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In this communication, we offer a theoretical explanation for the results of recent experiments that examine the stress response of a dilute suspension of bacteria (wild-type E. coli) subjected to step changes in the shear rate (Lopez et al., Phys. Rev. Lett., vol. 115, 2015, 028301). The observations include a regime of negative apparent shear viscosities. We start from a kinetic equation that describes the evolution of the single-bacterium orientation probability density under the competing effects of an induced anisotropy by the imposed shear, and a return to isotropy on account of stochastic relaxation mechanisms (run-and-tumble dynamics and rotary diffusion). We then obtain analytical predictions for the stress response, at leading order, of a dilute bacterial suspension subject to a weak but arbitrary time-dependent shear rate profile. While the predicted responses for a step-shear compare well with the experiments for typical choices of the microscopic parameters that characterize the swimming motion of a single bacterium, use of actual experimental values leads to significant discrepancies. The incorporation of a distribution of run times leads to a better agreement with observations.

This paper follows a recent article of Nambiar et al. (2017) on the linear rheological response of a dilute bacterial suspension (e.g. E. coli) to impulsive starting and stopping of simple shear flow. Here, we analyse the time dependent non-linear rheology for a pair of impulsively started linear flows -simple shear (a canonical weak flow) and uniaxial extension (a canonical strong flow). 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 ≡γτ , which dictates the importance of flow-induced orientation anisotropy, and τ D r , which quantifies the relative importance of the two intrinsic orientation decorrelation mechanisms (tumbling and rotary diffusion). Here, τ 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 andγ is the characteristic magnitude of the imposed velocity gradient. The solution of the kinetic equation is obtained numerically using a spectral Galerkin method, that yields the rheological properties (the shear viscosity, the first and second normal stress differences for simple shear, and the extensional viscosity for uniaxial extension) over the entire range of Pe. For simple shear, we find that 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. Our study suggests courses for future experimental and analytical studies that will help understand relaxation phenomena in active suspensions.

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