Skip to main content Accessibility help
×
Home

Out-of-plane buckling in two-dimensional glass drawing

  • D. O’Kiely (a1), C. J. W. Breward (a1), I. M. Griffiths (a1), P. D. Howell (a1) and U. Lange (a2)...

Abstract

We derive a mathematical model for the drawing of a two-dimensional thin sheet of viscous fluid in the direction of gravity. If the gravitational field is sufficiently strong, then a portion of the sheet experiences a compressive stress and is thus unstable to transverse buckling. We analyse the dependence of the instability and the subsequent evolution on the process parameters, and the mutual coupling between the weakly nonlinear buckling and the stress profile in the sheet. Over long time scales, the sheet centreline ultimately adopts a universal profile, with the bulk of the sheet under tension and a single large bulge caused by a small compressive region near the bottom, and we derive a canonical inner problem that describes this behaviour. The large-time analysis involves a logarithmic asymptotic expansion, and we devise a hybrid asymptotic–numerical scheme that effectively sums the logarithmic series.

Copyright

Corresponding author

Email address for correspondence: okiely@maths.ox.ac.uk

References

Hide All
Batty, C., Uribe, A., Audoly, B. & Grinspun, E. 2012 Discrete viscous sheets. ACM Trans. Graph. 31 (4), 113.
Buckmaster, J. D., Nachman, A. & Ting, L. 1975 The buckling and stretching of a viscida. J. Fluid Mech. 69 (01), 120.
Burke, S.2016 This glass can bend in half without shattering (online video clip). http://edition.cnn.com/videos/cnnmoney/2016/03/18/bendable-glass-schott-burke-pkg.cnn-money/video/playlists/technology/, CNN Money.
Chiu-Webster, S. & Lister, J. R. 2006 The fall of a viscous thread onto a moving surface: a ‘fluid-mechanical sewing machine’. J. Fluid Mech. 569, 89111.
Cruickshank, J. O. 1988 Low-Reynolds-number instabilities in stagnating jet flows. J. Fluid Mech. 193, 111127.
Cruickshank, J. O. & Munson, B. R. 1981 Viscous fluid buckling of plane and axisymmetric jets. J. Fluid Mech. 113, 221239.
Davidovitch, B., Schroll, R. D., Vella, D., Adda-Bedia, M. & Cerda, E. A. 2011 Prototypical model for tensional wrinkling in thin sheets. Proc. Natl Acad. Sci. USA 108 (45), 1822718232.
Filippov, A. & Zheng, Z. 2010 Dynamics and shape instability of thin viscous sheets. Phys. Fluids 22 (2), 023601.
Horvatitsch, T.2016 Ultra-thin glass. Schott Solutions 1/2016, pp. 6–11. Schott AG.
Howell, P. D.1994 Extensional thin layer flows. PhD thesis, University of Oxford.
Howell, P. D. 1996 Models for thin viscous sheets. Eur. J. Appl. Maths 7 (04), 321343.
Kropinski, M. C. A., Ward, M. J. & Keller, J. B. 1995 A hybrid asymptotic–numerical method for low Reynolds number flows past a cylindrical body. SIAM J. Appl. Maths 55 (6), 14841510.
Mahadevan, L., Ryu, W. S. & Samuel, A. D. T. 1998 Fluid ‘rope trick’ investigated. Nature 392 (6672), 140140.
Matovich, M. A. & Pearson, J. R. A. 1969 Spinning a molten threadline. Steady-state isothermal viscous flows. Ind. Engng Chem. Fundam. 8 (3), 512520.
Morris, S. W., Dawes, J. H. P., Ribe, N. M. & Lister, J. R. 2008 Meandering instability of a viscous thread. Phys. Rev. E 77 (6), 066218.
O’Kiely, D.2018 Mathematical models for the glass sheet redraw process. PhD thesis, University of Oxford.
O’Kiely, D., Breward, C. J. W., Griffiths, I. M., Howell, P. D. & Lange, U. 2015 Edge behaviour in the glass sheet redraw process. J. Fluid Mech. 785, 248269.
O’Kiely, D., Breward, C. J. W., Griffiths, I. M., Howell, P. D. & Lange, U. 2018 Glass sheet redraw through a long heater zone. IMA J. Appl. Maths 83 (5), 799820.
Perdigou, C.2015 Stability of viscous sheets in open flow. PhD thesis, Université Pierre et Marie Curie-Paris VI.
Ribe, N. M. 2001 Bending and stretching of thin viscous sheets. J. Fluid Mech. 433, 135160.
Srinivasan, S., Wei, Z. & Mahadevan, L. 2017 Wrinkling instability of an inhomogeneously stretched viscous sheet. Phys. Rev. Fluids 2 (7), 074103.
Taylor, G. I. 1969 Instability of jets, threads, and sheets of viscous fluid. In Applied Mechanics, pp. 382388. Springer.
Tchavdarov, B., Yarin, A. L. & Radev, S. 1993 Buckling of thin liquid jets. J. Fluid Mech. 253, 593615.
Ward, M. J., Heshaw, W. D. & Keller, J. B. 1993 Summing logarithmic expansions for singularly perturbed eigenvalue problems. SIAM J. Appl. Maths 53 (3), 799828.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Out-of-plane buckling in two-dimensional glass drawing

  • D. O’Kiely (a1), C. J. W. Breward (a1), I. M. Griffiths (a1), P. D. Howell (a1) and U. Lange (a2)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed