Modelling the transport and pattern formation of gyrotactic microswimmer suspensions

Abstract

Gyrotaxis describes the tendency of bottom-heavy motile micro-organisms to swim sideways under the balancing of gravitational and viscous torque. A suspension of gyrotactic microswimmers will self-focus into a plume in a pipe with a shearing downflow. This work utilises the dilute assumption and model microswimmers, such as the gyrotactic Chlamydomonas and Dunaliella, as hydrodynamically contributing (but not interacting) particles in a flowing suspension that are also subjected to rotational noise. In this framework, the suspension is governed by the Smoluchowski equation and the Navier-Stokes equation. We then further reduce the Smoluchowski equation into a transport equation for the swimmers using the Fokker-Planck model and the generalised Taylor dispersion (GTD) model. Both models are used to calculate the formation of the gyrotactic plume and the subsequent blip instability via a stability analysis. The Fokker-Planck model is found to be not as accurate as the GTD model. The calculation of the gyrotactic plume also results in the discovery of a series of imperfect transcritical bifurcations in the uniform solution and a singular solution when the Richardson number approaches a certain threshold. The transcritical bifurcations are likely linked to the formation of bioconvective patterns, whereas the singular solution is found to be an extension of Kessler (!986). Despite encouraging results from the GTD model, it cannot be applied in a general flow field or capture shear trapping in inhomogeneous shear flows. Therefore, a new model using a novel transformation and local approximation of the Smoluchowski equation is proposed. The resulting new transport model performs better than the GTD model due to its ability to capture shear trapping. It also exposes many new drifts and dispersions arising from the interaction between the orientational and spatial dynamics. Nevertheless, this new framework can be further improved by including the semi-dilute effect and better modelling at the wall.