9R45. Nonlinear Flow using Dual Reciprocity. Topics in Engineering, Vol 38. - WF Florez (Pontifical Bolivariana Univ, Colombia). WIT Press, Southampton, UK. 2001. 132 pp. ISBN 1-85312-860-0. $97.00.Reviewed by PM Adler (IPGP, 4 Place Jussieu, Paris Cedex 05, 75252, France).The purpose of this concise short book is to develop a numerical technique to solve nonlinear problems in fluid mechanics using the Boundary Element Method (BEM). The book starts by an introductory chapter that surveys the BEM applied to the solution of nonlinear flow problems and the related literature. It provides the objectives and motivations of this volume. An overview of the book ends this introduction. One can roughly distinguish three major parts; a first one that includes Chapters 2, 3, and 4, introduces some general tools. Chapters 5, 6, and 7 present the multi-domain formulation of the Navier-Stokes equations and of the Stokes equations for non-Newtonian fluids, and a numerical solution of nonlinear systems of algebraic equations. A last part composed by Chapters 8 and 9 provides two examples of application.Chapter 2 is a useful derivation and compilation of classical formula such as Green’s identities for scalar fields, which are immediately applied to the linear Poisson equation (1)∇2u=bIt is hard to follow the reasoning if this elementary example is not detailed. It can be shown that u at the point ξ can be expressed as (2)cξuξ=∫Γν∂u∂ndΓ−∫Γu∂ν∂n dΓ-∫ΩνbdΩwhere ν is a solution of the equation (3)∇2v=−δx−ξc(ξ) is a constant which depends on the position of the point ξ. Equation (2) clearly shows that u depends on the boundary values of u and of its derivatives along the boundary Γ, but also on values within the domain Ω because of the presence of the last integral. Such a term can be avoided either if one knows a particular solution of Eq. (1) which is not often the case, or if b can be expanded in a series of functions, which is the principle of the Dual Reciprocity Method (DRM). The rest of Chapter 2 is devoted to the derivation of the Green’s identities for a Newtonian incompressible fluid. Basically, the resulting equations have the same form as Eq. (2); it is important to note that the nonlinear terms due to inertia generate an integral over the domain Ω. The classical Stokeslets and Stresslets are defined. Chapter 3 presents the Boundary Element Method; this name stems from the fact that the boundary integrals are evaluated numerically over segments or elements into which the boundary is divided. In order to avoid difficulties due to the integral over the domain itself, the methodology is illustrated by the Stokes equations in two dimensions. The discretized equations are put under a matrix form and a series of technical difficulties such as the numerical evaluation of weakly singular integrals and the discontinuities at corner points are addressed. The chapter is ended by a section on the multi-domain technique where a complex domain can be decomposed into several domains; at the boundaries between the domains, continuity of the velocities and of the stresses is required. Since the Navier-Stokes equations contain nonlinear terms, the equivalent of Eq. (2) will contain a volume integral. The Dual Reciprocity Approximation consists of expanding the inertial terms as a series of known interpolating functions; the possible choices which have been made in the literature, are briefly reviewed. A series of technical problems are then addressed such as the treatment of the pseudo-body forces term and the approximation of derivatives. Finally, a mass conservative interpolation for the velocity derivatives is proposed in two dimensions. Chapters 5 and 6 start the multi-domain formulation of the Navier-Stokes equations and of the Stokes equations for non-Newtonian fluids. Basically, these two chapters end up with an elegant matrix formulation of the equations to be solved. Chapter 6 contains unexpected details about the physicochemical aspects of non-Newtonian fluids, and still more surprising about their industrial applications. Chapter 7 provides a numerical technique which is well suited to the solution of nonlinear systems of algebraic equations that arise from the boundary element method. In a few pages, the author succeeds in giving a simple presentation of the main ideas and then in applying them to the BEM equations. Finally, the last two chapters are devoted to numerical applications of the techniques developed so far; Chapter 8 adresses, in a detailed manner, driven cavity flows. Chapter 9 first successfully compares analytical solutions for power law fluids to numerical data for slit flow and Couette flow; then, flows in mixers are determined for the same fluids. This chapter is ended by a long section which should stand alone as a separate chapter in order to provide an overview over the whole work. This short book presents in a condensed way how to implement the BEM to Navier-Stokes equations and power law fluids. It is reasonably self-contained, and as such, it is a valuable tool for any beginner in the field. It should have been very useful to find in this book the basic routines in one language or another. This book is written in a pleasant way with an apparently very small number of errors; this reviewer could only find one in formula (2.29) where S obviously stands for Γ. A minor criticism is the use of component notation for vectors and tensors; a dyadic formulation of all the formula would have provided a much more efficient framework. Another minor criticism is the lack of relation of the presented material with what has been done in the literature; for instance, is the technique presented in Chapter 7 comparable to the well-known Picard technique alluded to in page 97. In the same vein, an index would have probably proved to be useful. Moreover, a finite volume formulation could be made to discretize the integrals over the boundary; has this been done? How does it compare with the proposed approach? This reviewer also finds the absence of any physical comments whatsoever sometimes puzzling. For instance, Eq. (2.29) is a very intuitive result. The terms of Stokeslets and Stresslets should have been commented more thoroughly in physical terms since there was a large use of these concepts in the field of low Reynolds numbers hydrodynamics in the seventies. To conclude, Nonlinear Flow Using Dual Reciprocity represents a valuable tool that should be present on the shelves of any beginner in the field. This reviewer also strongly recommends it as a basis for a specialized course on the subject.