# Mathematics

## ECM3707 - Fluid Dynamics (2015)

MODULE TITLE CREDIT VALUE Fluid Dynamics 15 ECM3707 Prof Andrew Gilbert (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 weeks 0 0
 Number of Students Taking Module (anticipated) 80
DESCRIPTION - summary of the module content

The aim of this module is to provide you with a further understanding of the basic concepts of fluid dynamics associated with flow of incompressible (constant density) fluids with both viscosity and inertia. You will learn to translate a physical problem into an appropriate mathematical system. Furthermore, you will learn about the many important applications of fluid dynamics in different branches of science and why solutions of fluid dynamics for many real physical problems cannot be obtained.

This module deals with the flow of incompressible fluids with both viscosity and inertia. The governing equations - the Navier-Stokes equations - admit an incredible variety of solutions, some of which will be presented. Topics covered will include some exact solutions of the NS equation in a variety of coordinate systems, together with similarity solutions and an introduction to boundary layer theory.

Prerequisite module: ECM2706 or equivalent

AIMS - intentions of the module

This module is mainly concerned with the flow of viscous fluids, and it aims to provide you with a further understanding of the basic concepts of fluid dynamics associated with real fluids; to show you that there are many important applications of fluid dynamics in different branches of science and, at the same time, to show why solutions of fluid dynamics for many real physical problems cannot be obtained.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 explain the basic concepts and equations of viscous fluid flow;
2 prove some key theorems and appreciate solutions of the Navier-Stokes equations in simple geometries.
Discipline Specific Skills and Knowledge:
3 translate a physical problem into an appropriate mathematical system;
4 interpret solutions of these equations in physical terms.
Personal and Key Transferable/ Employment Skills and  Knowledge:
5 demonstrate enhanced ability to formulate and analyse real physical problems using a variety of tools of applied mathematics.

SYLLABUS PLAN - summary of the structure and academic content of the module

- fundamentals and basic examples: introduction to module;

- introduction to Navier-Stokes equation, continuity equation, density, mass flux;

- plane Poiseuille and plane Couette flow;

- cylindrical polars, Poiseuille and Couette flow;

- vector calculus revision;

- derivation I: Navier-Stokes equation, acceleration, continuity equation;

- derivation II: pressure, strain, viscous stress;

- similarity solutions and boundary layers: Rayleigh problem;

- Stokes layer;

- boundary layer controlled by suction;

- boundary layer equation;

- Blasius boundary layer;

- other boundary layers, separation.

- Stokes flow:  Stokes equation;

- flow round cylinder and sphere;

- corner eddies;

- introduction to swimming;

- Taylor's swimming sheet;

- vorticity and vortex dynamics: vorticity equation in 3-D and in 2-D;

- Helmholtz laws, Kelvin circulation theorem;

- vortex stretching and alignment;

- Burgers vortex and 2-D analogue;

- axisymmetric flow with swirl, Hills spherical vortex and similar solutions;

- introduction to turbulence.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 30 Lectures Scheduled learning and teaching activities 3 Examples classes Guided independent study 20 Coursework Guided independent study 97 Reading, revision, preparation

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework - examples and exercises 10 hours/ 4 to 6 questions per problem sheet (2 sheets) 1-5 General comments uploaded to ELE, solutions uploaded to ELE, individual feedback on request.

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours 1-5 Oral on request
Coursework – examples and exercises 20 10 hours/4-6 questions, per problem sheet (2 sheets) 1-5 Comments on each script, general comments uploaded to ELE, solutions uploaded to ELE

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

Referred and deferred assessment will be by examination.  For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed.  For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE: http://vle.exeter.ac.uk