Elementary Fluid Mechanics

by Kambe, Tsutomu

• ISBN13: 9789812565976
• ISBN10: 9812565973
• Format: Paperback
• Copyright: 2007-01-04
• Publisher: World Scientific Pub Co Inc

Summary

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. It also emphasizes the dynamical aspects of fluid motions rather than the static aspects, illustrating vortex motions, waves, geophysical flows, chaos and turbulence. Beginning with the fundamental concepts of the nature of flows and the properties of fluids, the book presents fundamental conservation equations of mass, momentum and energy, and the equations of motion for both inviscid and viscous fluids. In addition to the fundamentals, water waves and sound waves, vortex motions, geophysical flows, nonlinear instability and chaos, turbulence, and the gauge theory of fluid flows are also covered. The material in the book emerged from the lecture notes for an intensive course on Elementary Fluid Mechanics for both undergraduate and postgraduate students of theoretical physics given in 2003 and 2004 at the Nankai Institute of Mathematics (Tianjin) in China. Hence, each chapter may be presented separately as a single lecture. Contents: Flows; Fluids; Fundamental Equations of Ideal Fluids; Viscous Fluids; Flows of Ideal Fluids; Water Waves and Sound Waves; Vortex Motions; Geophysical Flows; Instability and Chaos; Turbulence; Gauge Principle for Ideal-Fluid Flows; Appendices: Vector Analysis; Velocity Potential and Stream Function; Ideal Fluid and Ideal Gas; Curvilinear Reference Frames: Differential Operators. Key Features Textbook with a new and modern approach Includes concise and compact presentations of important new areas in fluid mechanics from the viewpoint of physics, such as flows in rotating frames, stratified flows, recent results of the Earth Simulator, Lorenz chaotic system, fully developed turbulence, and the gauge theory of fluid flows Readership: Undergraduate and postgraduate students in physics, mathematics, and engineering as well as scientists and engineers who are not specialists in fluid mechanics.

Table of Contents

Preface	p. v
Flows	p. 1
What are flows?	p. 1
Fluid particle and fields	p. 2
Stream-line, particle-path and streak-line	p. 6
Stream-line	p. 6
Particle-path (path-line)	p. 7
Streak-line	p. 8
Lagrange derivative	p. 8
Relative motion	p. 11
Decomposition	p. 11
Symmetric part (pure straining motion)	p. 13
Anti-symmetric part (local rotation)	p. 14
Problems	p. 15
Fluids	p. 17
Continuum and transport phenomena	p. 17
Mass diffusion in a fluid mixture	p. 18
Thermal diffusion	p. 21
Momentum transfer	p. 22
An ideal fluid and Newtonian viscous fluid	p. 24
Viscous stress	p. 26
Problems	p. 28
Fundamental equations of ideal fluids	p. 31
Mass conservation	p. 32
Conservation form	p. 35
Momentum conservation	p. 35
Equation of motion	p. 36
Momentum flux	p. 38
Energy conservation	p. 40
Adiabatic motion	p. 40
Energy flux	p. 42
Problems	p. 44
Viscous fluids	p. 45
Equation of motion of a viscous fluid	p. 45
Energy equation and entropy equation	p. 48
Energy dissipation in an incompressible fluid	p. 49
Reynolds similarity law	p. 51
Boundary layer	p. 54
Parallel shear flows	p. 56
Steady flows	p. 57
Unsteady flow	p. 58
Rotating flows	p. 62
Low Reynolds number flows	p. 63
Stokes equation	p. 63
Stokeslet	p. 64
Slow motion of a sphere	p. 65
Flows around a circular cylinder	p. 68
Drag coefficient and lift coefficient	p. 69
Problems	p. 70
Flows of ideal fluids	p. 77
Bernoulli's equation	p. 78
Kelvin's circulation theorem	p. 81
Flux of vortex lines	p. 83
Potential flows	p. 85
Irrotational incompressible flows (3D)	p. 87
Examples of irrotational incompressible flows (3D)	p. 88
Source (or sink)	p. 88
A source in a uniform flow	p. 90
Dipole	p. 91
A sphere in a uniform flow	p. 92
A vortex line	p. 94
Irrotational incompressible flows (2D)	p. 95
Examples of 2D flows represented by complex potentials	p. 99
Source (or sink)	p. 99
A source in a uniform flow	p. 100
Dipole	p. 101
A circular cylinder in a uniform flow	p. 102
Point vortex (a line vortex)	p. 103
Induced mass	p. 104
Kinetic energy induced by a moving body	p. 104
Induced mass	p. 107
d'Alembert's paradox and virtual mass	p. 108
Problems	p. 109
Water waves and sound waves	p. 115
Hydrostatic pressure	p. 115
Surface waves on deep water	p. 117
Pressure condition at the free surface	p. 117
Condition of surface motion	p. 118
Small amplitude waves of deep water	p. 119
Boundary conditions	p. 119
Traveling waves	p. 121
Meaning of small amplitude	p. 122
Particle trajectory	p. 123
Phase velocity and group velocity	p. 123
Surface waves on water of a finite depth	p. 125
KdV equation for long waves on shallow water	p. 126
Sound waves	p. 128
One-dimensional flows	p. 129
Equation of sound wave	p. 130
Plane waves	p. 135
Shock waves	p. 137
Problems	p. 139
Vortex motions	p. 143
Equations for vorticity	p. 143
Vorticity equation	p. 143
Biot-Savart's law for velocity	p. 144
Invariants of motion	p. 145
Helmholtz's theorem	p. 147
Material line element and vortex-line	p. 147
Helmholtz's vortex theorem	p. 148
Two-dimensional vortex motions	p. 150
Vorticity equation	p. 151
Integral invariants	p. 152
Velocity field at distant points	p. 154
Point vortex	p. 155
Vortex sheet	p. 156
Motion of two point vortices	p. 156
System of N point vortices (a Hamiltonian system)	p. 160
Axisymmetric vortices with circular vortex-lines	p. 161
Hill's spherical vortex	p. 162
Circular vortex ring	p. 163
Curved vortex filament	p. 165
Filament equation (an integrable equation)	p. 167
Burgers vortex (a viscous vortex with swirl)	p. 169
Problems	p. 173
Geophysical flows	p. 177
Flows in a rotating frame	p. 177
Geostrophic flows	p. 181
Taylor-Proudman theorem	p. 183
A model of dry cyclone (or anticyclone)	p. 184
Rossby waves	p. 190
Stratified flows	p. 193
Global motions by the Earth Simulator	p. 196
Simulation of global atmospheric motion by AFES code	p. 198
Simulation of global ocean circulation by OFES code	p. 198
Problems	p. 200
Instability and chaos	p. 203
Linear stability theory	p. 204
Kelvin-Helmholtz instability	p. 206
Linearization	p. 206
Normal-mode analysis	p. 208
Stability of parallel shear flows	p. 209
Inviscid flows (v = 0)	p. 210
Viscous flows	p. 212
Thermal convection	p. 213
Description of the problem	p. 213
Linear stability analysis	p. 215
Convection cell	p. 219
Lorenz system	p. 221
Derivation of the Lorenz system	p. 221
Discovery stories of deterministic chaos	p. 223
Stability of fixed points	p. 225
Lorenz attractor and deterministic chaos	p. 229
Lorenz attractor	p. 229
Lorenz map and deterministic chaos	p. 232
Problems	p. 235
Turbulence	p. 239
Reynolds experiment	p. 240
Turbulence signals	p. 242
Energy spectrum and energy dissipation	p. 244
Energy spectrum	p. 244
Energy dissipation	p. 246
Inertial range and five-thirds law	p. 247
Scale of viscous dissipation	p. 249
Similarity law due to Kolmogorov and Oboukov	p. 250
Vortex structures in turbulence	p. 251
Stretching of line-elements	p. 251
Negative skewness and enstrophy enhancement	p. 254
Identification of vortices in turbulence	p. 256
Structure functions	p. 257
Structure functions at small s	p. 259
Problems	p. 260
Superfluid and quantized circulation	p. 263
Two-fluid model	p. 264
Quantum mechanical description of superfluid flows	p. 266
Bose gas	p. 266
Madelung transformation and hydrodynamic representation	p. 267
Gross-Pitaevskii equation	p. 268
Quantized vortices	p. 269
Quantized circulation	p. 270
A solution of a hollow vortex-line in a BEC	p. 271
Bose-Einstein Condensation (BEC)	p. 273
BEC in dilute alkali-atomic gases	p. 273
Vortex dynamics in rotating BEC condensates	p. 274
Problems	p. 275
Gauge theory of ideal fluid flows	p. 277
Backgrounds of the theory	p. 278
Gauge invariances	p. 278
Review of the invariance in quantum mechanics	p. 279
Brief scenario of gauge principle	p. 281
Mechanical system	p. 282
System of n point masses	p. 282
Global invariance and conservation laws	p. 284
Fluid as a continuous field of mass	p. 285
Global invariance extended to a fluid	p. 286
Covariant derivative	p. 287
Symmetry of flow fields I: Translation symmetry	p. 288
Translational transformations	p. 289
Galilean transformation (global)	p. 289
Local Galilean transformation	p. 290
Gauge transformation (translation symmetry)	p. 291
Galilean invariant Lagrangian	p. 292
Symmetry of flow fields II: Rotation symmetry	p. 294
Rotational transformations	p. 294
Infinitesimal rotational transformation	p. 295
Gauge transformation (rotation symmetry)	p. 297
Significance of local rotation and the gauge field	p. 299
Lagrangian associated with the rotation symmetry	p. 300
Variational formulation for flows of an ideal fluid	p. 301
Covariant derivative (in summary)	p. 301
Particle velocity	p. 301
Action principle	p. 302
Outcomes of variations	p. 303
Irrotational flow	p. 304
Clebsch solution	p. 305
Variations and Noether's theorem	p. 306
Local variations	p. 307
Invariant variation	p. 308
Noether's theorem	p. 309
Additional notes	p. 311
Potential parts	p. 311
Additional note on the rotational symmetry	p. 312
Problem	p. 313
Vector analysis	p. 315
Definitions	p. 315
Scalar product	p. 316
Vector product	p. 316
Triple products	p. 317
Differential operators	p. 319
Integration theorems	p. 319
[delta] function	p. 320
Velocity potential, stream function	p. 323
Velocity potential	p. 323
Stream function (2D)	p. 324
Stokes's stream function (axisymmetric)	p. 326
Ideal fluid and ideal gas	p. 327
Curvilinear reference frames: Differential operators	p. 329
Frenet-Serret formula for a space curve	p. 329
Cylindrical coordinates	p. 330
Spherical polar coordinates	p. 332
First three structure functions	p. 335
Lagrangians	p. 337
Galilei invariance and Lorentz invariance	p. 337
Lorentz transformation	p. 337
Lorenz-invariant Galilean Lagrangian	p. 338
Rotation symmetry	p. 340
Solutions	p. 343
References	p. 373
Index	p. 377
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