The orientation of gyrotactic spheroidal micro-organisms in a homogeneous isotropic turbulent flow

Abstract

This paper studies the problem of the orientation of micro–organisms, with asymmetric mass distributions, when swimming through a homogeneous isotropic turbulent flow. In such a flow, the micro–organism'swimming direction p (where p is a unit vector governed by two Euler angles θ and ϕ ) is a random variable, and it will be assumed that the associated probability density function, f ( θ , ϕ ), satisfies a Fokker–Planck equation. In the Fokker–Planck equation, the viscous and gravitational torques on the cell'body are balanced by a diffusion term governed by an unknown rotary diffusivity D eff . An exact formulation for the Reynolds average of f ( θ , ϕ ), denoted by F ( θ , ϕ ), is obtained, which depends on a series of unknown terms derived from a randomly fluctuating contribution, f '( θ , ϕ ), to the total probability density function. A simple closure scheme for f '( θ , ϕ ) is presented, which allows one to show that in the long–time limit F ( θ , ϕ ) tends to a stationary oneparameter Fisher distribution; plus certain time–dependent terms derived from the closure assumption. Numerical results, obtained by calculating the orientations of an ensemble of swimming gyrotactic spheroidal micro–organisms in a series of kinematic simulations of turbulent–like flow fields, are then presented. These appear to show that the time–dependent terms in the equation for F ( θ , ϕ ) are negligibly small, and that the long–time limit for F ( θ , ϕ ) is well approximated by the Fisher distribution, independent of the closure assumption. The latter represents an equilibrium distribution, in which the gravitational couple trying to direct the swimming motion upwards is counterbalanced by the random torque exerted by the flow field, as well as other intrinsic random fluctuations in orientation. They manifest themselves in the value of the effective rotary diffusivity constant D eff . If this result could be replicated experimentally, it would have important implications for modelling populations of swimming micro–organisms living in a turbulent environment.