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.
What you will study
In simple terms, we think of a fluid as a flowing substance. Familiar examples are air (a gas) and water (a liquid). All fluids are liquids or gases. The analysis of the forces in and the 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 modelled by differential equations. The mathematical methods arise from (and are interpreted in) the context of fluid-flow problems. However, 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. Air flow over objects is of fundamental importance to the aerodynamicist in designing aeroplanes and the motor industry in designing 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 mass continuity equation 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 to describe 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 investigates phenomena such as flows through pipes and apertures, 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. It investigates some of their exact solutions and 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, diffusion equation and Laplace’s equation, which frequently arise in fluid mechanics and 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 the 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, fluid flow past a cylinder and a sphere.
Block 4
In this block, you’ll return to applications of mathematics to fluid flows.
Unit 12 Water waves uses some of the theory developed in Block 3 to investigate various types of water waves, and discusses several practical examples of these waves.
Unit 13 Boundary layers and turbulence looks at the effects of turbulence (chaotic fluid flow) and the nature of boundary layers within a flow, introducing models to describe these phenomena.
The full content list is on the Open mathematics and statistics website.
Entry requirements
You must have passed one of the following:
Mathematical methods (MST224)
Mathematical methods, models and modelling (MST210)
Mathematical methods and models (MST209)
You can check you’re ready for MST326 and see the topics it covers here.
Preparatory work
You should aim to be confident and fluent with the concepts covered in the Are you ready? quiz 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.
Teaching and assessment
Support from your tutor
You’ll get help and support from an assigned tutor throughout your module.
They’ll help by:
marking your assignments and offering detailed feedback to help you improve
providing individual guidance, whether that’s for general study skills or specific module content
guiding you to additional learning resources
facilitating online discussions between your fellow students in the dedicated forums.
Online tutorials run throughout the module. While they’re not compulsory, we strongly encourage you to participate. Where possible, we’ll make recordings available.
Assessment
Course work includes:
4 Tutor-marked assignments (TMAs)
Examination
For this module, we may use an assessment verification process to meet the requirements of relevant accrediting, professional, statutory or regulatory bodies. As part of this process, you may be asked to attend a short post-assessment video discussion, lasting around 15 minutes. During the discussion, you’ll need to show a photo ID and talk through your answers to a small number of questions with a tutor or a member of the module team. The discussion is not graded and is only used to confirm that you completed the assessment yourself.
We regularly review the assessments in our modules, so we may update the examination method used for this module to an in-person exam or an online invigilated exam. When we are making such a change, we will make it clear on this page. If we need to make a change after you have registered, we will notify you as soon as possible. If you have additional requirements, we will support you to complete your assessments.
What's included
You’ll have access to a module website, which includes:
a week-by-week study planner
course-specific module materials
audio and video content
assessment details and submission section
online tutorial access
access to student forums.
Additionally, the website includes:
supplementary items.
You will need
a scientific calculator.
Computing requirements
Primary device – A desktop or laptop computer with at least 8 GB of RAM and a quad-core processor (2.4 GHz minimum speed). It’s possible to access some materials on a mobile phone, tablet or Chromebook; however, they will not be suitable as your primary device or for taking your online exam.
Peripheral devices – Headphones/earphones with a built-in microphone for online tutorials, plus an external webcam and a microphone.
Operating systems – Windows 11 or the latest supported macOS.
Internet access – A broadband or mobile connection with at least 3 Mbps upload and download speed.
Browser – Google Chrome and Microsoft Edge are recommended. Mozilla Firefox and Safari may be suitable. You may need to use Google Chrome specifically for some activities.
Our OU Study app operates on supported versions of Android and iOS.
Software – Any additional software will be provided or is generally freely available.
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 website.
Future availability
Mathematical methods and fluid mechanics (MST326) starts once a year – in October.
It will next start in October 2027.
We expect it to start for the last time in October 2032.