Biological mixing and chaos

We consider a problem from the field of biological fluid mechanics which considers the flow associated with the motion of a flagellum on a sessile micro-organism. Motivation is taken from the movement of fluid around a species of choanoflagellate, \(Salpingoeca\) \(Amphoridium\). Choanoflagellates are a class of organism in the phylum Protozoa. Because the length scales and velocities are very low, the flow is one dominated by viscous forces and the environment is characterised by a low Reynolds number. The flow caused by the flagellum is initially modelled via a point force. These microorganisms operate in more than one location and the motion they create is modelled in a qualitative sense by using two stokeslets (appropriate to Stokes' flow) whose orientation and position is varied with time. The sessile micro-organism resides above a boundary which is modelled, most generally, as an interface between two fluids possessing different properties. Efficiency of feeding currents generated by the flagellum motion is studied. The resulting dynamics are investigated using chaotic measures, which examine the stretching and consequent mixing of elements within the fluid. Different point force locations lead to various eddy structures such that their superposition results in chaotic advection. The model is developed to examine the flow of particles around a three-dimensional realisation of a micro-organism which involves a flagellum and a cell body attached to a substrate. Green's functions are used to satisfy a number of boundary conditions simultaneously. Particle paths of a tracer introduced into the fully three-dimensional model are investigated. Comparisons with experimental data illustrate good agreement between theoretical and experimental results. Further extensions to the model are suggested