Instabilities, pattern formation, and mixing in active suspensions

Abstract

Suspensions of self-propelled particles, such as swimming micro-organisms, are known to undergo complex dynamics as a result of hydrodynamic interactions. To elucidate these dynamics, a kinetic theory is developed and applied to study the linear stability and the nonlinear pattern formation in these systems. The evolution of a suspension of self-propelled particles is modeled using a conservation equation for the particle configurations, coupled to a mean-field description of the flow arising from the stress exerted by the particles on the fluid. Based on this model, we first investigate the stability of both aligned and isotropic suspensions. In aligned suspensions, an instability is shown to always occur at finite wavelengths, a result that extends previous predictions by Simha and Ramaswamy [“Hydrodynamic fluctuations and instabilities in ordered suspensions of self-propelled particles,” Phys. Rev. Lett. 89, 058101 (2002)]. In isotropic suspensions, we demonstrate the existence of an instability for the active particle stress, in which shear stresses are eigenmodes and grow exponentially at long scales. Nonlinear effects are also investigated using numerical simulations in two dimensions. These simulations confirm the results of the stability analysis, and the long-time nonlinear behavior is shown to be characterized by the formation of strong density fluctuations, which merge and breakup in time in a quasiperiodic fashion. These complex motions result in very efficient fluid mixing, which we quantify by means of a multiscale mixing norm.