A Kinetic Model for Semidilute Bacterial Suspensions

Suspensions of self-propelled microscopic particles, such as swimming bacteria, exhibit collective motion leading to remarkable experimentally-observable macroscopic properties. Rigorous mathematical analysis of this emergent behavior can provide significant insight into the mechanisms behind these experimental observations; however, there are many theoretical questions remaining unanswered. In this paper, we study a coupled PDE/ODE system first introduced in the physics literature and used to investigate numerically the effective viscosity of a bacterial suspension. We then examine the kinetic theory associated to the coupled system, which is designed to capture the long-time behavior of a Stokesian suspension of point force dipoles (infinitesimal spheroids representing self-propelled particles) with Lennard-Jones-type repulsion. A planar shear background flow is imposed on the suspension through the novel use of Lees-Edwards quasiperiodic boundary conditions applied to a representative volume. We show the existence and uniqueness of solutions for all time to the equations of motion for particle configurations – dipole orientations and relative positions. This result follows from first establishing the regularity of the solution to the fluid equations. The existence and uniqueness result allows us to define the Liouville equation for the probability density of configurations. We show that this probability density defines the average bulk stress in the suspension underlying the definition of many macroscopic quantities of interest, in particular the effective viscosity. These effective properties are determined by microscopic interactions highlighting the multiscale nature of this work.