Orientation dynamics of a spheroidal particle settling in a simple shear flow

In a viscous simple shear flow, a force-free, torque-free spheroid executes a periodic rotary motion - known as Jeffery orbits. However, the orientation of a sedimenting axisymmetric particle is indeterminate in the Stokesian regime. The indeterminacy is removed with the inclusion of fluid inertia, wherein an inertial torque leads to a spheroid falling with a broadside-on orientation. In the current study, we explore the competition of the two torques, shear and inertial, for a spheroid settling in a simple shear flow. Two non-dimensional quantities span the parameter space - the ratio of flow to settling time-scale (S_F) and the angles subtended by the vorticity axis of the flow to gravity (α, β). The orientation phase-space exhibits a rich dynamical behavior as we scan through S_F, α, and β values. The bifurcation diagrams reveal the transition from an oscillating orientation angle to a constant fixed angle predicted from larger sedimentation torques.