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

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