Perturbations of the coupled Jeffery-Stokes equations

TL;DR

This paper investigates how hydrodynamic interactions influence fiber orientation in suspensions, showing that coupling Jeffery's and Stokes' equations explains deviations from classical predictions, supported by numerical spectral analysis.

Contribution

It introduces a coupled Jeffery-Stokes model to explain experimental deviations and demonstrates the growth of non-uniformities through spectral analysis.

Findings

Hydrodynamic interactions cause significant orientation deviations.

Coupling leads to growth of initial non-uniformities.

Spectral analysis supports the instability mechanism.

Abstract

This paper seeks to provide clues as to why experimental evidence for the alignment of slender fibers in semi-dilute suspensions under shear flows does not match theoretical predictions. This paper posits that the hydrodynamic interactions between the different fibers that might be responsible for the deviation from theory, can at least partially be modeled by the coupling between Jeffery's equation and Stokes' equation. It is proposed that if the initial data is slightly non-uniform, in that the probability distribution of the orientation has small spacial variations, then there is feedback via Stokes' equation that causes these non-uniformities to grow significantly in short amounts of time, so that the standard uncoupled Jeffery's equation becomes a poor predictor when the volume ratio of fibers to fluid is not extremely low. This paper provides numerical evidence, involving spectral analysis of the linearization of the perturbation equation, to support this theory. This paper differs from the published version in that it contains a corrogendum at the end.