# Mathematics / Applied Mathematics MSc

## 5E: Fluid Mechanics MATHS5068

**Academic Session:**2024-25**School:**School of Mathematics and Statistics**Credits:**10**Level:**Level 5 (SCQF level 11)**Typically Offered:**Semester 1**Available to Visiting Students:**No**Collaborative Online International Learning:**No

### Short Description

This course introduces students to the mathematical theory of fluids. In particular, the course develops theory and explores several important applications, such as classical aerofoil theory and viscous boundary layers, employing methods from complex analysis, differential equations and asymptotic analysis.

### Timetable

17 x 1 hr lectures and 6 x 1 hr tutorials in a semester

### Excluded Courses

4H: Fluid Mechanics (MATHS4102)

### Assessment

Assessment

90% Examination, 10% Coursework.

Reassessment

Resit available to MSc students.

**Main Assessment In:** April/May

### Course Aims

The main aim of this course is to introduce students to the many important concepts in the mathematical study of fluids and to arm them with a range of techniques to study the wealth of interesting problems that arise. Fluid dynamics has application in a huge range of disciplines, including the study of stars and aircraft to blood flow and swimming micro-organisms, and has led to the development of a large part of applied mathematics. This course will present some preliminary ideas of fluid flow and go on to explore aspects of classical aerofoil theory, viscous fluid flow, including the analysis of boundary layers. Additionally, the students will be given reading material and must engage in private study on the stability of viscous parallel flow. The course will aim to apply and develop the students' knowledge of complex analysis, differential equations and asymptotic methods to facilitate investigation of solution behaviour.

### Intended Learning Outcomes of Course

By the end of this course students will be able to:

a) Define and calculate streamlines, particle paths and streaklines;

b) Derive momentum balance laws and discuss Cauchy's hypothesis on surface forces; prove Chauchy's Theorem (existence of stress);

c) Define ideal fluids, vorticity, irrotational flows, and circulation, and use these concepts appropriately;

d) Derive Euler's equations, Bernoulli's streamline theorem and the vorticity equation, and apply them to given problems;

e) Define elastic fluids; derive acustic equations and define Mach number;

f) Define the Reynolds number and employ the Navier-Stokes equations with appropriate boundary conditions to find solutions for simple problems, such as for an impulsively moved plane boundary, flow down an inclined plane, flow between two boundaries;

g) Establish theorems of uniqueness and stability for the Navier-Stokes equations;

h) Apply complex potential methods to study problems in classical aerofoil theory;

i) Employ the method of images, conformal mapping, the Milne-Thompson circle theorem, Blasius's theorem, and the Kutta-Joukowski lift theorem to find solutions to the problems in e);

j) Establish governing equations for surface waves in a fluid of finite depth, define wave and group velocity, and determine the dispersion relation and particle paths;

k) Derive equations for high Reynolds number flow and the boundary layer approximation, and find appropriate similarity solutions.

### Minimum Requirement for Award of Credits

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.