Introduction

Abstract

A prime characteristic of many flows of multicomponent materials is that there is uncertainty in the exact locations of the particular constituents at any particular time. For some predictions, this is not important. Often, we are concerned with more gross features of the motion. This means that, for equivalent macroscopic flows, there will be uncertainty in the locations of particular constituents for all times. For instance, consider a suspension of small particles in a liquid. If such a suspension is to be used in, for example, a falling ball viscometer, it might be mixed outside the viscometer, then placed in that device prior to the conduction of an experiment with a falling ball. In a properly conducted experiment, the particles would be approximately uniformly distributed in the fluid. However, there would be no assurance that a particular point in the fluid would contain a particle or not. Experiments conducted with nominally identical fluids in identical viscometers would yield identical gross results, even though the exact initial locations of the suspended particles in the two experiments were noticeably different. As another example, in a sedimenting suspension, the exact distribution of the locations of the particles is immaterial as long as they are reasonably “spread out.” We would not allow the particles to be lumped in some way that is not consistent with the initial conditions appropriate for the flow; neither would we allow the particles to be “paired” so that very near each particle is exactly one neighbor. A more interesting case is the set of all experiments with the same boundary conditions, and initial conditions with some (undefined) properties that we would like to associate with the mean and distribution of the particles and their velocities. We call this set an ensemble. Such ensembles are reasonable sets over which to perform averages because variations in the details of the flows are assured in all situations, while at the same time variations in the gross flows cannot occur.