  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.
