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A new continuum model for suspensions of gyrotactic micro-organisms

Published online by Cambridge University Press:  26 April 2006

T. J. Pedley  and
J. O. Kessler
T. J. Pedley
Affiliation:
Department of Applied Mathematical Studies, University of Leeds, Leeds LS2 9JT. UK
J. O. Kessler
Affiliation:
Department of Physics, University of Arizona, Tucson, AZ 85721, USA
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Abstract

A new continuum model is formulated for dilute suspensions of swimming microorganisms with asymmetric mass distributions. Account is taken of randomness in a cell's swimming direction, p, by postulating that the probability density function for p satisfies a Fokker–Planck equation analogous to that obtained for colloid suspensions in the presence of rotational Brownian motion. The deterministic torques on a cell, viscous and gravitational, are balanced by diffusion, represented by an isotropic rotary diffusivity Dr, which is unknown a priori, but presumably reflects stochastic influences on the cell's internal workings. When the Fokker-Planck equation is solved, macroscopic quantities such as the average cell velocity Vc, the particle diffusivity tensor D and the effective stress tensor Σ can be computed; Vc and D are required in the cell conservation equation, and Σ in the momentum equation. The Fokker-Planck equation contains two dimensionless parameters, λ and ε; λ is the ratio of the rotary diffusion time D-1r to the torque relaxation time B (balancing gravitational and viscous torques), while ε is a scale for the local vorticity or strain rate made dimensionless with B. In this paper we solve the Fokker–Planck equation exactly for ε = 0 (λ arbitrary) and also obtain the first-order solution for small ε. Using experimental data on Vc and D obtained with the swimming alga, Chamydomonas nivalis, in the absence of bulk flow, the ε = 0 results can be used to estimate the value of λ for that species (λ ≈ 2.2; Dr ≈ 0.13 s−1). The continuum model for small ε is then used to reanalyse the instability of a uniform suspension, previously investigated by Pedley, Hill & Kessler (1988). The only qualitatively different result is that there no longer seem to be circumstances in which disturbances with a non-zero vertical wavenumber are more unstable than purely horizontal disturbances. On the way, it is demonstrated that the only significant contribution to Σ, other than the basic Newtonian stress, is that derived from the stresslets associated with the cells’ intrinsic swimming motions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 212 , March 1990 , pp. 155 - 182
Copyright
© 1990 Cambridge University Press

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