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The asymptotic structure of a slender dragged viscous thread

Published online by Cambridge University Press:  23 March 2011

MAURICE J. BLOUNT  and
JOHN R. LISTER
MAURICE J. BLOUNT*
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
JOHN R. LISTER
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: maurice.blount@cantab.net
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Abstract

The behaviour of a viscous thread as it falls onto a moving belt is analysed in the asymptotic limit of a slender thread. While the bending resistance of a slender thread is small, its effects are dynamically important near the contact point with the belt, where it changes the curvature and orientation of the thread. Steady flows are shown to fall into one of three distinct regimes, depending on whether the belt is moving faster than, slower than or close to the same speed as the free-fall velocity of the thread. The key dynamical balances in each regime are explained and the role of bending stresses is found to be qualitatively different. The asymptotic solutions exhibit the ‘backward-facing heel’ observed experimentally for low belt speeds, and provide the leading-order corrections to the stretching catenary in theory previously developed for high belt speeds. The asymptotic stability of the thread to the onset of meandering is also analysed. It is shown that the entire thread, rather than the bending boundary layer alone, governs the stability. A balance between the destabilising reaction forces near the belt and the restoring force of gravity on the remainder of the thread determines the onset of meandering, and an analytic estimate for the meandering frequency is thereby obtained. At leading order, neutral stability occurs with the belt moving a little more slowly than the free-fall velocity of the thread, not when the lower part of the thread begins to be under compression, but when the horizontal reaction force at the belt begins to be slightly against the direction of belt motion. The onset of meandering is the heel ‘losing its balance’.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 674 , 10 May 2011 , pp. 489 - 521
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Barnes, G. & MacKenzie, J. 1959 Height of fall versus frequency in liquid rope-coil effect. Am. J. Phys. 27, 112–115.Google Scholar
Barnes, G. & Woodcock, R. 1958 Liquid rope-coil effect. Am. J. Phys. 26, 205–209.CrossRefGoogle Scholar
Buckmaster, J. D., Nachman, A. & Ting, L. 1978 The buckling and stretching of a viscida. J. Fluid Mech. 69, 1–20.Google Scholar
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, 89–111.Google Scholar
Clarke, N. S. 1968 Two-dimensional flow under gravity in a jet of viscous liquid. J. Fluid Mech. 31, 481–500.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, 361–389.Google Scholar
Decent, S. P., King, A. C., Simmons, M. J. H, Părău, E. I., Wallwork, I. M., Gurney, C. J. & Uddin, J. 2009 The trajectory and stability of a spiralling liquid jet: viscous theory. Appl. Math. Mod. 33, 4283–4302.Google Scholar
Dewynne, J., Ockendon, J. R. & Wilmott, P. 1989 On a mathematical model for fiber tapering. SIAM J. Appl. Math. 49, 983–990.Google Scholar
Dewynne, J. N., Howell, P. D. & Wilmott, P. 1994 Slender viscous fibres with inertia and gravity. Q. J. Mech. Appl. Math. 47, 541–555.Google 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.Google Scholar
Doedel, E. J. & Oldeman, B. E. 2009 Auto-07p: continuation and bifurcation software for ordinary differential equations. Available at: http://indy.cs.concordia.ca/auto/.Google Scholar
Dyson, R. J. 2007 Mathematical modelling of curtain coating. PhD thesis, University of Oxford.Google Scholar
Entov, V. M. & Yarin, A. L. 1984 The dynamics of thin liquid jets in air. J. Fluid Mech. 140, 91–111.Google Scholar
Goren, S. L. & Wronski, S. 1966 The shape of low-speed capillary jets of Newtonian liquids. J. Fluid Mech. 25, 185–198.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
Habibi, M., Maleki, M., Golestanian, R., Ribe, N. M. & Bonn, D. 2006 Dynamics of liquid rope coiling. Phys. Rev. E 74 (6), 066306.Google Scholar
Hlod, A. 2009 Curved jets of viscous fluid: interactions with a moving wall. PhD thesis, Eindhoven University of Technology.Google Scholar
Hlod, A., Aarts, A. C. T., Van De Ven, A. A. F. & Peletier, M. A. 2007 Mathematical model of falling of a viscous jet onto a moving surface. Eur. J. Appl. Math. 18, 659–677.Google Scholar
Mahadevan, L., Ryu, W. S. & Samuel, A. D. T. 1998 Fluid ‘rope trick’ investigated. Nature 392, 140.Google Scholar
Mahadevan, L., Ryu, W. S. & Samuel, A. D. T. 2000 Correction: fluid ‘rope trick’ investigated. Nature 403, 502.Google Scholar
Maleki, M., Habibi, M., Golestanian, R., Ribe, N. M. & Bonn, D. 2006 Dynamics of liquid rope coiling. Phys. Rev. E 74, 066306.Google Scholar
Marheineke, N. & Wegener, R. 2009 Asymptotic model for the dynamics of curved viscous fibres with surface tension. J. Fluid Mech. 622, 345–369.Google Scholar
Matovich, M. A. & Pearson, J. R. A. 1969 Spinning a molten threadline: steady-state isothermal viscous flows. Ind. Engng Chem. Fundam. 8, 512–520.CrossRefGoogle Scholar
Morris, S. W., Dawes, J. H. P., Ribe, N. M. & Lister, J. R. 2008 The meandering instability of a viscous thread. Phys. Rev. E 77, 066218.Google Scholar
Ribe, N. M. 2004 Coiling of viscous jets. Proc. R. Soc. Lond. A 460, 3223–3239.CrossRefGoogle Scholar
Ribe, N. M., Habibi, M. & Bonn, D. 2006 a Stability of liquid rope coiling. Phys. Fluids 18, 084102.Google Scholar
Ribe, N. M., Huppert, H. E., Hallworth, M. A., Habibi, M. & Bonn, D. 2006 b Multiple coexisting states of liquid rope coiling. J. Fluid Mech. 555, 275–297.Google Scholar
Ribe, N. M., Lister, J. R. & Chiu-Webster, S. 2006 b Stability of a dragged viscous thread: onset of ‘stitching’ in a fluid-mechanical ‘sewing machine’. Phys. Fluids 18, 124105.Google Scholar
Taylor, G. I. 1968 Instability of jets, threads and sheets of viscous fluid. In Proc. 12th Intl Cong. Appl. Math. (ed. Hetényi, M. & Vincenti, W. G.), pp. 382–389. Springer.Google Scholar
Yarin, A. L. 1993 Free Liquid Jets and Films: Hydrodynamics and Rheology. Wiley.Google Scholar