This page is for messages which are no longer important to keep on the main page. Whereas the main page has messages in inverse chronological order, I keep them in plain chronological order here.
First lecture: Monday 22 August.
First exercise: Wednesday 31 August.
2005-07-01: If there are any foreign students taking the class, it will be taught in English. This page will be in English until I know for sure.
We have a new textbook this year, just out from the publisher. The modelling course has been plagued for years by the lack of a good textbook. There are many textbooks available, but all the others have been either too elementary or too advanced. Sam Howison's book seems exactly right for us.
2005-08-09: The textbook is now available at Tapir. Price: NOK 430.
The first lecture of the course, on Monday 22 August, will be on Buckingham's pi theorem. I first designed this as a colloquium talk, and it shows. You can have a look at last year's slides as pdf: with one slide per page, or two slides per A4 page. I don't expect major revisions this year.
Before diving into Buckingham's theorem, however, I'll spend some time on practical matters, and trying to give an idea what the course is all about.
(By the way, the schedules were hard to find in the NTNU web pages this year.)
2005-08-22: The slides from today's introductory lecture are available: For viewing on a screen, or for printing. In case you already printed out last year's version, you should know that the only difference is that a portrait of Edgar Buckingham is in this year's version.
Harald Krogstad will lecture on Friday, while I am in a meeting. The topic will be a more thorough introduction to the practice of non-dimensionalisation and use of Buckingham's theorem.
2005-08-31: There is only one book left in the bookstore. Maybe there aren't enough to go around? If you haven't got a copy yet, send me an email. I'll count the number of requests and ask the bookstore to order more books.
2005-09-05: Today's lecture is cancelled. My cold, which I had started on Friday, is now messing with my throat so that I am unable to speak without a lot of coughing. It's too late to ask someone else to step in. I'm staying home today. I expect to be fully recovered by Friday.
2005-09-05: More books to arrive. I spoke to the person responsible at the bookstore on Friday. She would order ten more copies of the textbook. They should arrive in two weeks, perhaps less.
2005-09-08: I have realized that we need to talk now about something that used to be talked about much later in the course, namely conservation laws on integral and differential form, and how to move between them (namely, using the divergence theorem). So this will be a big part of tomorrow's lecture, together with some applications to continuum mechanics and traffic. A simple but useful theorem sometimes known as the transport theorem allows us to use this kind of thinking also to write up Newton's laws of motion for a continuum, and perform many other useful stunts. So this will be a brief break from dimensional analysis and scaling, but we shall return to those topics soon enough.
As you may have understood from all this, I am still improvising a bit while I work out how to use the new textbook. I'll try to keep you informed as I plan ahead.
Oh, and check out the list of dimensionless numbers on Wikipedia. And even that one is not very comprehensive.
2005-09-15: I am going to jump around in the book a bit, and lecture it linearly from one end to the other. We'll jump into part III next, with chapters 12 and 13 coming up. I'll have a few more remarks on conservation laws and continuum mechanics first, however.
I apologize for a change of notation: In the lectures, I used T (really, T with a double arrow ↔ on top) for the stress tensor. But really, the usual notation, and the one used in the book, is σ, with the components σ_{ij}. If I had done this from the start, I could still have used T for the stress forces on a surface, T=n·σ, without the ambiguituous notation I forced myself into in the lectures. In component form, T_{j}=∑_{i}n_{i}σ_{ij}.
(I notice that I my web browser shows bold σ and plain σ the same. I meant to use the bold variant for the tensor and the plain one for the components. Mathematics on the web is still a problem.)
And while we're on the topic of notation: Note that the book uses a hat (like ^, but lower down) for the cross product, in accordance with British tradition – whereas the American, as well as the Norwegian, tradition uses a cross × for this product.
2005-09-19: The book is available in the bookstore, for those who didn't already get a copy. The books were probably there this morning but they had been put in a different place than previously, so I missed them. Remember the author's name – Howison – and follow the alphabet.
I will not be here on Friday, so Harald Krogstad will lecture in my place. I'll ask him to lecture on sections 13.4 and 13.5, more or less. (I will be in Bodø this Friday.)
As you can see above, I made a small note on conservation laws and continuum mechanics. It is really brief, but tries to summarize material I lectured on earlier that is not in the book. When I find more time, I will try to expand those notes a bit.
The misprint in the book that I mentioned in class today: It's on page 185, line 5 from the top. There are two equations side by side. The zero on the righthand side of the left equation should have been GMr_{1}/a^{3}=2ω^{2}r_{1}. Of course, I scaled the equations first, so that GM, a and ω got replaced by 1, and then the missing term is just r_{1}.
A challenge. I can't solve Exercise 2 of Ch 12 (on p. 180). Can you? (You may be glad I didn't try to assign it without trying to solve it first.)
2005-09-26: Today's lecture was on a model of enzyme kinetics, which is not in the book. I've put together a note instead (see above). I will have more to say on singular perturbation and matching on Friday. See the first part of chapter 16 in the book.
2005-10-09: I really should make a web page summarizing what has been lectured upon, with future plans as well. While we wait for this to happen, let me quickly summarize the last few lectures by stating that we have looked at singular perturbation problems with boundary layers, as in chapter 16. I went on to discuss lubrication theory, or long thin domains, sections 18.1–18.3.
Next up for discussion is the single nonlinear conservation law in one space dimension, with traffic flow (ch. 8) as the motivating example. We need to pick up some theory from ch. 7: basically sections 7.1 and 7.3.
The modelling seminar will happen in weeks 42 and 43. During week 42 there will be no ordinary lectures. We start on Monday 17 October in the usual lecture room with a brief presentation of the problem and the practical matter of creating the groups. The remaining time, for that lecture period and Friday's lecture period as well, will be spent working in groups. In week 43 there will again be lectures, and the groups put their reports together. On Monday 31 October the groups get to present their results. A final report is due later in that week.
2005-10-13: A slight change of plans. While I lost most of this week to packing and moving my office (I am now in room 954), I had forgotten about a meeting I must attend that collides with part of the modelling project. I cannot in good conscience ask Harald K to stand in for me in advicing on the project, so instead I am going to ask him to give a lecture on Friday the 21^{st}. Group work on the modelling project, with advice available, will then happen in the lecture hours on two Mondays instead: the 17^{th} and the 24^{th}. The project will start with a presentation of the problem on Monday, at the usual location. I am sorry about the confusion, and hope it doesn't cause too much trouble.
2005-10-18: No regular exercises this week. The modelling project should be sufficent work for you all. I will be present during the exercise hour on Wednesday to answer questions about the current modelling project or older exercises (or anything else you care to ask).
In connection with the modelling problem, many of the groups are likely to end up with an unsolvable initial value problem for a singular second order differential equation. I have made a tiny note showing what the problem is, for a much simpler equation.
2005-10-19: The friction factor. The friction factor can be read off from a Moody diagram. As you can see, it is fairly constant for large Reynolds numbers. It depends also on the roughness called ε/D in the referenced diagram: The size of the surface irregularities divided by the pipe diameter. A typical tunnel of the kind we're discussing can be expected to have fairly rough walls, so a friction factor around 0.05 does not seem unlikely. (But beware that there are slightly different conventions for the friction factor, so it can vary by a factor of 4. Check the reference to make sure you have the right one.) You may also wish to look at the Darcy–Weisbach equation. The head loss h is popular among hydraulic engineers. It is related to the pressure drop: ΔP=ρgh.
2005-10-19: Using Maple. You can use either Maple or Matlab to solve an differential equation numerically, and to plot the result. I am more familiar with Maple myself, so I haven't tried this in Matlab. I have a Maple worksheet available that shows some ways to approach the problem. The ODE in that worksheet bears some very vague resemblance to the equations of our current problem. Don't trust that resemblance too much.
2005-10-26: What values? Some students have asked what values were used in the original situation that gave rise to our problem. Actually, I am not too sure about this, but it appears that a tunnel diameter of 2 m, a length of 1000 m, and a water depth of 100 m at the inlet could be representative. But it is of interest to see what happens with a range of different values. Note that restricting your attention to dimensionless quantities will reduce the number of different variables you need to vary.
2005-11-02: It has come to my attention that some parts of problem set 8 can be misunderstood. Point (a) represents a super simplification, some aspects of which is justified in (c). But for (e) and (f), you are not supposed to use the simplification of (d)! So by "the equations" in (f), I mean the heat equation (between (a) and (b)) plus the initial and boundary conditions of (b) and (c). (Are any more equations needed?)
As for turning this one and the group project, details are provided on the exercises page.
2005-11-06: Friday was spent looking at the thermistor case study (chapters 5, 17). At the top of p. 238, the author makes what seems like a leap of logic; it certainly took me a while to figure out. As it turns out, if I got it right, the conclusion should be as in the third equation on p. 238, only with −8/λ on the righthand side. The practical consequence of this is less dramatic than you might think, due to the rapid change of σ with u. I'll present this result together with a matching of inner and outer solutions (exercise 1, really).
After that, I intend to look at the analysis of equilibrium points of a dynamical system from the perturbation viewpoint (section 13.3).
2005-10-28: Problem set 8 is one that will count 10 %. To be handed in on Friday 11 November.
2005-11-10: Dealing with the thermistor problem took more time than expected last Monday. I wrote up the important points in a brief note: (pdf – A5/A4)
2005-11-18: Finally, I have finished the tiny little note on dynamical systems. (pdf – A5/A4)
2005-11-21: Another tiny note, this one on bifurcations: (pdf – A5/A4)
2005-11-24: Yes, I know that the suggested solutions for the exercises are woefully late. They remaining ones should all appear this weekend.
2005-12-07: I am currently away from Trondheim, and will not be back before the exam. Arne Kvarving will be available to answer questions on 13 December. (He has an exam himself on the 12^{th}, so please do not bother him unnecessarily before then.) I will answer questions as well as I can by email. I know discussing mathematics by email can be hard, but it is the best I can do under the circumstances.
You can actually send me handwritten messages by scanning them on the copying machine in the mail room. (See separate instructions.) You don't have to type in my email address: Just select the button named Åpne adressebok and pick my name from the menu, then use the Lagre button twice. Be sure to select the PDF format from the menu, especially if your question runs to more than one page. Otherwise the machine produces a multipage TIFF that I cannot decode. Be sure to include your email address on the question. To be safe, follow up with a regular email explaining that you sent a scan.
Bjarte Hægland will be available during the exam. But remember: He cannot answer questions on how to solve the exam, only on possible ambiguities in the problem text. He will be able to contact me by phone in case there are questions he cannot answer.