Inertia Drives a Flocking Phase Transition in Viscous Active Fluids

Abstract

How fast must an oriented collection of extensile swimmers swim to escape the instability of viscous active suspensions? We show that the answer lies in the dimensionless combination R=ρv_{0}^{2}/2σ_{a}, where ρ is the suspension mass density, v_{0} the swim speed, and σ_{a} the active stress. Linear stability analysis shows that, for small R, disturbances grow at a rate linear in their wave number q and that the dominant instability mode involves twist. The resulting steady state in our numerical studies is isotropic hedgehog-defect turbulence. Past a first threshold R of order unity, we find a slower growth rate, of O(q^{2}); the numerically observed steady state is phase turbulent: noisy but aligned on average. We present numerical evidence in three and two dimensions that this inertia-driven flocking transition is continuous, with a correlation length that grows on approaching the transition. For much larger R, we find an aligned state linearly stable to perturbations at all q. Our predictions should be testable in suspensions of mesoscale swimmers [D. Klotsa, Soft Matter 15, 8946 (2019)SMOABF1744-683X10.1039/C9SM01019J].