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.

NOTE: The various links on this page (probably) won't work from the mirrored version on the school's undergraduate course pages. Please go to the original at https://personalpages.manchester.ac.uk/staff/matthias.heil/Lectures/Fluids/index.html before trying to download any of the handouts.

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.

Vortex shedding caused by the flow past a flat plate (snapshot 2)

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. Coursework (you will be expected to hand in homework on a regular basis) will account for 20% of the final mark.

Coursework:

Please hand in your coursework by Thursday 12 noon of the week following the one in which the sheet was covered in the examples class (so you'll have a second chance to ask questions about a example sheet if you can't get all the coursework done during the examples class). Please place your solutions into the envelope at my office door (room 2.224 in the Alan Turing building). I will return the marked homework (with solutions) in the following week (or so....).

Handouts:

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

Solutions:

Old exams:

This course only used to be given every two years so there's only a limited collection of previous exams available online. I believe you are able to locate the most recent ones (from 2006 and 2007). I don't have a copy of the actual exam paper from 2004 but here's the pdf version of the final draft that I submitted to the exam office who filled in dates etc.

Draft of 2004 exam (pdf) .

...and, just in case you're about to send me an email: "No, you can't have the solutions!".

Page last modified: December 18, 2008

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