Flexible loops settling under gravity in a viscous fluid

For sedimenting flexible loops, recent stability analysis and numerical simulations, based on the elastica approximation (R. Waszkiewicz et al., J. Fluid Mech. 919 (2021) A14), predict modes of the dynamics different from the previous results obtained by applying the bead model with the pairwise-additive Rotne-Prager mobility matrices (M. Gruziel-Słomka et al., Soft Matter 15 (2019) 7262). Therefore, we revisit the problem using a more precise theoretical method - the bead model and a fast-convergent, higher order multipole expansion of the Stokes equations, corrected for lubrication and implemented in the HYDROMULTIPOLE numerical codes of a high accuracy. As in the previous studies, initially the loop is circular and tilted at a small angle with respect to the horizontal plane, and the dynamics is determined for a wide range of values of the elasto-gravitation number B. We recover the essential findings of the simpler Rotne-Prager approximation. We determine the approach of more stiff loops to a unique stationary orientation, dependent on B. For more flexible loops we find a very similar phase diagram of sedimentation periodic orbits, with one new dynamical mode that we describe in details.