Mathematical Modelling of Fibre Fabrication; Coupling Heat Transport with Fluid Flow in Extrusion

Abstract

We consider microstructured optical fibre fabrication by a two-step process: preform extrusion and fibre drawing. During preform extrusion a molten glass bulk billet is pushed through a die to give a preform, the macroscopic version of the fibre, which is stretched either by pulling with a given force or by gravitational stretching. The extruded preform is then reheated as it is fed slowly into the furnace region of a draw tower, from which it is pulled at a fast speed to yield a fibre, which cools and is wound onto a spool some distance from the furnace region. Throughout both processes, surface tension, gravity, and any pulling force cause the glass to deform while it is sufficiently hot, and the temperature of the glass is important. The fibre drawing process is fast compared to preform extrusion and the e↵ect of the heat conduction along the glass is negligible, so that mathematical models of fibre drawing have only included advective heat transport; however, in preform extrusion, both advective and conductive transport are likely to be important. Moreover, to date temperature has not been included in mathematical modelling of preform extrusion. This thesis focuses on adding temperature to the flow model of extrusion of Tronnolone (2016), with inclusion of both advection and conduction. With our focus on preform extrusion, we explore two problems, namely the steady problem of pulling a preform from a hot die and the unsteady problem of extrusion from a hot die with gravity stretching. The first problem, the steady problem of pulling a preform, is similar to fibre drawing excepting that the glass starts hot and cools. There is no modelling of heating before cooling, as there is in fibre drawing in Stokes et al. (2019). The asymptotic heat and flow models of this steady fibre drawing problem, in which advective heat transport dominates conduction that is neglected, are relatively straightforward to solve: the model is a firstorder vector initial value-problem. In contrast, inclusion of the second-order conduction term gives rise to a boundary value problem that is surprisingly difficult to solve. An iterative finite di↵erence method is used to solve this boundary value problem. The solution method is verified in the limit that conduction becomes unimportant by comparison with the results from a model that neglects conduction. The second problem, unsteady extrusion with gravity stretching, has di↵erent gravitational force acting on each cross-section of the glass due to the varying mass below each cross-section. Because of this, the Eulerian spatial variable is converted to a Lagrangian variable, which simplifies the expression for the force due to gravity, and the problem may be solved using the method of lines. The solution leads to a better understanding of the influence of temperature on the geometry of an extruded glass preform.