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Gyrotaxis in uniform vorticity

Published online by Cambridge University Press:  04 December 2014

T. J. Pedley
T. J. Pedley*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: t.j.pedley@damtp.cam.ac.uk
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Abstract

Analysis of bioconvection in dilute suspensions of bottom-heavy but randomly swimming micro-organisms is commonly based on a model introduced in 1990. This couples the Navier–Stokes equations, the cell conservation equation and the Fokker–Planck equation (FPE) for the probability density function for a cell’s swimming direction $\boldsymbol{p}$, which balances rotational diffusion against viscous and gravitational torques. The results have shown qualitative agreement with observation, but the model has not been subjected to direct quantitative testing in a controlled experiment. Here, we consider a simple configuration in which the suspension is contained in a circular cylinder of radius $R$, which rotates at angular velocity ${\rm\Omega}$ about a horizontal axis. We solve the FPE and calculate the cells’ mean swimming velocity, which proves to be horizontal when $B{\rm\Omega}\gg 1$, where $B$ is the gyrotactic reorientation time scale. Then we compute the cell concentration distribution, which is non-uniform only in a thin boundary layer near the cylinder wall when ${\it\beta}^{2}={\rm\Omega}R^{2}/D\gg 1$, where $D$ is the cells’ translational diffusivity. The fact that cells are denser than water means that this concentration distribution drives a perturbation to the underlying solid-body rotational flow which can be calculated analytically. The predictions of the theory are evaluated in terms of a proposed experimental realisation of the configuration, using suspensions of the alga Chlamydomonas nivalis or Chlamydomonas reinhardtii or the algal colony Volvox.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Biological Fluid Dynamics: Micro-organism dynamics
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 762 , 10 January 2015 , R6
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Copyright
© 2014 Cambridge University Press 

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