Talk abstract:

FLUID MECHANICS OF CILIARY PROPULSION

Dedicated to the memory of Sir James Lighthill

John Blake
School of Mathematics and Statistics
The University of Birmingham
Edgbaston, Birmingham B15 2TT, U.K.

Cilia have many functions in the animal kingdom, some of these being cleansing, feeding, excretion, locomotion and reproduction. They occur in all phyla of the animal kingdom with the possible exception of the class Nematoda. This lecture will discuss the development of fluid mechanical models and theories that help with our understanding and interpretation of locomotion of protozoa, mucous transport in the lung, filter feeding in bivalve molluscs and gamete transport.

The theoretical development requires obtaining the fundamental singularities and image systems pertinent to the system under study, the physical interpretation of them and their constructive use to model the flow fields generated by fields of cilia. Examples will also be given where basic phenomena are isolated and studied in greater detail, such as the motion of a slender body close to, and penetrating, the interface between two viscous fluids.

These theories allow estimates of the flow fields in the cilia sublayer and for a greater understanding of propulsive mechanisms in both micro-organisms and mucous transport in the lower respiratory tract. The sophisticated models in turn allow us to develop better approximations for simplified models that provide an improved understanding of filter feeding in bivalve molluscs and with ovum transport in the oviduct.

Finally, studies of possible filter feeding strategies in the sessile organism, Vorticella, which alters the length of its stalk periodically, has led to the development of some interesting non-linear mathematics in a simplified `blinking stokeslet' model of this filter feeding phenomenon. We shall demonstrate that this can lead to chaotic dynamics, which has been shown to enhance mixing and hence improve the efficiency of feeding currents. The continuous system is reduced to an area-preserving map, which allows for greater analytical progress to be made in this inertia-free system. Poincar\'{e} sections and Lyapunov exponents are used alongside other chaotic measures to determine the nature and extent of the chaos. Effects of molecular diffusion are mimicked via the incorporation of white noise in the map and enhanced feeding levels are predicted.

The author of this paper acknowledged with gratitude the enormous influence that Sir James Lighthill has had on his life and academic career. This presentation is dedicated to his memory.

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1998-1999 Mathematics in Biology