21 The Governing Equations

For a parallel two-dimensional flow, the basic steady flow is of the form

where i denotes a unit vector in the x _*-direction and the asterisks denote dimensional quantities. For an inviscid fluid U _* (z _* ) can be an arbitrary function of z _*. The flow is assumed to be bounded by the two planes z _*= z _{1 *} and z _2* which may be either rigid or free. On a rigid boundary the normal component of the velocity must vanish and on a free boundary the pressure must be constant. More generally, one of the boundaries may be at infinity as in the case of boundary layers or they may both be at infinity as in the case of shear layers, jets and wakes.

It is convenient, as usual, to write the governing equations in terms of dimensionless quantities, and for this purpose we introduce a characteristic length L and a characteristic velocity V associated with the basic flow. The choice of L and V is, of course, not unique; considerable variation in their definition exists in the literature and some care is required theref ore in comparing the results of diff erent writers. In the present discussion, however, we will usually take

and, for flows in a channel, . If we now let

where ? is the (constant) density of the fluid, then the Euler equations of motion and...