Hostname: page-component-7c8c6479df-xxrs7 Total loading time: 0 Render date: 2024-03-29T11:28:11.868Z Has data issue: false hasContentIssue false

Stretching of viscous threads at low Reynolds numbers

Published online by Cambridge University Press:  19 August 2011

Jonathan 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
Huaxiong Huang
Affiliation:
Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA Department Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3
Robert M. Miura
Affiliation:
Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
*
Email address for correspondence: mawylie@cityu.edu.hk

Abstract

We investigate the classical problem of the extension of an axisymmetric viscous thread by a fixed applied force with small initial inertia and small initial surface tension forces. We show that inertia is fundamental in controlling the dynamics of the stretching process. Under a long-wavelength approximation, we derive leading-order asymptotic expressions for the solution of the full initial-boundary value problem for arbitrary initial shape. If inertia is completely neglected, the total extension of the thread tends to infinity as the time of pinching is approached. On the other hand, the solution exhibits pinching with finite extension for any non-zero Reynolds number. The solution also has the property that inertia eventually must become important, and pinching must occur at the pulled end. In particular, pinching cannot occur in the interior as can happen when inertia is neglected. Moreover, we derive an asymptotic expression for the extension.

Type
Papers
Copyright
Copyright © Cambridge University Press 2011

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

1. Bluman, G. & Kumei, S. 1980 On a remarkable nonlinear diffusion equation. J. Math. Phys. 21, 10191023.CrossRefGoogle Scholar
2. 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. Math. 67, 11661182.CrossRefGoogle Scholar
3. Eggers, J. 2005 Drop formation: an overview. Z. Angew. Math. Mech. 85, 400410.Google Scholar
4. 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
5. Forest, M. G., Zhou, H. & Wang, Q. 2000 Thermotropic liquid crystalline polymer fibers. SIAM J. Appl. Math. 60, 11771204.Google Scholar
6. Gallacchi, R., Kolsch, S., Kneppe, H. & Meixner, A. J. 2001 Well-shaped fibre tips by pulling with a foil heater. J. Microsc. 202, 182187.CrossRefGoogle ScholarPubMed
7. 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. Math. 63, 14991519.Google Scholar
8. Huang, H., Wylie, J. J., Miura, R. M. & Howell, P. D. 2007 On the formation of glass microelectrodes. SIAM J. Appl. Math. 67, 630666.CrossRefGoogle Scholar
9. Kaye, A. 1991 Convective coordinates and elongational flow. J. Non-Newtonian Fluid Mech. 40, 5577.CrossRefGoogle Scholar
10. Matta, J. E. & Tytus, R. P. 1990 Liquid stretching using a falling cylinder. J. Non-Newtonian Fluid Mech. 35, 215229.CrossRefGoogle Scholar
11. Stokes, Y. M. & Tuck, E. O. 2004 The role of inertia in extensional fall of a viscous drop. J. Fluid Mech. 498, 205225.CrossRefGoogle Scholar
12. Stokes, Y. M., Tuck, E. O. & Schwartz, L. W. 2000 Extensional fall of a very viscous fluid drop. Q. J. Mech. Appl. Math. 53, 565582.CrossRefGoogle Scholar
13. Wilson, S. D. R. 1988 The slow dripping of a viscous fluid. J. Fluid Mech. 190, 561570.CrossRefGoogle Scholar
14. Wylie, J. J. & Huang, H. 2007 Extensional flows with viscous heating. J. Fluid Mech. 571, 359370.CrossRefGoogle Scholar
15. Yin, Z. & Jaluria, Y. 2000 Neck down and thermally induced defects in high-speed optical fiber drawing. Trans. ASME: J. Heat Transfer 122, 351362.CrossRefGoogle Scholar