Diffusive property

 Understanding the diffusive behaviour of swimming micro-organisms is important in order to obtain a better continuum model for a cell suspen-sion. The translational diffusivity, which is a measure of the increasing dis-placements between pairs of particles, can be calculated from the mean square displacement. If the mean square displacement grows more rapidly than linearly in time, then the spread is not diffusive (if proportional to /^, it is as if the relative velocity of two squirmers is constant), but if it be-comes linear in time then the spread is diffusive. The diffusive property in a suspension of squirmers was discussed in Ishikawa and Pedley, (2004, 2007b), in which the movement of 27 identi-cal squirmers in a cubic region of fluid otherwise at rest was computed for random initial positions and orientations. The instantaneous positions of the squirmers and their trajectories during five time intervals are shown in figure 7. It is found that the trajectories of squirmers are not straight, be-cause the hydrodynamic interaction between squirmers generates transla-tional-rotational velocities between them. In the case of non-bottom-heavy squirmers, square displacement of cells becomes proportional to the time interval as shown in figure 8. The results indicate that the spreading of squirmers was correctly described as a diffusive process over a sufficiently long time scale, even though all the movements of the squirmers were deterministically calculated. Scaling of the results on the assumption that the squirmer trajectories were unbiassed random walks was shown to capture some but not all of the main features of the results. In the case of bottom-heavy squirmers, the diffusive behaviour in squirmers' orientations could be described by a biased random walk model, but only when the effect of hydrodynamic interaction dominated that of the
nstantaneous position of 27 identical squirmers. Solid lines are trajectories of the squirmers during five time interval. No background flow field is imposed.

bottom-heaviness. The spreading of bottom-heavy squirmers in the hori-zontal directions showed diffusive behaviour, and that in the vertical direc-tion also did provided that the average upward velocity was subtracted. The rotational diffusivity in this case, at a volume fraction c = 0.1, was shown to be at least as large as that previously measured in very dilute populations of randomly swimming algal cells {Chlamydomonas nivalis