Course overview

The course introduces differential operators and integral theorems which are extensively used in studies of various scalar and vector fields arising, for example, in fluid and electro dynamics. The course involves a significant applied component and contains a large number of computer and in-class experimental demonstrations which distinguish this course from the majority of typical pen-and-paper mathematical courses. The course builds students' mathematical skills with the help of enlightening geometrical illustrations of concepts and uses many physical examples of fluid flows to "visualise" the mathematics. The goal of the course is not only to provide technical knowledge but also to encourage students' independent creative thinking via extensive communications with the lecturer who guides their progress throughout the semester. This course is offered only in even-numbered years.

Pre-requisite prior studies

You must be completely comfortable with the knowledge and skills developed in MAT2100 and MAT3105.

Course queries

All matters related to the course must be directed to the examiner: Sergey Suslov, mailto: ssuslov@usq.edu.au, Fax: (07) 4631 5550

Course support

Extensive support for the course is provided via the home page http://www.sci.usq.edu.au/courses/MAT3106. If you have any difficulty accessing the home page, then contact the course team immediately.

The Introductory Book has much information, in particular, the assignments and the Study Chart to guide your pace of learning.
The Study Guide will guide your learning either by itself or with the aid of a text book.
The Submit Assignments page says how to do precisely that.
At the end of the course, tell us your experience via the Feedback form.

Residential school

There is no scheduled residential school for this course. However, if you are attending a residential school for another course, you are welcome to make appointments with the lecturers to discuss any problems with the course and/or content.

2008 Lecture audio recordings
Below you will find links to the MP3 audiofiles containing the voice recordings of selected lectures.

2006 Lecture audio recordings

Review Lecture on Differential Operators (6.6Mb)

Review Lecture on Integral Theorems and Modelling Inviscid Fluid (2.75 Mb)

Review Lecture on Navier-Stokes Equations (1.51 Mb)

Boundary Layer 1 (10.9 Mb)

Boundary Layer 2 (12.4 Mb)

Far Wake 1 (6.1 Mb)

Far Wake 2, Review of Boundary Layer and Slow Flows 1 (10.1 Mb)

Slow Flows (6.6 Mb)

Lubrication Theory 1 (6.2 Mb)

Lubrication Theory 2 (11.8 Mb)

Squeeze films (11.8 Mb)

Review 1 (6.2 Mb)

Review 2 (10.7 Mb)

Review 3 (6.5 Mb)

Important messages

Below you will find the copies of the most important messages the teaching team will be sending to the class during the semester. Make sure to check them regularly.

24/07/08. Welcome to MAT3106!

30/07/08. Homework policy
06/08/08. Assignment 1 Question 3
24/09/08. Assignment 2 hints.

24.07.08. Message 1.

Welcome to MAT3106 Vector Calculus and Mathematical Modelling of Fluid Flows. My name is Dr Sergey Suslov and I will be your lecturer and examiner in this course. You can contact me at one of the following addresses:

Room D218, Department of Mathematics and Computing, University of Southern Queensland, Toowoomba, Queensland 4350, Australia
or
Phone: +61-7-4631-5542
or
Fax: +61-7-4631-5550
or
Email: ssuslov@usq.edu.au

Personal (if you are on campus), email or phone contacts are preferable options. If you are in Australia but off campus and would like to discuss any issue over the phone you can email me your phone number and I will call you at a mutually convenient time so that you do not spend a fortune on phone bills. At this stage I require all of you to confirm your enrolment in this course by emailing me within the next few days at the address above. Please let me know if you still did not receive/buy your text books and other study materials and what you believe the reson for the delay is.

I would like to say a few words about this course and its assessment structure. The course consists of two big parts: Vector Calculus and Flow Modelling. The first part requires your fluency in Algebra and Calculus 2 (MAT2100), the second one requires solid knowledge of Partial Differential Equations (MAT3105). Therefore I suggest you keep the study materials for these two courses handy for your quick review and reference as you study MAT3106. In particular, the first concept used in MAT3106 will be the Taylor series expansions and linearisation. Partial derivatives and separation of variables for ordinary and partial differential equations will be the next. As we proceed with the course I will be reminding you which other previously studied mathematical concepts need to be reviewed. The first part of the study book (Vector Calculus) is more or less self-contained and uses the text (Kreyszig) mostly as a source of exercises and additional examples. In contrast, the second part is heavily based on the second text (Ockendon²) and should be read as its complement.
The course assessment contains three standard items: two assignments with standard extension policy (see course specs) which you work on fully on your own, and an open final examination. Please, do not be misled by the word "open". Even if you bring all the right books with you to the exam you simply will not have time to find/read the necessary topics unless you know for sure what you are looking for and know exact textbook page numbers. For this reason I strongly advise you to keep summarising all important concepts and formulae on a sheet of paper as you study the course. Endeavour to put something on your summary sheet every week. This will be much more helpful for you in the exam than all the clever books in the world. I would like to pay a very special attention to the fourth "nonstandard" assessment item -- regular homework. Its purpose is to encourage your active learning and involvement with the course throughout the semester rather than to assess your knowledge. The way it works is:

1. Each week identify the problems in the Activity sections of the study book to be submitted. See the study schedule in the Introductory book for the list of topics and submission deadlines.

2. Attempt solving these problems long before the submission deadline (the first one is next Monday, see the study schedule).

3. If you have any difficulties with the attempted problems you must contact me at your earliest opportunity. Upon realising what your specific difficulty is I will provide you with individual hints and explanations so that you will be able to obtain and submit a fully correct solution.

4. Once you are sure that your solutions are correct submit them by the specified deadline. As you can see, those of you who take homework seriously and follow the above instructions punctually are guaranteed 20% towards the final grade, so do not neglect this unique opportunity! On the other hand, given this very supportive "marks for free" policy and the amount of time I have to devote to individual consultations with each of you there will be no extensions granted on discussion sessions. For example, if you are not able to work on your homework this week and notify me of this fact ahead of time I might grant you an extension for homework submission, but I will not be discussing with you any of the difficulties you might find after the submission deadline (e.g. after 31/07 for the first lot). So pace yourself carefully. In order to guarantee that each of you receives a prompt feedback on your homework questions please see me during my office hours, email or phone me or send me a facsimile message. Clearly, regular mail is too slow to be used as a way of receiving a feedback before the deadline. Your homework will be marked normally out of 3 marks per problem. You will see where you went wrong, but no detailed comments on your errors and no solutions will be provided. So discussing homework problems before the deadline is the only way you can get a full feedback and advice. Please, note that each of you should have received a CD-ROM with video fragments illustrating nicely the studied concepts. They are taken from videotapes available from the USQ library. These tapes (referenced in the study book) contain much more relevant material and I encourage you to view them on your own. Also, a few live demonstrations will be included in my classroom lectures, especially in the second part of the course, so that I invite all students to attend my lectures regularly. My invitation extends to external students who happen to be on campus.
Finally, the course has a web page containing all course information
at http://www.sci.usq.edu.au/courses/MAT3106
This page will be updated during the semester, so please visit it regularly. In particular, any corrections/additions to the study and intro books will be reported there and audio recordings of lectures will be posted there weekly. Welcome to MAT3106 and start studying it from the very first day to guarantee your overall success.
To the message list.

30.07.08. Message 2.

Now when the first HW submission deadline has passed I would like to re-iterate a few related issues.

Firstly, not every one has submitted the first lot of problems. Whatever your reasons were you should take into account the following facts.

1. There were 12 problems each would take you a few minutes to solve and write the solution if you only read and understood the relevant course material.

2. The cost of the first HW lot was 2.4% which would go directly towards your final grade.

3. Those of you who had difficulties with these problems, but contacted me before the deadline, got comprehensive individual help and as a result
received the full mark for this set of problems. Those of you who did have difficulties, but decided not to contact me just wasted 2.4% of the final
grade which otherwise would be yours guaranteed.

Secondly, let me repeat that HW problems play a different role in this course than in any other course you studied so far. HW is not to assess you,
but to guide you through the material, to give you the opportunity to identify your own difficulties, to initiate a discussion with the lecturer, to get an
individual help and after all that to get guaranteed marks towards your final grade. Can you really afford not to take such an offer?

Thirdly, despite the relevance of HW problems to those in the Assignments and in the Exam, only those of you who attempts HW problems on your
own can request the full solution from me. Please, keep in mind that HW solutions exist only in a hand written form and thus you will need either to
pick them up from me personally or to give me your mailing address or fax number where I can send them to.

Lastly, my past years experience with this course shows that there is a 100% correlation between the student activity with HW and their final exam
mark and the final grade. Therefore those of you who will not become active with HW submissions by mid-August will be recommended to drop
this course. Unless you have very serious health or other reasons for delays with HW submission I strongly recommend you to take HW very seriously.
Please remember that although you will still be able to submit your HW late you can only discuss any difficulties with me prior to the specified deadlines.
Sorry, too late for questions re HW1, but HW2 questions are welcome!

PS. External students who choose to mail their HW to me: please use my direct address
To the message list

06.08.08 Message 3.

Correction to Question 3 of Assignment 1:

Question 3.

Do questions 8, 14, 24, 34 from Kreyszig's 9th edition Chapter 10 Review pages 473-474.

To the message list

24.09.08 Message 4.

Hints for Assignment 2:

Q1.

(i) Divide eqn (1.21) on page 11 in the textbook by density and subtract it from the first of equations given in the problem definition. Since it is clamed that these equations are equivalent you need to show that the difference is 0. Do this using term by term comparison and vector calculus identities from Appendix A in the study book. Once the equivalence is shown, take curl of first of the given equations to obtain the second one. Formulae from the Appendix and homework solutions you received so far will be of help.

(ii) Integrate the first of given equations over the closed contour keeping in mind that int(u.ds) is the circulation.

Q2.

This question refers to Exercise 1.20, not 1.19, in the 2008 edition of the study book.

Q3.

The assumption made in the text is that the pressure gradient dp/dx=c is constant. Note that in steady flows fluid does not accelerate. Therefore all forces must balance each other. In particular, the drag force arising due to the friction between the fluid and the pipe wall must be balanced by (i.e. be equal to) the force due to the applied pressure difference between inlet and outlet of the channel. Express this pressure force in terms of the given quantities taking into account that the crossectional area of the pipe is constant to obtain the answer.

In order to obtain the flow rate for a circular convert the given equation written in cartesian coordinates to cylindrical coordinates using Study book appendices, show that it reduces to and ordinary differential equation in radial direction, solve it to obtain an expression for a velocity profile and finally integrate the velocity to obtain the flow rate.

Q4.

Start with approximate non-dimensional boundary layer equations given in the study book, re-introduce the scales used in the study book to convert them to the dimensional form, then use scales given in the text book to non-dimensionalize them again and to obtain their required form.