Mathematical methods and fluid mechanics
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Half of this course 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 course will give you a good understanding of modelling in the context of fluids. You should have a sound knowledge of ordinary differential equations, vector calculus, and multiple integrals; basic particle mechanics; and some knowledge of partial differential equations and Fourier series.
Fees 2012
See fees and funding options for study from September 2012.
Course facts
An undergraduate course in Mathematics and Statistics and Science.
| About this course: | |
|---|---|
| Course code | MST326 |
| Credits | 30 |
| OU Level | 3 |
| SCQF level | 10 |
| FHEQ level | 6 |
| Course work includes: |
|---|
| 4 Tutor-marked assignments (TMAs) |
| Examination |
| No residential school |
Register for the course
This course is available for study in the countries shown. Fees and financial support may vary by country.
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 course 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 course is arranged in 13 units within four blocks.
Block 1 is the foundation on which the rest of the course 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 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 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 returns 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.
You will learn
Successful study of this course should enhance your skills in communicating mathematical ideas clearly and succinctly, expressing problems in mathematical language and interpreting mathematical results in real-world terms.
Entry
This is a Level 3 course. Level 3 courses build on study skills and subject knowledge acquired from studies at Levels 1 and 2. They are intended only for students who have recent experience of higher education in a related subject, preferably with the OU. The Level 2 course Mathematical methods and models (MST209) is ideal preparation for this course. You are more likely to complete this course successfully if you have acquired your prerequisite knowledge through passing MST209.
You must have 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.
If you have any doubt about the suitability of the course, please contact our Student Registration & Enquiry Service.
Preparatory work
To help you to revise the necessary mathematical methods and to check that you are ready to study MST326, a Revision Booklet, with worked examples and exercises is available to download.
Revising the later units of our Level 2 course Mathematical methods and models (MST209) on partial differential equations, vector calculus, multiple integrals and Fourier series would also be a good preparation. Your regional or national centre will be able to tell you where you can see reference copies, or you can buy selected texts from Open University Worldwide Ltd.
Your local library should be able to suggest books that cover the topics mentioned in Entry.
If you have a disability or additional requirement
The course relies heavily on visual representation of flows and manipulation of complicated mathematical symbols that are difficult to describe verbally. Written transcripts of any audio components and Adobe Portable Document Format (PDF) versions of printed material are available. Some transcripts or Adobe PDF components may not be available or fully accessible using a screen reader and mathematical and scientific materials may be particularly difficult to read in this way. Other alternative formats of the course materials may be available in the future. Our Services for disabled students website has the latest information about availability.
If you have a form of colour-blindness, then you may need help from a person with standard colour perception to get full use of the DVD. However, even if some or all of the facilities on the DVD prove to be inaccessible, the learning outcomes of the course should still mostly be achievable.
If you are a new student, or new to courses using a computer or the internet, you will need to inform us of your particular needs as soon as possible, as some of our support services may take several weeks to arrange. Details of how to do this and our range of support services are described in our publication Meeting Your Needs.
You can also find information about accessible study materials, financial support and the Disabled Students' Allowance, equipment and other services, on our Services for disabled students website. It also includes our contact details for advice and support both before you register and while you are studying.
Study materials
What's included
Course texts, audio-visual materials, access to a website from which all supplementary items are to be downloaded.
You will need
A scientific calculator. An audio CD player is not essential but could provide an alternative to playing CDs on your computer.
You will need access to the internet at least once a week during the course to download online materials including the course assignments, and keep up to date with news items.
Computing requirements
This course includes online computer activities – you can access these using a web browser that can play Flash and Shockwave.
You will need internet access and a computer. If you have purchased a new computer since 2005 it should meet your course computing requirements. Check our Technical Requirements section if your computer is older than this or is otherwise unusual.
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 course. Contact our Student Registration & Enquiry Service if you want to know more about study with The Open University before you register.
Assessment
The assessment details for this course can be found in the facts box above.
Please note that TMAs for all undergraduate mathematics and statistics courses must be submitted on paper as – due to technical reasons – we are unable to accept TMAs via our eTMA system.
Professional recognition
This course may help you to gain recognition from a professional body. You can view or download our Recognition leaflets 3.3 Professional Engineering Institutions, 3.6 Institute of Mathematics and its Applications, 3.7 Computing and 3.8 Scientific Institutions for further information.
Students also studied
Students who studied this course also studied at some time:
Future availability
The details given here are for the course that starts in October 2012. We then expect it to be available in October 2014, 2015 and 2016.
Fees 2012
See fees and funding options for study from September 2012.
Course facts
An undergraduate course in Mathematics and Statistics and Science.
| About this course: | |
|---|---|
| Course code | MST326 |
| Credits | 30 |
| OU Level | 3 |
| SCQF level | 10 |
| FHEQ level | 6 |
| Course work includes: |
|---|
| 4 Tutor-marked assignments (TMAs) |
| Examination |
| No residential school |
Study explained
- Financial support
- - find out if you qualify for support with your fees with our eligibility checker.
- Study explained
- - all you need to know about distance learning with the OU.
Student Reviews
This was my first Level 3 course after having completed MST209 the previous year. The first Blocks A and B ...
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A very challenging course. The course notes were excellent. The TMA's were enjoyable and again, challenging. However, it is one ...
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