Investigating the perturbed geometries of vortex rings in free fall through another liquid

The goal of this paper is to theoretically investigate the origin of the standing wave-like perturbation observed on the vortex rings falling through another liquid. We simplified the Navier-Stokes equation based on observational evidence from related research, and showed that there exists a steady-state velocity field with the required wave-like characteristics. For better visualisation we also plotted the velocity field.