Elementary Fluid Mechanics

by Tsutomu (Jixin) Kambe

Overview

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, this book also covers water waves and sound waves, vortex motions, geophysical flows, nonlinear instability, chaos, and turbulence. Furthermore, it includes the chapters on superfluids and the gauge theory of fluid flows.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.

Product Details

ISBN-13:	9789812565976
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	01/05/2007
Edition description:	New Edition
Pages:	404
Product dimensions:	6.00(w) x 8.90(h) x 1.10(d)

Table of Contents

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