  Suspensions of aerobic bacteria often develop flows from the interplay of
chemotaxis and buoyancy, which is so-called the chemotaxis-Navier-Stokes flow.
In 2004, Dombrowski et al. observed that Bacterial flow in a sessile drop
related to those in the Boycott effect of sedimentation can carry bioconvective
plumes, viewed from below through the bottom of a petri dish, and the
horizontal "turbulence" white line near the top is the air-water-plastic
contact line. In pendant drops such self-concentration occurs at the bottom. On
scales much larger than a cell, concentrated regions exhibit transient,
reconstituting, high-speed jets straddled by vortex streets. It's interesting
to verify these turbulent phenomena mathematically. In this note, we
investigate the Hausdorff dimension of these vortices (singular points) by
considering partial regularity of weak solutions of the three dimensional
chemotaxis-Navier-Stokes equations, and showed that the singular dimension is
not larger than $1$, which seems to be consistent with the linear singularity
in the experiment.

Analysis of PDEs