Elementary fluid mechanics / Tsutomu Kambe.

Format:

Book

Author/Creator:

Kambe, Tsutomu.

Language:

English

Subjects (All):

Fluid mechanics.

Physical Description:

xv, 386 pages : illustrations ; 23 cm

Place of Publication:

Hackensack, N.J. : World Scientific, [2007]

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, 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. Book jacket.

Contents:

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

2 Fluids 17

2.1 Continuum and transport phenomena 17

2.2 Mass diffusion in a fluid mixture 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

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 of motion 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 of motion 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

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

5.6 Examples of irrotational incompressible flows (3D) 88

5.6.1 Source (or sink) 88

5.6.2 A source in a uniform flow 90

5.6.3 Dipole 91

5.6.4 A sphere in a uniform flow 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 uniform flow 100

5.8.3 Dipole 101

5.8.4 A circular cylinder in a uniform flow 102

5.8.5 Point vortex (a line vortex) 103

5.9 Induced mass 104

5.9.1 Kinetic energy induced by a moving body 104

5.9.2 Induced mass 107

5.9.3 d'Alembert's paradox and virtual mass 108

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

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

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 of motion 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

8 Geophysical flows 177

8.1 Flows in a rotating frame 177

8.2 Geostrophic flows 181

8.3 Taylor-Proudman theorem 183

8.4 A model 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

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 (v = 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

10 Turbulence 239

10.1 Reynolds experiment 240

10.2 Turbulence signals 242

10.3 Energy spectrum and 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

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

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

12.6.2 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

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

Appendix A Vector analysis 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 [delta] function 320

Appendix B Velocity potential, stream function 323

B.1 Velocity potential 323

B.2 Stream function (2D) 324

B.3 Stokes's stream function (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

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.

Notes:

Includes bibliographical references (pages 373-375) and index.

ISBN:

9789812565976

9812565973

OCLC:

85779367