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Thermal instability in drawing viscous threads

Published online by Cambridge University Press:  14 October 2021

Jonathan J. Wylie
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
Huaxiong Huang
Department Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3
Robert M. Miura
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA


We consider the stretching of a thin viscous thread, whose viscosity depends on temperature, that is heated by a radiative heat source. The thread is fed into an apparatus at a fixed speed and stretched by imposing a higher pulling speed at a fixed downstream location. We show that thermal effects lead to the surprising result that steady states exist for which the force required to stretch the thread can decrease when the pulling speed is increased. By considering the nature of the solutions, we show that a simple physical mechanism underlies this counterintuitive behaviour. We study the stability of steady-state solutions and show that a complicated sequence of bifurcations can arise. In particular, both oscillatory and non-oscillatory instabilities can occur in small isolated windows of the imposed pulling speed.

Research Article
Copyright © Cambridge University Press 2007

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Blyler, L. L. & Gieniewski, C. 1980 Melt spinning and draw resonance studies on a poly(α-methyl styrene/silicone) block compound. Polym. Engng Sci. 20, 140148.CrossRefGoogle Scholar
Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. & Knuth, D. E. 1996 On the Lambert W function. Adv. Comp. Maths 5, 329359.Google Scholar
Cummings, L. J. & Howell, P. D. 1999 On the evolution of non-axisymmetric viscous fibers with surface tension, inertia and gravity. J. Fluid Mech. 389, 361389.CrossRefGoogle Scholar
Denn, M. M. 1980 Continuous drawing of liquids to form fibers. Annu. Rev. Fluid Mech. 12, 365387.CrossRefGoogle 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, 323338.CrossRefGoogle Scholar
Fisher, R. J. & Denn, M. M. 1977 Mechanics of nonisothermal polymer melt spinning AIChE. J. 23, 2328.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, 19241931.CrossRefGoogle Scholar
Forest, M. G. & Zhou, H. 2001 Unsteady analysis of thermal glass fiber drawing processes. Eur. J. Appl. Maths 12, 479496.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.Google Scholar
Gupta, G. & Schultz, W. W. 1998 Non-isothermal flows of Newtonian slender glass fibers. Intl J. Nonlinear Mech. 33, 151163.CrossRefGoogle Scholar
Han, C. D. & Apte, S. M. 1979 Studies on melt spinning. VIII. The effects of molecular-structure and cooling conditions on the severity of draw resonance. J. Appl. Polym. Sci. 24, 6187.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, 14991519.Google Scholar
Iooss, G. & Joseph, D. D. 1980 Elementary Stability and Bifurcation Theory. Springer.CrossRefGoogle Scholar
Matsumoto, T. & Bogue, D. C. 1978 Draw resonance involving rheological transitions. Polym. Engng Sci. 18, 564571.CrossRefGoogle Scholar
Pearson, J. R. A. & Shah, Y. T. 1973 Stability analysis of the fibre spinning process. Trans. Soc. Rheol. 16, 519533.CrossRefGoogle Scholar
Pyrex Glass Code 1987 7740, Material Properties Pyrex Glass Code Brochure Pyrex B-87.Google Scholar
Shah, Y. T. & Pearson, J. R. A. 1972a On the stability of nonisothermal fibre spinning. Ind. Engng Chem. Fundam. 11, 145149.CrossRefGoogle Scholar
Shah, Y. T. & Pearson, J. R. A. 1972b On the stability of nonisothermal fibre spinning–general case. Ind. Engng Chem. Fundam. 11, 150153.CrossRefGoogle Scholar
Vassilatos, G., Knox, B. & Frankfort, H. 1985 Dynamics, structure development, and fiber properties in high-speed spinning of polyethylene terephthalate. In High Speed Fiber Spinning (ed. Ziabicki, A. & Kawai, H.). J. Wiley and Sons.Google Scholar
Whitehead, J. A. & Helfrich, K. R. 1991 Instability of flow with temperature-dependent viscosity: A model of magma dynamics. J. Geophys. Res. 96 (B3) 41454155.CrossRefGoogle Scholar
Wylie, J. J. & Lister, J. R. 1995 The effects of temperature-dependent viscosity on flow in a cooled channel with applications to basaltic fissure eruptions. J. Fluid Mech. 305, 239261.CrossRefGoogle Scholar
Yarin, A. L. 1986 Effect of heat removal on nonsteady regimes of fiber formation. J. Engng Phys. 50, 569575.CrossRefGoogle Scholar