Axial oscillations of a sphere in a rotating cylinder

The flow in a rapidly rotating cylinder forced by the harmonic oscillations of a small sphere along the rotation axis is explored numerically. For oscillation frequencies less than twice the cylinder rotation frequency, the forced response flows feature conical shear layers emitted from the critical latitudes of the sphere. These latitudes are where the characteristics of the hyperbolic system, arrived at by ignoring nonlinear, viscous and forcing terms in the governing equations, are tangential to the sphere. These conical shear layers vary continuously with the forcing frequency so long as it remains inertial. At certain values of the forcing frequency, linear inviscid inertial modes of the cylinder are resonated. Of all possible inertial modes, only those whose symmetries are compatible with the symmetry of the forced system are resonated. This all occurs even in the linear limit of vanishingly small forcing amplitude. As the forcing amplitude is increased, nonlinearity leads to non-harmonic oscillations and a non-zero mean flow which features a Taylor columnar structure extending from the sphere to the two endwalls in an axially invariant fashion.