Stress relaxation in a dilute bacterial suspension: The active-passive transition

We analyse the time dependent non-linear rheology of a dilute bacterial suspension (e.g. E. coli) for a pair of impulsively started linear flows - simple shear and uniaxial extension. The rheology is governed by the bacterium orientation distribution which satisfies a kinetic equation that includes rotation by the imposed flow, and relaxation to isotropy via rotary diffusion and tumbling. The relevant dimensionless parameters are the Peclet number $Pe\equiv \dot{\gamma}\tau$, which dictates the importance of flow-induced orientation anisotropy, and $\tau D_r$, which quantifies the relative importance of the two intrinsic orientation decorrelation mechanisms (tumbling and rotary diffusion). Here, $\tau$ is the mean run duration of a bacterium that exhibits a run-and-tumble dynamics, $D_r$ is the intrinsic rotary diffusivity of the bacterium and $\dot{\gamma}$ is the characteristic magnitude of the imposed velocity gradient. The solution of the kinetic equation is obtained numerically using a spectral Galerkin method, and yields the relevant rheological properties over the entire range of $Pe$. For simple shear, the stress relaxation predicted by our analysis at small $Pe$ is in good agreement with the experimental observations of Lopez et al. (2015). However, the analysis at large $Pe$ yields relaxations that are qualitatively different. The rheological response in the experiments corresponds to a transition from a nearly isotropic suspension of active swimmers at small $Pe$, to an apparently (nearly) isotropic suspension of passive rods at large $Pe$. In contrast, the computations yield the expected transition to a nearly flow-aligned suspension of passive rigid rods at high $Pe$. We probe this active-passive transition systematically, complementing the numerical solution with analytical solutions obtained from perturbation expansions about appropriate base states.