Fluid dynamics of viscoelastic liquids / Daniel D. Joseph

Elastic solids and viscous fluids are two extremes of material behavior. Viscoelastic fluids such as polymers show intermediate behavior; they have a temporary elastic response associated with a relaxation time. Actually, real fluids do not have a single relaxation time but a widely spread spectrum of relaxation times. If the time scale of the flow is appropriate, then the fluid can be approximated by a model with one or a few relaxation times; shorter relaxation times can be lumped into a viscous contribution. If this viscous contribution is negligible, the equations of motion are hyperbolic. The basic theme of this book is the study of effects such as wave propagation and change of type which result from the hyperbolic nature of the equations. The author has worked extensively on this subject since approximately 1983, and the material in the book is based primarily on work by himself and his collaborators.
The beginning three chapters of the book introduce the equations of motion and the constitutive relations used. This is followed by a discussion of the basic notions of Hadamard instability, characteristics and classification of type. The main part of the book is concerned with determining characteristics and criteria for change of type in specific flows. Possible relationships between change of type and experimentally observed effects are discussed. The flows that are studied include flow past bodies, in particular flow past a flat plate, sink flow, elongational flows, flow between rotating parallel plates, Poiseuille flow, delayed die swell and flow between rotating cylinders. The chapters that follow discuss more classical issues, namely perturbation expansions for constitutive laws and in particular second order fluids. This is followed by a general discussion of elasticity and viscosity of liquids, including many historic references. The final chapters are concerned with linear and nonlinear wave propagation.
The book ought to be of interest to two kinds of readers: to rheologists who want to learn about an important aspect of the mathematical theory of viscoelastic fluids and its implications for numerical simulation and experiments, and to mathematicians working in PDEs who are looking for good problems. (MSN)