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**Mathematical methods in fluid dynamics.**
*(English)*
Zbl 0819.76001

Pitman Monographs and Surveys in Pure and Applied Mathematics. 67. London: Longman Scientific & Technical. New York: Wiley, xiii, 657 p. (1993).

This book intends to fill a gap between highly-specialized mathematical monographs on flow problems, rather difficult for students and engineers, and a number of books treating fluid mechanics from purely engineering point of view. Since it is impossible to include in the book all the methods and approaches existing now in theoretical and computational fluid dynamics, the author restricts himself to models and methods which can be applied in technology, in particular, the design of turbines and profiles in subsonic, transonic and supersonic regimes. The presentation of material keeps a good balance between the strong derivation of fundamental equations and mathematical validation of their properties, and practical computational methods.

The book is subdivided in nine chapters. After the introductory chapter 1 dealing with basic laws of continuous media and thermodynamics, chapter 2 commences with irrotational incompressible flows. The main attention is paid here to plane flows; a whole range of methods for solution of corresponding boundary value problems is presented, namely complex variable technique, conformal mappings, method of integral equations, finite difference and finite element methods. The concept of weak solution is introduced, and some solvability problems are investigated.

Chapter 3 formulates, under simplifying assumptions, basic boundary value problems describing compressible and/or rotational flows in channels, past profiles and through cascades of profiles. The variational reformulation of these problems and detailed analysis of existence and uniqueness of weak solution are given in chapter 4. In addition, the reader is informed here about finite element discretization of the problems under consideration, convergence of finite element approximations, and numerical realization of the discrete problems. Chapter 5 discusses some complications introduced by Kutta-Joukowski trailing condition. Much of the material presented in chapters 3, 4 and 5 is based on original work by the author over the last ten years.

Chapter 6 is devoted to stationary inviscid transonic flows, mainly to the potential flow problems. The emphasis is placed on the entropy compactification of transonic potential flows and on their finite element simulations. The next chapter 7 treats inviscid gas flows with the use of complete systems of conservation laws by means of the theory of nonlinear first-order hyperbolic systems. The topics covered here include fundamental theoretical results, numerical solution of hyperbolic equations in one and two dimensions, measure-valued solutions to conservation laws.

Chapter 8 develops mathematical theory and numerical methods for viscous flows. The main attention is focused on the incompressible Navier-Stokes equations and their Oseen linearization as well as on the finite difference and finite element methods. Finally, chapter 9, thought as an appendix, contains necessary concepts and results of functional analysis which makes the book self-contained.

On balance, the material is well selected and clearly presented. The book encourages a systematic study of the subject and can be warmly recommended to all mathematically inclined students, practicians, and specialists working in fluid dynamics.

The book is subdivided in nine chapters. After the introductory chapter 1 dealing with basic laws of continuous media and thermodynamics, chapter 2 commences with irrotational incompressible flows. The main attention is paid here to plane flows; a whole range of methods for solution of corresponding boundary value problems is presented, namely complex variable technique, conformal mappings, method of integral equations, finite difference and finite element methods. The concept of weak solution is introduced, and some solvability problems are investigated.

Chapter 3 formulates, under simplifying assumptions, basic boundary value problems describing compressible and/or rotational flows in channels, past profiles and through cascades of profiles. The variational reformulation of these problems and detailed analysis of existence and uniqueness of weak solution are given in chapter 4. In addition, the reader is informed here about finite element discretization of the problems under consideration, convergence of finite element approximations, and numerical realization of the discrete problems. Chapter 5 discusses some complications introduced by Kutta-Joukowski trailing condition. Much of the material presented in chapters 3, 4 and 5 is based on original work by the author over the last ten years.

Chapter 6 is devoted to stationary inviscid transonic flows, mainly to the potential flow problems. The emphasis is placed on the entropy compactification of transonic potential flows and on their finite element simulations. The next chapter 7 treats inviscid gas flows with the use of complete systems of conservation laws by means of the theory of nonlinear first-order hyperbolic systems. The topics covered here include fundamental theoretical results, numerical solution of hyperbolic equations in one and two dimensions, measure-valued solutions to conservation laws.

Chapter 8 develops mathematical theory and numerical methods for viscous flows. The main attention is focused on the incompressible Navier-Stokes equations and their Oseen linearization as well as on the finite difference and finite element methods. Finally, chapter 9, thought as an appendix, contains necessary concepts and results of functional analysis which makes the book self-contained.

On balance, the material is well selected and clearly presented. The book encourages a systematic study of the subject and can be warmly recommended to all mathematically inclined students, practicians, and specialists working in fluid dynamics.

Reviewer: O.Titow (Berlin)

### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76B10 | Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76Mxx | Basic methods in fluid mechanics |

35Q30 | Navier-Stokes equations |