Ciliary propulsion, chaotic filtration and a ?blinking? stokeslet

Abstract

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 Poincar6 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.

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“…It may be worthwhile to point out that the toroidal eddy patterns shown in Fig. 9 are similar to the viscous eddies near planar boundaries [23,24]. The velocity components for a stokeslet-cylinder (with the stokeslet at (c, 0) and its axis in the x -direction) combination are…”

A control theoretic approach to the swimming of microscopic organisms

“…An important example of swimming microscopic organisms is furnished by ciliata (see, for instance, Blake [2] or Brennen [3]). We recall, following Galdi [8], [9], that these organisms can be seen as rigid bodies covered by a large number of hair-like organelles called cilia which move in a rather complicated way (see Blake and Otto [1] or Brennen and Winet [4]). In a commonly accepted model (the layer model), the rather complex motion of cilia is replaced by a velocity field on a surface enclosing the layer of cilia (see, for instance, Keller and Wu [11]).…”

“…(1) Suppose that ∂S is of class C 2 . Then the family F 1 = g (1) , g (2) , g (3) , G (1) , G (2) , G (3) is linearly independent in L 2 (Γ, R 3 ) and the family…”

Nearby boundaries create eddies near microscopic filter feeders

“…Closed feeding streamlines limit the feeder's access to new material (Blake & Otto 1996), so it is important to determine what circumstances lead to closed eddies. The eddy structure is typically attributed to the substrate to which the organism is attached.…”

“…The eddy structure is typically attributed to the substrate to which the organism is attached. For instance, Blake & Otto (1996) modelled the feeding eddy structure as a point force, or stokeslet, oriented perpendicular to a plane wall, and this model has been used to estimate feeding fluxes in several biological contexts (Pettitt et al 2002;Orme et al 2003;Hartmann et al 2007).…”