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Extrusion of fluid cylinders of arbitrary shape with surface tension and gravity

Published online by Cambridge University Press:  24 November 2016

Hayden Tronnolone*
Affiliation:
School of Mathematical Sciences, The University of Adelaide, North Terrace, Adelaide, SA 5005, Australia
Yvonne M. Stokes
Affiliation:
School of Mathematical Sciences, The University of Adelaide, North Terrace, Adelaide, SA 5005, Australia
Heike Ebendorff-Heidepriem
Affiliation:
ARC Centre of Excellence for Nanoscale BioPhotonics, Institute for Photonics and Advanced Sensing, School of Physical Sciences, The University of Adelaide, North Terrace, Adelaide, SA 5005, Australia
*
Email address for correspondence: hayden.tronnolone@adelaide.edu.au

Abstract

A model is developed for the extrusion in the direction of gravity of a slender fluid cylinder from a die of arbitrary shape. Both gravity and surface tension act to stretch and deform the geometry. The model allows for an arbitrary but prescribed viscosity profile, while the effects of extrudate swell are neglected. The solution is found efficiently through the use of a carefully selected axial Lagrangian coordinate and a transformation to a reduced-time variable. Comparisons between the model and extruded glass microstructured optical fibre preforms show that surface tension has a significant effect on the geometry but the model does not capture all of the behaviour observed in practice. Experimental observations are used in conjunction with the model to argue that some deformation, due neither to surface tension nor gravity, occurs in or near the die exit. Methods are considered to overcome deformation due to surface tension.

Type
Papers
Copyright
© 2016 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), 11011110.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, 538.Google 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, 137165.Google Scholar
Chen, M. J., Stokes, Y. M., Buchak, P., Crowdy, D. G., Foo, H. T. C., Dowler, A. & Ebendorff-Heidepriem, H. 2016 Drawing tubular fibres: experiments versus mathematical modelling. Opt. Mater. Express 6 (1), 166180.Google Scholar
Crowdy, D. G. 2002 Exact solutions for the viscous sintering of multiply-connected fluid domains. J. Engng Maths 42 (3–4), 225242.Google Scholar
Crowdy, D. G. & Tanveer, S. 1998 A theory of exact solutions for annular viscous blobs. J. Nonlinear Sci. 8 (4), 375400.Google Scholar
Crowdy, D. G., Tanveer, S. & Vasconcelos, G. L. 2005 On a pair of interacting bubbles in planar Stokes flow. J. Fluid Mech. 541, 231261.Google 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, 361389.Google Scholar
Ebendorff-Heidepriem, H. & Monro, T. M. 2007 Extrusion of complex preforms for microstructured optical fibers. Opt. Express 15 (23), 8692.Google Scholar
Ebendorff-Heidepriem, H. & Monro, T. M. 2012 Analysis of glass flow during extrusion of optical fiber preforms. Opt. Mater. Express 2 (3), 304320.Google Scholar
Ebendorff-Heidepriem, H., Moore, R. C. & Monro, T. M. 2008 Progress in the fabrication of the next-generation soft glass microstructured optical fibers. AIP Conf. Proc. 1055 (1), 9598.Google Scholar
Ebendorff-Heidepriem, H., Schuppich, J., Dowler, A., Lima-Marques, L. & Monro, T. M 2014 3D-printed extrusion dies: a versatile approach to optical material processing. Opt. Mater. Express 4 (8), 14941504.Google Scholar
Goursat, É. J.-B. 1898 Sur l’équation ΔΔu = 0. Bull. Soc. Maths France 26, 233237.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, 181208.Google Scholar
Griffiths, I. M. & Howell, P. D. 2008 Mathematical modelling of non-axisymmetric capillary tube drawing. J. Fluid Mech. 605, 181206.Google Scholar
Hopper, R. W. 1990 Plane Stokes flow driven by capillarity on a free surface. J. Fluid Mech. 213, 349375.Google Scholar
Knight, J. C. 2003 Photonic crystal fibres. Nature 424 (6950), 847851.Google Scholar
Manning, S.2011 A study of tellurite glasses for electro-optic optical fibre devices. PhD thesis, School of Chemistry and Physics, University of Adelaide.Google Scholar
Monro, T. M. & Ebendorff-Heidepriem, H. 2006 Progress in microstructured optical fibres. Annu. Rev. Mater. Res. 36 (1), 467495.Google Scholar
Richardson, S. 1992 Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension. Eur. J. Appl. Maths 3, 193207.Google 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, 176203.Google Scholar
Stokes, Y. M., Crowdy, D. G., Tronnolone, H., Ebendorff-Heidepriem, H. & Monro, T. M. 2012 Towards understanding of geometrical structure in microstructured optical fibres. In International Congress of Theoretical and Applied Mechanics (23rd: 2012: Beijing, China), (ed. Y. Bai, J. Wang & D. Fang), FM10-013. Available at http://fluid.ippt.pan.pl/ conference_proceedings/ICTAM2012/.Google Scholar
Trabelssi, M., Ebendorff-Heidepriem, H., Richardson, K. A., Monro, T. M. & Joseph, P. F. 2015 Computational modeling of hole distortion in extruded microstructured optical fiber glass preforms. J. Lightwave Technol. 33 (2), 424431.Google Scholar
Tronnolone, H.2016 Extensional and surface-tension-driven fluid flows in microstructured optical fibre fabrication. PhD thesis, School of Mathematical Sciences, University of Adelaide.Google 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, 308338.Google Scholar
van de Vorst, G. A. L. 1993 Integral method for a two-dimensional Stokes flow with shrinking holes applied to viscous sintering. J. Fluid Mech. 257, 667689.Google Scholar
Wilson, S. D. R. 1988 The slow dripping of a viscous fluid. J. Fluid Mech. 190, 561570.Google Scholar