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

Abstract

Slender liquid jets that have a curved trajectory have been examined in a range of papers using a method introduced in Wallworket al.(Proc. IUTAM Symp. on Free-Surface Flows, 2000, Kluwer;J. Fluid Mech., vol. 459, 2002, pp. 43–65) and Decentet 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. Wallworket al.(2000, 2002) and Decentet 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$O(1)$, in an asymptotic framework based upon the slenderness of the jet, and shows that the method of Wallworket al.(2000, 2002) and Decentet al.(2002) is accurate for parameters corresponding to scenarios previously examined and also when the torsion is$O(1)$. 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.