An exposition of the derivation and use of equations of motion for two-phase flow. The approach taken derives the equations of motion using ensemble averaging, and compares them with those derived from control volume methods. Closure for dispersed flows is discussed, and some fundamental solutions are given. The work focuses on the fundamental aspects of two-phase flow, and is intended to give the reader a background for understanding the dynamics as well as a system of equations that can be used in predictions of the behavior of dispersed two-phase flows. The exposition in terms of ensemble averaging is new, and combining it with modern continuum mechanics concepts makes this book unique. Intended for engineering, mathematics and physics researchers and advanced graduate students working in the field.

The purpose of this four volume series is to make available for college teachers and students samples of important and realistic applications of mathematics which can be covered in undergraduate programs. The goal is to provide illustrations of how modem mathematics is actually employed to solve relevant contemporary problems. Although these independent chapters were prepared primarily for teachers in the general mathematical sciences, they should prove valuable to students, teachers, and research scientists in many of the fields of application as well. Prerequisites for each chapter and suggestions for the teacher are provided. Several of these chapters have been tested in a variety of classroom settings, and all have undergone extensive peer review and revision. Illustrations and exercises are included in most chapters. Some units can be covered in one class, whereas others provide sufficient material for a few weeks of class time. Volume 1 contains 23 chapters and deals with differential equations and, in the last four chapters, problems leading to partial differential equations. Applications are taken from medicine, biology, traffic systems and several other fields. The 14 chapters in Volume 2 are devoted mostly to problems arising in political science, but they also address questions appearing in sociology and ecology. Topics covered include voting systems, weighted voting, proportional representation, coalitional values, and committees. The 14 chapters in Volume 3 emphasize discrete mathematical methods such as those which arise in graph theory, combinatorics, and networks.

This IMA Volume in Mathematics and its Applications PARTICULATE FLOWS: PROCESSING AND RHEOLOGY is based on the proceedings of a very successful one-week workshop with the same title, which was an integral part of the 1995-1996 IMA program on "Mathematical Methods in Materials Science." We would like to thank Donald A. Drew, Daniel D. Joseph, and Stephen L. Passman for their excellent work as organizers of the meeting. We also take this opportunity to thank the National Science Foun dation (NSF), the Army Research Office (ARO) and the Office of Naval Research (ONR), whose financial support made the workshop possible. A vner Friedman Robert Gulliver v PREFACE The workshop on Particulate Flows: Processing and Rheology was held January 8-12, 1996 at the Institute for Mathematics and its Applications on the University of Minnesota Twin Cities campus as part of the 1995- 96 Program on Mathematical Methods in Materials Science. There were about forty participants, and some lively discussions, in spite of the fact that bad weather on the east coast kept some participants from attending, and caused scheduling changes throughout the workshop. Heterogeneous materials can behave strangely, even in simple flow sit uations. For example, a mixture of solid particles in a liquid can exhibit behavior that seems solid-like or fluid-like, and attempting to measure the "viscosity" of such a mixture leads to contradictions and "unrepeatable" experiments. Even so, such materials are commonly used in manufacturing and processing."

An exposition of the derivation and use of equations of motion for two-phase flow. The approach taken derives the equations of motion using ensemble averaging, and compares them with those derived from control volume methods. Closure for dispersed flows is discussed, and some fundamental solutions are given. The work focuses on the fundamental aspects of two-phase flow, and is intended to give the reader a background for understanding the dynamics as well as a system of equations that can be used in predictions of the behavior of dispersed two-phase flows. The exposition in terms of ensemble averaging is new, and combining it with modern continuum mechanics concepts makes this book unique. Intended for engineering, mathematics and physics researchers and advanced graduate students working in the field.