MST322 Mathematical Methods and Fluid Mechanics Course Guide

MST322 Mathematical Methods and Fluid Mechanics

Course Guide

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Contents 1 Introduction to the course

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2 Prerequisites

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3 Study units

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4 Set textbook

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5 MST322 audio sections

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6 MST322 video

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7 Assessment 7.1 Continuous assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The final examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Regulation of late submission application . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 MST322 Handbook

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9 Calculator regulation

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10 MST322 OLE Website

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11 Student supports

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12 Study pattern

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13 How to obtain help 13.1 Your tutor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 The MST322 OLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Your course coordinator (C.C.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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14 Useful references

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15 Course was produced by Open University of United Kingdom

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Introduction to the course

MST322 provides an introduction to the subject of fluid mechanics, and teaches the methods required to solve simple flow problems. About half of the course consists of mathematical methods and the other half is devoted to fluid mechanics. However, some of the ‘methods’ units do refer to ideas in the ‘fluids’ units for motivation and interpretation of the solutions. The ‘methods’ half of the course extends the theory of ordinary differential equations begun in the prerequisite courses (MST204, MST207 and M22l), and introduces the theory of partial differential equations. The ‘fluids’ half of the course is about the mathematical models which describe physical situations involving fluid-like substances. These models often involve partial differential equations of the types investigated in the ‘methods’ units. Like most of our courses, MST322 combines the study of correspondence texts with some other activities. This guide is designed to introduce you to the various activities and components of the course, and offers advice on a suitable study pattern. We hope that you will enjoy the course and find it rewarding. We hope that by studying this course you will become aware of the many physical situations that involve the flow of a fluid, the variety of these flow problems, and the application of mathematical modelling in order to solve them.

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Prerequisites

The assumed prerequisite for this course is anyone who has studied MST204 or MST207 or M221. The knowledge assumed for the methods ‘half’ is also taught in M221; however, this course does not provide any training in mathematical models. Differential equations were introduced in MST204, MST207 and M22l. Familiarity with the methods of solution introduced in these courses is important, although we do not consider that knowledge of the totality of any of these courses is necessary. In the modelling of fluids, a good working knowledge of vector algebra and vector calculus is essential. MST204, MST207 and M221 introduced these subjects, and MST322 continues the theory in Unit 4. The following table indicates the units of the prerequisite courses that are considered to be specific academic prerequisites. You are likely to find this course difficult unless you are familiar with this material. Topic Ordinary differential equations Vector algebra Vector calculus Line, surface and volume integrals

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Source in prerequisite courses MST204 MST207 (M221 in brackets) Units 2, 6 Units 2,3 (Units 2, 3) Unit 14 Unit 4 (Unit 4) Unit 26 Unit 25 (Unit 13) Units 26, 27 Unit 26 (Unit 14)

Study units

The mains study material consists of 1. fourteen correspondence study units; 2. one Revision Booklet; 3. two Exercise Booklets and 4. one Video Booklet. The study units are divided into four blocks, with each block developing a different theme of the course. Block I Unit 1 Properties of a Fluid Unit 2 Boundary-value Problems Unit 3 Part I: First-order Partial Differential Equations. Part 11: Dimensional Methods Unit 4 Vector Field Theory

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Block II Unit 5 Kinematics of Fluids Unit 6 Bernoulli’s Equation Unit 7 Vorticity Unit 8 The Flow of a Viscous Fluid Block III Unit 9 Second-order Partial Differential Equations Unit 10 Fourier Series and Power Series Unit 11 Sturm-Liouville Theory Unit 12 Laplace’s Equation Block IV Unit 13 The Wave Equation Unit 14 Water Waves • Block I (Units 1-4) forms a solid foundation on which the rest of the course is built. Unit 1 introduces some of the physical properties of fluids, and the continuum model of a fluid. Unit 2 develops further the analytical and numerical methods of solving ordinary differential equations which were begun in the prerequisite courses. Unit 3 gives an introduction to the solution of firstorder partial differential equations, and introduces the method of dimensional analysis. Unit 4 links line, surface and volume integrals through Stokes’ Theorem and Gauss’ Theorem. • Block II (Units 5-8) starts by investigating the motion of a fluid that is assumed to be incompressible (i.e. it cannot be reduced into a smaller volume) and inviscid (i.e. there is no internal friction). All real fluids do exhibit some degree of compressibility and viscosity, and the effects of viscosity on the flow of fluids are investigated towards the end of the block. Unit 5 develops the equations of motion for the incompressible, inviscid model and the appropriate boundary conditions. Unit 6 solves these equations for flows in pipes and channels, and through apertures. Unit 7 introduces the idea of a vortex and the effects of viscosity on the flow of a real fluid. Unit 8 develops a mathematical model for the forces due to viscosity, and modifies the (differential) equations of motion by including these forces. The unit concludes by showing how the rather complicated equations of motion can be simplified by introducing dimensionless quantities such as the Reynolds number. • Block III investigates analytically the solutions of second-order partial differential equations. Unit 9 shows that a second-order partial differential equation can be classified into one of three types: elliptic, hyperbolic or parabolic. Examples of these types are given as models for fluid flows. The method of separation of variables for solving partial differential equations is then introduced. Unit 10 introduces and develops the properties of power series and Fourier series. Unit 11 extends and generalizes the ideas of Unit 10 with an introduction to Sturm-Liouville theory. The final unit in Block III (Unit 12) uses the ideas introduced in Units 9, 10 and 11 in the solution of Laplace’s equation. These solutions are then used to investigate, for example, the flow of a fluid past a cylinder and a sphere. • Block IV (Units 13 and 14) is about waves. Unit 13 investigates some of the methods of solution of the wave equation. Unit 14 applies these solutions to mathematical models of water waves. Advice on how to study each unit is given in the study guide which forms part of the introduction to the unit, and advice on a study pattern for the course is given in a separate section below. • The Revision Booklet contains worked examples and exercises on those mathematical methods in the prerequisite courses that we consider essential as preparation for MST322. The Revision Booklet is a workbook designed to revise certain topics and to give adequate preparation for the new methods that are developed in the course. • The Video Booklet contains brief descriptions of experiments and fluid flows that can be seen by viewing the video (see below), though it should not be regarded as an adequate substitute for viewing the video. The booklet also includes the preparatory work required for some of the video sessions, and problems to solve which form part of the video activity. 4

• The two Exercise Booklets provide problems on the material in Blocks I and II. These problems are designed to help you to consolidate the work in each of these blocks before moving on to the next block.

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Set textbook

There is no set text book for this course. All teaching is provided through the correspondence units and the audio and video CDs.

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MST322 audio sections

Some units contain an audio activity. In the original UK version of this course these audio sections were provided on audio-cassettes; however, you will have access to them through an audio CD. Instructions on the use of the audio sections appear in the relevant correspondence texts, which contain related visual material. You will need to allow for differences arising from the reference to audio-cassette rather than CD.

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MST322 video

The fluid flows on the video show many of the features of a fluid in motion that we describe mathematically in the correspondence texts. The course team felt that as well as being able to do the mathematics, you should be given the opportunity of ‘seeing’ what the mathematics is trying to model. The video is divided into four modules, and each module is best viewed at a particular place in the course. 1. Module 1 is a general introduction to fluid flow phenomena and ways of visualizing fluid flows and describing them pictorially. 2. Module 2 is associated with Unit 6, and contains some experiments which provide data for you to interpret. 3. Module 3 shows the effects of viscosity for the flow in a pipe. 4. Module 4 is associated with Unit 14, on water waves, and shows how we model the surface waves generated by a ship. Some of the video activity is interactive, so that you have to stop the video and carry out an exercise. As with the audio sections you will need to make allowances for the reference in the material to a video-cassette rather than the VCD that we will be supplying you with.

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Assessment Continuous assessment

This course consists of FOUR TMAs and each covers one block of the course units. The best three of TMAs scores are used to determine your final continuous assessment score; however, you are strongly encouraged to submit all of the assignments so that you can get feedback from your tutor on your understanding of the entire course. Each TMA will contribute 33.33% towards your overall continuous assessment score (OCAS). As part of the course’s OLE Website site you will have the option of submitting your assignment solutions electronically. This will require that you use a package suited to the typesetting of mathematics. The recommended software is Scientific Notebook and Microsoft Office Words. If you do decide to submit electronically, then you must first consult your tutor to discuss the format for your solution files to ensure that he/she can process them. If you do not wish to choose this option, then you can submit hardcopy solutions via the postal service as usual. However, note that in both cases your tutor will enter the marks into the OLE system so that you can access them directly.

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7.2

The final examination

There is a three-hour examination at the end of the course – this is based upon the whole course. A specimen examination paper with solutions will be provided and they are available on the MST322 OLE website. You should work through this paper carefully some time before the examination. You will be allowed to take your calculator and your MST322 Handbook into the examination. The weighting of each assessment components: TMAs Exam Total

weights 50% 50% 100%

Covers materials Each TMA covers one block whole course

You should refer to your Student Handbook for the method by which the University determines the final grade based on your continuous assessment score and examination mark.

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Regulation of late submission application

You should always try to get your TMAs to your tutor by the cut-off date. This is to ensure that you keep to the schedule and that you are ready to start the next block on time. When unusual circumstances arise that prevent you from submitting on time, then the University allows you to apply through the M121 OLE Web site for permission to submit your solutions late, except for the last TMA. Who you apply for permission depends on the amount of time that you require: • 1-7 days - your tutor • 8-21 days - the course coordinator • more than 21 days - the Dean of School The OLE system presents you with these three options and will automatically forward your application to the appropriate individual for consideration. You will need to check with your account in the OLE for the decision. Note that requesting an extension is meant to be an exception - any delay for one TMA will make it harder for you to meet the cut-off date for the following assignments. In some circumstances it may be better to omit one or two of the questions in a TMA so as to submit on time rather than make it harder for later assignments. In addition, only the best four TMA marks are used in determining your final grade so you can omit one TMA if necessary, although you are strongly urged to submit all five.

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MST322 Handbook

The course handbook is provided to give you a convenient source of basic definitions and formulae taught in all the units. The Handbook contains (i) basic definitions, (ii) formulas for use throughout the year and in the examination, (iii) a section-by-section summary of each unit, and (iv) an index for the course material. You are allowed to bring your Handbook to the examination. Free annotation of any text is allowed but no additional pages of notes. We recommend that you familiarize yourself with the contents of the Handbook, and get into the habit of using them as your principal source of reference when attempting the Tutor-marked Assignments and the exercises in the Exercise Booklets.

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Calculator regulation

You will also need a calculator for this course and your calculator must be the approved model. The approved list of calculators can be found in the MST322 OLE - the University has strict rules about which type of calculators are permitted. If you have any query, please contact the course coordinator. The calculator should have the basic mathematical functions such as ex, sin x, arcsin x, and so on, together with a memory. 6

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MST322 OLE Website

MST322 will have its own Web OLE within the University’s Open Learning Environment. In addition to submitting assignments (see above) you will be able to use this site for: • submitting requests for late submission. • accessing your score for each TMA. • discussing the course with your fellow students, tutor and course coordinator on the discussion board. • E-mail to and from participants in the course. • downloading any supplementary material such as assignments, tutorial schedules and errata. You must note that some information will only be provided through this OLE and so access to the OLE Web site is mandatory.

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Student supports

There are two types of sessions:

Surgery This course provides a limited surgeries to help student to solve intermediate study problems and to review TMAs. This is indicated on the presentation schedule included in this mailing.

Tutorials These will be classroom-based and held shortly before the cut-off date for one of the TMAs. There is also a final session for revision. The length of tutorials is last for two hours. This is indicated on the presentation schedule included in this mailing.

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Study pattern

You should begin by working through the Revision Booklet during the first study week shown on the course schedule. The Revision Booklet is not an essential component of the course; however, if you have a good working knowledge of the topics in the booklet, then you should find the course less demanding. Before studying Unit 1, we advise you to view the first video module, which shows the many and complicated features of fluids in motion. The units should be studied in the order in which they appear, and there is an assignment to send to your tutor at the end of the study period for each block. In the text, we have provided various types of questions for you to answer as part of the teaching process; these are of two types. Exercises are designed to give you practice in what the preceding text teaches. Often, an exercise follows a (worked) example, which allows you to repeat the steps in the solution to the example. You are not expected to try and do the examples yourself, without reference to the text. End-of-section exercises come at the end of each section of each unit. They are designed to give you an overview of the types of exercises that are in the text, and are similar to those that appear as assignment questions or examination questions. Before tackling an end-of-section exercise, you are advised to read the summary of the relevant section that appears in the Handbook. You should attempt the end-ofsection exercise with as little reference to the unit section as possible. You may also find these exercises useful for examination revision. Solutions to the exercises and end-of-section exercises are given at the end of each correspondence text.

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How to obtain help

In tutorials and surgeries you will certainly have opportunity to get help from your tutor if you have any problems. But students learning at a distance like yourself often need help at other times. There are a number of other ways for you to get the help you need to succeed in this course, no matter what point of the course you’re at.

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Your tutor

Your tutor is there to help you understand the ideas in the course. One of the best ways for him/her to do this is through comments on your TMAs. When your tutor returns your TMA scripts to you after marking, you should always go through them and take note of the comments your tutor has written. This advice will help you avoid similar errors in later assignments and in the examination. Also, of course, you should try to attend tutorials because there you will have the opportunity to talk to your tutor directly and, just as important, to talk to other students. Tutors are also available for you to telephone and E-mail for immediate help or advice. Contact details for your tutor will be sent to you at the start of the course. Your tutor should also let you know what hours he/she is available for telephone tutoring. Using E-mail overcomes the problem that often arises in telephone contact - expressing a mathematical problem clearly. Within reasonable limits your tutor will reply quickly.

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The MST322 OLE

The University provides the Online Learning Environment (OLE) for all courses. This provides a variety of features to support the presentation of a course: • Discussion board shared by all students and tutors on the course. Problems can be posted for anyone to offer help. In the past this has possibly been the most popular way of getting help. Often you can find that the answer to your own problem has already been supplied to another student with a similar problem. • E-mail. Each student and tutor has an E-mail account for direct communication between individuals and groups. Most of the news and comments from the course coordinator will be sent to you through this E-mail system. • Important news. Any stop press will be posted on the MST322 OLE site. An alert will be displayed to indicate any new items posted. • Study units. Soft copies of the units will be available through the site. • Assessment questions and late submission application. All assignments are provided through the MST322 OLE website. All TMAs may be submitted through this site. Requests for the delay in submitting assignments must be applied through this OLE website. • Check assignment score. You can check the marking score of any assignment and check the marking status. The MST322 OLE site should become a regular part of your study habits. Check the site regularly for updates or news as well as useful and interesting posting on the discussion board. In your first mailing of material you will have received a copy of the OLE user guide, which explains how to use the system.

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Your course coordinator (C.C.)

You will find an introduction to your course coordinator in the first mailing, which will include his contact details. Whilst your tutor should always be your first point of contact when seeking help the CC is always available to provide additional assistance when needed. The course coordinator of this course is Dr. Anita Wong. Office Tel. 2768 6810, E-mail: [email protected]. Office: Room A0920 of Homantin OUHK campus.

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Useful references

The following books are not essential to the course, but may be useful as background and further reading. They have been categorized into those which are mostly ‘fluids’ and those containing ‘methods’.

Fluids (a) Those with a ‘more’ mathematical approach. 1. Acheson, D. J., Elementary Fluid Dynamics (Oxford University Press, 1990) ISBN 0 19 859679 0. A modern textbook used for conventional fluid mechanics courses. 2. Batchelor, G. K., An Introduction to Fluid Dynamics (Cambridge University Press, 1980) ISBN 0 521 09817 3. A very full textbook, containing much more than MST322 attempts: perhaps difficult to read, but ideal for post-course reading.) 3. Chorin, A. J., and Marsden, J. E., A Mathematical Introduction to Fluid Mechanics (SpringerVerlag, 1993) ISBN 3 540 979918 2. About one third of this book is relevant to MST322, thought the rest makes interesting reading. The emphasis is on the ‘mathematical’, many of the results being presented as theorems or lemmas. 4. Lighthill, J., Sir, An Informal Introduction to Theoretical Fluid Mechanics (Oxford University Press, 1986) ISBN 0 19 853630 5. This text is more condensed and advanced than its title would suggest. 5. Patterson, A., A First Course in Fluid Mechanics (Cambridge University Press, 1983) ISBN 0 521 27424 9. A modern textbook on fluid mechanics, which includes a good modelling approach to the subject. At one time, this textbook was considered as a ‘set book’ for the course. This is strongly recommended as further reading. (b) Those with a ‘more’ engineering approach. 1. Elder, S. A., and Williams, J., Fluid Physics for Oceanographers and Physicists (Pergamon Press, 1989) ISBN 0 7506 2958 4. Contains some useful examples and exercises. 2. Massey, B. S., Mechanics of Fluids (Chapman and Hall, 6th edition 1989) ISBN 0 06 044926 8. This book presents fluid mechanics from an engineering viewpoint. Fundamental concepts are illustrated by simple applications of the theory to engineering problems of general interest. Specialized mathematical techniques are avoided. 3. Olson, R. M., and Wright, S. J., Essentials of Engineering Fluid Mechanics (Harper Collins College, 5th edition 1990) ISBN 0 06 044926 8. The emphasis of this book is on the physical aspects of fluid behaviour. It complements the MST322 approach. 4. Tritton, D. J., Physical Fluid Dynamics (Van Nostrand Reinhold, 19788) ISBN 0 19 854493 6. This book approaches the subject from a physicist’s point of view, with particular emphasis given to information obtained from experiment and observation.

Methods 1. Berry, J., Norcliffe, A., and Humble, S. Introductory Mathematics through Science Applications (Cambridge University Press, 1989) ISBN 0 521 28446 5 This book contains some prerequisite material needed for the course.

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2. Boyce, W. E., and DiPrima, R. C., Elementary Differential Equations and Boundary Value Problems (John Wiley, 1992) ISBN 0 471 87096 X. Most of the methods taught in MST322 are covered in this textbook. It is written from an applied mathematics point of view, and contains practical applications of the theory. This textbook includes the prerequisite theory of differential equations required for the course. 3. Powers, D. L., Elementary Differential Equations (Kent) ISBN 0 87150 093 0.

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Course was produced by Open University of United Kingdom

The course was produced by the following team: Keith Attenborough {Technology) John Berry {Course Team Chairman: Mathematics) Mick Bromilow {Mathematics) Paul Clark {Science) Pat Edwin {Mathematics) Roy Knight {Academic Editor: Mathematics) Alec Parkinson {Technology) Ted Phythian {Mathematics) Mike Simpson {Mathematics) Tom Smith {Science) Glanffrwd Thomas {BBC) Howard Thomas {Mathematics) Mike Thorpe {Mathematics) John Trapp {Mathematics) Anne Walton {Mathematics) with the assistance of: Gordon Burt {IET) Alison Cadle {Publishing) Roberta Cheriyan {Course Coordinator) Shirley Knight {Course Team Secretary) Geoff Manser {Graphic Design) Fran Osborne {Graphic Design) Tag Taylor {Graphic Design) The course team would like to acknowledge the contributions of the following, who read drafts of course units: David Brannan {Mathematics) David Burghes {Exeter University) Chris Haines {City University, OU part-time tutor) John Horlock {Vice-Chancellor) Claire Jacobs {OU part-time tutor) Mike Patrick {Exeter University) Doug Quinney (Keele University, OU part-time tutor) Ann Round {OU part-time tutor) Tony Richardson {Bristol University)

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