Weakly nonlinear analysis of pattern formation in active suspensions

Abstract

We consider the Saintillan–Shelley kinetic model of active rod-like particles in Stokes flow (Saintillan & Shelley, Phys. Rev. Lett., vol. 100, issue 17, 2008a, 178103; Saintillan & Shelley, Phys. Fluids, vol. 20, issue 12, 2008b, 123304), for which the uniform isotropic suspension of pusher particles is known to be unstable in certain settings. Through weakly nonlinear analysis accompanied by numerical simulations, we determine exactly how the isotropic steady state loses stability in different parameter regimes. We study each of the various types of bifurcations admitted by the system, including both subcritical and supercritical Hopf and pitchfork bifurcations. Elucidating this system's behaviour near these bifurcations provides a theoretical means of comparing this model with other physical systems that transition to turbulence, and makes predictions about the nature of bifurcations in active suspensions that can be explored experimentally.