Oscillatory flow through constricted or dilated channels and axisymmetric pipes

Abstract
Two types of oscillatory viscous flow are considered: the first inside a two-dimensional channel, the second inside an axisymmetric pipe. The walls of the channel or pipe are taken to be small perturbations of the straight and parallel case, these distortions being much smaller than the width of the channel or pipe, so that the equation of motion may be linearized to give an Orr–Sommerfield type of equation. It is assumed that the width of the channel or pipe is comparable with the Stokes layer thickness. For sinusoidal perturbations of the wall, the asymptotic solutions for the parameter (Reynolds number times wavenumber) being very small or very large are considered. The general method may also be applied numerically to obtain solutions for non-periodic dilations or constrictions at arbitrary Reynolds number, and as an illustration the distributions of shear and pressure gradient are given for a number of examples.

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