Partial Differential Equations in Fluid Dynamics (Cambridge 2008)

Isom H. Herron, Michael R. Foster0521888247, 9780521888240

This book is concerned with partial differential equations applied to fluids problems in science and engineering. This work is designed for two potential audiences. First, this book can function as a text for a course in mathematical methods in fluid mechanics in non-mathematics departments or in mathematics service courses. The authors have taught both. Second, this book is designed to help provide serious readers of journals (professionals, researchers, and graduate students) in analytical science and engineering with tools to explore and extend the missing steps in an analysis. The topics chosen for the book are those that the authors have found to be of considerable use in their own research careers. These topics are applicable in many areas, such as aeronautics and astronautics; biomechanics; chemical, civil, and mechanical engineering; fluid mechanics; and geophysical flows. Continuum ideas arise in other contexts, and the techniques included have applications there as well.

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Table of contents :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Contents……Page 9
Preface……Page 13
Acknowledgments……Page 15
1.2 Fundamentals of Complex Numbers……Page 17
1.2.1 Complex Roots and Logarithms……Page 18
1.3 Analytic Functions……Page 20
1.3.2 Differentiation……Page 21
1.3.3 Harmonic Functions……Page 23
1.4 Integration and Cauchy’s Theorem……Page 24
1.4.1 Cauchy’s Integral Formula……Page 26
Singularities……Page 28
Isolated Singularity Examples……Page 29
1.4.3 The Residue Theorem……Page 34
1.5 Application to Real Integrals……Page 35
1.5.1 Principal Values; Jordan’s Lemma……Page 41
Exercises……Page 46
2.1.1 Batchelor’s Trailing Vortex……Page 53
2.2 The Gamma Function……Page 55
2.2.1 Different Forms for gamma(z)……Page 57
2.3.1 Legendre Functions……Page 59
2.3.2 Bessel Functions……Page 60
2.3.3 Hypergeometric Functions……Page 62
2.3.5 Chebyshev Functions; Worked Example……Page 63
Worked Example……Page 65
2.3.6 Airy Functions……Page 66
2.4.1 Physical Problem: Heat Conduction……Page 67
2.4.2 Bessel Function Integrals……Page 68
Hankel Functions……Page 69
2.4.4 Airy Function Integrals……Page 70
Exercises……Page 71
3.2 Synge’s Setup for Rayleigh’s Criterion……Page 78
3.3 Sturm–Liouville Problems……Page 81
3.3.1 Prüfer’s Method……Page 83
3.3.2 Orthogonality of Eigenfunctions……Page 85
3.3.3 Synge’s Proof of Rayleigh’s Criterion……Page 86
3.4 Expansions in Eigenfunctions……Page 87
3.4.1 The Nonhomogeneous Problem; Solvability Condition……Page 88
3.5.1 Heat Conduction in a Nonuniform Rod……Page 90
3.5.2 Waves on Shallow Water……Page 91
3.6.1 ………Page 94
3.6.2 Sturm–Liouville Problems with Weight Changing Sign (Counterrotating Cylinders)……Page 98
Boundary Conditions……Page 99
3.7 Fourier–Bessel Series……Page 101
3.7.1 Worked Bessel Function Examples……Page 102
3.8 Continuous versus Discrete Spectra……Page 107
References……Page 110
Exercises……Page 111
4.1.1 Sources and Fundamental Solutions……Page 122
4.1.2 Conduction of Heat in a Spherical Shell with Sources……Page 124
4.2.1 Second-Order Problems……Page 126
4.2.2 Higher-Order Problems……Page 129
4.2.3 Adjoint and Self-Adjoint Problems……Page 131
Separated Boundary Conditions……Page 132
A Fourth-Order Example……Page 134
4.3 Connections with Distributions……Page 135
4.4 First-Order System: Green’s Matrices……Page 137
4.5 Generalized Green’s Functions……Page 138
4.6 Expansions in Eigenfunctions……Page 141
4.6.1 Delta Function Representation……Page 142
4.6.2 Worked Examples……Page 143
References……Page 148
Exercises……Page 149
4.7.1 Fundamental Solutions and the Wronskian Matrix……Page 158
4.7.2 Variation of Parameters……Page 159
4.7.3 The Adjoint Equation; Lagrange’s Identity……Page 160
Separated Boundary Conditions……Page 162
5.2 The Laplace Transform and Its Inverse……Page 164
Limiting Behavior: Initial-Value and Final-Value Theorems……Page 166
5.2.1 The Convolution……Page 167
5.2.2 Completion of the gamma Contour for Laplace Inversion……Page 168
5.2.3 Transform Problems……Page 169
5.3.1 An Ordinary Differential Equation……Page 172
5.3.2 Translating Plate in a Fluid……Page 174
5.3.3 Heat Conduction in a Strip……Page 176
5.3.4 Telegraph Equation……Page 177
5.3.5 A Scattering Problem……Page 180
5.3.6 Conduction of Heat in a Spherical Shell……Page 183
5.3.7 Boundary Layer Evolution for MHD Flow: Hartmann Layer……Page 185
5.4 Bilateral Laplace Transform……Page 187
A Modified Bessel Function……Page 188
Time-Dependent Boundary Layer with Suction……Page 190
References……Page 191
Exercises……Page 192
6.2 The Fourier Transform and Its Inverse……Page 199
6.2.1 Problems……Page 200
6.2.2 The Convolution……Page 201
6.2.3 Special Properties of Fourier Transforms……Page 202
6.2.4 Cosine and Sine Transforms……Page 203
6.3.1 Example 1: The Ekman Layer……Page 204
6.3.2 Example 2: Heat Conduction in a Strip……Page 206
6.3.3 Example 3: Heat Conduction in a Half-Plane……Page 207
6.3.4 Example 4: Sound Waves……Page 208
6.3.5 Example 5: Diffusion in a Force Field……Page 210
6.3.6 Example 6: An Integro-Differential Equation……Page 212
6.3.7 Example 7: Thermal Wake in a Small-Prandtl-Number Fluid……Page 213
6.3.8 Example 8: Fundamental Solution for Stokes Flow……Page 215
6.4 Mellin Transforms……Page 216
6.4.1 Properties……Page 218
6.5.1 Stability of Flow Near a Stagnation Point……Page 221
6.5.2 Stability of Jeffery–Hamel Flows……Page 222
References……Page 223
Exercises……Page 224
7.2 Lee Waves……Page 233
7.3 The Far Momentum Wake……Page 236
7.4 Kelvin–Helmholtz Instability……Page 239
Case 2 – Localized Initial Disturbance……Page 240
7.5 The Boundary Layer Signal Problem……Page 241
7.6 Stability of Plane Couette Flow……Page 244
7.7.1 A Model Problem……Page 247
7.7.2 A Boundary-Layer Example……Page 249
References……Page 251
Exercises……Page 252
8.2 Asymptotic Expansions……Page 260
8.3 Integration by Parts……Page 261
8.4 Laplace-Type Integrals; Watson’s Lemma……Page 262
8.4.1 Worked Examples……Page 264
8.4.2 Application: Early-Time Heat Transfer……Page 267
8.5 Generalized Laplace Integrals: Laplace’s Method……Page 269
8.5.1 Stirling’s Formula……Page 271
8.6 Method of Steepest Descent……Page 272
8.6.1 Application: A Special Function……Page 274
8.6.2 Application: The Oscillating Plate……Page 276
8.6.3 Application: Lee Waves……Page 278
8.6.4 Application: Sound Waves……Page 280
8.6.5 Application: Two-dimensional Laminar Wake……Page 281
8.7 Method of Stationary Phase; Kelvin’s Results……Page 282
Exercises……Page 284
Index……Page 295