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Coupled fluid and energy flow in fabrication of microstructured optical fibres

Published online by Cambridge University Press:  11 July 2019

Yvonne M. Stokes Open the ORCID record for Yvonne M. Stokes [Opens in a new window] ,
Jonathan J. Wylie  and
M. J. Chen Open the ORCID record for M. J. Chen [Opens in a new window]
Yvonne M. Stokes*
Affiliation:
School of Mathematical Sciences and Institute for Photonics and Advanced Sensing, The University of Adelaide, SA 5005, Australia
Jonathan J. Wylie
Affiliation:
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong SAR Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA
M. J. Chen
Affiliation:
School of Mathematical Sciences and Institute for Photonics and Advanced Sensing, The University of Adelaide, SA 5005, Australia
*
†Email address for correspondence: [email protected]
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Abstract

We consider the role of heating and cooling in the steady drawing of a long and thin viscous thread with an arbitrary number of internal holes of arbitrary shape. The internal holes and the external boundary evolve as a result of the axial drawing and surface-tension effects. The heating and cooling affects the evolution of the thread because both the viscosity and surface tension of the thread are assumed to be functions of the temperature. We use asymptotic techniques to show that, under a suitable transformation, this complicated three-dimensional free boundary problem can be formulated in such a way that the transverse aspect of the flow can be reduced to the solution of a standard Stokes flow problem in the absence of axial stretching. The solution of this standard problem can then be substituted into a system of three ordinary differential equations that completely determine the flow. We use this approach to develop a very simple numerical method that can determine the way that thermal effects impact on the drawing of threads by devices that either specify the fibre tension or the draw ratio. We also develop a numerical method to solve the inverse problem of determining the initial cross-sectional geometry, draw tension and, importantly, heater temperature to obtain a desired cross-sectional shape and change in cross-sectional area at the device exit. This precisely allows one to determine the pattern of air holes in the preform that will achieve the desired hole pattern in the stretched fibre.

JFM classification

Interfacial Flows (free surface): Capillary flows Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 874 , 10 September 2019 , pp. 548 - 572
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Copyright
© 2019 Cambridge University Press 

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References

Boyd, K., Ebendorff-Heidepriem, H., Monro, T. M. & Munch, J. 2012 Surface tension and viscosity measurement of optical glasses using a scanning CO2 laser. Opt. Mater. Express 2 (8), 1101–1110.10.1364/OME.2.001101Google Scholar
Bradshaw-Hajek, B. H., Stokes, Y. M. & Tuck, E. O. 2004 Computation of extensional fall of slender viscous drops by a one-dimensional Eulerian method. SIAM J. Appl. Maths. 67, 1166–1182.Google Scholar
Buchak, P., Crowdy, D. G., Stokes, Y. M. & Ebendorff-Heidepriem, H. 2015 Elliptical pore regularisation of the inverse problem for microstructured optical fibre fabrication. J. Fluid Mech. 778, 5–38.10.1017/jfm.2015.337Google Scholar
Chen, M. J., Stokes, Y. M., Buchak, P., Crowdy, D. G. & Ebendorff-Heidepriem, H. 2015 Microstructured optical fibre drawing with active channel pressurisation. J. Fluid Mech. 783, 137–165.10.1017/jfm.2015.570Google 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, 361–389.10.1017/S0022112099005030Google Scholar
Denn, M. M. 1980 Continuous drawing of liquids to form fibers. Annu. Rev. Fluid Mech. 12, 365–387.Google Scholar
Dewynne, J. N., Howell, P. D. & Wilmott, P. 1994 Slender viscous fibres with inertia and gravity. Q. J. Mech. Appl. Maths 47, 541–555.10.1093/qjmam/47.4.541Google Scholar
Dewynne, J. N., Ockendon, J. R. & Wilmott, P. 1992 A systematic derivation of the leading-order equations for extensional flows in slender geometries. J. Fluid Mech. 244, 323–338.10.1017/S0022112092003094Google 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, 201–227.Google Scholar
Forest, M. G. & Zhou, H. 2001 Unsteady analysis of thermal glass fiber drawing processes. Eur. J. Appl. Maths 12, 479–496.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, 181–208.10.1017/S0022112007008683Google Scholar
Griffiths, I. M. & Howell, P. D. 2008 Mathematical modelling of non-axisymmetric capillary tube drawing. J. Fluid Mech. 605, 181–206.10.1017/S002211200800147XGoogle Scholar
Gupta, G. & Schultz, W. W. 1998 Non-isothermal flows of Newtonian slender glass fibers. Intl J. Non-Linear Mech. 33, 151–163.10.1016/S0020-7462(96)00143-6Google Scholar
He, D., Wylie, J. J., Huang, H. & Miura, R. M. 2016 Extension of a viscous thread with temperature-dependent viscosity and surface tension. J. Fluid Mech. 800, 720–752.10.1017/jfm.2016.426Google Scholar
Kaye, A. 1991 Convected coordinates and elongational flow. J. Non-Newtonian Fluid Mech. 40, 55–77.10.1016/0377-0257(91)87026-TGoogle Scholar
Matovich, M. A. & Pearson, J. R. A. 1969 Spinning a molten threadline. I&EC Fundamentals 8, 512–520.Google Scholar
Modest, M. F. 2013 Radiative Heat Transfer, 3rd edn. Academic Press.10.1016/B978-0-12-386944-9.50023-6Google Scholar
Scherer, G. W. 1992 Editorial comments on a paper by Gordon S. Fulcher. J. Am. Ceram. Soc. 75, 1060–1062.Google Scholar
Shah, Y. T. & Pearson, J. R. A. 1972a On the stability of nonisothermal fibre spinning. Ind. Engng Chem. Fundam. 11, 145–149.10.1021/i160042a001Google Scholar
Shah, Y. T. & Pearson, J. R. A. 1972b On the stability of nonisothermal fibre spinning – general case. Ind. Engng Chem. Fundam. 11, 150–153.Google Scholar
Shartsis, L. & Spinner, S. 1951 Surface tension of molten alkali silicates. J. Res. Natl. Bur. Stand. 46, 385–390.Google Scholar
Stokes, Y. M., Bradshaw-Hajek, B. H. & Tuck, E. O. 2011 Extensional flow at low Reynolds number with surface tension. J. Engng Maths 70, 321–331.10.1007/s10665-010-9443-3Google Scholar
Stokes, Y. M., Buchak, P., Crowdy, D. G. & Ebendorff-Heidepriem, H. 2014 Drawing of micro-structured optical fibres: circular and non-circular tubes. J. Fluid Mech. 755, 176–203.10.1017/jfm.2014.408Google Scholar
Stokes, Y. M. & Tuck, E. O. 2004 The role of inertia in extensional fall of a viscous drop. J. Fluid Mech. 498, 205–225.Google Scholar
Stokes, Y. M., Tuck, E. O. & Schwartz, L. W. 2000 Extensional fall of a very viscous fluid drop. Q. J. Mech. Appl. Maths 53, 565–582.Google Scholar
Suman, B. & Kumar, S. 2009 Draw ratio enhancement in nonisothermal melt spinning. AIChE J. 55, 581–593.10.1002/aic.11707Google Scholar
Taroni, M., Breward, C. J. W., Cummings, L. J. & Griffiths, I. M. 2013 Asymptotic solutions of glass temperature profiles during steady optical fibre drawing. J. Engng Maths 80, 1–20.Google Scholar
Tronnolone, H., Stokes, Y. M. & Ebendorff-Heidepriem, H. 2017 Extrusion of fluid cylinders of arbitrary shape with surface tension and gravity. J. Fluid Mech. 810, 127–154.10.1017/jfm.2016.729Google Scholar
Tronnolone, H., Stokes, Y. M., Foo, H. T. C. & Ebendorff-Heidepriem, H. 2016 Gravitational extension of a fluid cylinder with internal structure. J. Fluid Mech. 790, 308–338.Google Scholar
Wylie, J. J., Bradshaw-Hajek, B. H. & Stokes, Y. M. 2016 The evolution of a viscous thread pulled with a prescribed speed. J. Fluid Mech. 795, 380–408.10.1017/jfm.2016.215Google Scholar
Wylie, J. J. & Huang, H. 2007 Extensional flows with viscous heating. J. Fluid Mech. 571, 359–370.Google Scholar
Wylie, J. J., Huang, H. & Miura, R. M. 2007 Thermal instability in drawing viscous threads. J. Fluid Mech. 570, 1–16.Google Scholar
Wylie, J. J., Huang, H. & Miura, R. M. 2011 Stretching of viscous threads at low Reynolds numbers. J. Fluid Mech. 683, 212–234.10.1017/jfm.2011.259Google Scholar
Wylie, J. J., Huang, H. & Miura, R. M. 2015 Asymptotic analysis of a viscous thread extending under gravity. Physica D 313, 51–60.Google Scholar
Yarin, A. L. 1986 Effect of heat removal on nonsteady regimes of fiber formation. J. Engng Phys. 50, 569–575.10.1007/BF00870716Google Scholar
Yarin, A. L., Gospodinov, P. & Roussinov, V. I. 1994 Stability loss and sensitivity in hollow fiber drawing. Phys. Fluids 6 (4), 1454–1463.10.1063/1.868260Google Scholar
Yarin, A. L., Rusinov, V. I., Gospodinov, P. & St. Radev 1989 Quasi one-dimensional model of drawing of glass micro capillaries and approximate solutions. Theor. Appl. Mech. 20 (3), 55–62.Google Scholar