Mathematical modeling is the art of describing key aspects of reality with the help of language of mathematics. Mathematical models of physical processes usually take the form of conservation laws written as the set of nonlinear partial differential equations. The solutions of these equations in complex real situations almost always require the usage of the various numerical methods.

The present book is intended to provide an introduction to the models used in fluid mechanics. This book is revised and enlarged by the addition of three chapters on Mathematical Models of Fluid Dynamics (Rainer Ansorge, WILEY-VCH GmBH & Co. KGaA, Weinheim, 2003) and is now titled Mathematical Models of Fluid Dynamics. The term model, as understood in this book, refers to the representation of a particular fluid flows using such categories as appropriate equations (Euler’s equations and the Navier–Stokes equations) and discretization procedures (finite difference schemes and the finite volume method).

The book is organized into ten chapters.

The objective of chapter 1 is to present the basic aspects of the theory of fluid flows, concentrating on the part of this theory related to inviscid (or ideal) fluids. Fundamental conservation equations (mass, momentum, energy) are derived from conservation principles. Emphasis is placed on systems of conservation laws. First the one-dimensional example of gas flow is discussed to elucidate the concept of hyperbolicity. Then the notions of characteristics and singularities are explained through analysis of the scalar Burgers’ equation. The second part of the chapter is devoted to the potential flows, buoyancy, statics and sound waves. Initially the Bernoulli equation is obtained and Kutta–Zhukovsky buoyancy formulae are derived. Next the method of hodographs is briefly sketched and the local speed of sound is introduced.

To enable dealing with the occurrence of discontinuities or singularities, Jean Leray in the early 1930s, suggested the concept of weak, nonclassical solutions to the partial differential equations. Chapter 2 is dedicated to suitable generalization of the definition of a conservation law solution. The subtle questions of existence, uniqueness, and regularity of solutions are briefly outlined. The examples studied are derived directly from physical considerations in applied problems, namely, Greenshields’ traffic flow model and more general model for arbitrary fluxes due to Lighthill and Whitham. The chapter ends with a discussion of the Rankine–Hugoniot (or the jump) condition.

It is well known that the transition from a classic conservation law partial differential equation to its weak representation leads to a loss of uniqueness of the solution. Therefore, the additional rationally motivated conditions are needed to pick out the physically admissible solution. Chapter 3 deals with entropy conditions which allow to select a unique among all the weak solutions. First, the second law of thermodynamics is briefly recalled. Next the generalization of the second law in spirit of Oleinik and Lax is thoroughly described for a one-dimensional gas flow. The following section discusses the uniqueness of the Lax entropy solution and an adequate generalization of the definition of a Lax entropy solution to systems of conservation laws. At last Kruzkov-type notion of entropy solution for one-dimensional scalar problems is presented and explained.

Chapter 4 addresses briefly the Riemann problem. This question is of permanent interest because of many researchers consider weak solutions of conservation laws, in which existence is proved by approximating the prescribed Cauchy problem by a sequence of Riemann problems. First the numerical aspects of the problem are shortly mentioned. Then the Riemann problem for linear systems with constant coefficients is formulated and analysed. The role of the Riemann problem in a second order traffic flow modelling is also elucidated shortly.

In fact, real fluids are never inviscid (or ideal). Generalization of Euler’s equations, needed to account for physical fluid dynamics, introduces additional components in the momentum transport equations, namely, viscous forces. This strategy leads to nonlinear partial differential equations which are referred to as the Navier–Stokes equations (NSE), and is described in chapter 5. The basic equations are derived and, in addition, equations are given for the conservation of energy for viscous flows. Next, the dimensionless form of NSE is presented and the concepts of similarity, as well as Reynolds and Prandtl numbers, are introduced. Thereafter, two examples of laminar viscous stationary flows are described including the spectacular Hagen–Poiseuille flow through a pipe. An idea for a procedure of solving the stationary Navier–Stokes problem, the method of artificial compressibility, comes next. An elegant proof that the appropriate solution of the problem asymptotically converges in the sense of the L 2-norm to the zero function is presented also. The following section contains the elements of the famous Prandtl boundary layer theory and presents specific problems relating to the separation of the boundary layer flow from a curved surface in the case of stationary two-dimensional flows. Subsequently, stability of laminar flows via the Orr–Sommerfeld equation is analysed, along with a short derivation of the equation of the heated real gas flow. Finally, a one-dimensional fluid-dynamic model derived to describe tunnel fires is numerically analysed and studied for several situations.

Chapter 6 concentrates on the elements of the functional analysis of discretization algorithms. The existence of entropy solutions is treated in general with the formalism considered here, and the problem of the existence of a weak solution to a scalar conservation law with the strictly convex flux on a upper (x, t) half-plane, \( x \in \mathbb{R} \), t ≥ 0, is analysed and proved.

The remaining four chapters of the book contain theoretical foundations of constructions of discretization algorithms for hyperbolic conservation laws.

In chapter 7, three main classes of discretization methods for generic scalar equation, namely the finite difference method (FDM), the finite element method (FEM) and the finite volume method (FVM), are briefly mentioned, and basic tools for investigating FDM are thoroughly presented. The finite difference calculus is studied first giving the general criteria of stability of difference schemes including the Courant, Friedrichs and Lèvy condition, the central equivalence theorem of the Lax–Richtmayer theory and the von Neumann stability criterion. Next, some aspects of finite-difference approximations to simple transport equations are analysed by deriving and studying the modified parabolic differential equation with positive diffusion coefficient. Subsequently, the usefulness of difference schemes in conservation form is demonstrated briefly. Finally, the idea of FVM on unstructured grids applied to the scalar two-dimensional conservation law and the concept of continuous convergence of relations are concisely exposed.

Chapter 8 continues, basing on remarks presented in chapter 4, with discussions of numerical tools indispensable in the description of Riemann problems. Included are, among them the monotonicity-preserving three-point schemes, total variation diminishing method, the famous Godunov scheme and the Lax–Friedrichs scheme and its relation with the Godunov approach. An interesting discussion of similarity solutions in a Riemann problem and the complete solution to the Riemann problem in one-dimensional gas dynamics are presented also. The chapter ends with an analysis of Godunov’s method and its various derivatives who have gained increasing popularity in solving the Euler equations.

In the chapter which follows, discrete models on the curvilinear coordinate systems are concisely discussed. The basic transformation relations applicable to the description of curvilinear grids are considered, including transformation of a conservation law in the physical domain into a conservation law in configuration space. The procedures for the generation of curvilinear grids and some remarks concerning grid adaptivity are also presented.

The last chapter concerns FVMs, which are discrete evolution equations for cell averages. The key idea in FVMs is to adjust FDMs to unstructured grids like triangulations and to use the computational possibilities from FDMs and FEMs. This approach is treated in an elegant mathematical framework. Included are the first-order finite volume method, a high-order, conservative finite volume schemes, polynomial and non-polynomial recovery procedures and some remarks referring to grid generation techniques for unstructured grids.

The book is very interesting and well written, and it should prove useful for graduate students and researchers in fluid mechanics, aerodynamics, engineering and technical physics. The level of presentation in this book is not elementary; some chapters are aimed at the specialists in these respective fields. Overall a solid background in linear algebra, theory of differential equations, functional analysis and numerical methods would be very helpful. Some parts of the book I would definitely recommend to anybody giving a basic course on fluid mechanics.