There is fluid in between two boundaries $\operatorname{Im} z = a$ and $\operatorname{Im} z = -a$, with a vortex of strength $Q$ at the origin.
I need to find the complex potential using method of images.
Now, when you have this question but with a line source strength $Q$ instead, you end up with an infinite series of sources of strength $Q$ as you attempt to balance each boundary. However, when I balance the vortex with an image vortex above at $2a$ with strength $-Q$ and the same at $-2a$, it seems to me that the system is balanced, as whichever boundary you look at, the centre vortex seems to balance both the images.
Is this correct, and so I get a potential of $$w(z) = -{iQ\over 2\pi}\log(z) + {iQ\over 2\pi}\log(z-2ai) + {iQ\over 2\pi}\log(z+2ai)?$$
I can't really get my head around how vortices intereact with each other, when there is another vortex directly between them.
Thanks for any help!
prove that the two-dimensional irrotational motion of a liquid bounded by the lines y=0,y=2a due to a source at the point(o,a) is given by the complex velocity potential W=-mlogcosh(z/2a)