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The evolution of a viscous thread pulled with a prescribed speed

Published online by Cambridge University Press:  14 April 2016

J. J. Wylie ,
B. H. Bradshaw-Hajek  and
Y. M. Stokes
J. 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
B. H. Bradshaw-Hajek
Affiliation:
Phenomics and Bioinformatics Research Centre, School of Information Technology and Mathematical Sciences, University of South Australia, Mawson Lakes, SA 5095, Australia
Y. M. Stokes
Affiliation:
School of Mathematical Sciences and Institute for Photonics and Advanced Sensing, University of Adelaide, Adelaide, SA 5005, Australia
*
†Email address for correspondence: mawylie@cityu.edu.hk
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Abstract

We examine the extension of an axisymmetric viscous thread that is pulled at both ends with a prescribed speed such that the effects of inertia are initially small. After neglecting surface tension, we derive a particularly convenient form of the long-wavelength equations that describe long and thin threads. Two generic classes of initial thread shape are considered as well as the special case of a circular cylinder. In these cases, we determine explicit asymptotic solutions while the effects of inertia remain small. We further show that inertia will ultimately become important only if the long-time asymptotic form of the pulling speed is faster than a power law with a critical exponent. The critical exponent can take two possible values depending on whether or not the initial minimum of the thread radius is located at the pulled end. In addition, we obtain asymptotic expressions for the solution at large times in the case in which the critical exponent is exceeded and hence inertia becomes important. Despite the apparent simplicity of the problem, the solutions exhibit a surprisingly rich structure. In particular, in the case in which the initial minimum is not at the pulled end, we show that there are two very different types of solution that exhibit very different extension mechanics. Both the small-inertia solutions and the large-time asymptotic expressions compare well with numerical solutions.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Lubrication theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 795 , 25 May 2016 , pp. 380 - 408
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Copyright
© 2016 Cambridge University Press 

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References

Al Khatib, M. A. M. 2003 The stretching of a viscoplastic thread of liquid. Trans. ASME J. Fluids Engng 125, 946–951.CrossRefGoogle Scholar
Balmforth, N. J., Dubashi, N. & Slim, A. C. 2010 Extensional dynamics of viscoplastic filaments: II. Drips and bridges. J. Non-Newtonian Fluid Mech. 165, 1147–1160.CrossRefGoogle Scholar
Berg, C. P., Dreyer, M. & Rath, H. J. 1999 A large fluid-bridge device to measure the deformation of drops in uniaxial extensional flow fields. Meas. Sci. Technol. 10, 956–964.CrossRefGoogle Scholar
Berg, C. P., Kröger, R. & Rath, H. J. 1994 Measurement of extensional viscosity by stretching large liquid bridges in microgravity. J. Non-Newtonian Fluid Mech. 55, 307–319.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
Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids. Clarendon.Google Scholar
Chen, Y. J. & Steen, P. H. 1997 Dynamics of inviscid capillary breakup: collapse and pinchoff of a film bridge. J. Fluid Mech. 341, 245–267.CrossRefGoogle Scholar
Day, R. F., Hinch, E. J. & Lister, J. R. 1998 Self-similar capillary pinchoff of an inviscid fluid. Phys. Rev. Lett. 80, 704–707.CrossRefGoogle Scholar
Demko, S., Moss, W. F. & Smith, P. W. 1984 Decay rates for inverses of band matrices. Maths Comput. 43, 491–499.CrossRefGoogle Scholar
DeWynne, J., Ockendon, J. R. & Wilmott, P. 1989 On a mathematical model for fiber tapering. SIAM J. Appl. Maths 49, 983–990.CrossRefGoogle Scholar
DeWynne, J., 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, 3458–3460.CrossRefGoogle ScholarPubMed
Eggers, J. 2005 Drop formation: an overview. Z. Angew. Math. Mech. 85, 400–410.CrossRefGoogle Scholar
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
Fontelos, M. A. & Li, J. 2004 On the evolution and rupture of filaments in giesekus and fene models. J. Non-Newton. Fluid Mech. 118, 1–16.CrossRefGoogle Scholar
Forest, M. G., Zhou, H. & Wang, Q. 2000 Thermotropic liquid crystalline polymer fibers. SIAM J. Appl. Maths 60, 1177–1204.Google Scholar
Gaudet, S., McKinley, G. H. & Stone, H. A. 1996 Extensional deformation of Newtonian liquid bridges. Phys. Fluids 8, 2568–2579.CrossRefGoogle Scholar
Gupta, G. & Schultz, W. W. 1998 Non-isothermal flows of Newtonian slender glass fibers. Intl J. Non-Linear Mech. 33, 151–163.CrossRefGoogle Scholar
Hassager, O., Kolte, M. I. & Renardy, M. 1998 Failure and nonfailure of fluid filaments in extension. J. Non-Newtonian Fluid Mech. 76, 137–151.CrossRefGoogle Scholar
Huang, H., Miura, R. M., Ireland, W. & Puil, E. 2003 Heat-induced stretching of a glass tube under tension: application to glass microelectrodes. SIAM J. Appl. Maths 63, 1499–1519.Google Scholar
Huang, H., Wylie, J. J., Miura, R. M. & Howell, P. D. 2007 On the formation of glass microelectrodes. SIAM J. Appl. Maths 67, 630–666.CrossRefGoogle Scholar
Kaye, A. 1991 Convective coordinates and elongational flow. J. Non-Newtonian Fluid Mech. 40, 55–77.CrossRefGoogle Scholar
Matovich, M. A. & Pearson, J. R. A. 1969 Spinning a molten threadline. Ind. Engng Chem. Fundam. 8, 512–520.CrossRefGoogle Scholar
Matta, J. E. & Titus, R. P. 1990 Liquid stretching using a falling cylinder. J. Non-Newtonian Fluid Mech. 35, 215–229.CrossRefGoogle Scholar
Olagunju, D. O. 1999 A 1-D theory for extensional deformation of a viscoelastic filament under exponential stretching. J. Non-Newtonian Fluid Mech. 87, 27–46.CrossRefGoogle Scholar
Papageorgiou, D. T. 1995 On the breakup of viscous liquid threads. Phys. Fluids 7, 1529–1544.CrossRefGoogle Scholar
Rayleigh, Lord 1879 On the instability of jets. Proc. R. Soc. Lond. 10, 4–9.Google Scholar
Rayleigh, Lord 1892 On the instability of a cylinder of viscous liquid under capillary force. Phil. Mag. 34, 145–154.CrossRefGoogle Scholar
Renardy, M. 1994 Some comments on the surface-tension driven break-up (or the lack of it) of viscoelastic jets. J. Non-Newtonian Fluid Mech. 51, 97–107.CrossRefGoogle Scholar
Renardy, M. 1995 A numerical study of the asymptotic evolution and breakup of Newtonian and viscoelastic jets. J. Non-Newtonian Fluid Mech. 59, 267–282.CrossRefGoogle Scholar
Renardy, M. 2002 Similarity solutions for jet breakup for various models of viscoelastic fluids. J. Non-Newtonian Fluid Mech. 104, 65–74.CrossRefGoogle Scholar
Renardy, M. 2004 Self-similar breakup of non-Newtonian liquid jets. In Rheology Reviews (ed. Binding, D. M. & Walters, K.), pp. 171–196. British Society of Rheology.Google Scholar
Spiegelberg, S. H., Ables, D. C. & McKinley, G. H. 1996 The role of end-effects on measurements of extensional viscosity in filament stretching rheometers. J. Non-Newtonian Fluid Mech. 64, 229–267.CrossRefGoogle Scholar
Sridhar, T., Tirtaatmadja, V., Nguyenand, D. A. & Gupta, R. K. 1991 Measurement of extensional viscosity of polymer solutions. J. Non-Newtonian Fluid Mech. 40, 271–280.Google 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
Wilkes, E. D., Phillips, S. D. & Basaran, O. A. 1999 Computational and experimental analysis of dynamics of drop formation. Phys. Fluids 11, 3577–3598.CrossRefGoogle Scholar
Wilson, S. D. R. 1988 The slow dripping of a viscous fluid. J. Fluid Mech. 190, 561–570.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. 2011 Stretching of viscous threads at low Reynolds numbers. J. Fluid Mech. 683, 212–234.CrossRefGoogle Scholar
Yao, M. & McKinley, G. H. 1998 Numerical simulation of extensional deformations of viscous liquid bridges in filament stretching devices. J. Non-Newtonian Fluid Mech. 49, 47–88.CrossRefGoogle Scholar
Yin, Z. & Jaluria, Y. 2000 Neck down and thermally induced defects in high-speed optical fiber drawing. Trans. ASME J. Heat Transfer 122, 351–362.CrossRefGoogle Scholar