#### Foreword

Writing a new fluid dynamics textbook is a challenging task. In 1895, Sir Horace Lamb established a very high standard with the first edition of Hydrodynamics. This classical presentation was followed by other excellent introductions into the field of fluid mechanics, among them Landau and Lifshitz, 1959, and Batchelor, 1967. The strength of M.Belevich’s book is in its rigorous and systematic approach to developing the mathematical model of fluid dynamics from the first principles. It carefully explains the underlying hypothesis and simplifications used to establish equations that govern motions of a fluid. Extensive use of vector and tensor analysis results in a compact and generalized narrative, without the restrictions of a particular coordinate system.

This textbook is by no means a comprehensive description of the field of fluid dynamics. Some of important problems (*e.g.* waves) were deliberately left out of the book’s framework. Since the text is based on a course that is taught to students who specialize in geophysical fluid dynamics, more engineering aspects of fluid mechanics (such as turbomachines and airfoils) are also not covered.

The book’s content not only provides a general description of fluid dynamics, but also teaches how to apply universal principles to build a mathematical model of a particular problem. The distinctive feature of M. Belevich’s book is a somewhat non-standard approach of describing the dynamics of fluid from the point of view of the observer (chapter 15). It allows to underline some physical aspects of fluid mechanics which are usually not explicitly established in most textbooks.

The book is complemented by a carefully selected set of exercises. It provides consistent and self-sustained introduction to fluid dynamics, giving enough details to be used either in class or for self-study. It can be used to acquire knowledge in particular aspects of hydromechanics, and also as a source of inspiration for students, researchers and teachers in the field of classical fluid mechanics.

**Ilya Rivin**

Environmental Modeling Center

National Weather Service

National Oceanic and Atmospheric Administration

USA

#### Preface

This book presents the basis of the classical fluid mechanics and its content corresponds to a one-semester course which I am teaching from past several years to the 2^{nd} year students of the Russian State Hydrometeorological University in St. Petersburg.

The goal of this book is twofold. Firstly, I wanted to provide a reader with a holistic idea of the fluid model and the way it is constructed. To show him, how the model of the fluid is developed, what main hypotheses lie in its basis and what general conclusions based on observations (the so-called laws of nature) make up the model. Secondly, I wished to demonstrate some possible modifications of the initial model which either make the model applicable in some special cases (viscous or turbulent fluid) or simplify it in accordance with peculiarity of a particular problem (hydrostatics, two-dimensional flows, boundary layers, *etc.*).

The whole theoretical material of the book naturally falls into two parts. The first part is fully dedicated to development of the model of the fluid in the Cauchy form. Here, the basic notions are introduced, main hypotheses are discussed and necessary postulates, which actually make up the model of continuum, are formulated. Non-coordinate tensor form of equations is actively used. This shortens formulas and makes results more readable. With that end in view, a brief introduction in tensor analysis is given in Ch.4. This part results in derivation of the perfect fluid model which turns out to be the simplest although quite efficient model.

In the second part of the book the most important modifications of the developed model are considered. First of all this concerns the redefinition of the stress tensor which is needed when viscosity is taken into account. Another important modification is connected with averaging of equations of the model which is necessary in case of turbulent flows. The concept of the boundary layer is also rather fruitful. Both laminar and turbulent boundary layers are discussed in Ch.14.

It is clear, that all this does not exhaust theoretical fluid mechanics, and that in the study of many important problems, it is necessary to refer to other books, at times rather special. However the basis of all such particular cases of the fluid mechanics is the same, and this book is aimed to discuss this topic.

Exercises and problems which are solved by students in practical classes are integral part of this book. They are chosen so as to teach students to work with complex systems of differential equations, since different fluid models are just such. We are training skills in writing equations in vector-matrix form, transition to component form of notation, applying of the index summation convention. Special attention is paid to formalizing of a verbal description of a problem (choice of coordinate systems, their orientation, accounting directions and symmetries inherent in the problem, *etc.*) as well as the mathematical problem posing. The system of fluid mechanics equations is quite complex and does not have analytic solutions in most interesting cases. Therefore, the main goal of these exercises is to elaborate the ability to see a particular problem from different viewpoints and to estimate its possible simplifications.

Theoretical fluid mechanics is very mathematized discipline, so the reader must meet certain requirements. Knowledge of the following topics of algebra and calculus is assumed: determinants, matrices, eigenvalue problem, vector spaces, calculus, vector analysis, differential equations.

All required information about tensors is given in Ch.4.

#### Conflict of interest

The author confirms that author has no conflict of interest to declare for this publication.

### Acknowledgements

Besides my explicit and implicit teachers, I would like to especially thank those of my friends and colleagues, who patiently and sometimes willingly discussed with me various aspects of this course. These are primarily Dr. S.A.Fokin and A.A.Tron’. Their remarks, comments and suggestions were very helpful.

**M.Belevich**

St.Petersburg

Russia

#### List of Contributors

##### Author(s):

**Michael Belevich **

St. Petersburg

Russia