Gravitational extension of a fluid cylinder with internal structure

Abstract

Motivated by the fabrication of microstructured optical fibres, a model is presented for the extension under gravity of a slender fluid cylinder with internal structure. It is shown that the general problem decouples into a two-dimensional surface-tension-driven Stokes flow that governs the transverse shape and an axial problem that depends upon the transverse flow. The problem and its solution differ from that obtained for fibre drawing, because the problem is unsteady and the fibre tension depends on axial position. Solutions both with and without surface tension are developed and compared, which show that the relative importance of surface tension depends upon both the parameter values and the geometry under consideration. The model is compared to experimental data and is shown to be in good agreement. These results also show that surface-tension effects are essential to accurately describing the cross sectional shape.

Extrusion of fluid cylinders of arbitrary shape with surface tension and gravity

“…The velocity scale U is taken to be a representative axial velocity at the die exit related to the characteristic area and flux by Q = SU, while the time scale is L/U = LS/Q. In a typical preform extrusion the temperature varies with both position and time; however, as in previous work on fibre drawing (Stokes et al 2014;Buchak et al 2015;Chen et al 2015) and gravitational extension of a fluid cylinder (Tronnolone et al 2016), we assume that the temperature profile is known. The viscosity depends only on temperature so is thus assumed known and scaled µ = µ 0 µ * for some characteristic viscosity µ 0 .…”

“…A consequence of this is that the region above this, which is retained, may be considered slender for all times. Applying a slenderness approximation to the governing equations (2.1) and boundary conditions (2.2), in a similar manner to that of Tronnolone et al (2016) for a stretching cylinder, results in a one-dimensional model for the axial flow and a two-dimensional model for the transverse flow.…”

“…Reduced time has also been used to study active channel pressurisation during fibre drawing (Chen et al 2015); however, the addition of pressure to the model meant that the axial and transverse problems did not decouple. Tronnolone et al (2016) extended the work of Stokes et al (2014) to model the stretching under gravity of fluid cylinders with internal structure and surface tension. It was shown that the model provided a good match to experimental data, suggesting that surface tension and gravity alone were responsible for the deformation observed in experiments.…”

“…It was shown that the model provided a good match to experimental data, suggesting that surface tension and gravity alone were responsible for the deformation observed in experiments. For a detailed review of the literature the reader is directed to the articles by Stokes et al (2014) and Tronnolone et al (2016).…”

“…In this study we extend previous work to develop a model for preform extrusion that includes both surface tension and gravity; however, we assume plug flow at the die outlet and hence neglect the transition region in which the axial velocity profile is not uniform. Our approach is based upon the model used by Tronnolone et al (2016) for gravitational stretching, but differs in that for the current problem it proves useful to combine the Lagrangian time co-ordinate used by Wilson (1988) to model dripping with a modified reduced-time variable. Once again, the transverse flow decouples from the axial flow, the former of which reduces to a classical two-dimensional moving-boundary Stokes flow with unit surface tension.…”