The Centrifugal Instability of a Slender Rotating Cone

In this study, we provide a mathematical description of the onset of counter-rotating circular vortices observed for a family of slender rotating cones (of half-angles 15 degrees or less) in quiescent fluid. In particular, we apply appropriate scalings in order to simplify the basic-flow profiles, which are subsequently perturbed, accounting for the effects of streamline curvature. A combined large Reynolds number and large vortex wavenumber analysis is used to obtain an estimate for the asymptotic right-hand branch of neutral stability for a slender rotating cone. Our results confirm our earlier predictions pertaining to the existence of the new Gortler mode and capture the effects of the governing centrifugal instability mechanism. Meanwhile, favourable comparisons are drawn with existing numerical neutral stability curve results.