Consequences of a double zero eigenvalue for the rotational motion of a prolate spheroid in shear flow

The rotational motion of a single prolate spheroidal particle in a linear shear flow provides fundamental knowledge necessary to understand both rheology and orientation distributions of suspensions including elongated particles. In this work, we present results from both direct numerical simulations using the lattice Boltzmann method and stability analysis using Comsol Multiphysics. It has been found that particle and fluid inertia cause different stable rotational states.\footnote{Ros\'{e}n \textit{et al.}, J. Fluid Mech. \textbf{738}, 563 (2013).}$^,$\footnote{Ros\'{e}n \textit{et al.}, J. Fluid Mech., submitted (2014).} The transitions between these originate from the fact that the Log-rolling particle (particle aligned with vorticity) has a double zero eigenvalue for a certain choice of particle Reynolds number $Re_p=Re_{PF}$ and solid-to-fluid density ratio $\alpha=\alpha_{DZ}$. The consequence is that particles with $\alpha>\alpha_{DZ}$, will always go to a planar rotation (symmetry axis perpendicular to vorticity), while lighter particles can assume a stable periodic or chaotic state which is non-planar. Since $\alpha_{DZ}$ is decreasing with aspect ratio, we find further that only planar states exist for particles of low aspect ratio (length/width).