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Elementary fluid mechanics. (English) Zbl 1118.76001

Hackensack, NJ: World Scientific (ISBN 978-981-256-416-0/hbk; 978-981-256-597-6/pbk). xv, 386 p. (2007).
Based on his lecture notes for courses on fluid mechanics given by the author to both undergraduate and graduate students at the Nankai Institute of Mathematics (Nankai University, Tianjin) in China, the author published this modern introduction to elementary fluid mechanics. In addition to the fundamental aspects of fluid mechanics, this book covers traditional material on water waves and sound waves, vortex dynamics, geophysical fluid flows, instability and chaos, and turbulence. Included are two chapters on non-traditional but modern aspects of superfluid flows and quantized circulation and gauge theory of ideal fluid flows. The book has twelve chapters, six appendices, solutions of problems at the end of each chapter, references and an index.
Chapter 1 and 2 deal with fundamental ideas of ideal and viscous fluid flows. The basic equations of mass conservation, momentum conservation and energy conservation are described briefly in Chapter 3. Chapter 4 is concerned with standard topics including Navier-Stokes equations of viscous fluids in both Cartesian and cylindrical polar coordinates, equations of boundary layer flows, steady and unsteady fluid flows, low-Reynolds number flows, flows around a circular cylinder, drag and lift coefficients. Flows of ideal fluids, Bernoulli’s equation, Kelvin’s circulation theorem, potential flows, two- and three-dimensional irrotational incompressible fluid flows, complex potentials, d’Alembert paradox and virtual mass are discussed in Chapter 5.
Chapter 6 is devoted to water waves and sound waves, surface gravity waves, KdV equation for long waves in shallow water and shock waves. Vortex motions, equations of vorticity, Helmholtz’s theorem, axisymmetric vortices with circular vortex lines, an integrable filament equations, and Burgers’ vortex with swirl are topics covered in Chapter 7. Chapter 8 is concerned with geophysical flows, Taylor-Proudman theorem, axisymmetric swirling flow in a rotating frame and Ekman boundary layer, Rossby wave equation and stratified fluid flows, and global motions by the Earth simulator.
Chapter 9 deals with instability and chaos. Included are linear stability theory, Kelvin-Helmholtz instability theory, stability of parallel flows, thermal convection and linear stability analysis. Special attention is given to derivation of Lorenz equations, Lorenz attractor and deterministic chaos. Turbulence, vortex structure in turbulence, energy spectrum and energy dissipation, Kolmogorov-Obukhov spectral law of turbulence are briefly discussed in Chapter 10.
Superfluid and quantized circulation, quantum mechanical description of superfluid flows, Bose gas, Gross-Pitaevskii equation, and quantized vortices are presented in Chapter 11. Special attention is given to Bose-Einstein condensation (BEC), BEC in dilute alkali-atomic gases, and vortex dynamics in rotating BEC condensates. It is shown that the realization of BEC of dilute alkali-atomic gases is the most remarkable experimental discovery in the last decades at the end of the 20th century.
The final Chapter 12 deals with a modern gauge theory of ideal fluid flows. Based on the flow field on the basis of the gauge principle, both translation symmetry and rotation symmetry of flow field are discussed. Considerable attention has been given to variational formulation for ideal fluid flows, Noether’s theorem and action principle.
In order to make this book self-contained, the author included six appendices including vector analysis, velocity potential and stream function, ideal fluid and ideal gas, curvilinear reference frames, differential operators, first structure theorem and Lagrangians.
In summary, the book contains an excellent basic presentation of theory, applications, and methods of solutions for various problems of classical and modern fluid dynamics. This book seems to be suitable as a senior undergraduate and graduate level text book in Asian universities. However, this is likely to be less successful as a text in universities in the western countries because of lack of sufficient worked examples of applications and exercises.
Finally, many selected text books, research monographs and research papers related to the subject have been included in a short list of references so that they may serve to stimulate new interest in future advanced study and research. However, the author omitted many important books and monographs related to the subject matter of this book that include: L. Debnath, Nonlinear water waves, Academic Press (1994; Zbl 0793.76001); M. J. Lighthill, Waves in fluids, Cambridge University Press, Cambridge (1978; Zbl 0375.76001).; M. J. Lighthill, An informal introduction to theoretical fluid mechanics, Oxford University Press, Oxford (1986; Zbl 0604.76002); G. B. Whitham, Linear and nonlinear waves. John Wiley, New York (1974; Zbl 0373.76001).

MSC:

76-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to fluid mechanics
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