# Flexible fibers in shear flow approach attracting periodic solutions

## TL;DR

This study numerically investigates the complex three-dimensional behaviors of flexible fibers in shear flow, revealing multiple attracting states including steady configurations and periodic motions, with transitions influenced by bending to hydrodynamic force ratios.

## Contribution

It identifies and characterizes new attracting periodic solutions and steady configurations of flexible fibers in shear flow, expanding understanding beyond extensional flow dynamics.

## Key findings

- Existence of attracting steady and periodic motions.
- Effective Jeffery orbits with exponential parameter drift.
- Identification of a three-dimensional meandering periodic mode.

## Abstract

The three-dimensional dynamics of a single non-Brownian flexible fiber in shear flow is evaluated numerically, in the absence of inertia. A wide range of ratios A of bending to hydrodynamic forces and hundreds of initial configurations are considered. We demonstrate that flexible fibers in shear flow exhibit much more complicated evolution patterns than in the case of extensional flow, where transitions to higher-order modes of characteristic shapes are observed when A exceeds consecutive threshold values. In shear flow, we identify the existence of an attracting steady configuration and different attracting periodic motions that are approached by long-lasting rolling, tumbling and meandering dynamical modes, respectively. We demonstrate that the final stages of the first and second modes are effective Jeffery orbits, with the constant parameter C replaced by an exponential function that either decays or increases in time, respectively, corresponding to a systematic drift of the trajectories. In the limit of C going to zero, the fiber aligns with the vorticity direction and in the limit of C going to infinity, the fiber periodically tumbles within the shear plane. For moderate values of A, a three-dimensional meandering periodic motion exists, which corresponds to intermediate values of C. Transient, close to periodic oscillations are also detected in the first stages of the modes.