Mathematics in the glass industry

Peter Howell has been working in Oxford on extensional thin layer flows with application to the glass industry for the last 20 years. He first studied the evolution of thin sheets and jets of viscous fluid by taking a mathematical limit where the aspect ratio (the ratio of the thickness to the length or width) is small, precisely the situation in the manufacture of glass windows, tubing and optical fibres, for example. The result of the research [1] is a systematic framework for reducing the full Navier-Stokes equations to a simplified lower-dimensional system, which gives greater insight into possible instabilities and allows for much more efficient computation. These simplified models allow glass processes to be more effectively controlled to produce flawless products with optimized properties and to avoid catastrophic process failures.

Another important facet of the University of Oxford’s glass modelling concerns the drawing of non-axisymmetric glass tubing. A key question is: what die shape is needed to make tubes of a given cross-sectional shape? Using ideas from perturbation theory and partial differential equations, Ian Griffiths and Peter Howell were able to solve this inverse problem explicitly [2,3], as shown in the figure below, which shows the die shape required to draw glass tubing with a square cross-section.

glass_tubes2

Figure 1: Cross-sectional profile during fabrication of a square glass tube.

Current research within Oxford involves a collaboration with Schott AG and Oxford researchers Doireann O’Kiely, Chris BrewardIan Griffiths, and Peter Howell, in which mathematical models are being developed for the redraw process. Here, prefabricated glass blocks are drawn through a furnace and stretched. These mathematical models are able to predict the operating regimes for producing glass sheets with uniform thickness [4]. The resulting thin sheets are currently used in smartphone cameras and tablet computers, and the improvements afforded by modelling assist in reducing the fabrication costs.

To find out more on glass research in Oxford click here.

References
[1] P.D. Howell 1998 Models for thin viscous sheets, Euro. J. Appl. Math., 7, 321-343.
[2] I.M. Griffiths & P.D. Howell 2007 The surface-tension-driven evolution of a two-dimensional annular viscous tube, J. Fluid Mech.593, 181-208.
[3] I.M. Griffiths & P.D. Howell 2008 Mathematical modelling of non-axisymmetric capillary tube drawing, J. Fluid Mech., 605, 181-206.
[4] D. O’Kiely, C.J.W. Breward, I.M. Griffiths, P.D. Howell & U. Lange 2015 Edge behaviour in the glass sheet redraw process, J. Fluid Mech. (in press).

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