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

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.Yvonne M. Stokes, Peter Buchak, Darren G. Crowdy and Heike Ebendor-Heideprie

THE SURFACE-TENSION-DRIVEN RETRACTION OF A VISCIDA

We consider the surface-tension-driven evolution of a thin two-dimensional sheet of viscous fluid whose ends are held a fixed distance apart. We find that the evolution is governed by a nonlocal nonlinear partial differential equation, which may be transformed, via a suitable change of time variable, to a simple linear equation. This possesses an interesting dispersion relation which indicates that it is well posed whether solved forwards or backwards in time, enabling us to determine which initial shapes will evolve to a given shape at a later time. We demonstrate that our model may be used to describe the global evolution of a viscida containing small regions of high curvature, and proceed to investigate the evolution of a profile which contains a corner. We show that the corner is not smoothed out but persists for forward and inverse time. The introduction of a pressure differential across the free surfaces is shown to provide a method of controlling the shape evolution. © by SIAM