## Sunday, 10 March 2013

### MST326 Fluids 2nd TMA

Well as promised here is my review of the 2nd Fluids TMA
Block 2 can be described as a crash course in the essentials of Fluid  dynamics it covers in 4  short units, Streamlines and path lines. The Euler equation, the equation of continuity, Bernoullis  equation and it's application to flows where the diameter changes, Vorticity and flow around shapes such as cylinders and last but not least the Navier Stokes equations which deals with viscous flow. As Feynman said in his lectures treating Fluid flow without consideration of viscosity is like treating the flow of dry fluids.

Anyway the questions covered the following topics

Question 1 We had to calculate the pathlines and streamlines for a two dimensional flow and sketch these for various questions. As the books say Straightforward but tedious especially the sketching. Got most of it out

Question 2. A relatively straightforward but again tedious  question involving the solution of Bernoullis equation for the flow of water over a hump. We had to calculate the variation in height along certain points. One slightly interesting aspect of all this is  that quite often one has to solve a cubic equation so a potential hint of Galois theory here, But the question usually involves solving the resultant equation by the Newton Raphson method. Unfortunately by the later half of the question I had lost the will to live so about 2/3 of the marks here

Question 3 Calculating the flow and vorticity for flow around a segment of a cylinder for a given stream function the vorticity is the curl of the velocity vector. The meat of the question involved the calculation of the force on the cylinder I got this out but again by the end of the question I had lost the wiil to live so again about 3/4 of the marks

Question 4 Involved the setting up of the Navier Stokes equation and the appropriate boundary conditions for the flow of two layers of liquid between two plates one of which was moving with constant velocity. This question was a test of both the physics and the maths. One had to justify the various approximations. Again the meat of the question involved the solving of the Navier Stokes equations for the problem this was straightforward but tedious especially as the form of the equation required in the final answer involved recasting some of the constants in terms of another constant. This just added to the tedium but I got there in the end. and got most of the marks I think

So overall I think I'll just get enough for a grade 2 pass. There was a lot of tedious curve sketching for this TMA and I think I prefer the mathematical aspects. Next block gets back to the nitty gritty of solving partial differential equations including a brief look at one of my favourite topics Sturm Liouville theory of which I've been having a little discussion on the quantum mechanics forum about.

In general Fluid mechanics is simply a matter of balancing equations and Newton's laws of motion. On the other hand who would have thought some thing quite conceptually straightforward would lead to the complications of something like the Navier Stokes equations

http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations

Fluid mechanics is just Conservation of Energy and Newtons second law but you have to know them really well.

As a footnote I got a respectable 72% for my last music TMA.

Thats all folks till next time. Will speak about the first quantum mechanics TMA next.