# MATHEMATICAL FLUID MECHANICS - 2024/5

Module code: MAT3041

## Module Overview

This module introduces the ideas of viscous fluid flows which build on the ideas some students would have seen in MAT2050: Inviscid Fluid Dynamics. By the end of the module, students should be familiar with the Navier-Stokes equations, and should be able to solve these equations in various simplified situations as well as in a variety of geometries.

### Module provider

Mathematics & Physics

### Module Leader

NOBILI Camilla (Maths & Phys)

### Number of Credits: 15

### ECTS Credits: 7.5

### Framework: FHEQ Level 6

### Module cap (Maximum number of students): N/A

## Overall student workload

Independent Learning Hours: 69

Lecture Hours: 33

Guided Learning: 15

Captured Content: 33

## Module Availability

Semester 1

## Prerequisites / Co-requisites

None

## Module content

Indicative content includes:

**Introduction**Definition of a fluid and examples of situations where fluids can be modelled.**Axisymmetric Inviscid Flows**Overview of inviscid flows, General solution of Laplace’s equation in spherical geometry, Flow around a sphere, Flow associated with a singing bubble.**The Navier-Stokes Equations**Stress/Strain relation, Derivation of Navier-Stokes equations, Boundary conditions, Dynamical Similarity.**Vorticity dynamics**Derivation of vorticity equation, Physical interpretation, Burger's vortex.**Exact solutions of the Navier-Stokes Equations**2D flow between plane parallel walls (steady/unsteady), Oscillating plate, Flow in rectangular channel, Pipe flow, Flow between rotating cylinders, The stirring problem, Unsteady line vortex.**Mathematical Boundary Layers**Asymptotic theory for algebraic equations, Matched asymptotic expansions.**Fluid Boundary layer Theory**Derivation of boundary layer equations, Blasius boundary layer, Falkner-Skan solutions.**Application of Boundary Layer Theory**Jets, Wakes.**Very Viscous Flow**Lubrication theory, Viscous flow past a sphere.

## Assessment pattern

Assessment type | Unit of assessment | Weighting |
---|---|---|

School-timetabled exam/test | In-semester test (50 min) | 20 |

Examination | End-of-Semester Examination (2 hours) | 80 |

## Alternative Assessment

N/A

## Assessment Strategy

The __assessment strategy__ is designed to provide students with the opportunity to demonstrate:

- Understanding of the methods required to solve complex fluid flow problems.
- Subject knowledge through the recall of definitions as well as explaining why certain simplifications to the velocity field can be made and under what conditions certain approximations breakdown, also through a physical understanding of fluid problems.
- Analytic ability through the solution of unseen and seen similar problems in the test and exam.

Thus, the

__summative assessment__for this module consists of:

- One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcomes 1 to 5.
- A synoptic examination (2 hours), worth 80% of the module mark, corresponding to Learning Outcomes 1 to 7.

__Formative assessment__

There are two formative unassessed courseworks over an 11 week period, designed to consolidate student learning.

__Feedback__

Students receive individual written feedback on the formative unassessed coursework and the in-semester test. The feedback is timed so that feedback from the first unassessed coursework assists students with preparation for the in-semester test. The feedback from both unassessed courseworks and the in-semester test assists students with preparation for the end-of-semester examination. This written feedback is complemented by verbal and written feedback given in tutorials. Students also receive verbal and written feedback in office hours.

## Module aims

- Introduce students to viscous fluids in various simple geometries, and with various boundary conditions.
- Enable students to solve the Navier-Stokes equations in simple situations.
- Illustrate how fluid mechanics is connected to various problems in the real world, such as in engineering, and how the techniques learnt in this course can be applied to these problems.

## Learning outcomes

Attributes Developed | ||

001 | Students will be able to solve inviscid fluid problems in spherical polar geometry, such as finding the frequency of an oscillating bubble or calculating the Stokes stream function given an axisymmetric velocity potential. | KC |

002 | Students will demonstrate understanding of the role of viscosity in a fluid, and to be able to calculate the viscous stress on a solid surface given the stress tensor. | KC |

003 | Students will be able to generate exact solutions to the Navier-Stokes equations in both Cartesian and cylindrical polar coordinates. | KC |

004 | Students will be able to non-dimensionalize the Navier-Stokes equations with and without the effect of gravity and define the Reynolds number and Froude number. Students will demonstrate an understanding of the concept of dynamical similarity. | KCT |

005 | Students will be able to calculate solutions to algebraic equations and simple ODEs which contain a small parameter or a boundary layer, and demonstrate an understanding of when these approximations breakdown. | KC |

006 | Students will be able to apply scale analysis to rescale the Navier-Stokes equations and obtain the boundary layer equations. Further, they will be able to use these equations to form solutions to problems involving jets and wakes. | KC |

007 | Students will be able to apply scale analysis to derive the thin film equations from the Navier-Stokes equations and generate solutions of these equations. | KC |

Attributes Developed

**C** - Cognitive/analytical

**K** - Subject knowledge

**T** - Transferable skills

**P** - Professional/Practical skills

## Methods of Teaching / Learning

The __learning and teaching__ strategy is designed to provide:

- A thorough account of exact solutions to the Navier-Stokes equations in a variety of geometries and under various simplifications.
- Experience (through demonstration) of the methods and techniques used to solve problems in fluid mechanics.

The

__learning and teaching__methods include:

- Three one-hour lectures per week for eleven weeks. Students will receive typeset notes to complement the lectures: these contain incomplete sections which will be filled in during lectures. The lectures provide a structured learning environment with opportunities for students to ask questions and to practice methods taught.
- Students will reinforce lectures by tackling a wide range of problems.
- There are two unassessed courseworks to provide students with further opportunity to consolidate learning. Students receive individual written feedback on these as guidance on their progress and understanding.
- Lectures may be recorded. Lecture recordings are intended to give students the opportunity to review parts of the session that they might not have understood fully and should not be seen as an alternative to attendance at lectures.

Students should also use the books listed as background reading on the subject.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

## Reading list

https://readinglists.surrey.ac.uk

Upon accessing the reading list, please search for the module using the module code: **MAT3041**

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

**Digital Capabilities**: The SurreyLearn page for MAT3041 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

**Employability:*** *Students taking MAT3041 develop their problem-solving capabilities and quantitative reasoning skills. These skills are valued by employers. In particular, proficiency in analysing fluid dynamics is sought after in engineering, aerospace and environmental sectors.

**Global and Cultural Capabilities: **Students enrolled in MAT3041 originate from various countries and possess a wide range of cultural backgrounds. During problem solving sessions in lectures, student engagement in discussions naturally cultivates the sharing of different cultures.

**Resourcefulness and Resilience**: Solving complex problems involving fluid dynamics and adapting to dynamic situations cultivates resilience in the students. Student resourcefulness in tackling challenges is enhanced, as the module develops critical thinking skills that are widely applicable.

**Sustainability: **The tools encountered in MAT3041 can be used to analyse fluid dynamics in environmental contexts. Understanding fluid behaviour is pivotal in addressing water resource management and pollution control, aligning with sustainable practices and advancements toward a more environmentally conscious world.

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.