Perturbations of the coupled Jeffery–Stokes equations

Abstract

This paper seeks to provide clues as to why experimental evidence for the alignment of slender fibres in semi-dilute suspensions under shear flows does not match theoretical predictions. This paper posits that the hydrodynamic interactions between the different fibres that might be responsible for the deviation from theory, can at least partially be modelled by the coupling between Jeffery's equation and Stokes' equation. It is proposed that if the initial data are slightly non-uniform, in that the probability distribution of the orientation has small spatial 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 fibres 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.