Stretching of heated threads with temperature-dependent viscosity: Asymptotic analysis

P. D. Howell; J. J. Wylie; Huaxiong Huang; Robert M. Miura

doi:10.3934/dcdsb.2007.7.553

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2007, Volume 7, Issue 3: 553-572. Doi: 10.3934/dcdsb.2007.7.553

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Stretching of heated threads with temperature-dependent viscosity: Asymptotic analysis

1.
Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, Oxford OX1 3LB
2.
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
3.
Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON M3J 1P3
4.
Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102

Received: July 31, 2006

Revised: December 31, 2006

Published: April 2007

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Abstract

We consider the stretching of a thin cylindrical thread with viscosity that depends on temperature. The thread is pulled with a prescribed force while receiving continuous heating from an external axially nonuniform heater. We use the canonical equations derived by Huang et al. (2007) and consider the limit of large dimensionless heating rate. We show that the asymptotic solution depends only on the local properties of the heating near its maximal heating value. We derive a uniformly valid asymptotic solution for the shape and the temperature profiles during the stretching process. We use a criterion to determine when breaking will occur and derive simple analytical expressions for the shape at breaking that clearly show the influence of heating strength and the degree of localization of the heating. The asymptotic shape profiles give good agreement with numerical simulations. These results are applied to the formation of glass microelectrodes.

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Mathematics Subject Classification: Primary: 76D99, 41A60; Secondary: 76D27.

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