  A dilute suspension of motile micro-organisms subjected to a strong ambient
flow, such as algae in the ocean, can be modelled as a population of
non-interacting, orientable active Brownian particles (ABPs). Using the
Smoluchowski equation (i.e. Fokker-Planck equation in space and orientation),
one can capture the non-trivial transport phenomena of ABPs such as taxes and
shear-induced migration. This work presents an analytical method and an
asymptotic approximation to transform the Smoluchowski equation into a
physically-relevant transport equation. The approximation recovers the
effective diffusivity given by the generalised Taylor dispersion (GTD) model in
a quiescent flow as a dispersion tensor, but the approximation does not
generate singularities in strain-dominant flows like the GTD model. The
transformation has also unveiled the shear trapping mechanism previously shown
by Bearon and Hazel (2015) and other new drift and dispersion mechanisms caused
by the interactions between the orientational dynamics and the passive
advection or diffusion of ABPs. Using gyrotactic suspensions in shear flow as
an example, we show that the approximation is more accurate than the GTD model
by accounting for shear trapping and the other new mechanisms for drift and
dispersion. Lastly, we show that the dispersion from the translational
diffusion can be negative, while the dispersion from the interaction between
the rotational diffusion and the particles' motility can be highly asymmetric,
in contrast to the positive definite effective diffusivity of the GTD model.
The results open a new perspective on the nature and physical origin of the
dispersion of ABPs in a general flow field.
