## Module and Programme Catalogue

#### Find information on

This module is not currently running in the selected year. The information shown below is for the academic year that the module was last running in, prior to the year selected.

# 2014/15 Taught Postgraduate Module Catalogue

### 20 creditsClass Size: 30

Module manager: Dr M Evans
Email: r.m.l.evans@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2014/15

### Pre-requisite qualifications

MATH2365 and MATH2375 and (MATH2620 or MATH3501), or (MATH2420 and MATH2431 and MATH3501), or equivalent.

### This module is mutually exclusive with

 MATH3451 Introduction to Polymeric Fluids

Module replaces

MATH5450M

This module is approved as an Elective

### Module summary

Many of the fluids we come across every day differ markedly in their properties from the 'classical' Newtonian fluid dealt with in most introductory courses. For example, consider the 'stringiness' of the cheese on a pizza, or the 'elasticity' of bread dough - both are materials that flow, but are certainly not Newtonian! This course gives an introduction to the rapidly growing subject area of 'non-Newtonian' fluids. Its approach is two-fold. Firstly it gives an introduction to the 'phenomenology' of the subject - what kind of things do these fluids do when they flow? Then, the course focuses on a particular class of fluids, those containing polymer molecules. It shows how a microscopic description of the motion of the molecules can give an understanding of why these fluids flow like they do, and leads to equations that predict much of the flow behaviour. In order to get to this level of description, we have to examine the mathematics of polymer molecules and how they move.

### Objectives

On completion of this module, students should be able to:
a) identify and categorise characteristic phenomena of Non-Newtonian Fluids;
b) calculate unidirectional flows using simple Non-Newtonian fluid models;
c) apply mathematical descriptions of diffusion;
d) explain the molecular origins of stress and relaxation in polymeric fluids.

### Syllabus

(1) What is a polymer, and what sort of fluids contain them?

(2) Introduction to flow phenomena for non-Newtonian fluids in simple flows: (i) definition of stress and strain rate; (ii) steady state shear, shear thickening and thinning, 'power-law' fluids, pipe flow; (iii) shear flow and normal stresses, the 'rod climbing' experiment; (iv) steady state extensional flow, extension hardening and apparent viscosity, the tubeless siphon; (v) 'memory' effects and linear viscoelasticity, time-dependent modulus, storage and loss modulus, recoil after steady shear, Deborah and Weissenberg numbers.

(3) What is a constitutive equation, and where do they come from?

(4) The origin of stress in polymers - rubber elasticity; statistical description of polymers - random walks; force due to thermal motion; derivation of stress; elastic constitutive equation for rubber.

(5) Mathematical description of diffusion and Brownian motion (i) analogy with random walks; (ii) the diffusion equation; (iii) Wiener processes and stochastic calculus; (iv) the Langevin equation; (v) diffusion in an external potential.

(6) Microscopic derivation and properties of constitutive equations for flow of polymeric fluids (i) 'Oldroyd B' dumbbell model for dilute solutions, behaviour in extension and shear; (ii) the Rouse model.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 44 1.00 44.00 Private study hours 156.00 Total Contact hours 44.00 Total hours (100hr per 10 credits) 200.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Problem sheets throughout the semester.

### Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 00 mins 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated