Introduction to Mathematical Fluid Dynamics

Richard E. Meyer

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486458878
• eBook ISBN: 9780486138947
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

Fluid dynamics, the behavior of liquids and gases, is a field of broad impact that encompasses aspects of physics, engineering, oceanography, and meteorology. Full understanding demands fluency in higher mathematics, the only language of fluid dynamics. This introductory text is geared toward advanced undergraduate and graduate students in applied mathematics, engineering, and the physical sciences. It assumes a knowledge of calculus and vector analysis.
Author Richard E. Meyer notes, "This core of knowledge concerns the relation between inviscid and viscous fluids, and the bulk of this book is devoted to a discussion of that relation." Dr. Meyer develops basic concepts from a semi-axiomatic foundation, observing that such treatment helps dispel the common impression that the entire subject is built on a quicksand of assorted intuitions. His topics include kinematics, momentum principle and ideal fluid, Newtonian fluid, fluids of small viscosity, some aspects of rotating fluids, and some effects of compressibility. Each chapter concludes with a set of problems.

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 t_ij, i, j = 1, 2, 3 (the “components”), dependent on the Cartesian coordinate system used in such a way that

t_ijb_j, i = 1, 2, or 3, are the components of a vector whenever b_j, 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 t_ij denote the tensor as well as its individual components. A particular tensor occurring frequently is Kronecker’s symbol δ_ij, defined by δ_ij = 0 for i ≠ j and δ_ij = 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 subscri...

Table of contents

• Title Page
• Dedication
• Copyright Page
• Table of Contents
• Preface
• Acknowledgments
• CHAPTER 1 - Kinematics
• CHAPTER 2 - Momentum Principle and Ideal Fluid
• CHAPTER 3 - Newtonian Fluid
• CHAPTER 4 - Fluids of Small Viscosity
• CHAPTER 5 - Some Aspects of Rotating Fluids
• CHAPTER 6 - Some Effects of Compressibility
• Bibliography
• Index