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Mathematical modelling of non-axisymmetric capillary tube drawing

Published online by Cambridge University Press:  23 May 2008

Mathematical Institute, 24–29 St Giles', Oxford, OX1 3LB, UK
Mathematical Institute, 24–29 St Giles', Oxford, OX1 3LB, UK


This paper concerns the manufacture of non-axisymmetric capillary tubing via the Vello process, in which molten glass is fed through a die and drawn off vertically. The shape of the cross-section evolves under surface tension as it flows downstream. The aim is to achieve a given desired final shape, typically square or rectangular, and our goal is to determine the required die shape.

We use the result that, provided the tube is slowly varying in the axial direction, each cross-section evolves like a two-dimensional Stokes flow when expressed in suitably scaled Lagrangian coordinates. This allows us to use a previously derived model for the surface-tension-driven evolution of a thin two-dimensional viscous tube. We thus obtain, and solve analytically, equations governing the axial velocity, thickness and circumference of the tube, as well as its shape. The model is extended to include non-isothermal effects.

Copyright © Cambridge University Press 2008

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Buckmaster, J. D. & Nachman, A. 1978 The buckling and stretching of a viscida II. Effects of surface tension. Q. J. Mech. Appl. Maths 31, 157168.Google Scholar
Buckmaster, J. D., Nachman, A. & Ting, L. 1975 The buckling and stretching of a viscida. J. Fluid Mech. 69, 120.Google Scholar
Cummings, L. J. & Howell, P. D. 1999 On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity. J. Fluid Mech. 389, 361389.Google Scholar
Denn, M. 1980 Process Fluid Mechanics. Prentice Hall.Google Scholar
Dewynne, J. N., Ockendon, J. R. & Wilmott, P. 1989 On a mathematical model for fibre tapering. SIAM J. Appl. Maths 49, 983990.Google Scholar
Fitt, A. D., Furusawa, K., Monro, T. M. & Please, C. P. 2001 Modeling the fabrication of hollow fibres: Capillary drawing. IEEE J. Lightwave Technol. 19, 19241931.Google Scholar
Fitt, A. D., Furusawa, K., Monro, T. M., Please, C. P. & Richardson, D. A. 2002 The mathematical modelling of capillary drawing for holey fibre manufacture. J. Engng. Maths 43, 201227.Google Scholar
Graham, S. J. 1987 Mathematical modelling of glass flow in container manufacture. PhD thesis, University of Sheffield.Google Scholar
Griffiths, I. M. 2008 Mathematical modelling of non-axisymmetric glass tube manufacture. PhD thesis, University of Oxford.Google Scholar
Griffiths, I. M. & Howell, P. D. 2007 The surface-tension-driven evolution of a two-dimensional annular viscous tube. J. Fluid Mech. 593, 181208.Google Scholar
Howell, P. D. 1996 Models for thin viscous sheets. Eur. J. Appl. Maths 7, 321343.Google Scholar
Howell, P. D., Wylie, J. J., Huang, H & Miura, R. M. 2007 Stretching of heated threads with temperature-dependent viscosity: asymptotic analysis. Discrete Continuous Dyn. Syst. 7, 553572.Google Scholar
Huang, H., Miura, R. M., Ireland, W. P. & Puil, E. 2003 Heat-induced stretching of a glass tube under tension: Application to glass microelectrodes. SIAM J. Appl. Maths 63, 14991519.Google Scholar
Huang, H., Wylie, J. J., Miura, R. M. & Howell, P. D. 2007 On the formation of glass microelectrodes. SIAM J. Appl. Maths 67, 630666.Google Scholar
Karapet'yants, M. Kh. 1960 The viscosity-temperature relationship for silicate glasses. Glass Ceramics 15, 2632.Google Scholar
Krause, D. & Loch, H. 2002 Mathematical Simulation in Glass Technology. Springer.Google Scholar
Lee, S. H.-K. & Jaluria, Y. 1997 Simulation of the transport process in the neck-down region of a furnace drawn optical fibre. Intl J. Heat Mass Transfer 40, 843856.Google Scholar
Myers, M. R. 1989 A model for unsteady analysis of preform drawing. AIChE J. 35, 592601.Google Scholar
National Institute of Standards & Technology 1991 Standard Reference Material 710a. Soda-Lime-Silica Glass. Scholar
Paek, U. C. & Runk, R. B. 1978 Physical behaviour of the neck-down region during furnace drawing of silica fibres. J. Appl. Phys. 49, 44174422.Google Scholar
Papamichael, H. & Miaoulis, I. N. 1991 Thermal behavior of optical fibers during the cooling stage of the drawing process. J. Mater. Res. 6, 159167.Google Scholar
Pearson, J. R. A. & Matovich, M. A. 1969 Spinning a molten threadline. Stability. Ind. Engng Chem. Fund. 8, 605609.Google Scholar
Pearson, J. R. A. & Petrie, C. J. S. 1970 a The flow of a tubular film. Part 1. Formal mathematical representation. J. Fluid Mech. 40, 119.Google Scholar
Pearson, J. R. A. & Petrie, C. J. S. 1970 b The flow of a tubular film. Part 2. Interpretation of the model and discussion of solutions. J. Fluid Mech. 42, 609625.Google Scholar
Pfaender, H. G. 1996 Schott Guide to Glass. Chapman and Hall, London.Google Scholar
Sarboh, S. D., Milinković, S. A. & Debeljković, D. L. J. 1998 Mathematical model of the glass capillary tube drawing process. Glass Technol. 39, 5367.Google Scholar
Sivko, A. P. 1976 The production of glass tubes using the Vello method. Glass Ceramics 33, 728730.Google Scholar
Uhlmann, D. R. & Kreidl, N. J. 1984 Glass. Science and Technology. Volume 2. Processing I. Academic.Google Scholar
Voyce, C. J., Fitt, A. D. & Monro, T. M. 2004 Mathematical model of the spinning of microstructured optical fibres. Optics Express 12, 58105820.Google Scholar
Wu, C. Y., Somervell, A. R. D. & Barnes, T. H. 1998 Direct image transmission through a multi-mode square optical fiber. Optics Commun. 157, 1722.Google Scholar
Wu, C. Y., Somervell, A. R. D., Haskell, T. G. & Barnes, T. H. 2000 Optical sine transformation and image transmission by using square optical waveguide. Optics Commun. 175, 2732.Google Scholar
Wylie, J. J., Huang, H. & Miura, R. M. 2007 Thermal instability in drawing viscous threads. J. Fluid Mech. 570, 116.Google Scholar