Journal of Engineering Mathematics

, Volume 43, Issue 2–4, pp 201–227 | Cite as

The mathematical modelling of capillary drawing for holey fibre manufacture

  • A.D. Fitt
  • K. Furusawa
  • T.M. Monro
  • C.P. Please
  • D.J. Richardson
Article

Abstract

Microstructured optical fibres (i.e. fibres that contain holes) have assumed a high profile in recent years, and given rise to many novel optical devices. The problem of manufacturing such fibres by heating and then drawing a preform is considered for the particularly simple case of annular capillaries. A fluid-mechanics model is constructed using asymptotic analysis based on the small aspect ratio of the capillary. The leading-order equations are then examined in a number of asymptotic limits, many of which give valuable practical information about the control parameters that influence the drawing process. Finally, some comparisons with experiment are performed. For a limited set of experiments where the internal hole is pressurised, the theoretical predictions give qualitatively accurate results. For a much more detailed set of experiments carried out with a high-grade silica glass where no hole pressurisation is used, the relevant asymptotic solution to the governing equations is shown to give predictions that agree remarkably well with experiment.

asymptotic analysis extensional flow holey fibres optical-fibre manufacture slow viscous flow 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    T.A. Birks, J.C. Knight and P. St. J. Russell, Endlessly single-mode photonic crystal fibre. Opt. Lett. 22 (1997) 961–963.Google Scholar
  2. 2.
    T.M. Monro, D.J. Richardson and P.J. Bennett, Developing holey fibres for evanescent field devices. Elect. Lett. 35 (1999) 1188–1189.Google Scholar
  3. 3.
    T.M. Monro, D.J. Richardson, N.G.R. Broderick and P.J. Bennett, Holey fibres: an efficient modal model. J. Lightwave Technol. 17 (1999) 1093–1102.Google Scholar
  4. 4.
    J.K. Ranka, R.S. Windeler and A. Stentz, Optical properties of high-delta air-silica microstructure optical fibres. Opt. Lett. 25 (2000) 796–798.Google Scholar
  5. 5.
    J.C. Knight, J. Broeng, T.A. Birks and P. St. J. Russell, Photonic band gap guidance in optical fibres. Science 282 (1998) 1476–1478.Google Scholar
  6. 6.
    M. Key, I.G. Hughes, W. Rooijakkers, B.E. Sauer, E.A. Hinds, D.J. Richardson and P.G. Kazansky, Propagation of cold atoms along a miniature magnetic guide. Phys. Rev. Lett. 84 (2000) 1371–1373.Google Scholar
  7. 7.
    M.R. Matovich and J.R.A. Pearson, Spinning a molten threadline – Steady-state isothermal viscous flows. Ind. Eng. Chem. Fundam. 8 (1969) 512–520.Google Scholar
  8. 8.
    J.R.A. Pearson and M.A. Matovich, Spinning a molten threadline-Stability. Ind. Eng. Chem. Fundam. 8 (1969) 605–609.Google Scholar
  9. 9.
    Y.T. Shah and J.R.A. Pearson, On the stability of nonisothermal fiber spinning. Ind. Eng. Chem. Fundam. 11 (1972) 145–149.Google Scholar
  10. 10.
    Y.T. Shah and J.R.A. Pearson, On the stability of nonisothermal fiber spinning-general case. Ind. Eng. Chem. Fundam. 11 (1972) 150–153.Google Scholar
  11. 11.
    J.A. Burgman, Liquid glass jets in the forming of continuous fibers. Glass Technol. 11 (1970) 110–116.Google Scholar
  12. 12.
    L.R. Glicksmann, The cooling of optical fibres. Glass Technol. 9 (1968) 131–138.Google Scholar
  13. 13.
    G. Manfre, Forces acting in the continuous drawing of glass fibres. Glass Technol. 10 (1969) 99–106.Google Scholar
  14. 14.
    F.T. Geyling and G.M. Homsy, Extensional instabilities of the glass fiber drawing process. Glass Technol. 21 (1980) 95–102.Google Scholar
  15. 15.
    F.T. Geyling, Basic fluid dynamic considerations in the drawing of optical fibres. Bell Sys. Tech. J. 55 (1976) 1011–1056.Google Scholar
  16. 16.
    J.N. Dewynne, P.D. Howell and P. Wilmott, Slender viscous fibres with inertia and gravity. Quart. J. Mech. Appl. Math. 47 (1994) 541–555.Google Scholar
  17. 17.
    J.N. Dewynne, J.R. Ockendon and P. Wilmott, On a mathematical model for fibre tapering. SIAM J. Appl. Math. 49 (1989) 983–990.Google Scholar
  18. 18.
    H. Papamichael and I.N. Miaoulis, Thermal behavior of optical fibres during the cooling stage of the drawing process. J. Mater. Res. 6 (1991) 159–167.Google Scholar
  19. 19.
    S.E. Rosenberg, H. Papamichael and I.N. Miaoulis, A two-dimensional analysis of the viscous problem of a glass preform during the optical fiber drawing process. Glass Technol. 35 (1994) 260–264.Google Scholar
  20. 20.
    Z. Yin and Y. Jaluria, Thermal transport and flow in high-speed optical fiber drawing. J. Heat Transfer, Trans ASME 120 (1998) 916–930.Google Scholar
  21. 21.
    P. Gospodinov and A.L. Yarin, Draw resonance of optical microcapillaries in non-isothermal drawing. Int. J. Multiphase Flow 23 (1997) 967–976.Google Scholar
  22. 22.
    A.L. Yarin, P. Gospodinov and V.I. Roussinov, Stability loss and sensitivity in hollow fiber drawing. Phys. Fluids 6 (1994) 1454–1463.Google Scholar
  23. 23.
    S.D. Sarboh, S.A. Milinkovic and D.L.J. Debeljkovic, Mathematical model of the glass capillary tube drawing process. Glass Technol. 39 (1998) 53–67.Google Scholar
  24. 24.
    M. Hucker, I. Bond, A.-A. Foreman and J. Hudd, Optimisation of hollow glass fibers and their composites. Adv. Composite Lett. 8 (1999) 181–189.Google Scholar
  25. 25.
    S. H-K. Lee and Y. Jaluria, Simulation of the transport processes in the neck-down region of a furnace drawn optical fibre. Int. J. Heat Mass Transfer 40 (1997) 843–856.Google Scholar
  26. 26.
    U.C. Paek and R.B. Runk, Physical behaviour of the neck-down region during furnace drawing of silica fibers. J. Appl. Phys. 49 (1978) 4417–4422.Google Scholar
  27. 27.
    N.P. Bansal and R.H. Doremus, Handbook of Glass Properties. New York: Academic Press (1986) 680pp.Google Scholar
  28. 28.
    W.W. Schultz and S.H. Davis, Effects of boundary conditions on the stability of slender viscous fibers. Trans. ASME, J. Appl. Mech. 51 (1984) 1–5.Google Scholar
  29. 29.
    G. Hall and J.M. Watt (eds), Numerical Methods for Ordinary Differential Equations. Oxford: Clarendon Press (1976) 336pp.Google Scholar
  30. 30.
    D. Gelder, The stability of the fiber drawing processes. Ind. Eng. Chem. Fund. 10 (1971) 534–535.Google Scholar
  31. 31.
    J.C. Chang, M.M. Denn and F.T. Geyling, Effects of inertia, surface tension and gravity on the stability of isothermal drawing of Newtonian fluids. Ind. Eng. Chem. Fund. 20 (1981) 147–149.Google Scholar
  32. 32.
    A.D. Fitt, K. Furusawa, T.M. Monro and C.P. Please, Modeling the Fabrication of Hollow Fibers: Capillary Drawing. J. Lightwave Technol. 19 (2001) 1924–1931.Google Scholar
  33. 33.
    R.H. Doremus, Glass Science. New York: Wiley Interscience (1973) 349pp.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • A.D. Fitt
    • 1
  • K. Furusawa
    • 2
  • T.M. Monro
    • 2
  • C.P. Please
    • 1
  • D.J. Richardson
    • 2
  1. 1.Faculty of Mathematical StudiesUniversity of SouthamptonSouthampton
  2. 2.Optoelectronics Research CentreUniversity of SouthamptonSouthampton

Personalised recommendations