Imperfect O(2) symmetry in counter-rotating split-cylinder flow

The effect of a small imperfection in the counter-rotating split-cylinder flow is studied numerically. The defect is characterized by a small parameter ϵ, corresponding to the difference in the magnitude of rotations in each half of the cylinder. With the two half cylinders not rotating exactly in counter rotation, the O(2) symmetry of the exact counter-rotating case (invariance to azimuthal rotations as well as to an involution consisting of reflections about the mid-plane composed of reflections about any meridional plane) is weakly broken. This small defect results in relevant variations in the flow. For slow rotations (characterized by a small Reynolds number), the system remains axisymmetric with the imperfection only breaking the reflection symmetry about the cylinder half-height. At larger Reynolds numbers, in the absence of the imperfection, axisymmetry is broken resulting in steady states with azimuthal wavenumber m. When axisymmetry is broken in the presence of the imperfection, a background rotation is introduced. Depending on the case and the level of imperfection, either rotating waves or slow-fast dynamics with mean background rotations are found instead. The interaction between azimuthal wavenumbers m = 2 and 3 plays a crucial role in the flow. The flow is analyzed in detail, varying ϵ from a very small value of 0.01%, typical of a natural imperfection in an experimental setup, to higher values corresponding to forced symmetry breaking. The ramifications of the imperfection on various solution states found in the exact counter-rotating case for a fixed aspect ratio are investigated.