Module Identifier MAM0220  
Module Title MATHEMATICAL PROBLEMS IN NONLINEAR VISCOELASTICITY  
Academic Year 2007/2008  
Co-ordinator Dr Robert J Douglas  
Semester Semester 1  
Other staff Professor Russell Davies, Dr Robert J Douglas  
Course delivery Lecture   16 Hours. (16 x 1 hour lectures)  
  Seminars / Tutorials   4 Hours. (4 x 1 hour example classes)  
Assessment
Assessment TypeAssessment Length/DetailsProportion
Semester Assessment coursework  100%
Supplementary Assessment coursework  100%

Learning outcomes

On completion of this module, students should be able to:
1. explain the concept of a weak derivative.
2. calculate the weak formulation of a partial differential equation.
3. state and apply the Lax-Milgram Theorem.
4. classify the set of partial differential equations arising from a range of differential constitutive equations.
5. explain what is meant by a bifurcation.
6. explore and describe the stability of a number of viscoelastic flows.
7. explain possible causes of material instability and melt fracture.
8. examine and describe the formation of elastic boundary layers.
9. analyse the nature of the singular behaviour of solutions in regions of the flow near reentrant corners.

Brief description

This module develops techniques for the analysis of mathematical problems in nonlinear viscoelasticity by first introducing concepts from the modern theory of partial differential equations, and then specialising to problems for complex fluids. The systems of governing partial differential equations will be analysed to provide information on existence and uniqueness of solutions, the classification of the system and possible change of type. In addition, topics selected from the formation of elastic boundary layers, the flow near a reentrant corner, the stability of viscoelastic flows, material instability and melt fracture, will be examined in some detail.

Aims

This module will introduce techniques from the modern theory of partial differential equations and apply them to a number of mathematical problems in nonlinear viscoelasticity including existence and uniqueness of solutions to the governing sets of partial differential equations, change of type, flow stabilities, corner singularities and stress boundary layers.

Reading Lists

Books
** Recommended Text
D.D.Joseph (1990) Fluid dynamics of viscoelastic liquids Springer-Verlag 0387971556
M Renardy (2000) Mathematical analysis of viscoelastic flows SIAM 0898714575
R. G. Owens and T. N. Phillips (2002) Computational Rheology Imperial College Press 1860941869
** Supplementary Text
Evans, Lawrence C. (2004) Partial Differential Equations 0821807722

Notes

This module is at CQFW Level 7