Sedimentation of Triple Twisted M\"{o}bius Objects

Screw-like objects like helix and M\"{o}bius strips exhibit both rotation and translation as they move in a fluid due to external forces such as gravity. It is expected that the orientation of the twist in the structure determines the direction of the rotational motion. Here, we conduct both computational and experimental studies on the sedimentation of rigid M\"{o}bius bands with a three-half twist. We vary the aspect ratio of the bands and analyze their trajectories. We use two different schemes to construct the bands corresponding ruled developable surfaces and ruled binormal-scroll surfaces. For developable bands, we observe that the spinning direction is consistently determined by the orientation of the band, independent of its aspect ratio. Remarkably, for binormal-scroll bands, the direction of spin depends not only on the twist orientation but also on aspect ratio. Experimentally, we track the sedimentation of 3D-printed polystyrene bands in water. Computationally, we use the particle-base method dissipative particle dynamics. The hydrodynamic properties of these objects offer exciting applications for mixing and separation in microfluidics and may also serve as prototypes for passive swimmers.