Coupled fluid and energy flow in fabrication of microstructured optical fibres

Abstract

We consider the role of heating and cooling in the steady drawing of a long and thin viscous thread with an arbitrary number of internal holes of arbitrary shape. The internal holes and the external boundary evolve as a result of the axial drawing and surface-tension effects. The heating and cooling affects the evolution of the thread because both the viscosity and surface tension of the thread are assumed to be functions of the temperature. We use asymptotic techniques to show that, under a suitable transformation, this complicated three-dimensional free boundary problem can be formulated in such a way that the transverse aspect of the flow can be reduced to the solution of a standard Stokes flow problem in the absence of axial stretching. The solution of this standard problem can then be substituted into a system of three ordinary differential equations that completely determine the flow. We use this approach to develop a very simple numerical method that can determine the way that thermal effects impact on the drawing of threads by devices that either specify the fibre tension or the draw ratio. We also develop a numerical method to solve the inverse problem of determining the initial cross-sectional geometry, draw tension and, importantly, heater temperature to obtain a desired cross-sectional shape and change in cross-sectional area at the device exit. This precisely allows one to determine the pattern of air holes in the preform that will achieve the desired hole pattern in the stretched fibre.

A two-dimensional asymptotic model for capillary collapse

“…In this reference frame we may consider the problem to be steady. The model derivation is described in detail by Stokes et al (2014Stokes et al ( , 2019 for a capillary of arbitrary geometry in the context of fibre drawing, and specifically applied to an axisymmetric tube.…”

“…As for the 1-D model, the surface tension γ of the glass is assumed to be constant. As shown in Stokes et al (2019), using coupled energy and flow models, the viscosity μ is, to leading order, uniform in any cross-section and, so, depends on x only.…”

“…Thus, 2-D models exist which are applicable to capillary collapse and which predate that of Klupsch & Pan (2017). For a radial length scale that is much smaller than the axial length scale, as assumed by Klupsch & Pan, the asymptotic fibre-drawing model of Stokes et al (2014Stokes et al ( , 2019) may be used for capillary collapse and yields analytic formulae relating the extent of collapse to the harmonic mean of the viscosity and the surface tension of the glass. The accuracy of this model is excellent for drawing of fibre from very slender preforms (Chen et al 2016b) and is expected to be excellent for capillary collapse which does not see the high draw ratios and sharp tapers seen in fibre drawing and which are known to cause discrepancies between models and experiments, especially when the radius of the preform is not sufficiently small relative to the neck-down length (Chen et al 2016a).…”

“…2015 a , b ; Chen et al. 2016 b ; Stokes, Wylie & Chen 2019), and the drawing of slender axisymmetric tubes is a particular case. Thus, 2-D models exist which are applicable to capillary collapse and which predate that of Klupsch & Pan (2017).…”

“…For a radial length scale that is much smaller than the axial length scale, as assumed by Klupsch & Pan, the asymptotic fibre-drawing model of Stokes et al. (2014, 2019) may be used for capillary collapse and yields analytic formulae relating the extent of collapse to the harmonic mean of the viscosity and the surface tension of the glass. The accuracy of this model is excellent for drawing of fibre from very slender preforms (Chen et al.…”