Gyrotaxis in uniform vorticity

Abstract

AbstractAnalysis 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.