1.5.1 Couette Poiseuille Flow

The Navier Stokes equations have a few exact solutions. They are for very simple, idealized geometries. These exact solutions can provide estimates of quantities like pressure drop or frictional drag. They can provide a framework for less simple cases, which are computed numerically. One rather useful class of exact solutions is that of parallel flows.

Mathematically, a parallel flow is one in which the velocity does not vary in the direction of the flow. For instance, if the flow is in the x direction, the derivative of u in the direction of the flow is ( u ) u = u ? u/ ?x. The operator u is the projection of the derivative into the direction of the flow. If u is only a function of y and z, this vanishes. The fully developed flow down a pipe is a velocity, u, as a function of radius, r, and hence is a parallel flow.

In steady flow, the particles move along streamlines. If the velocity does not change along the streamline, then the particle experiences no acceleration. The condition for the velocity to be constant along a streamline is that the distance to neighboring streamlines be constant: that is, that they be parallel. The reasoning is as follows: by definition, streamlines are in the direction of the fluid flow; there is no flow across them. Hence, the mass flow follows the...