We examine a spiralling slender inviscid liquid jet which emerges from a rapidly rotating orifice. The trajectory of this jet is determined using asymptotic methods, and the stability using a multiple scales approach. It is found that the trajectory of the jet becomes more tightly coiled as the Weber number is decreased. Unstable travelling wave modes are found to grow along the jet. The breakup length of the jet is calculated, showing good agreement with experiments.

Faraday waves arise on the surface of a liquid in a container that is undergoing vertical periodic oscillations. Hysteresis occurs when both finite-amplitude solutions and the flat-surface solution are available. We derive a nonlinear model of Faraday resonance, extending the Lagrangian method of Miles (1976). The model is used to investigate hysteresis. The theoretical results are compared to previous experimental studies and to some new observations. It is found necessary to retain damping and forcing terms up to third-order in wave amplitude, and also the fifth-order conservative frequency shift, in order to achieve agreement with experiments. The latter fifth-order term was omitted from all previous studies of Faraday waves. The lower hysteresis boundary in forcing-frequency space is found in most cases to be defined by the lower boundary above which non-trivial stationary points exist. However, the stability of stationary points and the existence of limit cycles are also found to be factors in determining the lower hysteresis boundary. Our results also suggest an indirect method for estimating the coefficient of cubic damping, which is difficult to obtain either experimentally or theoretically.

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The linear instability of a power law liquid emerging as a jet from an orifice on the surface of a rotating container is investigated, with applications to industrial prilling. Asymptotic methods are used to examine the growth rate and wavenumber of the most unstable traveling wave mode for different flow index numbers. Comparison with Newtonian liquids show that for small rotation rates shear thinning liquids are most stable to disturbances. In contrast for higher rotation rates we find shear thickening liquids are more stable than shear thinning liquids. The influence of viscosity, surface tension, and rotation rate on the growth rates and most unstable wavenumbers associated with both types of liquids are also examined.