England.

# Mathematical methods and fluid mechanics

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, the wave 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.

## Modules count towards OU qualifications

OU qualifications are modular in structure; the credits from this undergraduate module could count towards a certificate of higher education, diploma of higher education, foundation degree or honours degree.

## Module

MST326

### Credits

Credits

• Credits measure the student workload required for the successful completion of a module or qualification.
• One credit represents about 10 hours of study over the duration of the course.
• You are awarded credits after you have successfully completed a module.
• For example, if you study a 60-credit module and successfully pass it, you will be awarded 60 credits.
30

### Study level

Across the UK, there are two parallel frameworks for higher education qualifications, the Framework for Higher Education Qualifications in England, Northern Ireland and Wales (FHEQ) and the Scottish Credit and Qualifications Framework (SCQF). These define a hierarchy of levels and describe the achievement expected at each level. The information provided shows how OU module levels correspond to these frameworks.
Level of Study
OU SCQF FHEQ
3 10 6

### Student Reviews

MST326 is a very applied module which I thoroughly enjoyed. I would say that half of the course is mathematical...
Read more

## 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.

You can find the full content list on the Open mathematics and statistics website.

### 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.

### Vocational relevance

The modelling of fluid flows is of significant importance to a number of disciplines, and requires knowledge of a broad range of tools that are essential in applied mathematics. In this module, you’ll learn important aspects that govern fluid processes, including the necessary mathematical methods for their modelling and analysis, as well as the physical intuition. Mastering this material will help you develop skills that are desirable qualities in the profile of applied mathematicians, scientists and engineers working in industry and academia.

### 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.

## Teaching and assessment

### Support from your tutor

Throughout your module studies, you’ll get help and support from your assigned module tutor. They’ll help you by:

• Marking your assignments (TMAs) and providing detailed feedback for you to improve.
• Guiding you to additional learning resources.
• Providing individual guidance, whether that’s for general study skills or specific module content.
• Facilitating online discussions between your fellow students, in the dedicated module and tutor group forums.

Module tutors also run online tutorials throughout the module. Where possible, recordings of online tutorials will be made available to students. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part.

### Assessment

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

## Future availability

Mathematical methods and fluid mechanics (MST326) starts once a year – in October.

This page describes the module that will start in October 2025.

We expect it to start for the last time in October 2028.

## Regulations

As a student of The Open University, you should be aware of the content of the academic regulations which are available on our Student Policies and Regulations website.

### Course work includes:

4 Tutor-marked assignments (TMAs)
Examination

## Entry requirements

There is no formal pre-requisite study, but you must have the required mathematical skills.

Talk to an advisor if you’re not sure you’re ready.

## Preparatory work

You should aim to be confident and fluent with the concepts covered in the Are you ready? quiz here, and follow the advice in the quiz.

The key topics to revise include:

• functions, graphs and coordinate systems
• differentiation and functions of several variable
• linear algebra
• differential equations
• mechanics
• numerical methods and practical applications
• Fourier series.

Mathematical methods, models and modelling (MST210) is ideal preparation, otherwise Mathematical methods (MST224).

## Register

Start End England fee Register
05 Oct 2024 Jun 2025 -

Registration now closed

This module is expected to start for the last time in October 2028.

## Additional Costs

### Study costs

There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.

If your income is not more than £25,000 or you receive a qualifying benefit, you might be eligible for help with some of these costs after your module has started.

## Ways to pay for this module

### Open University Student Budget Account

The Open University Student Budget Accounts Ltd (OUSBA) offers a convenient 'pay as you go' option to pay your OU fees, which is a secure, quick and easy way to pay. Please note that The Open University works exclusively with OUSBA and is not able to offer you credit facilities from any other provider. All credit is subject to status and proof that you can afford the repayments.

You pay the OU through OUSBA in one of the following ways:

• Register now, pay later – OUSBA pays your module fee direct to the OU. You then repay OUSBA interest-free and in full just before your module starts. 0% APR representative. This option could give you the extra time you may need to secure the funding to repay OUSBA.
• Pay by instalments – OUSBA calculates your monthly fee and number of instalments based on the cost of the module you are studying. APR 5.1% representative.

#### Joint loan applications

If you feel you would be unable to obtain an OUSBA loan on your own due to credit history or affordability issues, OUSBA offers the option to apply for a joint loan application with a third party. For example, your husband, wife, partner, parent, sibling or friend. In such cases, OUSBA will be required to carry out additional affordability checks separately and/or collectively for both joint applicants who will be jointly and severally liable for loan repayments.

As additional affordability checks are required when processing joint loan applications, unfortunately, an instant decision cannot be given. On average the processing time for a joint loan application is five working days from receipt of the required documentation.

Read more about Open University Student Budget Accounts (OUSBA).

### Employer sponsorship

Studying with The Open University can boost your employability. OU courses are recognised and respected by employers for their excellence and the commitment they take to complete. They also value the skills that students learn and can apply in the workplace.

More than one in ten OU students are sponsored by their employer, and over 30,000 employers have used the OU to develop staff so far. If the module you’ve chosen is geared towards your job or developing your career, you could approach your employer to see if they will sponsor you by paying some or all of the fees.

• Your employer just needs to complete a simple form to confirm how much they will be paying and we will invoice them.
• You won’t need to get your employer to complete the form until after you’ve chosen your module.

### Credit/debit card

You can pay part or all of your tuition fees upfront with a debit or credit card when you register for each module.

We accept American Express, Mastercard, Visa and Visa Electron.

### Mixed payments

We know that sometimes you may want to combine payment options. For example, you may wish to pay part of your tuition fee with a debit card and pay the remainder in instalments through an Open University Student Budget Account (OUSBA).

Please note: your permanent address/domicile will affect your fee status and, therefore, the fees you are charged and any financial support available to you. The fee information provided here is valid for modules starting before 31 July 2025. Fees typically increase annually. For further information about the University's fee policy, visit our Fee Rules

This information was provided on 15/09/2024.

## Can you study an Access module for free?

Depending on eligibility and availability of places, you could apply to study your Access module for free.

To qualify, you must:

1. be resident in England
2. have a household income of not more than £25,000 (or be in receipt of a qualifying benefit)
3. have not completed one year or more on any full-time undergraduate programme at FHEQ level 4 or above or successfully completed 30 credits or more of OU study within the last 10 years

## How to apply to study an Access module for free

Once you've started the registration process, either online or over the phone, we'll contact you about your payment options. This will include instructions on how you can apply to study for free if you are eligible and funded places are still available.

If you're unsure if you meet the criteria to study for free, you can check with one of our friendly advisers on +44 (0)300 303 0069, or you can request a call back.

## Not eligible to study for free?

Don't worry! We offer a choice of flexible ways to help spread the cost of your Access module. The most popular options include:

• monthly payments through OUSBA
• part-time tuition fee loan (you'll need to be registered on a qualification for this option)

To explore all the options available to you, visit Fees and Funding.

## 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

You’ll need broadband internet access and a desktop or laptop computer with an up-to-date version of Windows (10 or 11) or macOS Ventura or higher.

Any additional software will be provided or is generally freely available.

To join in spoken conversations in tutorials, we recommend a wired headset (headphones/earphones with a built-in microphone).

Our module websites comply with web standards, and any modern browser is suitable for most activities.

Our OU Study mobile app will operate on all current, supported versions of Android and iOS. It’s not available on Kindle.

It’s also possible to access some module materials on a mobile phone, tablet device or Chromebook. However, as you may be asked to install additional software or use certain applications, you’ll also require a desktop or laptop, as described above.

## 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.

To find out more about what kind of support and adjustments might be available, contact us or visit our disability support pages.