A sequence of transcritical bifurcations in a suspension of gyrotactic microswimmers in vertical pipe

TL;DR

This study explores the emergence of multiple steady solutions and bifurcations in gyrotactic microswimmer suspensions within a vertical pipe, revealing complex stability and flow dynamics influenced by geometry and flow parameters.

Contribution

It generalizes previous findings by demonstrating an infinite sequence of solutions arising with increasing Richardson number, and analyzes their stability and nonlinear behavior.

Findings

Multiple steady solutions emerge as Richardson number increases.

Most solutions are unstable, influencing transient dynamics.

A maximum steady downward flow rate exists for each solution.

Abstract

Kessler (Nature, vol. 313, 1985, pp. 218-220) first showed that plume-like structures spontaneously appear from both stationary and flowing suspensions of gyrotactic microswimmers in a vertical pipe. Recently, it has been shown that there exist multiple numbers of steady axisymmetric axially uniform solutions to such a system (Bees, M. A. & Croze, O. A., Proc. R. Soc. A., vol. 466, 2010, pp. 2057-2077). In the present study, we generalise this finding by reporting that a countably infinite number of such solutions emerge as Richardson number increases. Linear stability, weakly nonlinear and fully nonlinear analyses are performed, revealing that each of the solutions arises from the destabilisation of uniform suspension. The discrete number of the solutions is due to the finite flow domain, while the transcritical nature of the bifurcation is because of the cylindrical geometry which breaks the horizontal symmetry of the system. It is further shown that there exists a maximum threshold of achievable downward flow rate for each solution if the flow is to remain steady, as varying the pressure gradient can no longer increase the flow rate from the solution. Except the one arising at the lowest Richardson number, all of the solutions found are unstable, implying that they would play a role in the transient dynamics in the route from a uniform suspension to the fully-developed gyrotactic pattern.