1 INTRODUCTION 1
1 Introduction 1
2 Mechanisms of instability 4
3 Fundamental concepts of hydrodynamic stability 8
4 Kelvin-Helmholtz instability 14
5 Break-up of a liquid jet in air 22
Problems for chapter 1 27
2 THERMAL INSTABILITY 32
6 Introduction 32
7 The equations of motion 34
The exact equations 34
The Boussinesq equations 35
8 The stability problem 37
The linearized equations 37
The boundary condi-tions 40
Normal modes 42
9 General stability characteristics 44
Exchange of stabilities 44
A variational principle 45
10 Particular stability characteristics 50
Free-free boundaries 50
Rigid-rigid boundaries 51
free-rigid boundaries 52
11 The cells 52
12 Experimental results 59
13 Some applications 62
Problems for chapter 2 63
3 CENTRIFUGAL INSTABILITY 69
14 Introduction 69
15 Instability of an inviscid fluid 71
Three-dimensional disturbances 73
Axisymmetric disturbances 77
Two-dimensional disturbances 80
16 Instability of Couette flow of an inviscid fluid 82
17 The Taylor problem 88
Axisymmetric disturbances 90
Two-dimensional disturbances 103
Three-dimensional disturbances 104
Some experimental results 104
18 The Dean problem 108
The Dean problem 108
The Taylor-Dean prob-lem 113
19 The G？rtler problem 116
Problems for chapter 3 121
4 PARALLEL SHEAR FLOWS 124
20 Introduction 124
The inviscid theory 126
21 The governing equations 126
22 General criteria for instability 131
23 Flows with piecewise-linear velocity profiles 144
Unbounded vortex sheet 145
Unbounded shear layer 146
Bounded shear layer 147
24 The initial-value problem 147
The viscous theory 153
25 The governing equations 153
26 The eigenvalue spectrum for small Reynolds numbers 158
A perturbation expansion 159
Sufficient conditions for stability 161
27 Heuristic methods of approximation 164
The reduced equation and the inviscid approxima-tions 165
The boundary-layer approximation near a rigid wall 167
The WKBJ approximations 167
The local turning-point approximations 171
The truncated equation and Tollmien's improved viscous approximations 175
The viscous correction to the singular inviscid solution 177
28 Approximations to the eigenvalue relation 180
Symmetrical flows in a channel 181
Flows of the boundary-layer type 183
The boundary-layer approximation to φ3(z) 184
The WKBJ approxi-mation to φ3(z) 185
The local turning-point approximation to φ3(z) 188
Tollmien's improved approximation to φ3(z) 191
29 The long-wave approximation for unbounded flows 196
30 Numerical methods of solution 202
Expansions in orthogonal functions 203
Finite-difference methods 206
Initial-value methods(shooting) 207
31 Stability characteristics of various basic flows 211
Plane Couette flow 212
Poiseuille flow in a circular pipe 216
Plane Poiseuille flow 221
Combined plane Couette and plane Poiseuille flow 223
The Blasius boundary-layer profile 224
The asymptotic suction boundary-layer profile 227
Boundary layers at separation 229
The Falkner-Skan profiles 231
The Bickley jet 233
The hyper-bolic-tangent shear layer 237
32 Experimental results 239
Problems for chapter 4 245
5 UNIFORM ASYMPTOTIC APPROXIMATIONS 251
33 Introduction 251
Plane Couette flow 256
34 The integral representations of the solutions 256
35 The differential equation method 263
General velocity profiles 265
36 A preliminary transformation 265
37 The inner and outer expansions 267
The inner expansions 268
The outer expansions 271
The central matching problem 276
Com-posite approximations 278
38 Uniform approximations 280
The solution of well-balanced type 280
The solu-tions of balanced type 280
The solutions of dominant-recessive type 283
39 A comparison with Lin's theory 285
40 Preliminary simplification of the eigenvalue relation 290
41 The uniform approximation to the eigenvalue relation 295
A computational form of the first approximation to the eigenvalue relation 299
Results for plane Poiseuille flow 301
42 A comparision with the heuristic approximations to the eigenvalue relation 305
The local turning-point approximation to φ3(z) 305
Tollmien's improved approximation to φ3(z) 306
The uniform approximation to φ3(z)based on the truncated equation 308
The uniform approxima-tion to φ3(z)based on the Orr-Sommerfeld equation 309
43 A numerical treatment of the Orr-Sommerfeld problem using compound matrices 311
Symmetrical flows in a channel 315
Boundary-layer flows 316
Problems for chapter 5 317
6 ADDITIONAL TOPICS IN LINEAR STABlLITY THEORY 320
44 Instability of parallel flow of a stratified fluid 320
Introduction 320
Internal gravity waves and Ray-leigh-Taylor instability 324
Kelvin-Helmholtz instability 325
45 Baroclinic instability 333
46 Instability of the pinch 339
47 Development of linear instability in time and space 345
Initial-value problems 345
Spatially growing modes 349
48 Instability of unsteady flows 353
Introduction 353
Instability of periodic flows 354
Instability of other unsteady basic flows 361
Problems for chapter 6 363
7 NONLINEAR STABILITY 370
49 Introduction 370
Landau's theory 370
Discussion 376
50 The derivation of ordinary differential systems governing stability 380
51 Resonant wave interactions 387
Internal resonance of a double pendulum 387
Resonant wave interactions 392
52 Fundamental concepts of nonlinear stability 398
Introduction to ordinary differential equations 398
Introduction to bifurcation theory 402
Structural stability 407
Spatial development of nonlinear stability 416
Critical layers in parallel flow 420
53 Additional fundamental concepts of nonlinear stability 423
The energy method 424
Maximum and minimum energy in vortex motion 432
Application of boun-dary-layer theory to cellular instability 434
54 Some applications of the nonlinear theory 435
Bénard convection 435
Couette flow 442
Parallel shear flows 450
Problems for chapter 7 458
APPENDIX.A CLASS OF GENERALIZED AIRY FUNCTIONS 465
A1 The Airy functions Ak(z) 465
A2 The functions Ak(z,p),B0(z,p)and Bk(z,p) 466
A3 The functions Ak(z,p,q)and Bk(z,p,q) 472
A4 The zeros of At(z,p) 477
Addendum:Weakly non-parallel theories for the Blasius boundary layer 479
Solutions 481
Bibliography and author index 559
Motion picture index 595
Subject index 597