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Stability of drawing of microstructured optical fibres

Published online by Cambridge University Press:  27 April 2023

Jonathan J. Wylie Open the ORCID record for Jonathan J. Wylie [Opens in a new window] ,
Nazmun N. Papri Open the ORCID record for Nazmun N. Papri [Opens in a new window] ,
Yvonne M. Stokes Open the ORCID record for Yvonne M. Stokes [Opens in a new window]  and
Dongdong He Open the ORCID record for Dongdong He [Opens in a new window]
Jonathan J. Wylie*
Affiliation:
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA
Nazmun N. Papri
Affiliation:
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
Yvonne M. Stokes
Affiliation:
School of Mathematical Sciences and Institute for Photonics and Advanced Sensing, The University of Adelaide, SA 5005, Australia
Dongdong He
Affiliation:
School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong, 518172, PR China
*
†Email address for correspondence: mawylie@cityu.edu.hk
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Abstract

We consider the stability of the drawing of a long and thin viscous thread with an arbitrary number of internal holes of arbitrary shape that evolves due to axial drawing, inertia and surface tension effects. Despite the complicated geometry of the boundaries, we use asymptotic techniques to determine a particularly convenient formulation of the equations of motion that is well-suited to stability calculations. We will determine an explicit asymptotic solution for steady states with (a) large surface tension and negligible inertia, and (b) large inertia. In both cases, we will show that complicated boundary layer structures can occur. We will use linear stability analysis to show that the presence of an axisymmetric hole destabilises the flow for finite capillary number and which answers a question raised in the literature. However, our formulation allows us to go much further and consider arbitrary hole structures or non-axisymmetric shapes, and show that any structure with holes will be less stable than the case of a solid axisymmetric thread. For a solid axisymmetric thread, we will also determine a closed-form expression that delineates the unconditional instability boundary in which case the thread is unstable for all draw ratios. We will determine how the detailed effects of the microstructure affect the stability, and show that they manifest themselves only via a single function that occurs in the stability problem and hence have a surprisingly limited effect on the stability.

JFM classification

Interfacial Flows (free surface): Capillary flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 962 , 10 May 2023 , A12
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Argyros, A. & Pla, J. 2007 Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared. Opt. Express 15, 7713–7719.CrossRefGoogle ScholarPubMed
Bechert, M. 2017 Influence of Process and Material Parameters on the Draw Resonance Instability. F.A.U. University Press.Google Scholar
Bechert, M. & Scheid, B. 2017 Combined influence of inertia, gravity, and surface tension on the linear stability of Newtonian fiber spinning. Phys. Rev. Fluids 2, 113905.CrossRefGoogle Scholar
Bradshaw-Hajek, B.H., Stokes, Y.M. & Tuck, E.O. 2007 Computation of extensional fall of slender viscous drops by a one-dimensional Eulerian method. SIAM J. Appl. Maths 67, 1166–1182.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Chen, M.J., Stokes, Y.M., Buchak, P., Crowdy, D.G. & Ebendorff-Heidepriem, H. 2016 Asymptotic modelling of a six-hole microstructured optical fibre. J. Lightwave Technol. 34 (24), 5651–5656.CrossRefGoogle 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.CrossRefGoogle Scholar
Denn, M.M. 1980 Continuous drawing of liquids to form fibers. Annu. Rev. Fluid Mech. 12, 365–387.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Eggers, J. 1993 Universal pinching of 3D axisymmetric free-surface flow. Phys. Rev. Lett. 71, 3458.CrossRefGoogle ScholarPubMed
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.CrossRefGoogle Scholar
Fitt, A.D., Furusawa, K., Monro, T.M. & Please, C.P. 2001 Modeling the fabrication of hollow fibers: capillary drawing. J. Lightwave Technol. 19, 1924–1931.CrossRefGoogle 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.CrossRefGoogle Scholar
Forest, M.G. & Zhou, H. 2001 Unsteady analysis of thermal glass fiber drawing processes. Eur. J. Appl. Maths 12, 479–496.CrossRefGoogle Scholar
Geyling, F.T. 1976 Basic fluid dynamic considerations in the drawing of optical fibres. Bell Syst. Tech. J. 55, 1011–1056.CrossRefGoogle Scholar
Geyling, F.T. & Homsy, G.M. 1980 Extensional instabilities of the glass fiber drawing process. Glass Technol. 21, 95–102.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.CrossRefGoogle Scholar
Griffiths, I.M. & Howell, P.D. 2008 Mathematical modelling of non-axisymmetric capillary tube drawing. J. Fluid Mech. 605, 181–206.CrossRefGoogle 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.CrossRefGoogle Scholar
Kaye, A. 1991 Convected coordinates and elongational flow. J. Non-Newtonian Fluid Mech. 40, 55–77.CrossRefGoogle Scholar
Liu, Z., Tam, H.-Y., Htein, L., Tse, M.-L.V. & Lu, C. 2017 Microstructured optical fiber sensors. J. Lightwave Technol. 35, 3425–3439.CrossRefGoogle Scholar
Matovich, M.A. & Pearson, J.R.A. 1969 Spinning a molten threadline. I&EC Fundamentals 8, 512–520.CrossRefGoogle Scholar
Papageorgiou, D.T. 1995 On the breakup of viscous liquid threads. Phys. Fluids 7 (7), 1529–1544.CrossRefGoogle Scholar
Pearson, J.R.A. & Matovich, M.A. 1969 Spinning a molten threadline. Stability. Ind. Engng Chem. Fundam. 8, 605–608.CrossRefGoogle Scholar
Philippi, J., Bechert, M., Chouffart, Q., Waucquez, C. & Scheid, B. 2022 Linear stability analysis of nonisothermal glass fiber drawing. Phys. Rev. Fluids 7, 043901.CrossRefGoogle Scholar
Shah, Y.T. & Pearson, J.R.A. 1972 a On the stability of nonisothermal fibre spinning. Ind. Engng Chem. Fundam. 11, 145–149.CrossRefGoogle Scholar
Shah, Y.T. & Pearson, J.R.A. 1972 b On the stability of nonisothermal fibre spinning – general case. Ind. Engng Chem. Fundam. 11, 150–153.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Stokes, Y.M., Wylie, J.J. & Chen, M.J. 2019 Coupled fluid and energy flow in fabrication of microstructured optical fibres. J. Fluid Mech. 874, 548–572.CrossRefGoogle Scholar
Suman, B. & Kumar, S. 2009 Draw ratio enhancement in nonisothermal melt spinning. AIChE J. 55, 581–593.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Wylie, J.J. & Huang, H. 2007 Extensional flows with viscous heating. J. Fluid Mech. 571, 359–370.CrossRefGoogle Scholar
Wylie, J.J., Huang, H. & Miura, R.M. 2007 Thermal instability in drawing viscous threads. J. Fluid Mech. 570, 1–16.CrossRefGoogle Scholar
Wylie, J.J., Huang, H. & Miura, R.M. 2011 Stretching of viscous threads at low Reynolds numbers. J. Fluid Mech. 683, 212–234.CrossRefGoogle Scholar
Wylie, J.J., Huang, H. & Miura, R.M. 2015 Asymptotic analysis of a viscous thread extending under gravity. Physica D 313, 51–60.CrossRefGoogle Scholar
Xue, S.C., Large, M.C.J., Barton, G.W., Tanner, R.I., Polidian, L. & Lwin., R. 2005 a Role of material properties and drawing conditions in the fabrication of microstructured optical fibres. J. Lightwave Technol. 24, 853–860.CrossRefGoogle Scholar
Xue, S.C., Tanner, R.I., Barton, G.W., Lwin, R., Large, M.C.J. & Polidian, L. 2005 b Fabrication of microstructured optical fibres – part I: problem formulation and numerical modelling of transient draw process. J. Lightwave Technol. 23, 2245–2254.CrossRefGoogle Scholar
Xue, S.C., Tanner, R.I., Barton, G.W., Lwin, R., Large, M.C.J. & Polidian, L. 2005 c Fabrication of microstructured optical fibres – part II: numerical modelling of steady-state draw process. J. Lightwave Technol. 23, 2255–2266.CrossRefGoogle Scholar
Yarin, A.L. 1986 Effect of heat removal on nonsteady regimes of fiber formation. J. Engng Phys. 50, 569–575.CrossRefGoogle Scholar
Yarin, A.L., Gospodinov, P. & Roussinov, V.I. 1994 Stability loss and sensitivity in hollow fiber drawing. Phys. Fluids 6 (4), 1454–1463.CrossRefGoogle Scholar
Yarin, A.L., Rusinov, V.I., Gospodinov, P. & Radev, S. 1989 Quasi one-dimensional model of drawing of glass micro capillaries and approximate solutions. Theor. Appl. Mech. 20 (3), 55–62.Google Scholar