MAS286: Mathematics and Statistics in Action Assignment 3
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MAS286: Mathematics and Statistics in Action
Assignment 3: Vortices in Fluids
March 12, 2023
Administrative notes
• Your assignment should be uploaded as a PDF file to BlackBoard by the deadline of 10am, 1st May, 2023 (Monday in Week 10). Please include your registration number in the file name, with the convention 123456789_mas286_topic3.
• Any word processing software may be used, though LaTex is preferred. A sample LaTex file will be provided. Do not forget that the end-product of your assignment must be a single PDF file.
• The assignment should take the form of a technical report. It should, therefore, contain mathematical working but should also include written explanations of what you are doing and why. If you are not sure, aim for something similar, in style, to the lecture notes.
• The bulk of the marks will be for the mathematical work, but marks will also be awarded for clear explanations and justifications.
• You may refer to and use all the numbered equations used in the lecture notes without re-deriving them from scratch.
• For the assignment you will need to derive equations and a reasonable number of intermediate stages should be shown.
• Figures, if any, should be embedded into the body of the text.
• For an easier marking, please structure your submission according to the questions listed below.
• This is an individual assignment and must be your own work. You must not copy work from other students. You can ask for help on the MAS286 discussion board, but must not ask for help on any other
internet discussion forum, or email list or anything of a similar nature. Where we judge that plagiarism has occurred or two pieces of work are too similar, we may award zero for those assignments.
Assignment
1. Consider a system with two point vortices. Let C2 = κ 1z1 (t)+κ2z2 (t) and I2 = κ 1z1z 1(∗) +κ2z2z2(∗) . Calculate the time derivatives of C2 and I2 . [20 marks]
2. Consider a system with three point vortices. Let C3 = κ 1z1 (t) + κ2z2 (t) + κ3z3 (t) and I3 = κ 1z1z 1(∗) + κ2z2z2(∗) + κ3z3z3(∗) . Calculate the time derivatives of C3 and I3 . [30 marks]
3. As you might have suspected, the results of the above two questions can be generalised. State the results you believe to be true for a system of N point vortices. You do NOT need to prove your conjecture. [10 marks]
4. The above results are very useful in the discussion of point vortex systems. In particular, C2 and I2 can be used to find the movement of two point vortices. Consider again a system of two point vortices with κ 1 > 0 and κ2 > 0. Determine the centres and radii of the circular orbits of the vortices in terms of C2 and I2 and other parameters of the system. [20 marks]
5. Are C2 and I2 the only constants of motion for a system of two point vortices? The answer is no, as will be demonstrated in this question. Consider a system of N point vortices. Let H(t) be defined as
H = − 
\κℓκjln(|zℓ(t) − zj(t)|),
or equivalently,
H = − 
\κℓκjln [(xℓ(t) − xj(t))2 + (yℓ(t) − yj(t))2] .
dxj ∂H dyj ∂H
from which it can be shown further that dH /dt = 0. Prove the statements for the case where N = 2. [10 marks]
6. As we have seen in Question 4, the motion of two point vortices is completely determined by C2 and I2 . This raises a question: since H is another constant of motion, why does it not play a role in the solution? The reason is, for a system with two point vortices, H can be written in terms of C2 and I2 and other parameters of the system. Find this relation. [10 marks]
2023-04-11
Vortices in Fluids