In Partial Differential Equations in Fluid Dynamics, authors Isom Herron and Michael Foster sought to “create a book that would include many of the techniques that [they] have learned one way or another but are quite simply not in books.” The result is a rigorous and refreshing text appropriate for graduate students in mathematics, as well as upper-level undergraduate students in math and engineering.
Each of the book’s eight chapters begin with a rather informal preamble; here Herron and Foster give the reader a sense of the discussion to follow, but more importantly, place in context the interplay between physical phenomena and underlying mathematical theory. The first sentence of the first preamble leaves the authors’ priorities in no doubt: “Complex analysis is the foundation for everything in this book.” Thus chapter one provides a solid theoretical framework for the analytical techniques introduced later. Harmonic functions, integration and Cauchy’s Theorem, residues and Jordan’s lemma are highlighted early. Like the rest of the book, this section is rich with worked examples, exercises and references. Graphical representations of problem-specific paths of integration are introduced here, and I was pleasantly surprised to find they run throughout the text.
In each of the five following chapters, Herron and Foster isolate one specific technique essential for understanding fluid dynamics. These are: special functions, eigenvalue problems, Green’s Functions, and Laplace and Fourier transforms. In each discussion, the authors give a largely constructive approach to finding solutions of PDE. Their careful treatment, especially in the chapter on special functions, replaces a conceit too common in applied mathematics texts: if we look for solutions of the form… The book is richer for its abandonment of these “rabbit-out-of-a-hat” solutions; the extra discussion it takes to construct early-principle methods pays off in historical accuracy. This approach may well help foster in students a respect for some of the giants — Euler, Legendre, Fourier, Laplace and Chebyshev — behind it all.
The authors focus on the treatment of particular physical problems in fluid dynamics in the final two chapters. With a primer on asymptotic expansions and the application of techniques developed so far, problems like lee waves, Kelvin-Helmholtz instability, plane Couette flow and the boundary layer signal problem receive noteworthy attention.
In the classroom, a book with this level of difficulty, rigor and scope would find perhaps its best audience among first-year graduate students in applied mathematics. However, undergraduate math and engineering students (with a solid background in advanced calculus) could also benefit from the book. With its easy narrative and ample references directing further study, the text could also serve as a springboard in a project-oriented undergraduate course.
Matthew Glomski is assistant professor of mathematics at Marist College in Poughkeepsie, New York, and a Project NExT fellow (red dot '08). He can be reached at Matthew.Glomski@marist.edu.
1. Review of analytic function theory; 2. Special functions; 3. Eigenvalue problems and eigenfunction expansions; 4. Green’s functions for boundary-value problems; 5. Laplace transform methods; 6. Fourier transform methods; 7. Particular physical problems; 8. Asymptotic expansions of integrals.