Continuum modelling of Active Brownian Particles (ABP)

Active Brownian Particle (ABP) is a class of models for particles whose trajectories depend on their noise orientation. It is a particularly popular model for swimming microorganisms such as bacterial, motile and sinking phytoplanktons. In a dilute suspension where these particles only interact hydrodynamically through their disturbance to the fluid bulk, they can be modelled as a continuum phase through the Fokker-Planck equation, which governs the probability distribution of the particles in both the orientational and physical space. However, the two-way coupling between the Fokker-Planck equation and the momentum equation governing the flow, i.e. the Navier-Stokes equation, remains a difficult problem, largely owing to the high number of dimensions in the Fokker-Planck equation.

In this work, we present a new model to reduce the high-dimensional Fokker-Planck equation into a low-dimensional transport equation for the number density of the particles. This model is based on a novel transformation of the Fokker-Planck equation, which, even without any approximation, can reveal physical insight into how the orientational motility of ABPs affects their macro-transport. Compared to previous models such as those based on the generalised Taylor dispersion theory, which only works in simple flows such as homogeneous shear flow, our model is shown to be less restrictive in applications, as it works on a general flow field. It can also capture phenomena arising from the inhomogeneity of the flow, such as the high- and low-shear trapping of active spheroids in inhomogeneous shear flow.