MST326 Fluids TMA01

Well I've just finished the first TMA for MST326 Fluids although the course is on Fluid dynamics much of Block 1 is devoted to developing mathematical techniques

The first unit does cover fluids essentially hydrostatics with an interesting application of how lock gates work and also development of simple atmospheric models. The first question of this TMA covered this I confess I couldn't quite see how to do the later half of the first part which involved working out stability conditions for an inverted half open cylinder immersed in a liquidso left it, The second part was based on developing a simple atmospheric model of the pressure in the mesosphere. This I found tedious but quite straightforward.

The second unit covers techniques for solving second order differential equations specifically for those differential equations which have polynomial coefficeints in front of the derivatives not just constants . These included the Cauchy Euler method, the method of variation of parameters which enables you to generate a second solution to a differential equation if you know the first solution, order reduction another technique for generating the second solution to the differential equation from the first, Finally solution by series essentially you substitute a trial solution in the form of a polynomial into the differential equation and then generate recurrence relations for the coefficients. This method is probably the most powerful method for solving second order differential equations with polynomial coefficients. It leads to the so called special functions such as Bessel functions Legendre polynomials etc associated with the classic differential equations which crop up all the time in physics.

 The question covered all these techniques and I think I got most of it out apart from the last part which was quite tricky algebraically and I couldn't get the equation for the reduction of order to resemble anything like 
the equation in the hint so I gave up

The third unit is on techniques for solving first order partial differential equations this involves the method of characteristics. The question on this I found quite straightforward and am reasonably confident of getting a good mark for this one.

Finally the fourth unit on block 4 is Vector calculus, whilst this will be familiar to those who have done MST209 this is covered in more depth in this unit and the question in the TMA was correspondingly a lot more tricky. One part in particular involved calculating the curl of a vector field in spherical polar coordinates to show that it was zero. In order to do this involved quite a bit of playing around with trig identities. I was gratified to get it out, as also the last part which involved verifying Gauss's divergence theorem by calculating both a volume integral and a surface integral I was really pleased with myself when I showed that the volume integral was indeed the same as the surface integral

So overall two full questions more or less correct, and 4/5th s of another question and 3/5ths of the first question should be enough for a respectable grade 2 on this one.

In general the TMA is quite challenging and not just a repetition of exercises with slight modifications in the book. I enjoyed the intensity of it all even though slightly frustrated in parts. I hope by June to have got in a lot more practice on the past exam papers than I did for MS324 and realistically I can expect grade 2. But I need to spread the load a bit rather than let it all pile up in the last few days before the deadline.

Best wishes to you all Chris