## Elementary Fluid MechanicsThis 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. |

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### Contents

1 Flows | 1 |

2 Fluids | 17 |

3 Fundamental equations of ideal fluids | 31 |

4 Viscous fluids | 45 |

5 Flows of ideal fluids | 77 |

6 Water waves and sound waves | 115 |

7 Vortex motions | 143 |

8 Geophysical flows | 177 |

12 Gauge theory of ideal fluid flows | 277 |

Appendix A Vector analysis | 315 |

Appendix B Velocity potential stream function | 323 |

Appendix C Ideal fluid and ideal gas | 327 |

Differential operators | 329 |

Appendix E First three structure functions | 335 |

Appendix F Lagrangians | 337 |

Solutions | 343 |

9 Instability and chaos | 203 |

10 Turbulence | 239 |

11 Superfluid and quantized circulation | 263 |

373 | |

377 | |

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### Common terms and phrases

axis boundary condition boundary layer called coeﬃcient component consider const constant continuity equation convection coordinate covariant derivative cylinder deﬁned denotes density derivative diﬀerent diﬀusion divv Earth Simulator energy entropy equation of motion expression ﬁrst ﬁxed point flow ﬂow ﬁeld fluid ﬂuid ﬂows ﬂuid motion ﬂuid particle ﬂux frame Galilean transformation gauge given grad Hence horizontal ideal ﬂuid incompressible inﬁnitesimal integral invariant inviscid irrotational Lagrangian linear Lorenz attractor mass momentum normal obtain perturbation plane pressure Problem radius represented respect Reynolds number right-hand side rotation satisﬁes solution space steady stream function stream-line superﬂuid Suppose surface surface integral symmetry temperature tensor term theorem thermal thermal conduction tion transformation turbulence uniform ﬂow vanishes variables variation vector velocity ﬁeld velocity potential viscous viscous ﬂuid vortex vortex ring vortex-line vorticity vorticity equation wave wavenumber