كتاب Elementary Fluid Mechanics

تذكير بمساهمة فاتح الموضوع :

معى اليوم احبتى فى الله
كتاب

Elementary Fluid Mechanics
Contents
Preface v
1. Flows 1
1.1. What are flows? . . . . . . . . . . . . . . . . . . . . 1
1.2. Fluid particle and fields . . . . . . . . . . . . . . . . 2
1.3. Stream-line, particle-path and streak-line . . . . . . 6
1.3.1. Stream-line . . . . . . . . . . . . . . . . . . . 6
1.3.2. Particle-path (path-line) . . . . . . . . . . . 7
1.3.3. Streak-line . . . . . . . . . . . . . . . . . . . 8
1.3.4. Lagrange derivative . . . . . . . . . . . . . . 8
1.4. Relative motion . . . . . . . . . . . . . . . . . . . . 11
1.4.1. Decomposition . . . . . . . . . . . . . . . . . 11
1.4.2. Symmetric part (pure straining motion) . . . 13
1.4.3. Anti-symmetric part (local rotation) . . . . . 14
1.5. Problems . . . . . . . . . . . . . . . . . . . . . . . . 15
2. Fluids 17
2.1. Continuumand transport phenomena . . . . . . . . 17
2.2. Mass diffusion in a fluidmixture . . . . . . . . . . . 18
2.3. Thermal diffusion . . . . . . . . . . . . . . . . . . . 21
2.4. Momentum transfer . . . . . . . . . . . . . . . . . . 22
2.5. An ideal fluid and Newtonian viscous fluid . . . . . 24
2.6. Viscous stress . . . . . . . . . . . . . . . . . . . . . 26
2.7. Problems . . . . . . . . . . . . . . . . . . . . . . . . 28
vii
viii Contents
3. Fundamental equations of ideal fluids 31
3.1. Mass conservation . . . . . . . . . . . . . . . . . . . 32
3.2. Conservation form. . . . . . . . . . . . . . . . . . . 35
3.3. Momentum conservation . . . . . . . . . . . . . . . 35
3.3.1. Equation ofmotion . . . . . . . . . . . . . . 36
3.3.2. Momentum flux . . . . . . . . . . . . . . . . 38
3.4. Energy conservation . . . . . . . . . . . . . . . . . . 40
3.4.1. Adiabatic motion . . . . . . . . . . . . . . . 40
3.4.2. Energy flux . . . . . . . . . . . . . . . . . . . 42
3.5. Problems . . . . . . . . . . . . . . . . . . . . . . . . 44
4. Viscous fluids 45
4.1. Equation ofmotion of a viscous fluid . . . . . . . . 45
4.2. Energy equation and entropy equation . . . . . . . 48
4.3. Energy dissipation in an incompressible fluid . . . . 49
4.4. Reynolds similarity law . . . . . . . . . . . . . . . . 51
4.5. Boundary layer . . . . . . . . . . . . . . . . . . . . 54
4.6. Parallel shear flows . . . . . . . . . . . . . . . . . . 56
4.6.1. Steady flows . . . . . . . . . . . . . . . . . . 57
4.6.2. Unsteady flow . . . . . . . . . . . . . . . . . 58
4.7. Rotating flows . . . . . . . . . . . . . . . . . . . . . 62
4.8. Low Reynolds number flows . . . . . . . . . . . . . 63
4.8.1. Stokes equation . . . . . . . . . . . . . . . . 63
4.8.2. Stokeslet . . . . . . . . . . . . . . . . . . . . 64
4.8.3. Slow motion of a sphere . . . . . . . . . . . . 65
4.9. Flows around a circular cylinder . . . . . . . . . . . 68
4.10. Drag coefficient and lift coefficient . . . . . . . . . . 69
4.11. Problems . . . . . . . . . . . . . . . . . . . . . . . . 70
5. Flows of ideal fluids 77
5.1. Bernoulli’s equation . . . . . . . . . . . . . . . . . . 78
5.2. Kelvin’s circulation theorem . . . . . . . . . . . . . 81
5.3. Flux of vortex lines . . . . . . . . . . . . . . . . . . 83
5.4. Potential flows . . . . . . . . . . . . . . . . . . . . . 85
5.5. Irrotational incompressible flows (3D) . . . . . . . . 87
Contents ix
5.6. Examples of irrotational incompressible
flows (3D) . . . . . . . . . . . . . . . . . . . . . . . 88
5.6.1. Source (or sink) . . . . . . . . . . . . . . . . 88
5.6.2. A source in a uniformflow . . . . . . . . . . 90
5.6.3. Dipole . . . . . . . . . . . . . . . . . . . . . 91
5.6.4. A sphere in a uniformflow . . . . . . . . . . 92
5.6.5. A vortex line . . . . . . . . . . . . . . . . . . 94
5.7. Irrotational incompressible flows (2D) . . . . . . . . 95
5.8. Examples of 2D flows represented by complex
potentials . . . . . . . . . . . . . . . . . . . . . . . 99
5.8.1. Source (or sink) . . . . . . . . . . . . . . . . 99
5.8.2. A source in a uniformflow . . . . . . . . . . 100
5.8.3. Dipole . . . . . . . . . . . . . . . . . . . . . 101
5.8.4. A circular cylinder in a uniformflow. . . . . 102
5.8.5. Point vortex (a line vortex) . . . . . . . . . . 103
5.9. Inducedmass . . . . . . . . . . . . . . . . . . . . . 104
5.9.1. Kinetic energy induced by a
moving body . . . . . . . . . . . . . . . . . . 104
5.9.2. Inducedmass . . . . . . . . . . . . . . . . . 107
5.9.3. d’Alembert’s paradox and virtual mass . . . 108
5.10. Problems . . . . . . . . . . . . . . . . . . . . . . . . 109
6. Water waves and sound waves 115
6.1. Hydrostatic pressure . . . . . . . . . . . . . . . . . 115
6.2. Surface waves on deep water . . . . . . . . . . . . . 117
6.2.1. Pressure condition at the free surface . . . . 117
6.2.2. Condition of surface motion . . . . . . . . . 118
6.3. Small amplitude waves of deep water . . . . . . . . 119
6.3.1. Boundary conditions . . . . . . . . . . . . . 119
6.3.2. Traveling waves . . . . . . . . . . . . . . . . 121
6.3.3. Meaning of small amplitude . . . . . . . . . 122
6.3.4. Particle trajectory . . . . . . . . . . . . . . . 123
6.3.5. Phase velocity and group velocity . . . . . . 123
6.4. Surface waves on water of a finite depth . . . . . . 125
6.5. KdV equation for long waves on
shallow water . . . . . . . . . . . . . . . . . . . . . 126
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6.6. Sound waves . . . . . . . . . . . . . . . . . . . . . . 128
6.6.1. One-dimensional flows . . . . . . . . . . . . . 129
6.6.2. Equation of sound wave . . . . . . . . . . . . 130
6.6.3. Plane waves . . . . . . . . . . . . . . . . . . 135
6.7. Shock waves . . . . . . . . . . . . . . . . . . . . . . 137
6.8. Problems . . . . . . . . . . . . . . . . . . . . . . . . 139
7. Vortex motions 143
7.1. Equations for vorticity . . . . . . . . . . . . . . . . 143
7.1.1. Vorticity equation . . . . . . . . . . . . . . . 143
7.1.2. Biot–Savart’s law for velocity . . . . . . . . . 144
7.1.3. Invariants ofmotion . . . . . . . . . . . . . . 145
7.2. Helmholtz’s theorem . . . . . . . . . . . . . . . . . 147
7.2.1. Material line element and vortex-line . . . . 147
7.2.2. Helmholtz’s vortex theorem. . . . . . . . . . 148
7.3. Two-dimensional vortex motions . . . . . . . . . . . 150
7.3.1. Vorticity equation . . . . . . . . . . . . . . . 151
7.3.2. Integral invariants . . . . . . . . . . . . . . . 152
7.3.3. Velocity field at distant points . . . . . . . . 154
7.3.4. Point vortex . . . . . . . . . . . . . . . . . . 155
7.3.5. Vortex sheet . . . . . . . . . . . . . . . . . . 156
7.4. Motion of two point vortices . . . . . . . . . . . . . 156
7.5. System of N point vortices (a Hamiltonian
system) . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.6. Axisymmetric vortices with circular
vortex-lines . . . . . . . . . . . . . . . . . . . . . . 161
7.6.1. Hill’s spherical vortex . . . . . . . . . . . . . 162
7.6.2. Circular vortex ring . . . . . . . . . . . . . . 163
7.7. Curved vortex filament . . . . . . . . . . . . . . . . 165
7.8. Filament equation (an integrable equation) . . . . . 167
7.9. Burgers vortex (a viscous vortex with swirl) . . . . 169
7.10. Problems . . . . . . . . . . . . . . . . . . . . . . . . 173
8. Geophysical flows 177
8.1. Flows in a rotating frame . . . . . . . . . . . . . . . 177
8.2. Geostrophic flows . . . . . . . . . . . . . . . . . . . 181
Contents xi
8.3. Taylor–Proudman theorem . . . . . . . . . . . . . . 183
8.4. Amodel of dry cyclone (or anticyclone) . . . . . . . 184
8.5. Rossby waves . . . . . . . . . . . . . . . . . . . . . 190
8.6. Stratified flows . . . . . . . . . . . . . . . . . . . . . 193
8.7. Global motions by the Earth Simulator . . . . . . . 196
8.7.1. Simulation of global atmospheric motion by
AFES code . . . . . . . . . . . . . . . . . . . 198
8.7.2. Simulation of global ocean circulation by
OFES code . . . . . . . . . . . . . . . . . . . 198
8.8. Problems . . . . . . . . . . . . . . . . . . . . . . . . 200
9. Instability and chaos 203
9.1. Linear stability theory . . . . . . . . . . . . . . . . 204
9.2. Kelvin–Helmholtz instability . . . . . . . . . . . . . 206
9.2.1. Linearization . . . . . . . . . . . . . . . . . . 206
9.2.2. Normal-mode analysis . . . . . . . . . . . . . 208
9.3. Stability of parallel shear flows . . . . . . . . . . . . 209
9.3.1. Inviscid flows (ν =0) . . . . . . . . . . . . . 210
9.3.2. Viscous flows . . . . . . . . . . . . . . . . . . 212
9.4. Thermal convection . . . . . . . . . . . . . . . . . . 213
9.4.1. Description of the problem . . . . . . . . . . 213
9.4.2. Linear stability analysis . . . . . . . . . . . . 215
9.4.3. Convection cell . . . . . . . . . . . . . . . . . 219
9.5. Lorenz system . . . . . . . . . . . . . . . . . . . . . 221
9.5.1. Derivation of the Lorenz system . . . . . . . 221
9.5.2. Discovery stories of deterministic chaos . . . 223
9.5.3. Stability of fixed points . . . . . . . . . . . . 225
9.6. Lorenz attractor and deterministic chaos . . . . . . 229
9.6.1. Lorenz attractor . . . . . . . . . . . . . . . . 229
9.6.2. Lorenz map and deterministic chaos . . . . . 232
9.7. Problems . . . . . . . . . . . . . . . . . . . . . . . . 235
10. Turbulence 239
10.1. Reynolds experiment . . . . . . . . . . . . . . . . . 240
10.2. Turbulence signals . . . . . . . . . . . . . . . . . . . 242
xii Contents
10.3. Energy spectrumand energy dissipation . . . . . . 244
10.3.1. Energy spectrum . . . . . . . . . . . . . . . 244
10.3.2. Energy dissipation . . . . . . . . . . . . . . 246
10.3.3. Inertial range and five-thirds law . . . . . . 247
10.3.4. Scale of viscous dissipation . . . . . . . . . 249
10.3.5. Similarity law due to Kolmogorov
and Oboukov . . . . . . . . . . . . . . . . . 250
10.4. Vortex structures in turbulence . . . . . . . . . . . 251
10.4.1. Stretching of line-elements . . . . . . . . . . 251
10.4.2. Negative skewness and enstrophy
enhancement . . . . . . . . . . . . . . . . . 254
10.4.3. Identification of vortices in turbulence . . . 256
10.4.4. Structure functions . . . . . . . . . . . . . . 257
10.4.5. Structure functions at small s . . . . . . . . 259
10.5. Problems . . . . . . . . . . . . . . . . . . . . . . . . 260
11. Superfluid and quantized circulation 263
11.1. Two-fluid model . . . . . . . . . . . . . . . . . . . . 264
11.2. Quantum mechanical description of
superfluid flows . . . . . . . . . . . . . . . . . . . . 266
11.2.1. Bose gas . . . . . . . . . . . . . . . . . . . . 266
11.2.2. Madelung transformation and hydrodynamic
representation . . . . . . . . . . . . . . . . 267
11.2.3. Gross–Pitaevskii equation . . . . . . . . . . 268
11.3. Quantized vortices . . . . . . . . . . . . . . . . . . 269
11.3.1. Quantized circulation . . . . . . . . . . . . 270
11.3.2. A solution of a hollow vortex-line in
a BEC . . . . . . . . . . . . . . . . . . . . . 271
11.4. Bose–Einstein Condensation (BEC) . . . . . . . . . 273
11.4.1. BEC in dilute alkali-atomic gases . . . . . . 273
11.4.2. Vortex dynamics in rotating BEC
condensates . . . . . . . . . . . . . . . . . . 274
11.5. Problems . . . . . . . . . . . . . . . . . . . . . . . . 275
Contents xiii
12. Gauge theory of ideal fluid flows 277
12.1. Backgrounds of the theory . . . . . . . . . . . . . . 278
12.1.1. Gauge invariances . . . . . . . . . . . . . . 278
12.1.2. Review of the invariance in quantum
mechanics . . . . . . . . . . . . . . . . . . . 279
12.1.3. Brief scenario of gauge principle . . . . . . 281
12.2. Mechanical system . . . . . . . . . . . . . . . . . . 282
12.2.1. System of n point masses . . . . . . . . . . 282
12.2.2. Global invariance and conservation laws . . 284
12.3. Fluid as a continuous field of mass . . . . . . . . . 285
12.3.1. Global invariance extended to a fluid . . . . 286
12.3.2. Covariant derivative . . . . . . . . . . . . . 287
12.4. Symmetry of flow fields I: Translation symmetry . . 288
12.4.1. Translational transformations . . . . . . . . 289
12.4.2. Galilean transformation (global) . . . . . . 289
12.4.3. Local Galilean transformation . . . . . . . . 290
12.4.4. Gauge transformation (translation
symmetry) . . . . . . . . . . . . . . . . . . 291
12.4.5. Galilean invariant Lagrangian . . . . . . . . 292
12.5. Symmetry of flow fields II: Rotation symmetry . . . 294
12.5.1. Rotational transformations . . . . . . . . . 294
12.5.2. Infinitesimal rotational transformation . . . 295
12.5.3. Gauge transformation (rotation symmetry) 297
12.5.4. Significance of local rotation and the
gauge field . . . . . . . . . . . . . . . . . . 299
12.5.5. Lagrangian associated with the rotation
symmetry . . . . . . . . . . . . . . . . . . . 300
12.6. Variational formulation for flows of an ideal fluid . 301
12.6.1. Covariant derivative (in summary) . . . . . 301
12.6.2. Particle velocity . . . . . . . . . . . . . . . 301
12.6.3. Action principle . . . . . . . . . . . . . . . 302
12.6.4. Outcomes of variations . . . . . . . . . . . . 303
12.6.5. Irrotational flow . . . . . . . . . . . . . . . 304
12.6.6. Clebsch solution . . . . . . . . . . . . . . . 305
xiv Contents
12.7. Variations and Noether’s theorem . . . . . . . . . . 306
12.7.1. Local variations . . . . . . . . . . . . . . . . 307
12.7.2. Invariant variation . . . . . . . . . . . . . . 308
12.7.3. Noether’s theorem . . . . . . . . . . . . . . 309
12.8. Additional notes . . . . . . . . . . . . . . . . . . . . 311
12.8.1. Potential parts . . . . . . . . . . . . . . . . 311
12.8.2. Additional note on the rotational symmetry 312
12.9. Problem . . . . . . . . . . . . . . . . . . . . . . . . 313
Appendix A Vector analysis 315
A.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 315
A.2. Scalar product . . . . . . . . . . . . . . . . . . . . . 316
A.3. Vector product . . . . . . . . . . . . . . . . . . . . 316
A.4. Triple products . . . . . . . . . . . . . . . . . . . . 317
A.5. Differential operators . . . . . . . . . . . . . . . . . 319
A.6. Integration theorems . . . . . . . . . . . . . . . . . 319
A.7. δ function . . . . . . . . . . . . . . . . . . . . . . . 320
Appendix B Velocity potential, stream function 323
B.1. Velocity potential . . . . . . . . . . . . . . . . . . . 323
B.2. Streamfunction (2D) . . . . . . . . . . . . . . . . . 324
B.3. Stokes’s streamfunction (axisymmetric) . . . . . . 326
Appendix C Ideal fluid and ideal gas 327
Appendix D Curvilinear reference frames:
Differential operators 329
D.1. Frenet–Serret formula for a space curve . . . . . . . 329
D.2. Cylindrical coordinates . . . . . . . . . . . . . . . . 330
D.3. Spherical polar coordinates . . . . . . . . . . . . . . 332
Appendix E First three structure functions 335
Contents xv
Appendix F Lagrangians 337
F.1. Galilei invariance and Lorentz invariance . . . . . . 337
F.1.1. Lorentz transformation . . . . . . . . . . . . 337
F.1.2. Lorenz-invariant Galilean Lagrangian . . . . 338
F.2. Rotation symmetry . . . . . . . . . . . . . . . . . . 340
Solutions 343
References 373

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