TABLE OF CONTENTS In Chapter 1 we examined the motion of fluid parcels, material vectors, material lines, and material surfaces in a specified field of flow. The velocity field was assumed to be known either in terms of Eulerian variables, including space and time, or Lagrangian variables, including point-particle labels and time. In the present chapter we shall discuss alternative methods of describing the flow in terms of secondary scalar or vectorial fields. By definition, the velocity field is related to a secondary field through a differential or integral relationship.
Examples of secondary fields are the vorticity and rate of expansion introduced in Chapter 1. Additional secondary fields, to be introduced in the present chapter, are the velocity potential for irrotational flow, the vector potential for incompressible flow, the stream function for two-dimensional flow, the Stokes stream function for axisymmetric flow, and a pair of stream functions for a general three-dimensional incompressible flow. Certain secondary fields, such as the rate of expansion, the vorticity, and the stream functions, have a clear physical significance, but others are mathematical entities introduced for mere analytical convenience.
Describing a flow in terms of a secondary field is motivated by two reasons. The first one has to do with the fact that the number of scalar secondary fields that are necessary in order to describe an incompressible flow is less than the dimensionality of the flow by one unit, and this results in analytical and computational simplifications. For instance, we shall see in this chapter...
