# MATH35001: Viscous Fluid Flow

This course is concerned with the mathematical theory of viscous fluid flows. Fluid mechanics is one of the major areas for the application of mathematics and has obvious practical applications in many important disciplines (aeronautics, meteorology, geophysical fluid mechanics, biofluid mechanics, and many others). Using a general continuum mechanical approach, we will first derive the governing equations (the famous Navier-Stokes equations) from first principles. We will then apply these equations to a variety of practical problems and examine appropriate simplifications and solution strategies.

Many members of staff in the department have research interests in fluid mechanics and this course will also lay the foundations for possible future postgraduate work in this discipline.

#### Vortex shedding caused by the flow past a flat plate (snapshot 1)

This course is currently taught by Prof Matthias Heil. This page provides online access to the lecture notes, example sheets and other handouts and announcements.

Please note that the lecture notes only summarize the main results and will generally be handed out after the material has been covered in the lecture.

If you have any questions about the lecture, please see me in my office (2.224 in the Alan Turing building), contact me by email ( M.Heil@maths.man.ac.uk) or catch me after the lecture.

## Syllabus

• 1. Introduction; overview of the course; introduction to index notation. [2]
• 2. The kinematics of fluid flow: The Eulerian velocity field; the rate of strain tensor and the vorticity vector; the equation of continuity. [3]
• 3. The Navier-Stokes equations: The substantial derivative; the stress tensor; Cauchy's equation; the constitutive equations for a Newtonian fluid. [4]
• 4. Boundary and initial conditions; surface traction and the conditions at a free surface. [1].
• 5. One-dimensional flows: Couette/Poiseuille flow; flow down an inclined plane; the vibrating plate. [3]
• 6. The equations in curvilinear coordinates; Hagen-Poiseuille flow; circular Couette flow. [2]
• 7. Dimensional analysis and scaling; the dimensionless Navier-Stokes equations and the importance of the Reynolds number; limiting cases and their physical meaning; lubrication theory. [3]
• 8. The streamfunction/vorticity equations [2]
• 9. Stokes flow (zero Reynolds number flow) [2]
• 10. High-Reynolds number flow; boundary layers; the Blasius boundary layer. [2]

## Assessment:

The course will be examined in a two hour exam in January. "Formative feedback" will be provided via discussions in the weekly examples class and/or during my office hour (Thu 1-2; feel free to send me an email to arrange a specific meeting if you're not free during this time). In my experience the above arrangements are perfectly satisfactory for the students (if they aren't, then tell me!), but, sadly not for the people whose job it is to tell me how to teach. To please the latter, I am offering to provide written comments on the "starred question" (Question 1) on Example Sheet VI. Please hand in your write-up of the solution at the end of the Friday lecture following the examples class in which this sheet will be discussed (i.e. on Friday Nov 17th). I will then annotate your solution and return it in the following week. However, please note that the purpose of this exercise is purely "formative feedback" and does not bear any credit.

# Anti-cramming policy

This course used to have a 20% coursework component, introduced in order to force the students to work continuously. The weekly coursework resulted in a fairly heavy workload during term-time (which the students hated) but made exam revision very easy (which they loved). Overall, the coursework element was perceived to be an excellent feature of the course -- judging by the replies on the student questionnaires handed out at the end of the course.

Unfortunately, various constraints made it impossible to continue this very successful setup, facing me with the question of how to get you to work for the course throughout term, rather than adopting the "I don't have to work for this course because I can simply cram at the end of term and revise by looking at past exam papers" attitude that all lecturers (and me in particular) detest.

So, here is the deal:

• You are hereby told (yet again) that it is essential to work continuously on the material presented in this (and any other) lecture course. If you don't understand the concepts presented in week 1 you will not understand what I talk about in week 2, etc. The best (only?) way to achieve this is to work through the example sheets before the example/feedback class, to make sure you can bombard me with questions about any issues that you don't understand. There is little point in turning up for the example/feedback class without having looked at the problem sheet beforehand -- there's not enough time to do all the work in class.

• On each problem sheet I will identify a small number of (sometimes substantial) questions that used to form the coursework. The logic behind the selection of the questions is that they'll force you to understand concepts that are essential to the understanding of subsequent material.

• The (detailed) solutions to each problem sheet will be made available at some suitable time after the example/feedback class (probably a week later). At that point you should swap your nicely-written-up solutions (written up at least as nicely as if you were to hand them in to me) with another student and mark each other's work. No need for a marking scheme, just check what's right and what's wrong (and why!). In my experience, most students tend to work best in small groups anyway so I don't think you'll have problems finding people to swap work with. If you really have no mates, let me know and I'll find you somebody (we'll do triangular swaps if there's an odd number of students).

• I can see panic developing: Will the "grade" awarded by your mates "count towards the exam"? No! There won't be a grade. You're simply supposed to give each other feedback, and if there are any disputes, I'm happy to act as judge. In fact, I'll be delighted to go back over material in subsequent example/feedback classes if it turns out that many of you struggled with the same problem (assuming I didn't realise that during the example/feedback class itself). However, I can only do this if I know what you struggled with and this obviously requires you to have done the work.

• Now that the panic is gone, you'll obviously ask yourself why you should bother with this. Surely it'll be easier to use the example/feedback class to read the metro newspaper and exchange gossip, and then use the tried-and-tested "cramming for the exam" technique to "revise". I obviously can't stop you from doing that but I can assure you that, following the end of term, I will refuse to answer any questions (about the course material or previous exam papers) from students who do not have an (at least nearly) complete set of written-up "coursework", with signs that the work was seen by and discussed with somebody else. If you don't work, I won't either.

• Final question: "Hang on, you want us to do the questions, write up the answers, exchange them with another student, mark each other's work, and then discuss any mistakes/misconceptions with him/her (and the lecturer)? This will take a lot of time!" Answer: "Indeed -- I expect you to work hard for this course!" If you check the UG handbook you'll find that you're supposed to anticipate at least two hours of private study per lecture hour. Furthermore, whatever time you invest on this during term time, you'll save during the exam revision. (Note that this will free up valuable time to catch up with any past issues of the metro newspaper that you didn't have time to read during the examples class!).

## Scans of visualiser sheets:

My handwriting isn't exactly famous for being super-clear so here's another chance to stare at what I wrote on the visualiser. Keep shouting at me in the lecture (yes, really) if you can't read something and/or catch me before/after the next lecture if even close inspection of these scans leaves you at a loss as to what I could possibly have meant by all these scribbles...

## Corrections:

Please note a few corrections for previous handouts (the files above have already been corrected).

## Exam feedback:

I'm supposed to provide "feedback on the exam", so, for what it's worth, here's what I thought after marking the exam However, rather than wasting your time reading this (in an attempt to improve your grade by "improving your exam technique"), I suggest you have another read through the Anti-Cramming Policy above. Concentrate on actually understanding the material rather than trying to memorise likely exam questions and how to tackle them. Pleeeeease!

## Feedback on feedback:

I'm now also supposed to give you feedback on your feedback. Rant omitted. Here's my The inofficial one is: Thank you, I enjoyed it too!