Course Information

Lecturer:  

Associate Professor Yvonne Stokes, University of Adelaide

Synopsis:

Asymptotic methods are used to investigate problems featuring a small (or large) parameter. By using appropriate scalings and writing dependent variables as power series of a small parameter, complicated PDE models may be simplified to ODE and/or simper PDE models which provide considerable insight and understanding of problems. Accuracy may be improved by solving for higher-order terms.

In this course we focus on extensional flows having a small geometric parameter or aspect ratio. These are ubiquitous in nature and industry and include honey dripping from a spoon, ink-jet printing, the float glass process used for making sheet glass, and a spider spinning a web. We consider the work of Trouton (1906) on the so-called ‘Trouton viscosity’, modelling of the spinning of polymer threads by Pearson and coworkers in the late 60s and early 70s, through to modern research particularly relating to the fabrication of optical fibres. We see the power of Lagrangian coordinate and other transformations, and consider such issues as solution stability and finite-time blowup.

Course Overview:

  • Trouton models and the Trouton viscosity,
  • Scaling
  • Asymptotic methods,
  • The Reynolds transport theorem,
  • The equations of Newtonian fluid flow,
  • 1D model derivation, neglecting inertia and surface tension,
  • Lagrangian coordinate systems,
  • Finite-time ‘blowup’,
  • Drop ‘pinch-off’,
  • Draw stability,
  • Inclusion of surface tension and the cross-plane problem,
  • The ‘reduced-time’ transformation,
  • Extension of a solid rod,
  • Extension of an axisymmetric tube.

Contact Hours:

28 hours.

Prerequisites:

The required background for this course is some understanding of PDEs, ODEs and vector calculus. An undergraduate course on fluid dynamics and/or other mathematical modelling courses will be an advantage but is not essential. Ability to write a basic Matlab (or equivalent) program is also assumed.

 Assessment:

  • Mid-School assignment: 40%
  • Final examination: 60%

 Resources:

  • Irgens (2008) Continuum Mechanics, Springer Verlag Berlin Heidelberg.
  • R. Munson, D.F. Young, T.H. Okiishi (2002) Fundamentals of Fluid Mechanics, 4th ed. John Wiley & Sons Inc.

Lecturer Biography

Mathematics of extensional flows

Associate Professor Yvonne Stokes, University of Adelaide

Yvonne is an Australian Research Council Future Fellow, an Associate Professor at the University of Adelaide and the current Chair of the Women in Mathematics Special Interest Group (WIMSIG) of AustMS. Yvonne enjoys the application of mathematics to solving problems and gaining basic knowledge and understanding of the natural and physical world. Her research interests lie in the broad field of continuum mechanics, using ordinary and partial differential equations to model problems, and analytic and numerical methods to solve the models. Her research focus is on viscous fluid mechanics and mathematical biology.

More information is available at http://www.maths.adelaide.edu.au/yvonne.stokes/

This course is proudly sponsored by

Mathematics of extensional flows
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