Introduction

We begin our study by considering suspensions so dilute that each particle behaves independently of others. In this situation, the dynamic behavior of a particle is determined by its physical properties and by the pressure, velocity, and temperature fields in the fluid surrounding it. When no particles are present, those fields may be regarded as known. But particles introduce generally unknown disturbances that modify them and that obey the same equations as the main, or background, field. These are the equations of fluid mechanics; they will be needed throughout the book and are presented herein without derivation.

Equations of Motion

The equations of fluid mechanics are based on conservation laws of mass, momentum, and energy – supplemented by an equation of state and by a relation between the stress and the rate of strain. To express these laws mathematically, we will usually use the Eulerian, or field, description, in which the quantities describing the motion of the fluid are specified at a fixed point in space. For example, pf(x, t), uf(x, t), pf(x, t), and Tf(x, t) denote the fluid's density, velocity, pressure, and temperature, respectively, at time t, at a point whose position vector – relative to the origin of a system of coordinates – is x. When necessary, the Lagrangian description will also be used. Here, the conservation equations are written for an element of fluid composed of the same molecules, whose volume, momentum, and energy; but, not its mass, can change as the element moves.