Instabilities and dynamics in active suspensions: direct numerical

simulations and kinetic theory

David Saintillan

Courant Institute of Mathematical Sciences, New York University

Suspensions of swimming microorganisms are characterized by complex dynamics involving strong fluctuations and large-scale correlated motions. These motions, which result from the many-body interactions between the particles, are biologically relevant as they impact mean particle transport, mixing and diffusion, with possible consequences for nutrient uptake.

Using direct numerical simulations, I first investigate aspects of the dynamics and microstructure in suspensions of interacting self-locomoting rods at low Reynolds number. A detailed model is developed that accounts for hydrodynamic interactions based on slender-body theory and encompasses both biological and non-biological locomotion mechanisms. It is first shown that aligned suspensions of swimming particles are unstable as a result of hydrodynamic fluctuations. In spite of this instability, a local nematic ordering persists in the suspensions over short length scales and has a significant impact on the mean swimming speed. Consequences of the large-scale orientational disorder for particle dispersion are discussed and explained in the context of generalized Taylor dispersion theory. Dynamics in thin liquid films are also presented, and are characterized by a strong particle migration towards the interfaces.

The results from direct numerical simulations are complemented by a kinetic model, in which the dynamics are captured using a continuity equation for the particle configurations, coupled to a mean-field description of the flow arising from the active stress exerted by the particles on the fluid. Based on this model, the linear stability of both aligned and isotropic suspensions is revisited. In aligned suspensions, the instability observed in the simulations is predicted to occur at all wavelengths, a result that extends previous predictions by Simha and Ramaswamy (2002). In isotropic suspensions, an instability for the active particle stress is also found to occur, in which shear stresses are eigenmodes and grow exponentially at long scales. Non-linear effects in these systems are also investigated using numerical simulations in two-dimensions. The results of the stability analysis are confirmed, and the long-time non-linear behavior is shown to be characterized by the formation of strong density fluctuations, which appear to be driven by the active stress instability.