Effective viscosity of bacterial suspensions: A three-dimensional pde model with stochastic torque

We present a PDE model for dilute suspensions of swimming bac- teria in a three-dimensional Stokesian uid. This model is used to calculate the statistically-stationary bulk deviatoric stress and active viscosity of the suspension from the microscopic details of the interaction of an elongated body with the background ow. A bacterium is modeled as an impenetrable prolate spheroid with 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 reorienta- tion). Due to a bacterium's asymmetric shape, interactions with prescribed generic planar background ows, such as a pure straining or planar shear ow, cause the bacterium to preferentially align in certain directions. Due to the stochastic torque, the steady-state distribution of orientations is unique for a given background ow. Under this distribution of orientations, self-propulsion produces a reduction in the effective viscosity. For suffciently weak background ows, the effect of self-propulsion on the effective viscosity dominates all other contributions, leading to an effective viscosity of the suspension that is lower than the viscosity of the ambient uid. This is in qualitative agreement with recent experiments on suspensions of Bacillus subtilis.