Inertia Drives a Flocking Phase Transition in Viscous Active Fluids

How fast must an oriented collection of extensile swimmers swim to escape the instability viscous active suspensions? We show that answer lies in dimensionless combination $R=\rho v_0^2/2\sigma_a$, where $\rho$ is suspension mass density, $v_0$ speed and $\sigma_a$ stress. Linear stability analysis shows for small $R$ disturbances grow at a rate linear their wavenumber $q$, dominant mode involves twist. The resulting steady state our numerical studies isotropic hedgehog-defect turbulence. Past first threshold order unity we find slower growth rate, $O(q^2)$; numerically observed {\it phase-turbulent}: noisy but aligned} on average. present evidence three two dimensions this inertia driven flocking transition continuous, with correlation length grows approaching transition. For much larger aligned linearly stable perturbations all $q$. Our predictions should be testable suspensions mesoscale [D Klotsa, Soft Matter \textbf{15}, 8946 (2019)].