Spatial temporal distribution of helical gyrotactic swimmers

Abstract

We consider a spherical swimmer that undergoes helical motion due to the existence of a propulsive torque which is not parallel to a propulsive force that pulls the cell through the fluid. In addition, the cell is bottom-heavy; the centre of gravity is offset from the centre of buoyancy which generates a gravitational torque. In the presence of shear, fluid viscosity generates a further torque. Because cells swim at low Reynolds number, these torques are balanced. This thesis extends the model developed in Bearon (2013) in two distinct directions. Firstly, we consider an extension to the case of a flow where the shear varies with position. We consider a downward flow in a vertical channel. We observe that depending on the parameters, cells may exhibit the classical accumulation towards the centre of the channel or display a new focussing away from the centre. Secondly, we develop the model to describe randomness associated with changes in cell orientation. This is done by developing a Fokker-Planck equation for helical swimmers in terms of Euler angles. The classical Fokker-Planck equation obtained by Pedley and Kessler (1992) is a special case of the equation derived in this thesis. To implement this model numerically as an individual based model, we derive the corresponding stochastic differential equations. The Fokker-Planck equation and stochastic differential equation are extended to examine the spatial-temporal distribution of helical swimmers. We explore in detail how the horizontal distribution of cells in channel flow evolves to an equilibrium state, and how the evolution depends on the model parameters. For non-helical swimmers, we compare the result of the model to the recent experiments of Croze et al (2017).