Using the continuum model of Pedley, Hill & Kessler (1988) for bioconvection in
a suspension of swimming, gyrotactic micro-organisms, we investigate the existence
and stability of a two-dimensional plume in tall, narrow chambers with stress-free
sidewalls. The system is governed by the Navier–Stokes equations for an incompressible
fluid coupled with a micro-organism conservation equation. These equations are
solved numerically using a conservative finite-difference scheme. In sufficiently deep
chambers, the plume is always unstable to both varicose and meandering modes. A
linear stability analysis for an infinitely long plume predicts the growth rates of these
instabilities, explains the mechanisms, and is in good agreement with the numerical
results.