Table Of ContentEELLEEMMEENNTTAARRYY
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MMEECCHHAANNIICCSS
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ELEMENTARY
F L U I D
MECHANICS
Tsutomu Kambe
Institute of Dynamical Systems, Tokyo, Japan
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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British Library Cataloguing-in-Publication Data
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ELEMENTARY FLUID MECHANICS
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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ISBN-13 978-981-256-416-0
ISBN-10 981-256-416-0
ISBN-13 978-981-256-597-6 (pbk)
ISBN-10 981-256-597-3 (pbk)
Typeset by Stallion Press
Email: [email protected]
Printed in Singapore.
Lakshmi - ElemFluidMechanics.pmd 1 2/27/2007, 9:38 AM
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Preface
This book aims to provide an elementary interpretation on physical
aspects of fluid flows for beginners of fluid mechanics in physics,
mathematics and engineering from the point of view of modern
physics. Original manuscripts were prepared as lecture notes for
intensive courses on Fluid Mechancis given to both undergraduate
and postgraduate students of theoretical physics in 2003 and 2004 at
the Nankai Institute of Mathematics (Nankai University, Tianjin) in
China.
Beginning with introductory chapters of fundamental concepts of
the nature of flows and properties of fluids, the text describes basic
conservation equations ofmass,momentumandenergyinChapter3.
Themotionsofviscousfluidsandthoseofinviscidfluidsarefirstcon-
sidered in Chapters 4 and 5. Emphasizing the dynamical aspects of
fluid motions rather than static aspects, the text describes, in subse-
quent chapters, various important behaviors of fluids such as waves,
vortex motions, geophysical flows, instability and chaos, and tur-
bulence. In addition to those fundamental and basic chapters, this
text incorporates a new chapter on superfluidand quantized vortices
because it is an exciting new area of physics, and another chapter on
gauge theory of fluid flows since it includes a new fundamental for-
mulation of fluid flows on the basis of the gauge theory of theoretical
physics. The materials in this book are taken from the lecture notes
of intensive courses, so that each chapter in the second half may be
read separately, or handled chapter by chapter.
v
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vi Preface
Thisbookiswrittenwiththeviewthatfluidmechanicsisabranch
of theoretical physics.
June 2006 Tsutomu Kambe
Former Professor (Physics), University of Tokyo
Visiting Professor, Nankai Institute
of Mathematics (Tianjin, China)
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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. 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
2.7. Problems . . . . . . . . . . . . . . . . . . . . . . . . 28
vii
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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 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
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
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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 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
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
