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In the fabrication of optical fibres, the viscosity of the glass varies dramatically with temperature so that heat transfer plays an important role in the deformation of the fibre geometry. Surprisingly, for quasi-steady drawing, with measurement of pulling tension, the applied heat can be adjusted to control the tension and temperature modelling is not needed. However, when pulling tension is not measured, a coupled heat and fluid flow model is needed to determine the inputs required for a desired output. In the fast process of drawing a preform to a fibre, heat advection dominates conduction so that heat conduction may be neglected. By contrast, in the slow process of extruding a preform, heat conduction is important. This means that solving the coupled flow and temperature modelling is essential for prediction of preform geometry. In this paper we derive such a model that incorporates heat conduction for the extensional flow of fibres. The dramatic variations in viscosity with temperature means that this problem is extremely challenging to solve via standard numerical techniques and we therefore develop a novel finite-difference numerical solution method that proves to be highly robust. We use this method to show that conduction significantly affects the size of internal holes at the exit of the device., Comment: 32 pages, 6 figures, 1 table