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Mathematical models for fibre drawing
Mathematical models for fibre drawing
In fibre drawing, a thread of viscous fluid is stretched and elongated as it cures or solidifies so that it is very long and thin. In manufacturing contexts, this can be a polymer or molten glass in the production of e.g. telecommunications fibres, but the mathematics is also relevant for honey falling from a spoon.
Mathematical modelling of this process goes back as far as 1906, when Trouton developed an empirical relationship between applied tension and rate of elongation:
Tension = C x Area x Viscosity x Elongation rate,
where the constant of proportionality C depends on the shape, and is 3 for a circular fibre and 4 for a sheet.
In the 1980 and 1990s mathematicians used the tool of asymptotic analysis to study the same problem with more mathematical rigour, and determined that this model describes the leading-order behaviour in the limit where the aspect ratio is small.
Niall Hanevy and I worked on the circular fibre problem in 2022, and dermined the correction to this leading-order behaviour using a combination of asymptotic analysis and finite element methods. One reason this was important is the pre-existing literature contained an incorrect prediction for the higher-order correction, based on an incorrect assumption about the appropriate choice of gauge functions in asymptotic expansions. The challenge of choosing the coorect gauge function had also arisen in previous work in my PhD, where we studied sheets (see below), and more recently has proven to be relevant in metal sheet rolling (see Metals).
Asymptotic analysis for fiber drawing processes, N. Hanevy & D. O'Kiely, SIAM Journal on Applied Mathematics (2023)
Niall carried out this research as a student on our Mathematical Modelling MSc, and is now completing a PhD at Aston University.
Mathematical models for glass sheets
Mathematical models for glass sheets
There are a number of glass manufacture processes where a sheet of molten glass is elongated to create a very thin screen. Challenges arise in these manufacture processes which can be aided by mathematical modelling. From a mathematics point of view, the long thin geometry of these sheets and threads corresponds to a small aspect ratio which can be exploited to derive very simplified models. For example, highly complex three-dimensional systems can be accurately described with one- or two-dimensional equations.
During my PhD I developed mathematical models for the redraw process, used to manufacture very thin (<100 μm) glass sheets. In the redraw process, a prefabricated glass block is fed into a furnace where it is reheated and stretched, making it thinner and also causing the edges to neck in. This stretching process gives rise to a final product that is thicker at the edges than in the centre, and these "thick edges" are highly undesirable. We developed mathematical models to describe the redraw process and hence identify the source of the thick edges, as well as a method to reduce or eliminate them. My PhD was supervised by Ian Griffiths, Chris Breward and Peter Howell at the University of Oxford and was in collaboration with glass-manufacture company Schott AG. Read about our research in more detail in:
Edge behaviour in the glass sheet redraw process, D. O'Kiely, C. J. W. Breward, I. M. Griffiths, P. D. Howell & U. Lange, Journal of Fluid Mechanics (2016)
Glass sheet redraw through a long heater zone, D. O'Kiely, C. J. W. Breward, I. M. Griffiths, P. D. Howell & U. Lange, IMA Journal of Applied Mathematics (2018)
This research has also been featured in the IMA magazine Mathematics Today and in the LMS newsletter.
Out-of-plane buckling in glass manufacture
Out-of-plane buckling in glass manufacture
In addition to the thickness variations described above, glass sheets undergoing redraw may also buckle out of plane, forming ripples that render the final product unusable. These ripples form due to localized regions of lateral compression that occur when the sheet is stretched along its length. We investigated the growth of ripples due to compression in a glass sheet, and the feedback between the out-of-plane deformation and the consequent change in stresses.
Read about glass buckling in more detail in:
Out-of-plane buckling in two-dimensional glass drawing, D. O'Kiely, C. J. W. Breward, I. M. Griffiths, P. D. Howell & U. Lange, Journal of Fluid Mechanics (2019)
or read about models for three-dimensional buckling in my thesis
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