Spiralling liquid jets: verifiable mathematical framework, trajectories and peristaltic waves

Abstract

The dynamics of a jet of an inviscid incompressible liquid spiralling out under the action of centrifugal forces is considered with both gravity and the surface tension taken into account. This problem is of direct relevance to a number of industrial applications, ranging from the spinning disc atomization process to nanofibre formation. The mathematical description of the flow by necessity requires the use of a local curvilinear non-orthogonal coordinate system centred around the jet’s baseline, and we present the general formulation of the problem without assuming that the jet is slender. To circumvent the inconvenience inherent in the non-orthogonality of the local coordinate system, the orthonormal Frenet basis is used in parallel with the local non-orthogonal basis, and the equation of motion, with the velocity considered with respect to the local coordinate system, is projected onto the Frenet basis. The variation of the latter along the baseline is then described by the Frenet equations which naturally brings the baseline’s curvature and torsion into the equations of motion. This technique allows one to handle different line-based non-orthogonal curvilinear coordinate systems in a straightforward and mathematically transparent way. An analysis of the slender-jet approximation that follows the general formulation shows how a set of ordinary differential equations describing the jet’s trajectory can be derived in two cases: $\mathit{We}=O(1)$ and $\unicode[STIX]{x1D716}\mathit{We}=O(1)$ as $\unicode[STIX]{x1D716}\rightarrow 0$, where $\unicode[STIX]{x1D716}$ is the ratio of characteristic length scales across and along the jet and $\mathit{We}$ is the Weber number. A one-dimensional model for the propagation of nonlinear peristaltic disturbances along the jet is derived in each of these cases. A critical review of the work published on this topic is presented showing where errors typically occur and how to identify and avoid them.

A comprehensive mathematical model for nanofibre formation in centrifugal spinning methods

“…As can be seen, our local bases are neither orthogonal (due to existence of off-diagonal elements) nor normalized (due to non-unity of diagonal elements) as pointed out by Shikhmurzaev A comprehensive mathematical model for nanofibre formation 892 A26-11 & Sisoev (2017). It is noted that some of the relations presented in this and the next subsections have been derived with alternative methods by Shikhmurzaev & Sisoev (2017), which we re-derive for the sake of completeness and verification. Using (3.17), we can simply compute the conjugate metric tensor g ij so that g ik • g kj = δ i j , in which δ i j is the Kronecker delta; therefore, we find…”

“…in which {I 1 , I 2 , I 3 } and {J 1 , J 2 , J 3 } are the first and second fundamental forms (Marheineke & Wegener 2007;Nguyen-Schäfer & Schmidt 2014;Shikhmurzaev & Sisoev 2017). It is noted that to obtain the deviatoric stress tensor Π, one has to use the projection of the stress tensor components (3.30) onto the Frenet basis.…”

“…When using the asymptotic methods to derive the curved jet equations, most of the previous studies assumed that the jet baseline torsion can be ignored, making it possible to use the orthogonal curvilinear coordinate system. Alternatively, Shikhmurzaev & Sisoev (2017) considered the effect of torsion when deriving the equations, leading to non-orthogonal basis vectors the terms of which they projected onto the orthonormal Frenet basis. Ignoring the jet cross-section deformation due to torsion, Panda (2006) and Marheineke & Wegener (2009) used the Bishop frame (Bishop 1975) to provide coordinates of the curved jet and thereby avoid a cumbersome derivation of the governing equations in the non-orthogonal curvilinear coordinate system.…”

Oblique liquid curtains with a large Froude number

“…Consider a set of curvilinear coordinates (l, τ ) such that the Cartesian coordinates are related to them by 1a,b) where t is the time. The curve τ = 0 is assumed to coincide with the curtain's centreline, and l on this curve is the centreline's arc length -but other than that, (l, τ ) are not related to the coordinates used in Decent et al (2002), Wallwork et al (2002) and Shikhmurzaev & Sisoev (2017). Let (l, τ ) be orthogonal and with a unit Jacobian, i.e.…”

Spinning disk atomization: Theory of the ligament regime

“…As reviewed in Ref. 37, this purely technical element turned out to be a major hurdle for many theoretical studies. The second element of difficulty is that the waves excited at the beginning of the jet in the SDA process then have to propagate over a spatially varying base flow, a particular class of problems reviewed in Ref.…”