Lectures on Fluid Mechanics

Overview


A readable and user-friendly introduction to fluid mechanics, this high-level text is geared toward advanced undergraduates and graduate students. Mathematicians, physicists, and engineers will also benefit from this lucid treatment.
The book begins with a derivation of the equations of fluid motion from statistical mechanics, followed by examinations of the classical theory and a portion of the modern mathematical theory of viscous, incompressible fluids. A considerable part ...
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Lectures on Fluid Mechanics

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Overview


A readable and user-friendly introduction to fluid mechanics, this high-level text is geared toward advanced undergraduates and graduate students. Mathematicians, physicists, and engineers will also benefit from this lucid treatment.
The book begins with a derivation of the equations of fluid motion from statistical mechanics, followed by examinations of the classical theory and a portion of the modern mathematical theory of viscous, incompressible fluids. A considerable part of the final chapters is devoted to the Navier-Stokes equations. The text assumes a familiarity with functional analysis and some complex variables, and it includes an especially valuable discussion of the modern function theoretic approach to solving partial differential equations. Numerous exercises appear throughout the text.
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Product Details

  • ISBN-13: 9780486488172
  • Publisher: Dover Publications
  • Publication date: 7/19/2012
  • Series: Dover Books on Physics Series
  • Edition description: Reprint
  • Pages: 242
  • Sales rank: 543,058
  • Product dimensions: 6.00 (w) x 9.10 (h) x 0.60 (d)

Table of Contents

Preface vii

Part I Setting the Scene 1

Introduction 3

Chapter 1 The equations of motion 7

1 Notation 7

2 The transport theorem 8

3 Conservation of probability 10

Exercises 17

4 The conservation equations. Definition of a fluid 17

Exercises 25

5 The Stokes hypothesis 25

Exercise 26

6 Boundary conditions. A theorem of Grad 27

Exercise 32

7 Fluid mechanical derivation of the conservation equations 32

Chapter 2 Potential flow 36

1 Ideal fluids 36

2 The good fairy strikes 37

3 Lagrange's theorem 39

Exercises 41

4 Some examples of potential flow 41

Exercise 46

Chapter 3 Some properties of potential flows 47

1 Introduction 47

2 Gauss's theorem 48

Exercise 49

3 The maximum principle 49

4 The minimum principle for the pressure 50

5 A variational principle for potential flows 52

6 Uniqueness of potential flows 53

Exercises 57

7 Uniqueness of ideal fluid flows 58

Exercise 60

Chapter 4 Potential flows in two dimensions 61

1 Introduction 61

2 Examples of two-dimensional potential flows 63

3 Existence of potential flows 67

Exercises 71

4 Examples of two-dimensional potential flows (continued) 72

Exercises 76

Chapter 5 d'Alembert's paradox and early attempts at its resolution 77

1 Introduction 77

2 The d'Alembert paradox 77

3 Cavity flows 80

Exercises 85

4 Discussion of the result 85

5 Water waves 86

Chapter 6 Flows with circulation 89

1 The stream function 89

2 Circulation 90

3 Circulatory flow past an airfoil 94

4 Lift. Blasius' theorem 96

5 Joukowski's hypothesis. Goodbye to all that 99

Exercise 101

Chapter 7 Viscous fluids 102

1 The Stokes hypothesis again 102

Exercise 107

2 The equations of motion 107

3 The stream function 108

4 The energy equation 110

5 The existence question 111

Chapter 8 Examples of viscous fluid flow 113

1 Introduction 113

2 Steady flow between parallel planes 113

3 Steady flow in a pipe 114

4 Steady flow past a moving plane 115

5 Unsteady flow past a moving plane 116

6 Viscous flow due to a source 117

Exercises 123

Chapter 9 Various approximations 124

1 Introduction 124

2 Stokes low 124 Exercise 129

3 Oseen flow 129

4 Boundary layer flow 131

Part II A Taste of the Modern Theory 135

Introduction 137

Chapter 10 Preliminaries 139

1 To start 139

2 Spaces of functions 139

Exercises 149

3 Functions of time 149

Table of Spaces Defined 153

Chapter 11 The weak solution 155

1 Definition of the weak solution 155

Exercise 157

2 The Lax-Milgram lemma 158

3 The quantized Navier-Stokes equations 159

4 The Navier-Stokes equations in a bounded domain 164

Exercises 173

5 Some properties of weak solutions 174

Exercises 179

Chapter 12 Uniqueness of weak solutions 180

1 Introduction 180

2 Uniqueness of classical solutions 181

3 Uniqueness of weak solutions 182

Exercises 192

Chapter 13 Strong solutions 193

1 Introduction 193

2 Some preliminaries 194

3 Uniqueness of strong solutions 199

4 Some a priori inequalities 199

5 Existence of strong solutions 203

Exercise 210

6 Smoothness and uniqueness of weak solutions 211

Chapter 14 A reproductive property of the Navier-Stokes equations 213

1 Preliminary remarks 213

2 The reproductive property 214

Exercise 217

3 Periodic solutions and stability 217

Index 220

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