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 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 $O(1)$, 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 $O(1)$. 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.

Oblique liquid curtains with a large Froude number

“…the slender-jet equations derived by Wallwork (2001), and Wallwork et al (2002). Those papers employed curvilinear coordinates linked to the jet's centreline and assumed them to be orthogonal -which they were not, as it turned out (Entov & Yarin 1984; Shikhmurzaev & Sisoev 2017) -although the asymptotic equations derived by Wallwork, Decent and collaborators still happened to be correct, as the non-orthogonality of the coordinates for slender jets is weak (Decent et al 2018). For two-dimensional flows, however, the coordinates of Decent et al (2002) and Wallwork et al (2002) are orthogonal -so, in principle, they could be used in the present work.…”

The trajectory of slender curved liquid jets for small Rossby number

“…Shikhmurzaev & Sisoev (2017) showed that the torsion of the centreline of a jet, denoted here by κ 2 , can influence the flow of the jet, and also produced equations that describe a curved jet which are valid even when the jet is not asymptotically slender by using differential geometry. Following that paper, Decent et al (2018) showed that the method used by and is accurate for slender jets when the torsion is small or O (1), and also showed that the methods developed by , and Shikhmurzaev & Sisoev (2017) produce slender jet equations that agree at leading order in ε (based upon asymptotic expansions of the physical equations in ε). Decent et al (2018) also showed that their slender jet model may break-down if the torsion is asymptotically large, and in particular argued that this may occur if κ 2 = O ( ε −1 ) or larger.…”

“…Following that paper, Decent et al (2018) showed that the method used by and is accurate for slender jets when the torsion is small or O (1), and also showed that the methods developed by , and Shikhmurzaev & Sisoev (2017) produce slender jet equations that agree at leading order in ε (based upon asymptotic expansions of the physical equations in ε). Decent et al (2018) also showed that their slender jet model may break-down if the torsion is asymptotically large, and in particular argued that this may occur if κ 2 = O ( ε −1 ) or larger. This asymptotically large torsion corresponds to the basis vectors of the coordinate system no longer being orthogonal and this non-orthogonality having an effect at leading order on the jet, based upon asymptotic expansions using ε (see Decent et al 2018): in other words, the coordinate system used by and may become inappropriate to describe the jet effectively when κ 2 is sufficiently asymptotically large.…”

“…Decent et al (2018) also showed that their slender jet model may break-down if the torsion is asymptotically large, and in particular argued that this may occur if κ 2 = O ( ε −1 ) or larger. This asymptotically large torsion corresponds to the basis vectors of the coordinate system no longer being orthogonal and this non-orthogonality having an effect at leading order on the jet, based upon asymptotic expansions using ε (see Decent et al 2018): in other words, the coordinate system used by and may become inappropriate to describe the jet effectively when κ 2 is sufficiently asymptotically large. However, in all previous situations examined numerically or in laboratory experiments, the torsion has been small (and often zero) or at most O (1) (e.g.…”

“…(see Decent et al 2018). A numerical solution can be found to equations (1.1) to (1.3) for O (1) values of the parameters.…”