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Mathematical modelling is used to examine the unsteady problem of heating and pulling an axisymmetric cylindrical glass tube with an over-pressure applied within the tube to form tapers with a near uniform bore and small wall thickness at the tip. To allow for the dependence of viscosity on temperature, a prescribed axially varying viscosity is assumed. Our motivation is the manufacture of emitter tips for mass spectrometry which provide a continuous fluid flow and do not become blocked. We demonstrate, for the first time, the feasibility of producing such emitters by this process and examine the influence of the process parameters, in particular the pulling force and over-pressure, on the geometry. For small values of these parameters a more uniform bore may be obtained but the geometry is highly sensitive to small fluctuations. Higher values of the parameters result in more variation in the bore size but less sensitivity to fluctuations. The best parameters depend on the accuracy of the puller used to manufacture the tapers and the permissible tolerances on the geometry. The model has wider application to the manufacture of other devices., Comment: 13 pages, 9 figures