Non-linear Instability of periodic orbits of suspensions of thin fibers in fluids

TL;DR

This paper investigates the nonlinear instability of fiber orientations in suspensions, revealing limitations of Jeffery's equation and providing rigorous proofs of instability using Floquet Theory, with implications for predicting fiber behavior in fluids.

Contribution

The paper offers rigorous proofs of nonlinear instability of fiber orientation models in suspensions, extending understanding beyond Jeffery's equation limitations.

Findings

Instability can occur even when spectral radius is one.

Nonlinear effects cause deviations from Jeffery's predictions.

Floquet Theory effectively demonstrates instability mechanisms.

Abstract

This paper is concerned with difficulties encountered by engineers when they attempt to predict the orientation of fibers in the creation of injection molded plastic parts. It is known that Jeffery's equation, which was designed to model a single fiber in an infinite fluid, breaks down very badly when applied, with no modifications, to this situation. In a previous paper, the author described how interactions between the fiber orientation and the viscosity of the suspension might cause instability, which could result in the simple predictions from Jeffery's equation being badly wrong. In this paper, we give some rigorous proofs of instability using Floquet Theory. We also give an example where the spectal radius of the monodromy operator is one, but there is still non-linear instability.