On mathematical approaches to modelling slender liquid jets with a curved trajectory

© 2018 Cambridge University Press. Slender liquid jets that have a curved trajectory have been examined in a range of papers using a method introduced in Wallwork et al. (Proc. IUTAM Symp. on Free-Surface Flows, 2000, Kluwer; J. Fluid Mech., vol. 459, 2002, pp. 43-65) and Decent et al. (J. Engng Maths, vol. 42, 2002, pp. 265-282), for jets that emerge from an orifice on the surface of a rotating cylindrical container, successfully comparing computational results to measurements arising from laboratory experiments. Wallwork et al. (2000, 2002) and Decent et al. (2002) based their analyses on the slenderness of the jet, and neglected the torsion of the centreline of the jet, which is valid since in most situations examined the torsion is zero or small. Shikhmurzaev & Sisoev (J. Fluid Mech., vol. 819, 2017, pp. 352-400) used differential geometry and incorporated the torsion. This paper shows that these two methods produce identical results at leading order when the torsion is zero or when the torsion is , in an asymptotic framework based upon the slenderness of the jet, and shows that the method of Wallwork et al. (2000, 2002) and Decent et al. (2002) is accurate for parameters corresponding to scenarios previously examined and also when the torsion is . It is shown that the method of Shikhmurzaev & Sisoev (2017) should be used when the torsion is asymptotically large or when the jet is not slender

The trajectory of slender curved liquid jets for small Rossby number

© The Author(s) 2018. 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. FluidMech., 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

Sideband instability and modulations of Faraday waves.

Three-mode interaction equations are derived for Faraday waves in a long rectangular container. Two water depths are studied, revealing very different behaviour. Our equations include conservative and non-conservative cubic nonlinear terms, and quintic conservative nonlinear terms. Instability of a single standing wave to neighbouring modes is examined. Resultant three-mode interactions display rich structure, with fast and slow timescales. For the smaller depth, but not the greater, recurrent nearly calm intervals are separated by strong wave activity. Our results agree quite well with experimental findings of Craik and Armitage [Faraday excitation, hysteresis and wave instability in a narrow rectangular wave tank, Fluid Dyn. Res. 15 (1995) 129-143] and with unpublished observations by Armitage, Craik and Sterratt, here briefly described. (C) 1999 Elsevier Science B.V. All rights reserved.</p

General string theory for dynamic curved viscida with surface tension

This work deals with the asymptotic derivation and numerical investigation of a model for the dynamics of curved inertial viscous fibres under surface tension, as they occur in rotational spinning processes. The resulting string model accounts for the inner viscous transport and places no restriction on either motion or shape of the fibre centre-line. The boundary conditions for the free end of the fibre yield a description for its temporal evolution, depending on the ratio of viscous and surface tension (capillary number). The behaviour of the fibre is studied numerically as function of the effects of viscosity, gravity, rotation and surface tension