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A two-dimensional asymptotic model for capillary collapse

Published online by Cambridge University Press:  17 December 2020

Yvonne M. Stokes Open the ORCID record for Yvonne M. Stokes [Opens in a new window]
Yvonne M. Stokes*
Affiliation:
School of Mathematical Sciences and Institute for Photonics and Advanced Sensing, The University of Adelaide, SA5005, Australia
*
†Email address for correspondence: [email protected]
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Abstract

The collapse under surface tension of a long axisymmetric capillary, held at both ends and softened by a travelling heater, is used to determine the viscosity or surface tension of silica glasses. Capillary collapse is also used in the manufacture of some optical fibre preforms. Typically, a one-dimensional (1-D) model of the closure of a concentric fluid annulus is used to relate a measure of the change in the cross-sectional geometry, for example the external radius, to the desired information. We here show that a two-dimensional (2-D) asymptotic model developed for drawing of optical fibres, but with a unit draw ratio, may be used and yields analytic formulae involving a single dimensionless parameter, the scaled heater speed $V$, equivalently a capillary number. For a capillary fixed at both ends, this 2-D model agrees with the 1-D model and offers the significant benefit that it enables determination of both the surface tension and viscosity from a single capillary-collapse experiment, provided the pulling tension in the capillary during collapse is measured. The 2-D model also enables our investigation of the situation where both ends of the capillary are not fixed, so that the capillary cannot sustain a pulling tension. Then the collapse of the capillary is markedly different from that predicted by the 1-D model and the ability to determine both surface tension and viscosity is lost.

JFM classification

Interfacial Flows (free surface): Capillary flows Low-Reynolds-number flows: Slender-body theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 909 , 25 February 2021 , A5
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Boyd, K., Ebendorff-Heidepriem, H., Monro, T. M. & Munch, J. 2012 Surface tension and viscosity measurement of optical glasses using a scanning $\textrm {CO}_2$ laser. Opt. Mater. Express 2, 1101–1110.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., Foo, H. T. C., Dowler, A. & Ebendorff-Heidepriem, H. 2016 a Drawing tubular fibres: experiments versus mathematical modelling. Opt. Mater. Express 6, 166–180.CrossRefGoogle Scholar
Chen, M. J., Stokes, Y. M., Buchak, P., Crowdy, D. G. & Ebendorff-Heidepriem, H. 2016 b Asymptotic modelling of a six-hole MOF. J. Lightwave Technol. 34, 5651–5656.CrossRefGoogle Scholar
Das, S. K. & Gandhi, K. S. 1986 A model for thermal collapse of tubes: application to optical glass fibres. Chem. Engng Sci. 41, 73–81.CrossRefGoogle Scholar
Fitt, A. D., Furusawa, K., Monro, T. M., Please, C. P. & Richardson, D. J. 2002 The mathematical modelling of capillary drawing for holey fibre manufacture. J. Engng Maths 43, 201–227.CrossRefGoogle Scholar
Geyling, F. T., Walker, K. L. & Csencsits, R. 1983 The viscous collapse of thick-walled tubes. J. Appl. Mech. 50, 303–310.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
Kirchhof, J. 1980 A hydrodynamic theory of the collapsing process for the preparation of optical waveguide preforms. Phys. Status Solidi A 60, K127–K131.CrossRefGoogle Scholar
Kirchhof, J. 1985 a Reactor problems in modified chemical vapour deposition (I). The collapse of quartz glass tubes. Cryst. Res. Technol. 20, 705–712.CrossRefGoogle Scholar
Kirchhof, J. 1985 b Reactor problems in modified chemical vapour deposition (II). The mean viscosity of quartz glass reactor tubes. Cryst. Res. Technol. 21, 763–770.CrossRefGoogle Scholar
Kirchhof, J. & Unger, S. 2017 Viscous behavior of synthetic silica glass tubes during collapsing. Opt. Mater. Express 7, 386–400.CrossRefGoogle Scholar
Klupsch, T. & Pan, Z. 2017 Collapsing of glass tubes: analytic approaches in a hydrodynamic problem with free boundaries. J. Engng Maths 106, 143–168.CrossRefGoogle Scholar
Lewis, J. A. 1977 The collapse of a viscous tube. J. Fluid Mech. 81, 129–145.CrossRefGoogle Scholar
Makovetskii, A. A., Zamyatin, A. A. & Ivanov, G. A. 2013 Technique for estimating the viscosity of molten silica glass on the kinetics of the collapse of the glass capillary. Glass Phys. Chem. 40, 526–530.CrossRefGoogle Scholar
Stokes, Y. M., Buchak, P., Crowdy, D. G. & Ebendorff-Heidepriem, H. 2014 Drawing of micro-structured fibres: circular and non-circular tubes. J. Fluid Mech. 755, 176–203.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
Xue, S. C., Tanner, R. I., Barton, G. W., Lwin, R., Large, M. C. J. & Poladian, L. 2015 a Fabrication of microstructured optical fibers. Part 1. Problem formulation and numerical modeling 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. & Poladian, L. 2015 b Fabrication of microstructured optical fibers. Part 2. Numerical modeling of steady-state draw process. J. Lightwave Technol. 23, 2255–2266.CrossRefGoogle Scholar
Yarin, A., Rusinov, Vl., Gospodinov, P. & Radev, St. 1989 Quasi one-dimensional model of drawing of glass microcapillaries and approximate solutions. Theor. Appl. Mech. 20, 55–62.Google Scholar
Yarin, A., Gospodinov, P. & Rusinov, Vl. 1994 Stability loss and sensitivity in hollow fibre drawing. Phys. Fluids 6, 1454–1463.CrossRefGoogle Scholar