The basis of this monograph is a series of lectures that the author, a renowned expert on mathematical rheology, presented at the University of Delaware in 1999 as part of an NSF-CBMS Regional Research Conference. The book starts with a collection of wonderful examples that set the tone for the rest of the manuscript by exhibiting pictures of experiments that crystallize the main differences between Newtonian and non-Newtonian fluid dynamics. Chapter 2 presents the governing and constitutive equations of a variety of models that are currently under investigation by a host of researchers. Chapter 3 follows the plan of standard texts in fluid dynamics by investigating the mathematical impact of the various models presented in the previous chapter in the context of simple flows. The state of existence theory for initial value problems as well as steady flows is the subject of chapter 4. The contents of chapter 5 are built on these existence theories; state-of-the-art numerical methods are presented for the special nonlinear structures one confronts in non-Newtonian fluids. In chapters 6 through 10, the author returns to the original theme of the book by addressing concrete examples of flows, such as reentrant corner singularities, and motivates his discussion on instabilities. A short but complete treatment of the center manifold theorem and Hopf bifurcation makes chapter 8 an invaluable source for these important modern tools of mathematical analysis. In fact, this monograph takes the reader on a tour of some of the fundamental problems mathematicians have addressed in the past few decades.

I found the book enjoyable to read and easy to follow. The author’s style of writing and choice of topics are excellent. The book points to several areas for future research, and offers an important resource for graduate students in mathematics, mechanics, and physics.