The flow of a viscous fluid through axially symmetric pipes and symmetrical channels is investigated under the assumption that the Reynolds number is small enough for the Stokes flow approximations to be made. It is assumed that the cross-section of the pipe or channel varies sinusoidally along the length. The flow is produced by a prescribed pressure gradient and by the variation in cross-section that occurs during the passage of a prescribed sinusoidal peristaltic wave along the walls. The theory is applied in particular to two extreme cases, peristaltic motion with no pressure gradient and flow under pressure along a pipe or channel with fixed walls and sinusoidally varying cross-section. Perturbation solutions are found for the stream function in powers of the ratio of the amplitude of the variation in the pipe radius or channel breadth to the mean radius or breadth respectively. These solutions are used to calculate, in particular, the flux through the pipe or channel for a given wave velocity in the first case and for a given pressure gradient in the second case. With a suitable notation it is possible to combine the analysis required for the two cases of pipe and channel flow.

A simple elliptic model is developed for the spread of a fire front through grassland. This is used to predict theoretical fire fronts, which agree closely with those obtained in practice.