  Due to their long-lived nature, vortex rings are highly promising for
non-contact transportation of colloidal microparticles. However, they are
complex structures, and their description using rigorous, closed-form
mathematical expressions is challenging, particularly in the presence of
strongly inhomogeneous colloidal suspensions. This study presents
straightforward analytical approximations that reveal the dynamics of vortex
rings transporting microparticles. Our results were validated using
comprehensive simulations and experimental measurements.
