Instability of uniform micro-organism suspensions revisited

Uniform suspensions of bottom-heavy, upswimming (gyrotactic) micro-organisms that are denser than water are unstable, through a gravitational mechanism first described by Pedley, Hill & Kessler (J. Fluid Mech., vol. 195, 1988, p. 223). Suspensions of downswimming, head-heavy cells do not experience this instability. In the absence of gravity, a uniform suspension of swimming micro-organisms may be unstable because of the 'particle stresses' generated by the swimming cells themselves, each of which acts as a force-dipole or stresslet (Simha & Ramaswamy, Phys. Rev. Lett., vol. 89, 2002, p. 058101). The stresslet strength S is positive for 'pullers' such as algae and negative for 'pushers' such as bacteria or spermatozoa. In this paper, the combined problem is investigated, with attention being paid also to the effect of rotational diffusivity and to whether the probability density function f(e) for the cells' swimming direction e can be approximated as quasi-steady in calculations of the mean swimming direction, which arises in the cell conservation equation, and the particle stress tensor, which appears in the momentum equation. The existence of both the previous instabilities is confirmed at long wavelength. However, the non-quasi-steadiness or the orientation distribution means that the particle-stress-driven instability no longer arises for arbitrarily small |S|, in the Stokes limit, but requires that the dimensionless stresslet strength (proportional to the product of S and the basic state cell volume fraction n(o)) exceed a critical value involving both viscosity and rotational diffusivity. In addition, a new mode of gravitational instability is found for 'head-heavy' cells, even when they exert no particle stresses (S=0), in the form of weakly growing waves. This is a consequence of unsteadiness in the mean swimming direction, together with non-zero fluid inertia. For realistic parameter values, however, viscosity is expected to suppress this instability.