# BSc Mathematics with Finance

Year of entry: 2024

## Course unit details:Fluid Mechanics

Unit code MATH24412 10 Level 2 Semester 2 No

### Overview

The primary aim of this course unit is to provide students with a first introduction to continuum mechanics in general and theoretical fluid mechanics in particular. The material provides the student with an essential background to many third and fourth level courses on physical applied mathematics.

Fluid mechanics is concerned with understanding, and hence predicting, the properties (pressure, density, velocity etc.) of liquids and gases under external forces. This subject provides one of the major modern areas for the successful practical application of mathematics. Water, blood, air are all examples of fluids; of the many diverse fields where an understanding of the motion of fluids is important, one can mention oceanography and meteorology (in particular the dynamics of ocean circulation and weather forecasting), biological fluid dynamics (for example, blood flows through arteries), and aerodynamics.

The main physical focus at the end of the course is to calculate the forces on a body moving in a fluid e.g. aeroplane wing; the same study also relates to the behaviour of balls in football, cricket and golf, and of boomerangs and frisbees.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Partial Differential Equations and Vector Calculus A MATH20401 Pre-Requisite Compulsory
Partial Differential Equations and Vector Calculus B MATH20411 Pre-Requisite Compulsory
Partial Differential Equations & Vector Calculus MATH24420 Co-Requisite Compulsory

Students must have taken MATH11411, and MATH11422 or MATH11412

### Aims

This course aims to offer an introduction to the study of the motion of fluids (liquids and gases), in the important and widely-applicable case where the internal resistance of the fluid can be neglected.

The course starts by looking at how to visualise fluid flows, before building in such important concepts as conservation of mass, and deriving the equation of motion for fluid under the action of different types of forces. Integrating the equation of motion then leads to Bernoulli’s Equation which has varied applications.

After a short consideration of angular velocity in fluids, the course then considers flows which are two-dimensional, such as that past an aircraft wing section. A succession of interesting and powerful results follow, culminating in being able to calculate the lift on such a wing section.

### Learning outcomes

On completion of this unit successful students will be able to:

• Derive and apply identities involving Grad, Div and Curl, and alternative forms of the Divergence Theorem and Stokes' Theorem.
• Solve for the streamlines, particle paths and streaklines of a suitable given fluid flow.
• Define and use the Material Derivative.
• Identify the different types of forces acting on a fluid particle, derive the equation for Hydrostatic Equilibrium and apply this in simple situations.
• Derive and apply the Conservation of Mass equation and Euler's Equations of Motion in a fluid flow.
• Recognise the vorticity and the circulation in a fluid flow, and use the simplifications resulting from irrotational motion.
• Derive different forms of Bernoulli's Equation, under suitable sets of assumptions, and apply them in simple flow situations.
• In the case of irrotational, incompressible two-dimensional fluid flows, derive the streamfunction, the velocity potential and the complex potential, and use these to solve for the streamlines and the velocity components in suitably simple situations.
• State the Circle Theorem and Blasius' Theorem and apply these to find the forces on a body in a suitable simple flow.

### Syllabus

• Basic assumptions: Differences between fluids and solids. Differences between liquids and gases. Typical flow speeds and compressibility. Fluid particles and the continuum approximation.Lagrangian and Eulerian descriptions of a flow. Steady Flow.
• Vector calculus: Recap on Grad, Div and Curl, and the Divergence Theorem and Stokes’ Theorem.
• Visualising fluid flows: Streamlines, Stagnation points, Streaklines and Particle paths.
• Rates of change: The Material Derivative, and the acceleration of a fluid particle.
• Suffix Notation
• Modelling: Forces, Pressure and Hydrostatic Equilibrium. Conservation of Mass. Equations of Motion. Constitutive equations. Boundary Conditions.
• Energy and momentum: Bernoulli’s Equation for steady flow, and applications.
• Angular Velocity: Vorticity and Irrotational motion. Velocity potential. Laplace’s equation. Bernoulli’s Equation for irrotational flow.
• Two-dimensional motion: The stream-function and vorticity, in Cartesians and other co-ordinate systems. Equipotentials and streamlines. The complex potential and the complex velocity. (Some elementary complex analysis is discussed.) Some special 2-D flows. The Method of Images. Source in a uniform stream. Dipole in a uniform stream. The Circle Theorem and examples. Force on a cylinder. Blasius’ Theorem. The lift on a circular cylinder with circulation, and the lift on an aerofoil.

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Coursework; Weighting within unit 20%
• End of semester examination; Weighting within unit 80%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

All of these books, which are introductions to Fluid Mechanics, are to be found in Blue, Floor 2 532 or 532.5 or 532.6 in the John Rylands Library; there are many others in the same sections which may be worth browsing over (e.g. Schaum’s Outline Series).

1. Lighthill, M.J.

“An Informal Introduction to Theoretical Fluid Mechanics”

Library: 532/L16

2. Prandtl, L. & Tietjens, O.G.

“Fundamentals of Hydro- and Aeromechanics”

Library: 532/P46

3. Paterson, A.R.

“A first course in Fluid Dynamics”

Library: 532.5/P6

4. Batchelor, G.K.

“An Introduction to Fluid Dynamics”        (hard!)

Library: 532.5/B42

5. Currie, I.G.

“Fundamental Mechanics of Fluids”          (good but easy content)

Library: 532/C36

6. Milne-Thomson, L.M.

“Theoretical Aerodynamics” and “Theoretical Hydrodynamics”

Library: 532.6/M24 and 532.5/M49 respectively

7. Lamb, Sir H.

“Hydrodynamics”

Library: 532.5/L55

Publishers: 1: Oxford University Press

2 & 6: Dover

3, 4 & 7: Cambridge University Press

5: New York, Marcel Dekker

### Study hours

Scheduled activity hours
Lectures 22
Practical classes & workshops 6
Tutorials 6
Independent study hours
Independent study 66

### Teaching staff

Staff member Role
Mike Simon Unit coordinator