From sinking spheroids to chemotactic collapse: Continuum modelling of dilute active Brownian particle suspension

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 the first half of the talk, I am going to 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. We will also compare this new model with the more restrictive generalised Taylor dispersion model.

In the second half of the talk, we apply the novel transformation to the modelling of two classical problems in ABP suspensions – the formation of plumes in gyrotactic suspension and streamers in the sedimentation of spheroids. With the help of the transformation, we shall demonstrate how the system is analogous to the Keller-Segel model that governs many chemotactic phenomena. Also, similar to chemotactic collapse, the nonlinear solution to the plume structure exhibit a finite-time blow-up when extended from 2D to 3D.