Proudman Blocking.
Some Effects of Compressibility: Thermodynamics, Waves, Shock relations and structure, Navier-Stokes equations.
Dr. Meyer writes, "This core of our knowledge concerns the relation between inviscid and viscous fluids, and the bulk of this book is devoted to a discussion of that relation."

Product Details

ISBN-13:	9780486615547
Publisher:	Dover Publications
Publication date:	12/22/2010
Series:	Dover Books on Physics Series , #24
Pages:	190
Sales rank:	707,014
Product dimensions:	5.50(w) x 8.50(h) x (d)

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Introduction to Mathematical Fluid Dynamics

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By Richard E. Meyer

Dover Publications, Inc.

Copyright © 1971 Richard E. Meyer
All rights reserved.
ISBN: 978-0-486-61554-7

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CHAPTER 1

Kinematics

1. Introduction

Of the two possible didactic approaches to a physical subject, one proceeds from general definitions to their implications, leaving aside the problem of deciding which definitions are relevant to a particular, real situation. The other proceeds from specific situations and examples to definitions there relevant leaving the general principles of the subject to emerge only by and by. Fluid dynamics is too varied and subtle a subject for the first approach, but the second is too hard on the student, who has to persevere through an inordinate volume of work before the subject takes over-all shape. Like most other textbooks, this work therefore adopts largely the first approach.

One of its notable shortcomings is that the physical scales remain vague during the discussion of the general laws of fluid dynamics, because these laws can be made nondimensional only by reference to specific situations. An important part of the nature of the subject will therefore become manifest only in the second half of the book. For instance, a practical criterion for the applicability of continuum fluid dynamics — to which this book is restricted — will not be stated before Section 23. This long postponement arises from the subtlety of the considerations involved in any more than superficial discussion of the general definitions; Chapters 2 and 3 are needed, in part, to explain why nothing more simple-minded is adequate.

The definition of fluid motion will be presented in four stages (Chapters 1, 2, 3, and 6) of which the first two concern a very general continuum concept of fluid, which is necessarily incomplete. Of course, any model of a fluid that can be discussed mathematically is an idealization, and therefore incomplete. The term "ideal fluid," however, has acquired a specific, technical meaning (Section 13) denoting a radically simplified model. By antithesis, the Newtonian fluid is often called "real" because it is a highly realistic model of many common fluids under a wide range of circumstances. It is the fluid model primarily considered in this book, and the conventional term "real fluid" will be used for it on occasion, especially for comparison with the "ideal fluid" model in Chapter 2.

The physical definition of the (incompressible) Newtonian fluid will be given in the form of ten "postulates." Their purpose is to differentiate this definition from the manifold approximations (often most conveniently introduced as assumptions motivated intuitively) which are needed as aids to the description of fluid motion. Such differentiation is helpful in the earlier stages of a study of fluid dynamics, but the categorical appearance of the postulates is not intended to hide the fact that they become properly fruitful only when complemented by the nondimensional formulation of definite problems (Chapters 4 and 5). The further definitions needed for the compressible fluid are introduced more conventionally in Chapter 6.

Appendix 1

Notation. Equations, figures, problems, and statements cast for ease of reference in the form of theorems, lemmas, or corollaries are numbered by section. Appendices are numbered by the sections to which they are attached. Numbers in brackets [ ] refer to the bibliography at the end of the book, which aims only to serve the convenience of readers wishing to check on arguments for which there is no space in this book.

Bold letters denote vectors in a three-dimensional Euclidean space E3. Their components with respect to some particular Cartesian coordinate system are usually denoted by subscripts, for example, x = {x1, x2, x3}. On occasion, the alternative notation x = {x, y, z} and v = {u, v, w} is employed for convenience.

Tensors will play a minor role, and the reader not familiar with them will find it sufficient to interprete them as matrices with nine elements tij, i, j = 1, 2, 3 (the "components"), dependent on the Cartesian coordinate system used in such a way that Σ3j=1tijbj, i = 1, 2, or 3, are the components of a vector whenever bj, j = 1, 2, 3, are the components of a vector. No special symbol for tensors will be introduced, since no serious confusion will arise in the following from letting tij denote the tensor as well as its individual components. A particular tensor occurring frequently is Kronecker's symbol δij, defined by δij = 0 for i ≠ j and δij = 1 for i = j.

The summation convention will always be used unless the contrary is stated explicitly; it is to sum automatically over any repeated subscripts which are not separated by a +, -, or = sign; for instance, δii = 3.

Logic Notation. It will be convenient and will help to clarify the text to make fairly frequent use of the following symbols.

a [member of] B for the object a is an element of, or belongs to, the set B;

S1 [subset] S2 or S2 [contains] S1 for S1 is a subset of S2;

{a|β} for the set of all objects a to which the statement β applies;

S1 [union] S2 for the union of the sets S1 and S2 i.e., the set {a|a [member of] S1 or a [member of] S2};

S1 [intersection] S2 for the intersection of the sets S1 and S2, i.e., the set {a|both a [member of] S1 and a [member of] S2}.

Point Set Notation. For the purposes of this book, adequate definitions of the terminology used in Section 2 and, with varying frequency, also in later sections are as follows. If x0 denotes the position vector of any chosen point P in E3, the set of points with position vectors x such that |x - x0| < ε is called a neighborhood of P, if ε > 0. A point-set S is open means that P [member of] S implies P has a neighborhood [subset] S. S is bounded means that a spherical ball Σ of finite radius can be found such that S [subset] Σ. Q is a limit point of S means that every neighborhood of Q contains a point [member of] S distinct from Q. A set R is closed means that all its limit points also belong to R. On the other hand, the boundary [partial derivative] S of an open set S is the set of all those limit points of S which do not belong to S. The closure [bar.S] of an open set S is S [union] [partial derivative] S. A standard example of an open point set is S = {x|0 < |x| < 1}; its boundary [partial derivative] S consists of the sphere |x| = 1 and the point x = 0; its closure is [bar.S] = {x| |x| ≤ 1}. The distinction between points and their position vectors will usually be omitted.

Order Symbol. The asymptotic notation ψ(y, z) = O(χ(y, z)) as z -> ∞ (or -> 0) means that there are positive numbers D and M such that z > D (or < D) implies |ψ(y, z)| < M|χ(y, z)| for every fixed y in the domain of ψ or in some other, clearly specified domain. An o in the place of O means that the statement holds even for arbitrarily small M. Another useful notation is ψ ~ χ, meaning that |ψ - χ| -> 0 as z -> ∞ (or 0) for every fixed y. The word "fixed" here is meant to recall that such an asymptotic statement refers to a single limit process (with respect to z) in which ψ and χ are considered at a value of y independent of z.

2. Lagrangian Description

To cast our intuitive feeling for the motion of fluids (i.e., liquids and gases) into a definition on which mathematical argument can be based, is a more complicated task than might be thought at first. The attempt to be made now will not be completed until Section 17, even for the simplest fluids of our everyday experience. To make a start, the notion of some identifiable entity which might be called a "body of fluid" would appear to be near the core of our intuitive ideas. It is supported by the simple experiences of watching a puff of smoke in an air stream, or a drop of ink in a liquid stream, as they float along. A drop of ink, for instance, which is of the same density as the surrounding transparent liquid, may be plausibly considered a part of the liquid distinguished from the rest only by its color, but not by any dynamically relevant properties. If the ink is not soluble in the liquid and diffuses only slowly, and if the motion is not too involved, there is no difficulty in maintaining the visual identification of the colored liquid, as it moves and deforms, for a considerable time.

This leads naturally to the idea of associating fluid motion with a geometrical transformation represented by a function x = x (a, t) giving the position vectors x at various times t of the "bit" or "element" of fluid identified by the label a. A description of fluid motion thus based on identifying the individual bits of fluid is called Lagrangian. One simple way of labeling is to let a denote the position vector at some chosen time, which may then be called t = 0.

While this suffices for some purposes, the notion of "bit" or "element" is too confused for others. The underlying idea is expressed more clearly by the following

Definition. Let Ω0 be any open, bounded point-set in E3 "occupied by fluids" at time t = 0. "Fluid motion" shall mean a transformation Ht on the closure [bar.Ω]0 into E3 such that the point set HtΩ0 is that occupied by "the same fluid" at time t.

Readers unfamiliar with some of these terms will find definitions in Appendix 1. The notion that the set occupied by a fluid should be regarded as open is both mathematically necessary and suggested by physical intuition; it is difficult to think of a fluid as occupying a closed set, for that would imply, for instance, in the case of a solid spherical shell filled with water and surrounded by air, that the solid is restricted to an open point set. On the other hand, it is plausible that the set of mappings Ht describing how the body of fluid moves from Ω0 into the set HtΩ0 should be such that some statements are also possible regarding the mapping of the boundary points of Ω0. That is expressed in the definition regarding Ht as defined on the closure [bar.Ω]0.

The notion of identification may not at first appear natural for the description of processes such as the mixing of two chemically distinct fluids by diffusion, but it is actually quite convenient to regard the mixture itself as the fluid and the concentration of one chemical in the other as one of the fluid properties varying with a and t.

Any definition of fluid motion in terms of a geometrical transformation requires specification of the times t for which the transformation is defined, and most accounts of fluid dynamics tend to give the impression that Ht is defined on [bar.Ω]0 for all real t. Experience does not really support this. The difficulty becomes clear in any attempt to identify a blob of ink in a turbulent river for more than a few moments. The motion is too involved, and the blob of ink soon stretches into a ribbon so entwined and entangled with clear liquid that visual distinction between clear and colored liquid becomes prohibitively difficult. The study of such motions has raised serious doubt whether identification can ever be maintained indefinitely. On the other hand, no circumstances are known in which it cannot be maintained over a sufficiently short time, and experience therefore suggests

Postulate I. Ht is defined on [bar.Ω]0 over a time interval (T1, T2) such that T1< 0 < T2.

The first task, then, is to explore the nature of fluid motion during such an interval, and that provides ample scope for a book such as this. The further discussion will therefore be based on the assumption that |T1| and T2 are as large as may be required.

The notion of identification is closely related to the intuitive idea that two different bodies of fluid cannot simultaneously occupy the same portion of space. To express this with respect to the concept of fluid motion just defined requires

Postulate II. Ht has an inverse on HtΩ0.

Thus, if Ht is represented by the function x = Hta = x(a,t) on [bar.Ω]0, there exists an inverse function, a = a(x,t), mapping HtΩ0 onto Ω0. However (Appendix 2), a single-valued inverse cannot always be defined on the closure of HtΩ0.

Most general accounts of fluid dynamics do not cover processes involving drop formation and evaporation, and the same limitation will be adopted here. Common experience then indicates clearly that fluid motion is a continuous process in the sense of

Postulate III.x = x(a,t) is continuous with respect to a and t on its domain of definition, and its inverse is similarly continuous.

This expresses that it is impossible to break or cut a fluid — one of the basic points distinguishing it from a solid. The intuitive idea of a solid involves limitations on the possible extent (or rate) of continuous deformation. By contrast, a body of fluid is intuitively felt to be capable of changing shape continuously to an arbitrary extent. Moreover, an attempt to cut fluid results only in the fluid wrapping itself around the knife. A simple experiment is to move a knife through a liquid in which an ink filament has been laid across the path of the knife; it will be observed that the filament is stretched, but not severed, as indicated schematically in Figs. 2.1 and 2.2.

Observe the implications of Postulate III that HtΩ0 must, like Ω0, be a bounded set. Our basic definitions and postulates will be formulated only with reference to such point sets, since we cannot have direct experience or intuition with respect to any others. Of course, extrapolation to unbounded sets will often be mathematically convenient.

It follows [e.g., 1, Vol. 2, p. 97] from Postulates II and III that the set Ω0 [equivalent to] HtΩ0 is also open, and that its boundary is

(2.1) [partial derivative]Ωt = Ht[partial derivative]Ω0.

This is sometimes expressed by the phrase "the bounding surface of a body of fluid always consists of the same fluid particles" (see Appendix 2).

All experimental experience indicates that Postulate III does not fully reflect the continuity of fluid motion and should be amplied by

Postulate IV. The velocity v [equivalent to] [partial derivative]/[partial derivative]t x(a, t) is defined on the domain of x(a,t).

(Continues...)

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Excerpted from Introduction to Mathematical Fluid Dynamics by Richard E. Meyer. Copyright © 1971 Richard E. Meyer. Excerpted by permission of Dover Publications, Inc..
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Table of Contents

Preface
Acknowlegments
1 Kinematics
1 Introduction
2 Lagrangian description
3 Eulerian description
4 Conservation of mass
5 Circulation
6 Some potential flows
7 Vorticity
8 Line vortex
9 Vortex sheet
2 Momentum Principle and Ideal Fluid
10 Conservation of linear momentum
11 Mixing and lift
12 Equations of motion
13 D'Alembert's theorem
14 Kelvin's theorem
15 Conservation of angular momentum
3 Newtonian Fluid
16 The Couette experiment
17 Constitutive equation
18 Kinetic theory
19 Some viscous fluid motions
4 Fluids of Small Viscosity
20 Reynolds number
21 A singular perturbation example
22 Limit equations for the flat plate
23 Discussion of Blasius' solution
24 Limit equations with pressure gradient and wall curvature
25 Similarity solutions
26 Momentum integral
27 Separation
28 Wake
5 Some Aspects of Rotating Fluids
29 Bjerknes' theorem
30 Rossby number
31 Ekman layer
32 Taylor-Proudman theorem
6 Some Effects of Compressibility
33 Thermodynamic state
34 Flow initiation
35 Conservation of energy
36 Shock relations
37 Shock structure
38 Navier-Stokes equations
Bibliography
Index