# Flapping, swirling and flipping: Non-linear dynamics of pre-stressed active filaments

## TL;DR

This paper investigates the complex three-dimensional behaviors of pre-stressed active filaments in fluid, revealing how boundary conditions influence the transition from planar to swirling and flipping oscillations under follower forces.

## Contribution

It introduces a detailed analysis of non-linear instabilities and dynamics of pre-stressed filaments, highlighting the role of boundary constraints in pattern formation and oscillation behaviors.

## Key findings

- Clamped-pinned filaments exhibit stable planar flapping oscillations.
- Clamped-clamped filaments develop three-dimensional swirling and flipping patterns.
- Swirling reversal resembles relaxation oscillations with torsional jumps.

## Abstract

Initially straight slender elastic rods with geometrically constrained ends buckle and form stable two-dimensional shapes when compressed by bringing the ends together. It is also known that beyond a critical value of the pre-stress, clamped rods transition to bent, twisted three-dimensional equilibrium shapes. Recently, we showed that pre-stressed planar shapes when immersed in a dissipative fluid and animated by nonconservative follower forces exhibit stable large-amplitude flapping oscillations. Here, we use time-stepper methods to analyze the three-dimensional instabilities and dynamics of pre-stressed planar and non-planar filament configurations when subject to active follower forces and dissipative fluid drag. First, we find that type of boundary constraint determines the nature of the non-linear patterns following instability. When the filament is clamped at one end and pinned at the other with follower forces directed towards the clamped end, we observe only stable planar (flapping) oscillations termed flapping result. When both ends are clamped however, we observe a secondary instability wherein planar oscillations are destabilized by off-planar perturbations and result in fully three-dimensional swirling patterns characterized by two distinct time-scales. The first time scale characterizes continuous and unidirectional swirling rotation around the end-to-end axis. The second time scale captures the rate at which the direction of swirling reverses or flips. The overall time over which the direction of swirling flips is very short compared to the long times over which the filament swirls in the same direction. Computations indicate that the reversal of swirling oscillations resembles relaxation oscillations with each cycle initiated by a sudden jump in torsional deformation and then followed by a period of gradual decrease in net torsion until the next cycle of variations.