7R60. Transport Modeling in Hydrogeochemical Systems. Interdisciplinary Applied Math, Vol 15. - JD Logan (Dept of Math and Stat, Univ of Nebraska, Lincoln NE 68588-0323). Springer-Verlag, New York. 2001. 223 pp. ISBN 0-387-95276-4. $69.95.
Reviewed by J Koplik (Levich Inst, CCNY, 138 St and Convent Ave, New York NY 10031).
The book is a very well-written monograph which discusses the theory and some analytic solution methods for a number of the partial differential equations (PDEs) which arise in modeling transport processes in geological situations. The book is designed to appeal to both the mathematics and applications communities, and it does a good job of presenting interesting material on both aspects. While quite readable on its own, the book could be used for a special topics course in either applications of PDEs or the mathematics of transport phenomena, respectively, for its two intended audience groups.
The transport phenomena considered here are diffusion, advection, adsorption, and reaction, usually expressed in terms of a single PDE, or occasionally a coupled system of them.
The general theory covers boundary conditions, Greens’ functions, maximum principles, wave propagation, some aspects of stability and nonlinearity, and numerous tricks which prove useful in particular situations. Numerical methods are very briefly discussed in appendices. The principal applications are to groundwater transport, filtration processes, solution kinetics, and reactive fluid flows. The physical basis of the equations considered is given in enough detail to be convincing to all but the specialist, and in a way that allows the mathematical results to be interpreted physically. The motivation for the models studied, the relevant PDE theory, and the applications to specific problems are all presented in a unified fashion, rather than sequentially, and as a result one gets a feeling for the subject instead of a collection of facts.
The mathematical level is that of graduate students in mathematical sciences or engineering, meaning that the prerequisite is something along the lines of a standard graduate mathematical methods course, the basic theorems are proven in a convincing, but non-rigorous way, and some of the less trivial general results are left for the references. A mathematics student or a person who has previously studied partial differential equations might find these parts of the book elementary or familiar, but could still learn a great deal from the numerous examples included.
The book is not quite a practical guide, since it emphasizes problems which have an interesting PDE aspect to them and in addition focuses on “clean” mathematical problems, without delving into the effects of material heterogeneity and uncertainty in modeling parameters, which are often the major difficulties in real situations. However, the insight obtained from the basic mathematics is certainly presented clearly. The typography in the book is attractive, the figures are clear, and the price is typical for volumes of this sort. The general references to the mathematics literature are fine, but those to the applications seem somewhat limited to the author’s experience and interests. Transport Modeling in Hydrogeochemical Systems is certainly suitable for libraries and as a collection of problems for a general PDE course. It would be a useful reference and source of hints for engineers and scientists studying transport problems.