Periodic Motion of Sedimenting Flexible Knots

We study the dynamics of knotted deformable closed chains sedimenting in a viscous fluid. We show experimentally that trefoil and other torus knots often attain a remarkably regular horizontal toroidal structure while sedimenting, with a number of intertwined loops, oscillating periodically around each other. We then recover this motion numerically and find out that it is accompanied by a very slow rotation around the vertical symmetry axis. We analyze the dependence of the characteristic timescales on the chain flexibility and aspect ratio. It is observed in the experiments that this oscillating mode of the dynamics can spontaneously form even when starting from a qualitatively different initial configuration. In numerical simulations, the oscillating modes are usually present as transients or final stages of the evolution, depending on chain aspect ratio and flexibility, and the number of loops.

Figure 1

Examples of structures evolving to regular configurations ${\mathcal{T}}_{2,3}$: (a),(b) experiments, (c) simulations, (d) the notation used in this Letter, with ${\omega }_{\text{rot}}=2\pi /{\tau }_{\text{rot}}$ and ${\omega }_{\text{swirl}}=2\pi /{\tau }_{\text{swirl}}$. In all images gravity points downwards.

Figure 2

Snapshots of sedimenting ${\mathcal{T}}_{2,3}$ knots from experiments (left) and simulations (right).

Figure 3

Ratios of timescales for swirling, rotation, and sedimentation of ${\mathcal{T}}_{2,3}$ loops vs the number of beads $N$ (numerical results).

Figure 4

Swirling motion of a single bead of the ${\mathcal{T}}_{2,3}$ chain with $N=90$, in scaled cylindrical coordinates $\tilde{\rho }$, $\tilde{\alpha }$, $\tilde{z}$ (see text) represented with squares, circles, and triangles, respectively. Solid lines are fits to the simulation data (see Supplemental Material [41]).