Effective viscosity of bacterial suspensions: a three-dimensional PDE model with stochastic torque

We present a PDE model for dilute suspensions of swimming bacteria in a three-dimensional Stokesian fluid.This model is used tocalculate the statistically-stationary bulk deviatoric stress and effective viscosity of the suspensionfrom the microscopic details of the interaction of an elongated body with the background flow.A bacterium is modeled as an impenetrable prolate spheroidwith self-propulsion provided by a point force, which appears in the model as an inhomogeneous delta function in the PDE.The bacterium is also subject to a stochastic torque in order to model tumbling (random reorientation).Due to a bacterium's asymmetric shape, interactions with prescribed generic planar background flows, such as apure straining or planar shear flow, cause the bacterium to preferentially align incertain directions. Due to the stochastic torque, the steady-state distribution of orientations is unique for a givenbackground flow.Under this distribution of orientations, self-propulsion produces a reduction in the effective viscosity.For sufficiently weak background flows, the effect of self-propulsion on the effective viscosity dominates all other contributions, leading to aneffective viscosity of the suspension that is lower than the viscosity of the ambient fluid. This is in qualitative agreement with recent experimentson suspensions of Bacillus subtilis.