Stability of active suspensions.

We study theoretically the stability of "active suspensions," modeled here as a Stokesian fluid in which are suspended motile particles. The basis of our study is a kinetic model recently posed by Saintillan and Shelley [D. Saintillan and M. J. Shelley, Phys. Rev. Lett. 100, 178103 (2008); D. Saintillan and M. J. Shelley, Phys. Fluids 20, 123304 (2008)] where the motile particles are either "pushers" or "pullers." General considerations suggest that, in the absence of diffusional processes, perturbations from uniform isotropy will decay for pullers, but grow unboundedly for pushers, suggesting a possible ill-posedness. Hence, we investigate the structure of this system linearized near a state of uniform isotropy. The linearized system is nonnormal and variable coefficient, and not wholly described by an eigenvalue problem, in particular at small length scales. Using a high wave-number asymptotic analysis, we show that while long-wave stability depends on the particular swimming mechanism, short-wave stability does not and that the growth of perturbations for pusher suspensions is associated not with concentration fluctuations, as we show these generally decay, but with a proliferation of oscillations in swimmer orientation. These results are also confirmed through numerical simulation and suggest that the basic model is well-posed, even in the absence of translational and rotational diffusion effects. We also consider the influence of diffusional effects in the case where the rotational and translational diffusion coefficients are proportional and inversely proportional, respectively, to the volume concentration and predict the existence of a critical volume concentration or system size for the onset of the long-wave instability in a pusher suspension. We find reasonable agreement between the predictions of our theory and numerical simulations of rodlike swimmers by Saintillan and Shelley [D. Saintillan and M. J. Shelley, Phys. Rev. Lett. 99, 058102 (2007)].