I'm really stuck on the following question which is from a previous examination paper from years ago. I kind of get how to do part (a) but not the rest. The numbers on the side indicate how many marks that part is worth and thus indicating how much work is needed for each part. Any help is much appreciated.



(a) A 2D point vortex (PV) of circulation $\displaystyle \Gamma$ is located at the origin. Determine the fluid velocity $\displaystyle v_\phi (R) \underline{e_\phi}$ around the PV, and show that the flow field may be described in cylindrical polar coordinates by the stream function

$\displaystyle \psi (R) = \frac{\Gamma}{2\pi}ln(R) + C
$
where $\displaystyle C$ is a constant and $\displaystyle R^2 = x^2 + y^2$[4]

(b) A different system has a PV of circulation $\displaystyle \Gamma = \Gamma_0$ located at $\displaystyle (x,y) = (-l/2,0)$ and a second PV of circulation $\displaystyle \Gamma = - \Gamma_0$ at $\displaystyle (+l/2,0)$. $\displaystyle \Gamma_0$ is a positive constant.

Calculate the subsequent motion of the PV s, and deduce a transformation of the reference frame which ensures that the PVs do not move.[5]

(c) Write down the total stream function in the new reference frame and determine $\displaystyle v_x$ and $\displaystyle v_y$.[6]

(d) Determine the location of the stagnation points and give a rough sketch of the form of the stream lines.[6]


[$\displaystyle \Gamma = \oint v.ds$]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I know that $\displaystyle v_x = - \frac{\partial \psi}{\partial y}$ and $\displaystyle v_y = \frac{\partial \psi}{\partial x}$. I don't understand about the stream function in the new reference frame, it makes no sense to me. I would have thought the PVs would just move upwards in part(b)