Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-25T08:51:59.761Z Has data issue: false hasContentIssue false
  • English
  • Français

Extension of a viscous thread with temperature-dependent viscosity and surface tension

Published online by Cambridge University Press:  14 July 2016

Dongdong He ,
Jonathan J. Wylie ,
Huaxiong Huang  and
Robert M. Miura
Dongdong He*
Affiliation:
School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China
Jonathan J. Wylie
Affiliation:
Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Hong Kong Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA
Huaxiong Huang
Affiliation:
Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada Fields Institute for Research in Mathematical Sciences, Toronto, ON M5T 3J1, Canada Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA
Robert M. Miura
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
*
†Email address for correspondence: dongdonghe@tongji.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the evolution of a long and thin vertically aligned axisymmetric viscous thread that is composed of an incompressible fluid. The thread is attached to a solid wall at its upper end, experiences gravity and is pulled at its lower end by a fixed force. As the thread evolves, it experiences either heating or cooling by its environment. The heating affects the evolution of the thread because both the viscosity and surface tension of the thread are assumed to be functions of the temperature. We develop a framework that can deal with threads that have arbitrary initial shape, are non-uniformly preheated and experience spatially non-uniform heating or cooling from the environment during the pulling process. When inertia is completely neglected and the temperature of the environment is spatially uniform, we obtain analytic solutions for an arbitrary initial shape and temperature profile. In addition, we determine the criteria for whether the cross-section of a given fluid element will ever become zero and hence determine the minimum stretching force that is required for pinching. We further show that the dynamics can be quite subtle and leads to surprising behaviour, such as non-monotonic behaviour in time and space. We also consider the effects of non-zero Reynolds number. If the temperature of the environment is spatially uniform, we show that the dynamics is subtly influenced by inertia and that the location at which the thread will pinch is selected by a competition between three distinct mechanisms. In particular, for a thread with initially uniform radius and a spatially uniform environment but with a non-uniform initial temperature profile, pinching can occur either at the hottest point, at the points near large thermal gradients or at the pulled end, depending on the Reynolds number. Finally, we show that similar results can be obtained for a thread with initially uniform radius and uniform temperature profile but exposed to a spatially non-uniform environment.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Slender-body theory Drops and Bubbles: Thermocapillarity
Type
Papers
Information
Journal of Fluid Mechanics , Volume 800 , 10 August 2016 , pp. 720 - 752
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Argyros, A. 2013 Microstructures in polymer fibres for optical fibres, THz waveguides, and fibre-based metamaterials. ISRN Optics 785162.Google Scholar
Ashby, M. & Jones, R. H. D. 2013 Engineering Materials 2: An Introduction to Microstructures and Processing, 4th edn, pp. 393–473. Butterworth-Heinemann.Google Scholar
Bansal, N. P. & Doremus, R. H. 1986 Handbook of Glass Properties. Materials Engineering Department Rensselaer Polytechnic Institute Troy. Academic.Google Scholar
Bingham, P. A. 2010 Design of new energy-friendly compositions. In Fiberglass and Glass Technology: Energy-Friendly Compositions and Applications (ed. Wallenberger, F. T. & Bingham, P. A.), pp. 267–351. Springer.Google Scholar
Blyth, M. G. & Bassom, A. P. 2012 Flow of a liquid layer over heated topography. Proc. R. Soc. Lond. A 468, 4067–4087.Google 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.Google Scholar
Chen, J. C., Sheu, J. C. & Lee, Y. T. 1990 Maximum stable length of nonisothermal liquid bridges. Phys. Fluids 2, 1118–1123.CrossRefGoogle Scholar
Chen, Y. J., Abbaschian, R. & Steen, P. H. 2003 Thermocapillary suppression of the Plateau–Rayleigh instability: a model for long encapsulated liquid zones. J. Fluid Mech. 485, 97–113.Google Scholar
Chinnov, E. A. & Shatskiy, E. N. 2014 Thermocapillary instabilities in a falling liquid film at small Reynolds numbers. Tech. Phys. Lett. 40, 7–9.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.CrossRefGoogle 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.CrossRefGoogle Scholar
D’ALESSIO, S. J. D., Pascal, J. P. & Jasmine, H. A. 2010 Film flow over heated wavy inclined surfaces. J. Fluid Mech. 665, 418–456.Google Scholar
Denn, M. M. 1980 Continuous drawing of liquids to form fibers. Annu. Rev. Fluid Mech. 12, 365–387.Google Scholar
Denn, M. M. 2014 Polymer Melt Processing: Foundations in Fluid Mechanics and Heat Transfer. (Part of Cambridge Series in Chemical Engineering). Cambridge University Press.Google Scholar
Dewynne, J. N., Howell, P. D. & Wilmott, P. 1994 Slender viscous fibers with inertia and gravity. Q. J. Mech. Appl. Maths 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
Dijkstra, H. A. & Steen, P. H. 1991 Thermocapillary stabilization of the capillary break up of an annular film of fluid. J. Fluid Mech. 229, 205–228.CrossRefGoogle Scholar
Eggers, J. 1993 Universal pinching of 3D axisymmetric free-surface flow. Phys. Rev. Lett. 71, 3458–3460.Google Scholar
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.Google 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
Fluegel, A. 2007 Glass viscosity calculation based on a global statistical modeling approach. Glass Technol.: Eur. J. Glass Sci. Technol. 48, 13–30.Google Scholar
Forest, M. G. & Zhou, H. 2001 Unsteady analysis of thermal glass fiber drawing processes. Eur. J. Appl. Maths 12, 479–496.Google Scholar
Gallacchi, R., Kölsch, S., Kneppe, H. & Meixner, A. J. 2001 Well-shaped fibre tips by pulling with a foil heater. J. Microsc. 202, 182–187.Google Scholar
Gospodinov, P. & Yarin, A. L. 1997 Draw resonance of optical microcapillaries in non-isothermal drawing. Intl J. Multiphase Flow 23, 967–976.Google Scholar
Goussis, D. A. & Kelly, R. E. 1991 Surface wave and thermocapillary instability in a liquid film flow. J. Fluid Mech. 223, 25–45.Google Scholar
Griffiths, I. M. & Howell, P. D. 2008 Mathematical modelling of non-axisymmetric capillary tube drawing. J. Fluid Mech. 605, 181–206.Google 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
Hu, J., Ben Hadid, H. & Daniel, H. 2008 Linear temporal and spatio-temporal stability analysis of a binary liquid film flowing down an inclined uniformly heated plate. J. Fluid Mech. 599, 269–298.Google 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., Miura, R. M. & Wylie, J. J. 2008 Optical fiber drawing and dopant transport. SIAM J. Appl. Maths 69, 330–347.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.Google Scholar
Kabova, Y. O., Kuznetsov, V. V. & Kabov, O. A. 2012 Temperature dependent viscosity and surface tension effects on deformations of non-isothermal falling liquid film. Intl J. Heat Mass Transfer 55, 1271–1278.Google Scholar
Kaye, A. 1991 Convected coordinates and elongational flow. J. Non-Newtonian Fluid Mech. 40, 55–77.Google Scholar
Kalpakjian, S. & Schmid, S. R. 2007 Manufacturing Processes for Engineering Materials, 5th edn. Prentice Hall.Google Scholar
Kalliadasis, S., Kiyashko, A. & Demekhin, E. A. 2003a Marangoni instability of a thin liquid film heated from below by a local heat source. J. Fluid Mech. 475, 377–408.Google Scholar
Kalliadasis, S., Demekhin, E. A. & Ruyer-Quil, C. 2003b Thermocapillary instability and wave formation on a film falling down a uniformly heated plane. J. Fluid Mech. 492, 303–338.CrossRefGoogle Scholar
Kostecki, R., Ebendorff-Heidepriem, H., Warren-Smith, S. C. & Monro, T. M. 2014 Predicting the drawing conditions for microstructured optical fiber fabrication. Opt. Mater. Express 4, 29–40.Google Scholar
Kuhlmann, H. C. & Rath, H. J. 1993 Hydrodynamic instability in cylindrical thermocapillary liquid bridges. J. Fluid Mech. 247, 247–274.Google Scholar
Mashayek, F. & Ashgriz, N. 1995 Nonlinear instability of liquid jets with thermocapillarity. J. Fluid Mech. 283, 97–123.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.Google Scholar
Miladinova, S., Slavtchev, S. & Lebon, G. 2002 Long-wave instabilities of non-uniformly heated falling films. J. Fluid Mech. 453, 153–175.Google Scholar
Pearson, J. R. A. & Shah, Y. T. 1973 Stability analysis of the fibre spinning process. Trans. Soc. Rheol. 16, 519–533.CrossRefGoogle Scholar
Roe, R. J. 1968 Surface tension of polymer liquids. J. Phys. Chem. 72, 2013–2017.Google Scholar
Samanta, A. 2008 Stability of liquid film falling down a vertical non-uniformly heated wall. Physica D 237, 2587–2598.Google Scholar
Scheid, B., Ruyer-Quil, C., Thiele, U., Kabov, O. A., Legros, J. C. & Colinet, P. 2005 Validity domain of the Benney equation including the Marangoni effect for closed and open flows. J. Fluid Mech. 527, 303–335.Google Scholar
Seward, T. P. III & Vascott, T. (Eds) 2005 High Temperture Glass Melt Property Database for Process Modeling. The American Ceramic Society, Wiley.Google Scholar
Shah, Y. T. & Pearson, J. R. A. 1972a On the stability of nonisothermal fibre spinning. Ind. Engng Chem. Fundam. 11, 145–149.Google Scholar
Shah, Y. T. & Pearson, J. R. A. 1972b On the stability of nonisothermal fibre spinning-general case. Ind. Engng Chem. Fundam. 11, 150–153.Google 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.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.Google Scholar
Stokes, Y. M., Bradshaw-Hajek, B. H. & Tuck, E. O. 2011 Extensional flow at low Reynolds number with surface tension. J. Engng Maths 70, 321–331.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, 176–203.Google Scholar
Suman, B. & Kumar, S. 2009 Draw ratio enhancement in nonisothermal melt spinning. AIChE J. 55, 581–593.Google Scholar
Tilley, B. S. & Bowen, M. 2005 Thermocapillary control of rupture in thin viscous fluid sheets. J. Fluid Mech. 541, 399–408.Google Scholar
Taroni, M., Breward, C. J. W., Cummings, L. J. & Griffiths, I. M. 2013 Asymptotic solutions of glass temperature profiles during steady optical fibre drawing. J. Engng Maths 80, 1–20.Google Scholar
Vasilyev, O. V., Ten, A. A. & Yuen, D. A. 2001 Temperature-dependent viscous gravity currents with shear heating. Phys. Fluids 13, 3664–3674.Google Scholar
Vlachopoulos, J. 2003 The Role of Rheology in Polymer Extrusion, New Technologies for Extrusion Conference. SPIE Press.Google Scholar
Vlachopoulos, J. & Polychronopoulos, N. 2012 Basic concepts in polymer melt rheology and their importance in processing. In Applied Polymer Rheology: Polymeric Fluids with Industrial Applications (ed. Kontopoulou, M.), pp. 1–27. Wiley.Google Scholar
Wang, J. S. & Porter, R. S. 1995 On the viscosity temperature behavior of polymer melts. Rheol. Acta 34, 496–503.Google Scholar
Wei, H. H. 2005 Thermocapillary instability of core-annular flows. Phys. Fluids 17, 102102.Google Scholar
Wu, S. H. 1969 Surface and interfacial tensions of polymer melts: I. Polyethylene, polyisobutylene, and polyvinyl acetate. J. Colloid Interface Sci. 31, 153–161.Google Scholar
Wu, S. H. 1970 Surface and interfacial tensions of polymer melts. II. Poly(methyl methacrylate), poly(n-butyl methacrylate), and polystyrene. J. Phys. Chem. 74, 632–638.Google Scholar
Wilson, S. D. R. 1988 The slow dripping of a viscous fluid. J. Fluid Mech. 190, 561–570.Google Scholar
Wylie, J. J. & Huang, H. 2007 Extensional flows with viscous heating. J. Fluid Mech. 571, 359–370.Google Scholar
Wylie, J. J., Huang, H. & Miura, R. M. 2007 Thermal instability in drawing viscous threads. J. Fluid Mech. 570, 1–16.Google Scholar
Wylie, J. J., Huang, H. & Miura, R. M. 2011 Stretching of viscous threads at low Reynolds numbers. J. Fluid Mech. 683, 212–234.Google Scholar
Wylie, J. J., Huang, H. & Miura, R. M. 2015 Asymptotic analysis of a viscous thread extending under gravity. Physica D 313, 51–60.Google Scholar
Yarin, A. L. 1986 Effect of heat removal on nonsteady regimes of fiber formation. J. Engng Phys. 50, 569–575.Google Scholar
Yarin, A. L., Gospodinov, P. & Roussinov, V. I. 1994 Stability loss and sensitivity in hollow fiber drawing. Phys. Fluids 6, 1454–1463.Google Scholar
Yin, Z. L. & Jaluria, Y. 1998 Thermal transport and flow in high-speed optical fiber drawing. Trans. ASME J. Heat Transfer 120, 916–930.Google Scholar
Yin, Z. L. & Jaluria, Y. 2000 Neck down and thermally induced defects in high-speed optical fiber drawing. Trans. ASME J. Heat Transfer 122, 351–362.Google Scholar