Variable flow in pipes

The problem considered is that of finding a variable pressure gradient or forced motion of the pipe in a longitudinal direction which, at the time when it ceases to act, will have produced a prescribed distribution of velocity over the cross-section of the pipe. The subsequent changes in the distribution as the motion decays are also investigated and the cases examined point to the conclusion that an initial velocity profile with curvature of one sign will retain this property as it changes into the profiles for the different stages of the decaying motion.