The trajectory of slender curved liquid jets for small Rossby number

Wallwork et al. (2002, The trajectory and stability of a spiralling liquid jet. Part 1. Inviscid theory. J. Fluid Mech., 459, 43–65) and Decent et al. (2002, Free jets spun from a prilling tower. J. Eng. Math., 42, 265–282) developed an asymptotic method for describing the trajectory and instability of slender curved liquid jets. Decent et al. (2018, On mathematical approaches to modelling slender liquid jets with a curved trajectory. J. Fluid Mech., 844, 905–916.) showed that this method is accurate for slender curved jets when the torsion of the centreline of the jet is small or O(1), but the asymptotic method may become invalid when the torsion is asymptotically large. This paper examines the torsion for a slender steady curved jet which emerges from an orifice on the outer surface of a rapidly rotating container. The torsion may become asymptotically large, close to the orifice when the Rossby number
Rb≪1, which corresponds to especially high rotation rates. This paper examines this asymptotic limit in different scenarios and shows that the torsion may become asymptotically large inside a small inner region close to the orifice where the jet is not slender. Outer region equations which describe the slender jet are determined and the torsion is found not to be asymptotically large in the outer region; these equations can always be used to describe the jet even when the torsion is asymptotically large close to the orifice. It is in this outer region where travelling waves propagate down the jet and cause it to rupture in the unsteady formulation, and so the method developed by Wallwork et al. (2002, The trajectory and stability of a spiralling liquid jet. Part 1. Inviscid theory. J. Fluid Mech., 459, 43–65) and Decent et al. (2002, Free jets spun from a prilling tower. J. Eng. Math., 42, 265–282) can be used to accurately study the jet dynamics even when the torsion is asymptotically large at the orifice.