Time-periodic flows play important roles in many practical problems, with a wide range of examples occurring in engineering, physiology and ocean dynamics, among others fields. While the stability of spatially developing fluid flows is a relatively well-studied topic of research, there is a relative paucity of corresponding results that discuss purely time-periodic flows. Here we examine the stability of the classical plane Stokes layer: this is the flow generated in a layer of fluid that lies above a bounding plate that oscillates harmonically in its own plane. It has proved surprisingly difficult to understand the linear stability of this type of fluid flow, and here we trace the steps that have been taken to achieve this goal. There is a significant difference between the theoretical description of the stability of a Stokes layer and the corresponding practical measurements. We discuss possible reasons for this discrepancy and its possible implications for more general time-periodic motions.

Calculation of the flux through a two-dimensional channel with one sinusoidal wall is presented using a perturbation expansion. This gives a series for the flux up to $\mathcal {O}(\epsilon ^{160})$, where $\epsilon $ is the amplitude of the wall undulation, ranging from 0 (flat wall) to 2 (channel pinched off). The series appears to diverge near $\epsilon =0.95$, but is continued through the use of Padé approximants. The divergence is investigated and found to be caused by pole singularities lying at $\epsilon \approx \pm 0.977i$ on the imaginary axis. The continued series is validated against a purely numerical solution with excellent agreement for all values of the undulation amplitude.