Bifurcation and stability of downflowing gyrotactic micro-organism suspensions in a vertical pipe

• Published in: arXiv.org 2020-07
• Main Authors: Fung, Lloyd, Bearon, Rachel N, Hwang, Yongyun
• Format: Article
• Language: English
• Subjects: Axisymmetric flow; Bifurcations; Flow velocity; Pipes; Stability analysis; Swimming
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2001.08072

In the experiment that first demonstrated gyrotactic behaviour of bottom-heavy swimming microalgae (e.g. Chlamydomonas), Kessler (Nature, vol. 313, 1985, pp. 218-220) showed that a beam-like structure, often referred to as a gyrotactic plume, would spontaneously appear from a suspension of gyrotactic swimmers in a downflowing pipe. Such a plume is prone to an instability to form blips. This work models the gyrotactic plume as a steady parallel basic state and its subsequent breakdown into blips as an instability, employing both the Generalised Taylor Dispersion (GTD) theory and the Fokker-Planck model for comparison. Upon solving for the basic state, it is discovered that the steady plume solution undergoes sophisticated bifurcations. When there is no net flow, there exists a non-trivial solution of the plume structure other than the stationary uniform suspension, stemming from a transcritical bifurcation with the average cell concentration. When a net downflow is prescribed, there exists a cusp bifurcation. Furthermore, there is a critical concentration, at which the cell concentration at the centre would blow up for the GTD model. The subsequent stability analysis using the steady plume solution shows that the Fokker-Planck model is inconsistent with what was experimentally observed, as it predicts stabilisation of axisymmetric blips at high concentration of the plume and destabilisation of the first non-axisymmetric mode at low flow rates.