Elementary fluid mechanics

Tsutomu Kambe9812564160, 9789812706676, 9789812565976, 9789812564160, 9812565973

This textbook describes the fundamental physical aspects of fluid flows for beginners of fluid mechanics in physics, mathematics and engineering, from the point of view of modern physics.

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Table of contents :
2.5. An ideal fluid and Newtonian viscous fluid……Page 8
Preface……Page 6
1.1. What are flows?……Page 18
1.2. Fluid particle and fields……Page 19
1.3.1. Stream-line……Page 23
1.3.2. Particle-path (path-line)……Page 24
1.3.4. Lagrange derivative……Page 25
1.4.1. Decomposition……Page 28
1.4.2. Symmetric part (pure straining motion)……Page 30
1.4.3. Anti-symmetric part (local rotation)……Page 31
1.5. Problems……Page 32
2.1. Continuumand transport phenomena……Page 34
2.3. Thermal diffusion……Page 38
2.4. Momentum transfer……Page 39
2.6. Viscous stress……Page 43
2.7. Problems……Page 45
3. Fundamental equations of ideal fluids……Page 48
3.1. Mass conservation……Page 49
3.3. Momentum conservation……Page 52
3.3.1. Equation of motion……Page 53
3.3.2. Momentum flux……Page 55
3.4.1. Adiabatic motion……Page 57
3.4.2. Energy flux……Page 9
3.5. Problems……Page 61
4.1. Equation of motion of a viscous fluid……Page 62
4.2. Energy equation and entropy equation……Page 65
4.3. Energy dissipation in an incompressible fluid……Page 66
4.4. Reynolds similarity law……Page 68
4.5. Boundary layer……Page 71
4.6. Parallel shear flows……Page 73
4.6.1. Steady flows……Page 74
4.6.2. Unsteady flow……Page 75
4.7. Rotating flows……Page 79
4.8.1. Stokes equation……Page 80
4.8.2. Stokeslet……Page 81
4.8.3. Slowmotion of a sphere……Page 82
4.9. Flows around a circular cylinder……Page 85
4.10. Drag coeffcient and lift coeffcient……Page 86
4.11. Problems……Page 87
5. Flows of ideal fluids……Page 94
5.1. Bernoulli’s equation……Page 95
5.2. Kelvin’s circulation theorem……Page 98
5.3. Flux of vortex lines……Page 100
5.4. Potential flows……Page 102
5.5. Irrotational incompressible flows (3D)……Page 104
5.6.1. Source (or sink)……Page 105
5.6.2. A source in a uniform flow……Page 107
5.6.3. Dipole……Page 108
5.6.4. A sphere in a uniform flow……Page 109
5.6.5. A vortex line……Page 111
5.7. Irrotational incompressible flows (2D)……Page 112
5.8.1. Source (or sink)……Page 116
5.8.2. A source in a uniform flow……Page 117
5.8.3. Dipole……Page 118
5.8.4. A circular cylinder in a uniform flow……Page 119
5.8.5. Point vortex (a line vortex)……Page 120
5.9.1. Kinetic energy induced by a moving body……Page 121
5.9.2. Induced mass……Page 124
5.9.3. d’Alembert’s paradox and virtual mass……Page 125
5.10. Problems……Page 126
6.1. Hydrostatic pressure……Page 132
6.2.1. Pressure condition at the free surface……Page 134
6.2.2. Condition of surface motion……Page 135
6.3.1. Boundary conditions……Page 136
6.3.2. Traveling waves……Page 138
6.3.3. Meaning of small amplitude……Page 139
6.3.5. Phase velocity and group velocity……Page 140
6.4. Surface waves on water of a finite depth……Page 142
6.5. KdV equation for long waves on shallow water……Page 143
6.6. Sound waves……Page 145
6.6.1. One-dimensional flows……Page 146
6.6.2. Equation of sound wave……Page 147
6.6.3. Plane waves……Page 152
6.7. Shock waves……Page 154
6.8. Problems……Page 156
7.1.1. Vorticity equation……Page 160
7.1.2. Biot–Savart’s law for velocity……Page 161
7.1.3. Invariants of motion……Page 162
7.2.1. Material line element and vortex-line……Page 164
7.2.2. Helmholtz’s vortex theorem……Page 165
7.3. Two-dimensional vortex motions……Page 167
7.3.1. Vorticity equation……Page 168
7.3.2. Integral invariants……Page 169
7.3.3. Velocity field at distant points……Page 171
7.3.4. Point vortex……Page 172
7.4. Motion of two point vortices……Page 173
7.5. System of N point vortices (a Hamiltonian system)……Page 177
7.6. Axisymmetric vortices with circular vortex-lines……Page 178
7.6.1. Hill’s spherical vortex……Page 179
7.6.2. Circular vortex ring……Page 180
7.7. Curved vortex filament……Page 182
7.8. Filament equation (an integrable equation)……Page 184
7.9. Burgers vortex (a viscous vortex with swirl)……Page 186
7.10. Problems……Page 190
8.1. Flows in a rotating frame……Page 194
8.2. Geostrophic flows……Page 198
8.3. Taylor–Proudman theorem……Page 200
8.4. Amodel of dry cyclone (or anticyclone)……Page 201
8.5. Rossby waves……Page 207
8.6. Stratified flows……Page 210
8.7. Global motions by the Earth Simulator……Page 213
8.7.2. Simulation of global ocean circulation by OFES code……Page 215
8.8. Problems……Page 217
9. Instability and chaos……Page 220
9.1. Linear stability theory……Page 221
9.2.1. Linearization……Page 223
9.2.2. Normal-mode analysis……Page 225
9.3. Stability of parallel shear flows……Page 226
9.3.1. Inviscid flows (v = 0)……Page 227
9.3.2. Viscous flows……Page 229
9.4.1. Description of the problem……Page 230
9.4.2. Linear stability analysis……Page 232
9.4.3. Convection cell……Page 236
9.5.1. Derivation of the Lorenz system……Page 238
9.5.2. Discovery stories of deterministic chaos……Page 240
9.5.3. Stability of fixed points……Page 242
9.5.3.1. Stability of the point O……Page 243
9.5.3.2. Stability of the fixed points C and C……Page 244
9.6.1. Lorenz attractor……Page 246
9.6.2. Lorenz map and deterministic chaos……Page 249
9.7. Problems……Page 252
10. Turbulence……Page 256
10.1. Reynolds experiment……Page 257
10.2. Turbulence signals……Page 259
10.3.1. Energy spectrum……Page 261
10.3.2. Energy dissipation……Page 263
10.3.3. Inertial range and five-thirds law……Page 264
10.3.4. Scale of viscous dissipation……Page 266
10.3.5. Similarity law due to Kolmogorov and Oboukov……Page 267
10.4.1. Stretching of line-elements……Page 268
10.4.2. Negative skewness and enstrophy enhancement……Page 271
10.4.3. Identification of vortices in turbulence……Page 273
10.4.4. Structure functions……Page 274
10.4.5. Structure functions at small s……Page 276
10.5. Problems……Page 277
11. Superfluid and quantized circulation……Page 280
11.1. Two-fluid model……Page 281
11.2.1. Bose gas……Page 283
11.2.2. Madelung transformation and hydrodynamic representation……Page 284
11.2.3. Gross–Pitaevskii equation……Page 285
11.3. Quantized vortices……Page 286
11.3.1. Quantized circulation……Page 287
11.3.2. A solution of a hollow vortex-line in a BEC……Page 288
11.4.1. BEC in dilute alkali-atomic gases……Page 290
11.4.2. Vortex dynamics in rotating BEC condensates……Page 291
11.5. Problems……Page 292
12. Gauge theory of ideal fluid flows……Page 294
12.1.1. Gauge invariances……Page 295
12.1.2. Review of the invariance in quantum mechanics……Page 296
12.1.3. Brief scenario of gauge principle……Page 298
12.2.1. System of n point masses……Page 299
12.2.2. Global invariance and conservation laws……Page 301
12.3. Fluid as a continuous field of mass……Page 302
12.3.1. Global invariance extended to a fluid……Page 303
12.3.2. Covariant derivative……Page 304
12.4. Symmetry of flow fields I: Translation symmetry……Page 305
12.4.2. Galilean transformation (global)……Page 306
12.4.3. Local Galilean transformation……Page 307
12.4.4. Gauge transformation (translation symmetry)……Page 308
12.4.5. Galilean invariant Lagrangian……Page 309
12.5.1. Rotational transformations……Page 311
12.5.2. Infinitesimal rotational transformation……Page 312
12.5.3. Gauge transformation (rotation symmetry)……Page 314
12.5.4. Significance of local rotation and the gauge field……Page 316
12.5.5. Lagrangian associated with the rotation symmetry……Page 317
12.6.2. Particle velocity……Page 318
12.6.3. Action principle……Page 319
12.6.4. Outcomes of variations……Page 320
12.6.5. Irrotational flow……Page 321
12.6.6. Clebsch solution……Page 322
12.7. Variations and Noether’s theorem……Page 323
12.7.1. Local variations……Page 324
12.7.2. Invariant variation……Page 325
12.7.3. Noether’s theorem……Page 326
12.8.1. Potential parts……Page 328
12.8.2. Additional note on the rotational symmetry……Page 329
12.9. Problem……Page 330
A.1. Definitions……Page 332
A.3. Vector product……Page 333
A.4. Triple products……Page 334
A.6. Integration theorems……Page 336
A.7. function……Page 337
B.1. Velocity potential……Page 340
B.2. Stream function (2D)……Page 341
B.3. Stokes’s streamfunction (axisymmetric)……Page 343
Appendix C Ideal fluid and ideal gas……Page 344
D.1. Frenet–Serret formula for a space curve……Page 346
D.2. Cylindrical coordinates……Page 347
D.3. Spherical polar coordinates……Page 349
Appendix E First three structure functions……Page 352
F.1.1. Lorentz transformation……Page 354
F.1.2. Lorenz-invariant Galilean Lagrangian……Page 355
F.2. Rotation symmetry……Page 357
Solutions……Page 360
References……Page 390
Index……Page 394