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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*
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

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.

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Papers
Copyright
© 2016 Cambridge University Press 

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