Rotation of a spheroid in a simple shear at small Reynolds number

We derive an effective equation of motion for the orientational dynamics of a neutrally buoyant spheroid suspended in a simple shear flow, valid for arbitrary particle aspect ratios and to linear order in the shear Reynolds number. We show how inertial effects lift the degeneracy of the Jeffery orbits and determine the stabilities of the log-rolling and tumbling orbits at infinitesimal shear Reynolds numbers. For prolate spheroids we find stable tumbling in the shear plane, log-rolling is unstable. For oblate particles, by contrast, log-rolling is stable and tumbling is unstable provided that the aspect ratio is larger than a critical value. When the aspect ratio is smaller than this value tumbling turns stable, and an unstable limit cycle is born.