Drawing of micro-structured fibres: circular and non-circular tubes

Abstract

A general mathematical framework is presented for modelling the pulling of optical glass fibres in a draw tower. The only modelling assumption is that the fibres are slender; cross-sections along the fibre can have general shape, including the possibility of multiple holes or channels. A key result is to demonstrate how a so-called reduced time variable τ serves as a natural parameter in describing how an axial-stretching problem interacts with the evolution of a general surface-tension-driven transverse flow via a single important function of τ , herein denoted by H(τ ), derived from the total rescaled cross-plane perimeter. For any given preform geometry, this function H(τ ) may be used to calculate the tension required to produce a given fibre geometry, assuming only that the surface tension is known. Of principal practical interest in applications is the 'inverse problem' of determining the initial cross-sectional geometry, and experimental draw parameters, necessary to draw a desired final cross-section. Two case studies involving annular tubes are presented in detail: one involves a cross-section comprising an annular concatenation of sintering near-circular discs, the cross-section of the other is a concentric annulus. These two examples allow us to exemplify and explore two features of the general inverse problem. One is the question of the uniqueness of solutions for a given set of experimental parameters, the other concerns the inherent ill-posedness of the inverse problem. Based on these examples we also give an experimental validation of the general model and discuss some experimental matters, such as buckling and stability. The ramifications for modelling the drawing of fibres with more complicated geometries, and multiple channels, are discussed.

Collapsing of glass tubes: analytic approaches in a hydrodynamic problem with free boundaries

“…It has also been shown that the description for capillary geometries [9] provides practical insights into the drawing of more complex multi-hole geometries [29]. Based, in addition, on an earlier leadingorder analysis for extensional flows in slender geometries [32], a mathematical leading-order framework of fiber drawing, specified to arbitrary cross-plane geometries, is applied in [33] and subsequent papers [34,35].…”

“…Of course, the approaches used in the drawing theory are adjusted, first of all, to arrive at a basic understanding of influencing factors of the technical fiber drawing process, such as peculiarities of drawing of fibers with complicated cross-sections (see, e.g., [33][34][35]). The limiting case of equal, impressed axial velocities on both sides of the heating zone (collapsing limit), if mentioned at all [9,26,33], is considered being of minor interest, e.g., for preform fabrication (see, e.g., [33]).…”

“…The limiting case of equal, impressed axial velocities on both sides of the heating zone (collapsing limit), if mentioned at all [9,26,33], is considered being of minor interest, e.g., for preform fabrication (see, e.g., [33]). Known papers on collapsing only are not cited in the papers on drawing.…”

Asymptotic Modelling of a Six-Hole MOF

“…Our approach assumes the shape of the neck-down is slender, that is = √ S 0 /L 1, and this significantly simplifies the full three-dimensional Navier-Stokes equations. As outlined in detail in [3] and [4], we make various perturbation expansions in 2 and solve a coupled system of equations for the leading-order flow in the axial direction (that is, along the axis of the fibre) and for the flow in the cross-plane (in every cross-section along the length of the neck-down).…”

“…The modelling approach of the present study has been developed by the authors [3], [4] (building on earlier work in [5]- [7], among others) specifically for the problem of fibre drawing. This approach immediately has an advantage over a package which is, by nature, designed to be as general as possible in scope and not context specific.…”