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# Mathematical methods and fluid mechanics

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Half of this module is about modelling simple fluid flows; the other half is about mathematical methods. You'll learn how to solve ordinary and partial differential equations such as: Laplace’s equation, the wave equation and the diffusion equation; some vector field theory; and Fourier analysis. The fluid mechanical aspects of the module will give you a good understanding of modelling in the context of fluids. To study this module you should have a sound knowledge of ordinary differential equations, vector calculus, multiple integrals, basic particle mechanics and some knowledge of partial differential equations and Fourier series as provided by the appropriate OU level 2 study.

## What you will study

In simple terms, we think of a fluid as a substance that flows. Familiar examples are air (a gas) and water (a liquid). All fluids are liquids or gases. The analysis of the forces in ­and motion of ­liquids and gases is called fluid mechanics. This module introduces the fundamentals of fluid mechanics and discusses the solutions of fluid-flow problems that are modelled by differential equations. The mathematical methods arise from (and are interpreted in) the context of fluid-flow problems, although they can also be applied in other areas such as electromagnetism and the mechanics of solids.

Because of its many applications, fluid mechanics is important for applied mathematicians, scientists and engineers. The flow of air over objects is of fundamental importance to the aerodynamicist in the design of aeroplanes and to the motor industry in the design of cars with drag-reducing profiles. The flow of fluids through pipes and channels is also important to engineers. Fluid mechanics is essential to the meteorologist in studying the complicated flow patterns in the atmosphere.

The module is arranged in 13 units within four blocks.

#### Block 1

This is the foundation on which the rest of the module is built.

Unit 1 Properties of a fluid introduces the continuum model and many of the properties of a fluid, such as density, pressure and viscosity. The basic equation of fluid statics is formulated and used to find the pressure distribution in a liquid and to provide a model for the atmosphere.

Unit 2 Ordinary differential equations starts by showing how changes of variables (involving use of the Chain Rule) can be applied to solve certain non-constant-coefficient differential equations, and leads on to the topics of boundary-value and eigenvalue problems. It concludes with an introduction to the method of power-series for solving initial-value problems.

Unit 3 First-order partial differential equations extends the earlier version of the Chain Rule to cover a change of variables for functions of two variables, and shows how this leads to the method of characteristics for solving first-order partial differential equations.

Unit 4 Vector field theory relates line, surface and volume integrals through two important theorems – Gauss’ theorem and Stokes’ theorem – and formulates the equation of mass continuity for a fluid in motion.

#### Block 2

The second block starts by investigating the motion of a fluid that is assumed to be incompressible (its volume cannot be reduced) and inviscid (there is no internal friction).

Unit 5 Kinematics of fluids introduces the equations of streamlines and pathlines, develops the concept of a stream function as a method of describing fluid flows, and formulates Euler’s equation of motion for an inviscid fluid.

Unit 6 Bernoulli’s equation analyses an important equation arising from integrals of Euler’s equation for the flow of an inviscid fluid. It relates pressure, speed and potential energy, and is presented in various forms. Bernoulli’s equation is used to investigate phenomena such as flows through pipes and apertures, through channels and over weirs.

Unit 7 Vorticity discusses two important mathematical tools for modelling fluid flow, the vorticity vector (describing local angular velocity) and circulation. The effects of viscosity on the flow of a real (viscous) fluid past an obstacle are described.

Unit 8 The flow of a viscous fluid establishes the Navier–Stokes equations of motion for a viscous fluid, and investigates some of their exact solutions and some of the simplifications that can be made by applying dimensional arguments.

#### Block 3

This block looks at a class of differential equations typified by the wave equation, the diffusion equation and Laplace’s equation, which arise frequently in fluid mechanics and in other branches of applied mathematics.

Unit 9 Second-order partial differential equations shows how a second-order partial differential equation can be classified as one of three standard types, and how to reduce an equation to its standard form. Some general solutions (including d’Alembert’s solution to the wave equation) are found.

Unit 10 Fourier series reviews and develops an important method of approximating a function. The early sections refer to trigonometric Fourier series, and it is shown how these series, together with separation of variables, can be used to represent the solutions of initial-boundary value problems involving the diffusion equation and the wave equation. Later sections generalise to the Fourier series that arise from Sturm–Liouville problems (eigenvalue problems with the differential equation put into a certain standard format), including Legendre series.

Unit 11 Laplace’s equation is a particular second-order partial differential equation that can be used to model the flow of an irrotational, inviscid fluid past a rigid boundary. Solutions to Laplace’s equation are found and interpreted in the context of fluid flow problems, for example, the flow of a fluid past a cylinder and past a sphere.

#### Block 4

In this block you'll return to applications of the mathematics to fluid flows.

Unit 12 Water waves uses some of the theory developed in Block 3 to investigate various types of water wave, and discusses several practical examples of these waves.

Unit 13 Boundary layers and turbulence looks at the effects of turbulence (chaotic fluid flow) and at the nature of boundary layers within a flow, introducing models to describe these phenomena.

If you are considering progressing to The engineering project (T452), this is one of the OU level 3 modules on which you could base your project topic. Normally, you should have completed one of these OU level 3 modules (or be currently studying one) before registering for the project module.

Read the full content list here.

### You will learn

Successful study of this module should enhance your skills in communicating mathematical ideas clearly and succinctly, expressing problems in mathematical language and interpreting mathematical results in real-world terms.

### Professional recognition

This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.

## Entry requirements

You need a sound knowledge of first- and second-order ordinary differential equations; vectors and elementary vector calculus; partial differentiation; multiple integrals; dimensions; and basic particle (Newtonian) mechanics – check if you’re ready for MST326, with our self-assessment quiz.

You’d normally be prepared by completing OU level 2 study as part of one of our engineering or mathematics qualifications. For this module, we recommend that you’ve passed Mathematical methods, models and modelling (MST210).

If you’re not sure you’re ready, talk to an adviser.

## Preparatory work

To help you to revise the necessary mathematical methods and to check that you are ready to study MST326, you can download a Revision Booklet with worked examples and exercises.

Revising the units that focus on partial differential equations, vector calculus, multiple integrals and Fourier series in our level 2 module Mathematical methods, models and modelling (MST210) would also be good preparation.

If you have studied Mathematical methods (MST224), we recommend that you revise Book 1 and the first two chapters of Book 4. Revising the Newtonian mechanics sections of The physical world (S207) (now discontinued) would also be helpful preparation.

Your local library should be able to suggest books that cover the topics mentioned in Entry requirements.

## What's included

Module texts, audio-visual materials, access to a website from which all supplementary items are to be downloaded.

## You will need

A scientific calculator.

### Computing requirements

A computing device with a browser and broadband internet access is required for this module. Any modern browser will be suitable for most computer activities. Functionality may be limited on mobile devices.

Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.

• A desktop or laptop computer with Windows 7 or higher
• The screen must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.

To join in the spoken conversation in our online rooms we recommend a headset (headphones or earphones with an integrated microphone).

Our Skills for OU study website has further information including computing skills for study, computer security, acquiring a computer and Microsoft software offers for students.

## Teaching and assessment

### Support from your tutor

You will have a tutor who will help you with the study material and mark and comment on your written work, and whom you can ask for advice and guidance. We may also be able to offer group tutorials or day schools that you are encouraged, but not obliged, to attend. Where your tutorials are held will depend on the distribution of students taking the module.

Contact us if you want to know more about study with The Open University before you register.

### Assessment

The assessment details for this module can be found in the facts box above.

You can choose whether to submit your tutor-marked assignments (TMAs) on paper or online through the University’s online TMA/EMA service. You may want to use the online TMA/EMA service for some of your assignments but submit on paper for others. This is entirely your choice.

## If you have a disability

The OU strives to make all aspects of study accessible to everyone and this Accessibility Statement outlines what studying MST326 involves. You should use this information to inform your study preparations and any discussions with us about how we can meet your needs.

## Future availability

Mathematical methods and fluid mechanics (MST326) starts once a year – in October. This page describes the module that will start in October 2019. We expect it to start for the last time in October 2024.

### Course work includes:

4 Tutor-marked assignments (TMAs)
Examination
No residential school

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