Abstract

We construct a fluid mechanics model of the drawing of microstructured optical fibres (‘holey fibres’). This model can be used to understand and quantify methods for controlling the fibre geometry. The effects of preform rotation are included to examine methods for reducing fibre birefringence. Asymptotic numerical-solutions are obtained and applied to two typical microstructured-fibres and a number of practical suggestions are made for achieving sub-mm spin pitches without damaging the microstructure within.

© 2004 Optical Society of America

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References

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  1. R.S. Ranka and A.J. Stentz, “Optical properties of high-delta air-silica microstructure optical fibers,” Opt. Lett. 25, 796–798 (2000).
    [Crossref]
  2. T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P.J. Bennett, “Holey optical fibers: An efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999).
    [Crossref]
  3. T.M. Monro, D.J. Richardson, and P.J. Bennett, “Developing holey fibres for evanescent field devices,” Elect. Lett. 35, 1188–1189 (1999).
    [Crossref]
  4. A.J. Barlow, J.J. Ramskov-Hansen, and D.N. Payne, “Birefringence and polarization mode dispersion in spun single-mode fibres,” Applied Optics 30, 2962–68 (1981).
    [Crossref]
  5. M.J. Li and D.A. Nolan, “Fiber spin-profile designs for producing fibers with low polariazation mode dispersion,” Opt. Lett.,  23, 1659–1661 (1998).
    [Crossref]
  6. R.E. Schuh, X. Shan, and A. Shamim Siddiqui, “Polarization Mode Dispersion in Spun Fibers with Different Linear Birefringence and Spinning Parameters,” J. Lightwave Technol. 16, 1583–1588 (1998).
    [Crossref]
  7. M. Fuochi, J.R. Hayes, K. Furusawa, W. Belardi, J.C. Baggett, T. M. Monro, and D.J. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 9, 1972–1977 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1972.
    [Crossref]
  8. J.P. Gordon and H. Kogelnik, “PMD Fundamentals: Polarization mode dispersion in optical fibers,” PNAS 97 (9), 4541–4550 (2000).
    [Crossref] [PubMed]
  9. A. Ortigosa-Blanch, J.C. Knight, W.J. Wadsworth, J. Arriaga, B.J. Mangan, T.A. Birks, and St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 25, 1325–1327 (2000).
    [Crossref]
  10. J.R. Hayes, Optoelectronics Research Centre, University of Southampton, University Road, Southampton, Hampshire, SO17 1BJ, U.K. (personal communication, 2003).
  11. A.D. Fitt, K. Furusawa, T.M. Monro, and C.P. Please, “Modelling the fabrication of hollow fibers: capillary drawing,” J. Lightwave Technol. 31, 1924–31 (2001).
    [Crossref]
  12. A.D. Fitt, K. Furusawa, T.M. Monro, C.P. Please, and D.J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43, 201–227 (2002).
    [Crossref]
  13. C.J. Voyce, School of Mathematics, University of Southampton, Southampton, SO17 1BJ, U.K., A.D. Fitt and T.M. Monro are preparing a manuscript to be called “The mathematical modelling of spun capillaries.”
  14. R.H. Doremus, “Viscosity of silica,” J. Appl. Phys. 92, 7619–7629 (2002).
    [Crossref]
  15. T.M. Monro, K. Furusawa, J.H. Lee, J.H.V. Price, Z. Yusoff, J.C. Baggett, and D.J. Richardson, “Advances in holey fibers,” in Advances in Fiber Lasers,L.N. Durvasula, ed., Proc. SPIE4974, 83–95 (2003).
  16. P.K.A. Wai, W.L. Kath, C.R. Menyuk, and J.W. Zhang, “Nonlinear polarization-mode dispersion in optical fibers with randomly varying birefringence,” J. Opt. Soc. Am. B 14, 2967–2979 (1997).
    [Crossref]
  17. U.C. Paek, “Free Drawing and Polymer Coating of Silica Glass Optical Fibers,” ASME Journal of Heat Transfer 121, 774–789 (1999).
    [Crossref]
  18. P. Petropoulous, H. Ebendorff-Heidepriem, V. Finazzi, R.C. Moore, K. Frampton, D.J. Richardson, and T.M. Monro, “Highly nonlinear and anomalously dispersive lead silicate glass holey fibers,” Opt. Express 11, 3568–3573 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3568.
    [Crossref]
  19. C.J. Voyce, A.D. Fitt, and T.M. Monro“Mathematical modelling of the drawing of spun capillary tubes,” in Progress in Industrial Mathematics at ECMI 2002,A. Buikis, R. Ciegis and A.D. Fitt, eds. (Springer-Verlag, Berlin, 2004), pp. 387–391.

2004 (1)

M. Fuochi, J.R. Hayes, K. Furusawa, W. Belardi, J.C. Baggett, T. M. Monro, and D.J. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 9, 1972–1977 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1972.
[Crossref]

2003 (1)

2002 (2)

A.D. Fitt, K. Furusawa, T.M. Monro, C.P. Please, and D.J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43, 201–227 (2002).
[Crossref]

R.H. Doremus, “Viscosity of silica,” J. Appl. Phys. 92, 7619–7629 (2002).
[Crossref]

2001 (1)

A.D. Fitt, K. Furusawa, T.M. Monro, and C.P. Please, “Modelling the fabrication of hollow fibers: capillary drawing,” J. Lightwave Technol. 31, 1924–31 (2001).
[Crossref]

2000 (3)

1999 (3)

T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P.J. Bennett, “Holey optical fibers: An efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999).
[Crossref]

T.M. Monro, D.J. Richardson, and P.J. Bennett, “Developing holey fibres for evanescent field devices,” Elect. Lett. 35, 1188–1189 (1999).
[Crossref]

U.C. Paek, “Free Drawing and Polymer Coating of Silica Glass Optical Fibers,” ASME Journal of Heat Transfer 121, 774–789 (1999).
[Crossref]

1998 (2)

1997 (1)

1981 (1)

A.J. Barlow, J.J. Ramskov-Hansen, and D.N. Payne, “Birefringence and polarization mode dispersion in spun single-mode fibres,” Applied Optics 30, 2962–68 (1981).
[Crossref]

Arriaga, J.

Baggett, J.C.

M. Fuochi, J.R. Hayes, K. Furusawa, W. Belardi, J.C. Baggett, T. M. Monro, and D.J. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 9, 1972–1977 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1972.
[Crossref]

T.M. Monro, K. Furusawa, J.H. Lee, J.H.V. Price, Z. Yusoff, J.C. Baggett, and D.J. Richardson, “Advances in holey fibers,” in Advances in Fiber Lasers,L.N. Durvasula, ed., Proc. SPIE4974, 83–95 (2003).

Barlow, A.J.

A.J. Barlow, J.J. Ramskov-Hansen, and D.N. Payne, “Birefringence and polarization mode dispersion in spun single-mode fibres,” Applied Optics 30, 2962–68 (1981).
[Crossref]

Belardi, W.

M. Fuochi, J.R. Hayes, K. Furusawa, W. Belardi, J.C. Baggett, T. M. Monro, and D.J. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 9, 1972–1977 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1972.
[Crossref]

Bennett, P.J.

T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P.J. Bennett, “Holey optical fibers: An efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999).
[Crossref]

T.M. Monro, D.J. Richardson, and P.J. Bennett, “Developing holey fibres for evanescent field devices,” Elect. Lett. 35, 1188–1189 (1999).
[Crossref]

Birks, T.A.

Broderick, N.G.R.

Buikis, A.

C.J. Voyce, A.D. Fitt, and T.M. Monro“Mathematical modelling of the drawing of spun capillary tubes,” in Progress in Industrial Mathematics at ECMI 2002,A. Buikis, R. Ciegis and A.D. Fitt, eds. (Springer-Verlag, Berlin, 2004), pp. 387–391.

Ciegis, R.

C.J. Voyce, A.D. Fitt, and T.M. Monro“Mathematical modelling of the drawing of spun capillary tubes,” in Progress in Industrial Mathematics at ECMI 2002,A. Buikis, R. Ciegis and A.D. Fitt, eds. (Springer-Verlag, Berlin, 2004), pp. 387–391.

Doremus, R.H.

R.H. Doremus, “Viscosity of silica,” J. Appl. Phys. 92, 7619–7629 (2002).
[Crossref]

Ebendorff-Heidepriem, H.

Finazzi, V.

Fitt, A.D.

A.D. Fitt, K. Furusawa, T.M. Monro, C.P. Please, and D.J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43, 201–227 (2002).
[Crossref]

A.D. Fitt, K. Furusawa, T.M. Monro, and C.P. Please, “Modelling the fabrication of hollow fibers: capillary drawing,” J. Lightwave Technol. 31, 1924–31 (2001).
[Crossref]

C.J. Voyce, A.D. Fitt, and T.M. Monro“Mathematical modelling of the drawing of spun capillary tubes,” in Progress in Industrial Mathematics at ECMI 2002,A. Buikis, R. Ciegis and A.D. Fitt, eds. (Springer-Verlag, Berlin, 2004), pp. 387–391.

C.J. Voyce, A.D. Fitt, and T.M. Monro“Mathematical modelling of the drawing of spun capillary tubes,” in Progress in Industrial Mathematics at ECMI 2002,A. Buikis, R. Ciegis and A.D. Fitt, eds. (Springer-Verlag, Berlin, 2004), pp. 387–391.

Frampton, K.

Fuochi, M.

M. Fuochi, J.R. Hayes, K. Furusawa, W. Belardi, J.C. Baggett, T. M. Monro, and D.J. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 9, 1972–1977 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1972.
[Crossref]

Furusawa, K.

M. Fuochi, J.R. Hayes, K. Furusawa, W. Belardi, J.C. Baggett, T. M. Monro, and D.J. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 9, 1972–1977 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1972.
[Crossref]

A.D. Fitt, K. Furusawa, T.M. Monro, C.P. Please, and D.J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43, 201–227 (2002).
[Crossref]

A.D. Fitt, K. Furusawa, T.M. Monro, and C.P. Please, “Modelling the fabrication of hollow fibers: capillary drawing,” J. Lightwave Technol. 31, 1924–31 (2001).
[Crossref]

T.M. Monro, K. Furusawa, J.H. Lee, J.H.V. Price, Z. Yusoff, J.C. Baggett, and D.J. Richardson, “Advances in holey fibers,” in Advances in Fiber Lasers,L.N. Durvasula, ed., Proc. SPIE4974, 83–95 (2003).

Gordon, J.P.

J.P. Gordon and H. Kogelnik, “PMD Fundamentals: Polarization mode dispersion in optical fibers,” PNAS 97 (9), 4541–4550 (2000).
[Crossref] [PubMed]

Hayes, J.R.

M. Fuochi, J.R. Hayes, K. Furusawa, W. Belardi, J.C. Baggett, T. M. Monro, and D.J. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 9, 1972–1977 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1972.
[Crossref]

J.R. Hayes, Optoelectronics Research Centre, University of Southampton, University Road, Southampton, Hampshire, SO17 1BJ, U.K. (personal communication, 2003).

Kath, W.L.

Knight, J.C.

Kogelnik, H.

J.P. Gordon and H. Kogelnik, “PMD Fundamentals: Polarization mode dispersion in optical fibers,” PNAS 97 (9), 4541–4550 (2000).
[Crossref] [PubMed]

Lee, J.H.

T.M. Monro, K. Furusawa, J.H. Lee, J.H.V. Price, Z. Yusoff, J.C. Baggett, and D.J. Richardson, “Advances in holey fibers,” in Advances in Fiber Lasers,L.N. Durvasula, ed., Proc. SPIE4974, 83–95 (2003).

Li, M.J.

Mangan, B.J.

Menyuk, C.R.

Monro, T. M.

M. Fuochi, J.R. Hayes, K. Furusawa, W. Belardi, J.C. Baggett, T. M. Monro, and D.J. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 9, 1972–1977 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1972.
[Crossref]

Monro, T.M.

P. Petropoulous, H. Ebendorff-Heidepriem, V. Finazzi, R.C. Moore, K. Frampton, D.J. Richardson, and T.M. Monro, “Highly nonlinear and anomalously dispersive lead silicate glass holey fibers,” Opt. Express 11, 3568–3573 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3568.
[Crossref]

A.D. Fitt, K. Furusawa, T.M. Monro, C.P. Please, and D.J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43, 201–227 (2002).
[Crossref]

A.D. Fitt, K. Furusawa, T.M. Monro, and C.P. Please, “Modelling the fabrication of hollow fibers: capillary drawing,” J. Lightwave Technol. 31, 1924–31 (2001).
[Crossref]

T.M. Monro, D.J. Richardson, and P.J. Bennett, “Developing holey fibres for evanescent field devices,” Elect. Lett. 35, 1188–1189 (1999).
[Crossref]

T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P.J. Bennett, “Holey optical fibers: An efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999).
[Crossref]

T.M. Monro, K. Furusawa, J.H. Lee, J.H.V. Price, Z. Yusoff, J.C. Baggett, and D.J. Richardson, “Advances in holey fibers,” in Advances in Fiber Lasers,L.N. Durvasula, ed., Proc. SPIE4974, 83–95 (2003).

C.J. Voyce, A.D. Fitt, and T.M. Monro“Mathematical modelling of the drawing of spun capillary tubes,” in Progress in Industrial Mathematics at ECMI 2002,A. Buikis, R. Ciegis and A.D. Fitt, eds. (Springer-Verlag, Berlin, 2004), pp. 387–391.

Moore, R.C.

Nolan, D.A.

Ortigosa-Blanch, A.

Paek, U.C.

U.C. Paek, “Free Drawing and Polymer Coating of Silica Glass Optical Fibers,” ASME Journal of Heat Transfer 121, 774–789 (1999).
[Crossref]

Payne, D.N.

A.J. Barlow, J.J. Ramskov-Hansen, and D.N. Payne, “Birefringence and polarization mode dispersion in spun single-mode fibres,” Applied Optics 30, 2962–68 (1981).
[Crossref]

Petropoulous, P.

Please, C.P.

A.D. Fitt, K. Furusawa, T.M. Monro, C.P. Please, and D.J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43, 201–227 (2002).
[Crossref]

A.D. Fitt, K. Furusawa, T.M. Monro, and C.P. Please, “Modelling the fabrication of hollow fibers: capillary drawing,” J. Lightwave Technol. 31, 1924–31 (2001).
[Crossref]

Price, J.H.V.

T.M. Monro, K. Furusawa, J.H. Lee, J.H.V. Price, Z. Yusoff, J.C. Baggett, and D.J. Richardson, “Advances in holey fibers,” in Advances in Fiber Lasers,L.N. Durvasula, ed., Proc. SPIE4974, 83–95 (2003).

Ramskov-Hansen, J.J.

A.J. Barlow, J.J. Ramskov-Hansen, and D.N. Payne, “Birefringence and polarization mode dispersion in spun single-mode fibres,” Applied Optics 30, 2962–68 (1981).
[Crossref]

Ranka, R.S.

Richardson, D.J.

M. Fuochi, J.R. Hayes, K. Furusawa, W. Belardi, J.C. Baggett, T. M. Monro, and D.J. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 9, 1972–1977 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1972.
[Crossref]

P. Petropoulous, H. Ebendorff-Heidepriem, V. Finazzi, R.C. Moore, K. Frampton, D.J. Richardson, and T.M. Monro, “Highly nonlinear and anomalously dispersive lead silicate glass holey fibers,” Opt. Express 11, 3568–3573 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3568.
[Crossref]

A.D. Fitt, K. Furusawa, T.M. Monro, C.P. Please, and D.J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43, 201–227 (2002).
[Crossref]

T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P.J. Bennett, “Holey optical fibers: An efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999).
[Crossref]

T.M. Monro, D.J. Richardson, and P.J. Bennett, “Developing holey fibres for evanescent field devices,” Elect. Lett. 35, 1188–1189 (1999).
[Crossref]

T.M. Monro, K. Furusawa, J.H. Lee, J.H.V. Price, Z. Yusoff, J.C. Baggett, and D.J. Richardson, “Advances in holey fibers,” in Advances in Fiber Lasers,L.N. Durvasula, ed., Proc. SPIE4974, 83–95 (2003).

Russell, St. J.

Schuh, R.E.

Shamim Siddiqui, A.

Shan, X.

Stentz, A.J.

Voyce, C.J.

C.J. Voyce, School of Mathematics, University of Southampton, Southampton, SO17 1BJ, U.K., A.D. Fitt and T.M. Monro are preparing a manuscript to be called “The mathematical modelling of spun capillaries.”

C.J. Voyce, A.D. Fitt, and T.M. Monro“Mathematical modelling of the drawing of spun capillary tubes,” in Progress in Industrial Mathematics at ECMI 2002,A. Buikis, R. Ciegis and A.D. Fitt, eds. (Springer-Verlag, Berlin, 2004), pp. 387–391.

Wadsworth, W.J.

Wai, P.K.A.

Yusoff, Z.

T.M. Monro, K. Furusawa, J.H. Lee, J.H.V. Price, Z. Yusoff, J.C. Baggett, and D.J. Richardson, “Advances in holey fibers,” in Advances in Fiber Lasers,L.N. Durvasula, ed., Proc. SPIE4974, 83–95 (2003).

Zhang, J.W.

Applied Optics (1)

A.J. Barlow, J.J. Ramskov-Hansen, and D.N. Payne, “Birefringence and polarization mode dispersion in spun single-mode fibres,” Applied Optics 30, 2962–68 (1981).
[Crossref]

ASME Journal of Heat Transfer (1)

U.C. Paek, “Free Drawing and Polymer Coating of Silica Glass Optical Fibers,” ASME Journal of Heat Transfer 121, 774–789 (1999).
[Crossref]

Elect. Lett. (1)

T.M. Monro, D.J. Richardson, and P.J. Bennett, “Developing holey fibres for evanescent field devices,” Elect. Lett. 35, 1188–1189 (1999).
[Crossref]

J. Appl. Phys. (1)

R.H. Doremus, “Viscosity of silica,” J. Appl. Phys. 92, 7619–7629 (2002).
[Crossref]

J. Eng. Math. (1)

A.D. Fitt, K. Furusawa, T.M. Monro, C.P. Please, and D.J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43, 201–227 (2002).
[Crossref]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. B (1)

Opt. Express (2)

M. Fuochi, J.R. Hayes, K. Furusawa, W. Belardi, J.C. Baggett, T. M. Monro, and D.J. Richardson, “Polarization mode dispersion reduction in spun large mode area silica holey fibres,” Opt. Express 9, 1972–1977 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1972.
[Crossref]

P. Petropoulous, H. Ebendorff-Heidepriem, V. Finazzi, R.C. Moore, K. Frampton, D.J. Richardson, and T.M. Monro, “Highly nonlinear and anomalously dispersive lead silicate glass holey fibers,” Opt. Express 11, 3568–3573 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3568.
[Crossref]

Opt. Lett. (3)

PNAS (1)

J.P. Gordon and H. Kogelnik, “PMD Fundamentals: Polarization mode dispersion in optical fibers,” PNAS 97 (9), 4541–4550 (2000).
[Crossref] [PubMed]

Other (4)

J.R. Hayes, Optoelectronics Research Centre, University of Southampton, University Road, Southampton, Hampshire, SO17 1BJ, U.K. (personal communication, 2003).

C.J. Voyce, A.D. Fitt, and T.M. Monro“Mathematical modelling of the drawing of spun capillary tubes,” in Progress in Industrial Mathematics at ECMI 2002,A. Buikis, R. Ciegis and A.D. Fitt, eds. (Springer-Verlag, Berlin, 2004), pp. 387–391.

T.M. Monro, K. Furusawa, J.H. Lee, J.H.V. Price, Z. Yusoff, J.C. Baggett, and D.J. Richardson, “Advances in holey fibers,” in Advances in Fiber Lasers,L.N. Durvasula, ed., Proc. SPIE4974, 83–95 (2003).

C.J. Voyce, School of Mathematics, University of Southampton, Southampton, SO17 1BJ, U.K., A.D. Fitt and T.M. Monro are preparing a manuscript to be called “The mathematical modelling of spun capillaries.”

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Figures (4)

Fig. 1.
Fig. 1.

(Left) The cross-section of a type-one holey fibre. (Right) The cross-section of a type-two holey fibre.

Fig. 2.
Fig. 2.

Problem geometry and nomenclature.

Fig. 3.
Fig. 3.

The effects of preform rotation on outer capillary radius. The diagram shows the outer radius h 2 for fibre pulls with and without rotation. The thin-walled tube has h 1(0)=0.01m, h 2(0)=0.015m and the thick-walled tube has h 1(0)=0.01m, h 2(0)=0.02m. (Draw length L=0.03m, temperature T=2200C, draw speed Wd =25m/min, feed speed Wf =15mm/min, rotation rate Ω=35rad/s.)

Fig. 4.
Fig. 4.

The destructive effects of preform rotation on the microstructure of type-two fibres. Dashed lines show holey cladding and jacket radii of preform without rotation and solid lines show the radii with rotation.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

ρ ( h 2 2 h 1 2 ) ( w 0 t + w 0 w 0 z g ) = [ 3 μ ( h 2 2 h 1 2 ) w 0 z + γ ( h 1 + h 2 ) + ρ 4 ( h 2 4 h 1 4 ) B 2 ] z ,
( h 1 2 ) t + ( h 1 2 w 0 ) z = 2 p 0 h 1 2 h 2 2 2 γ h 1 h 2 ( h 1 + h 2 ) + ρ h 1 2 h 2 2 B 2 ( h 2 2 h 1 2 ) 2 μ ( h 2 2 h 1 2 ) ,
( h 2 2 ) t + ( h 2 2 w 0 ) z = 2 p 0 h 1 2 h 2 2 2 γ h 1 h 2 ( h 1 + h 2 ) + ρ h 1 2 h 2 2 B 2 ( h 2 2 h 1 2 ) 2 μ ( h 2 2 h 1 2 ) ,
μ ( ( h 2 4 h 1 4 ) B z ) z = ρ [ h 2 2 ( h 2 2 B ) t h 1 2 ( h 1 2 B ) t ] + ρ w 0 [ h 2 2 ( h 2 2 B ) z h 1 2 ( h 1 2 B ) z ]
ρ γ B μ ( h 1 2 h 2 + h 2 2 h 1 ) + ρ 2 B 3 2 μ ( h 1 2 h 2 4 h 2 2 h 1 4 ) + ρ μ p 0 B h 1 2 h 2 2 ,
d ( z ) = 2 π w 0 ( z ) ϕ t .
ϕ t + w 0 ( z ) ϕ z = B ( z ) ,
d ( L ) = 2 π W d ϕ t z = L = 2 π W d B ( 0 ) .
3 μ ̅ ( h ̅ 2 2 w ̅ 0 z ̅ ) z ̅ 1 4 R e S 2 ( h ̅ 2 4 B ̅ 2 ) Z ̅ ,
Ω 3 2 h 3 μ ̅ μ 0 W L ρ .

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