Undergraduate Programme and Module Handbook 2024-2025

# Module MATH3101: FLUID MECHANICS III

## Department: Mathematical Sciences

### MATH3101: FLUID MECHANICS III

Type | Open | Level | 3 | Credits | 20 | Availability | Available in 2024/2025 | Module Cap | Location | Durham |
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#### Prerequisites

- (Dynamics I (MATH1607) AND Analysis in Many Variables II (MATH2031) AND 20 additional credits of Level 2 mathematics modules) OR (Analysis in Many Variables II (MATH2031) AND Analysis I (MATH1051) (if taken in Year 2) AND Dynamics I (if taken in Year 2)) OR (Foundations of Physics I (PHYS1122) AND Analysis in Many Variables II (MATH2031) AND 20 additional credits of Level 2 mathematics modules.)

#### Corequisites

- One 20 credit Level 3 mathematics module.

#### Excluded Combination of Modules

#### Aims

- To introduce a mathematical description of fluid flow and other continuous media to familiarise students with the successful applications of mathematics in this area of modelling.
- To prepare students for future study of advanced topics.

#### Content

- Kinematics: Eulerian and Lagrangian pictures, velocity field, streamlines and stream functions.
- Compressibility, vorticity and circulation, integrals over material domains.
- Dynamics of ideal fluids: derivation of the incompressible Euler equations, energy, vorticity dynamics, circulation and rotating frames.
- Water waves: free-surface boundaries, linearisation, travelling and standing waves.
- Compressible flow: barotropic fluids, sound waves, nonlinearity.
- Hydrodynamic stability: normal mode analysis, Rayleigh-Taylor and Kelvin-Helmholtz instabilities.
- Dynamics of viscous fluids: Newtonian fluids, derivation of the Navier-Stokes equations, exact solutions, dynamical similarity, boundary layers.

#### Learning Outcomes

Subject-specific Knowledge:

- By the end of the module students will: be able to solve novel and/or complex problems in Fluid Mechanics.
- Have a systematic and coherent understanding of theoretical mathematics in the field of Fluid Mechanics.
- Have acquired coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas:
- Kinematics and dynamics of fluid flows, compressible flow, hydrodynamic stability and dynamics of viscous fluids.
- Equations of motion and their derivation for fluids.

Subject-specific Skills:

- In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Modelling.
- They will be able to formulate and use mathematical models in various situations.

Key Skills:

#### Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

- Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
- Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
- Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
- The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

#### Teaching Methods and Learning Hours

Activity | Number | Frequency | Duration | Total/Hours | |
---|---|---|---|---|---|

Lectures | 42 | 2 per week for 20 weeks and 2 in term 3 | 1 Hour | 42 | |

Problems Classes | 8 | Four in each of terms 1 and 2 | 1 Hour | 8 | |

Preparation and Reading | 150 | ||||

Total | 200 |

#### Summative Assessment

Component: Examination | Component Weighting: 100% | ||
---|---|---|---|

Element | Length / duration | Element Weighting | Resit Opportunity |

Written examination | 3 Hours | 100% |

#### Formative Assessment:

Eight assignments to be submitted.

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University