The chapter deals with the effect of instabilities on the paths of sedimenting or rising bodies in Newtonian fluids. To separate the effect of shape, we focus on axisymmetric objects. Spheres, discs, oblate spheroids and flat cylinders are all expected to follow vertical trajectories with axisymmetry axis aligned with the trajectory. This is, indeed, the case, however, this regime remains stable only if viscous effects are sufficiently strong. The loss of stability of the vertical regime leads to a large variety of trajectories depending on the details of shape, namely their flatness (expressed by aspect ratio for cylinders or spheroids), their inertia and viscous effects (expressed by some equivalent of Reynolds number). Defined in this manner, the problem is basically that of axisymmetry breaking.
Since a significant part of dynamics is expected to arise in the wake, we first focus on axisymmetry breaking of wakes of fixed axisymmetric bodies, the sphere being considered as a prototypical case. We show that the scenario is dominated by two bifurcations following systematically, with increasing Reynolds number, in the order of the regular one as primary and a Hopf one as secondary. The weakly non-linear analysis points out the relevance of Fourier azimuthal decomposition serving as an optimal numerical tool for all presented simulations. Next, the free body degrees of freedom are accounted for. The interplay of the regular and Hopf bifurcations still dominates, however, the scenario is significantly different for spheres and flat objects. The presented parametric study shows that trajectories of spheres become very rapidly chaotic. As an example of flat object, nominally infinitely thin disc is investigated. In this case the Hopf bifurcation is the primary one. The scenario is remarkable by strong subcritical effects due to the significant role of inertia of the combined motion of the solid and of the surrounding fluid.
