American Institute of Mathematical Sciences

Advanced
May  2007, 7(3): 553-572. doi: 10.3934/dcdsb.2007.7.553

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

P. D. Howell 1, , J. J. Wylie 2, , Huaxiong Huang 3, and  Robert M. Miura 4,

1. 

Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom
2. 

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China
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, United States

Received  August 2006 Revised  January 2007 Published  February 2007

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.
Keywords: Asymptotic analysis, incompressible fluids, temperature dependent viscosity., glass microelectrodes.
Mathematics Subject Classification: Primary: 76D99, 41A60; Secondary: 76D2.
Citation: P. D. Howell, J. J. Wylie, Huaxiong Huang, Robert M. Miura. Stretching of heated threads with temperature-dependent viscosity: Asymptotic analysis. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 553-572. doi: 10.3934/dcdsb.2007.7.553
[1]

M. Bulíček, P. Kaplický. Incompressible fluids with shear rate and pressure dependent viscosity: Regularity of steady planar flows. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 41-50. doi: 10.3934/dcdss.2008.1.41
[2]

Nicolas Crouseilles, Mohammed Lemou, SV Raghurama Rao, Ankit Ruhi, Muddu Sekhar. Asymptotic preserving scheme for a kinetic model describing incompressible fluids. Kinetic and Related Models, 2016, 9 (1) : 51-74. doi: 10.3934/krm.2016.9.51
[3]

Javier A. Almonacid, Gabriel N. Gatica, Ricardo Oyarzúa, Ricardo Ruiz-Baier. A new mixed finite element method for the n-dimensional Boussinesq problem with temperature-dependent viscosity. Networks and Heterogeneous Media, 2020, 15 (2) : 215-245. doi: 10.3934/nhm.2020010
[4]

Nurul Hafizah Zainal Abidin, Nor Fadzillah Mohd Mokhtar, Zanariah Abdul Majid. Onset of Benard-Marangoni instabilities in a double diffusive binary fluid layer with temperature-dependent viscosity. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 413-421. doi: 10.3934/naco.2019040
[5]

Jishan Fan, Fucai Li, Gen Nakamura. Regularity criteria for the Boussinesq system with temperature-dependent viscosity and thermal diffusivity in a bounded domain. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4915-4923. doi: 10.3934/dcds.2016012
[6]

Konstantina Trivisa. Global existence and asymptotic analysis of solutions to a model for the dynamic combustion of compressible fluids. Conference Publications, 2003, 2003 (Special) : 852-863. doi: 10.3934/proc.2003.2003.852
[7]

Chaoying Li, Xiaojing Xu, Zhuan Ye. On long-time asymptotic behavior for solutions to 2D temperature-dependent tropical climate model. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1535-1568. doi: 10.3934/dcds.2021163
[8]

Grigory Panasenko, Ruxandra Stavre. Asymptotic analysis of the Stokes flow with variable viscosity in a thin elastic channel. Networks and Heterogeneous Media, 2010, 5 (4) : 783-812. doi: 10.3934/nhm.2010.5.783
[9]

Pierluigi Colli, Gianni Gilardi, Paolo Podio-Guidugli, Jürgen Sprekels. An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 353-368. doi: 10.3934/dcdss.2013.6.353
[10]

Daoyuan Fang, Ting Zhang, Ruizhao Zi. Dispersive effects of the incompressible viscoelastic fluids. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5261-5295. doi: 10.3934/dcds.2018233
[11]

Jianwei Yang, Peng Cheng, Yudong Wang. Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity. Electronic Research Announcements, 2015, 22: 20-31. doi: 10.3934/era.2015.22.20
[12]

Takeshi Fukao, Nobuyuki Kenmochi. A thermohydraulics model with temperature dependent constraint on velocity fields. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 17-34. doi: 10.3934/dcdss.2014.7.17
[13]

Miroslav Bulíček, Eduard Feireisl, Josef Málek, Roman Shvydkoy. On the motion of incompressible inhomogeneous Euler-Korteweg fluids. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 497-515. doi: 10.3934/dcdss.2010.3.497
[14]

József Z. Farkas, Gary T. Smith, Glenn F. Webb. A dynamic model of CT scans for quantifying doubling time of ground glass opacities using histogram analysis. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1203-1224. doi: 10.3934/mbe.2018055
[15]

M. Bulíček, Josef Málek, Dalibor Pražák. On the dimension of the attractor for a class of fluids with pressure dependent viscosities. Communications on Pure and Applied Analysis, 2005, 4 (4) : 805-822. doi: 10.3934/cpaa.2005.4.805
[16]

Xiao-Dong Yang, Roderick V. N. Melnik. Accounting for the effect of internal viscosity in dumbbell models for polymeric fluids and relaxation of DNA. Conference Publications, 2007, 2007 (Special) : 1052-1060. doi: 10.3934/proc.2007.2007.1052
[17]

Shuji Yoshikawa, Irena Pawłow, Wojciech M. Zajączkowski. A quasilinear thermoviscoelastic system for shape memory alloys with temperature dependent specific heat. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1093-1115. doi: 10.3934/cpaa.2009.8.1093
[18]

Colette Guillopé, Zaynab Salloum, Raafat Talhouk. Regular flows of weakly compressible viscoelastic fluids and the incompressible limit. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1001-1028. doi: 10.3934/dcdsb.2010.14.1001
[19]

Pitágoras Pinheiro de Carvalho, Juan Límaco, Denilson Menezes, Yuri Thamsten. Local null controllability of a class of non-Newtonian incompressible viscous fluids. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021043
[20]

Tomáš Roubíček. From quasi-incompressible to semi-compressible fluids. Discrete and Continuous Dynamical Systems - S, 2021, 14 (11) : 4069-4092. doi: 10.3934/dcdss.2020414

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy