Mathematical methods and fluid mechanics

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.

Entry

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.

Study materials

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

• Primary device – A desktop or laptop computer. It's possible to access some materials on a mobile phone, tablet or Chromebook; however, they will not be suitable as your primary device.
• Peripheral device – Headphones/earphones with a built-in microphone for online tutorials.
• Our OU Study app operates on supported versions of Android and iOS.
• Operating systems – Windows 11 or latest supported macOS. Microsoft will no longer support Windows 10 as of 14 October 2025.
• Internet access – Broadband or mobile connection.
• Browser – Google Chrome and Microsoft Edge are recommended. Mozilla Firefox and Safari may be suitable.

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 module and tutor group forums.

Online tutorials run throughout the module. Where possible, we'll make recordings available. While they're not compulsory, we strongly encourage you to participate.

Assessment

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

If you have a disability

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

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.