Ciliary propulsion, chaotic filtration and a 'blinking' stokeslet

The paper discusses the fundamental singularity of Stokes flow (the stokeslet) in the context of applications to locomotion and feeding currents in micro-organisms. The image system for a stokeslet in a rigid plane boundary may be derived from Lorentz's mirror image technique [1] or by an appropriate limit of Oseen's solution for a sphere near a plane boundary [2]. An alternative derivation using Fourier transform methods [3] leads to an immediate physical interpretation of the image system in terms of a stokeslet and its multipole derivatives. The schematic illustration of a stokeslet and its image system in a plane boundary are exploited to explain the fluid dynamical principles of ciliary propulsion. For a point force oriented normal to the plane boundary, the resulting axisymmetric motion leads to a Stokes stream function representation which illustrates the toroidal eddy structure of the flow field. A similar eddy structure is also obtained for the two-dimensional system, although in this case, the toroidal structure is replaced by two eddies. This closed streamline model is developed to model chaotic filtration through the concept of a 'blinking stokeslet', a stokeslet alternating its vertical position according to a specific protocol. The resulting behaviour is illustrated via Poincare sections, particle dispersion and length of particle path tracings. Sessile micro-organisms may exploit a similar process so they can filter as large a volume of liquid as possible in search of food and nutrients.