Linear stability analysis of rotationally symmetric flow above an infinite rotating disk

A convective hydrodynamic instability analysis is presented for the axially symmetric flow of a rotating fluid above an infinite disk that is itself rotating, focusing on the role of relative sense and strength of rotation, quantified by the parameter s, the angular velocity ratio of the ambient fluid relative to that of the disk (or equivalently σ=1/s). This study further incorporates wall suction and surface roughness, key factors that not only influence the onset of instability in rotating-disk flows but also extend the range of s (or σ) for which base flow solutions exist. For a solid disk without suction and roughness, co-rotation (s>0 and σ>0) features a neutral curve with only an upper lobe, corresponding to an inviscid type-I mode that stabilizes as rotation strength increases, with transition delayed for s>0 relative to σ>0. In contrast, a counter-rotating solid disk exhibits both upper and lower lobes, the latter associated with a viscous type-II mode, which destabilizes with increasing counter-rotation. With suction, the base flow admits non-unique solutions in the counter-rotating disk regimes, of which only the first is physically admissible; consequently, the convective instability analysis is performed for this branch. For strong counter-rotation s=−0.8 with appropriate suction, neutral curves exist for negative wavenumbers in the Re−β (or Re−α) planes, showing only the upper lobe, while isotropic roughness adds the lower lobe, with the opposite trend observed for (σ=−0.8). The stability results are further validated through convective growth rates and energy analysis.