Asymptotic Analysis and Boundary Layers Jean Cousteix and Jacques Mauss Series: Scientific Computation Springer (2007), 432 pp., Price: US$139.00 ISBN 978-3-540-46488-4.
Reviewer: Ferdinand Verhulst, Mathematics Institute, University of Utrecht, The Netherlands

This is an extended version of the French version of this book that appeared in 2006. The novelty of this book lies in the use of thorough and advanced singular perturbation methods in combination with detailed applications to fluid mechanics. In the first chapters asymptotic methods are discussed, in particular an in-depth treatment of matched asymptotic expansions followed by the development of the so-called Successive Complementary Expansion Method (SCEM). This method, SCEM, constitutes (in chapter 5) a new approach producing uniformly valid approximations while circumventing many of the classical matching problems. Although various matching principles, like Van Dyke's or intermediate matching, work quite well in a large number of cases, all of them present occasional problems and pitfalls, showing that the fundamentals of matching techniques are still not in a satisfactory state.

Second order ODEs are the first examples of SCEM applications involving a number of notorious difficult problems. Other chapters are devoted to high Reynolds number flow, interactive boundary layers, various applications of boundary layer interaction, turbulent boundary layers and channel flow. Comparison of asymptotic, experimental and numerical results are often presented. Five appendices give additional technical details.

There are 116 references, a solid index and moreover the book contains many exercises; the section with solutions of the exercises consists of 85 pages. Still, it should be clear that this is not a standard introductory textbook. Although it is self-contained, students will be more comfortable with the book if they have studied an elementary book on perturbation theory first. In the combination of asymptotics and fluid mechanics in such a fundamental and detailed approach, the book is unique in the science literature.