Learn about the proof of Lamb-Chaplygin dipole

[gerar]

I learn about Lamb-Chaplygin dipole and try to find the proof of this topic.
so I found the next link (is named: link1): https://docs.google.com/file/d/0B3-t1lLIJWOLc2NmT1RIWV9HOWs/edit?usp=sharing&pli=1

In addition, I found another link (is named: link2): http://ics.org.ru/doc?pdf=875&dir=e

can anyone tell me please on which pages (and of which link) the proof is found?
I think the proof is found only at link1: 166-174 but I'm not sure.

any help appreciated!

 

[gerar]

explanation the next steps in other words (dipole)

I am trying to understand the lamb-chaplygin dipole solution.
can someone explain me the green lines in other words please?

this is the proof:

When introducing the stream function, the steps that you usually take are as follows:
1) replace u and v by the stream function.
2) Derive the horizontal momentum equation (for u) with respect to y and the other with respect to x.
3) Eliminate the pressure term, to end up with a single equation in ψ.

so:

v = -∂ψ/∂x, u = ∂ψ/∂y

(∂^2)ψ/∂x^2 + (∂^2)ψ/∂y^2 = f(ψ)

if we put f(ψ) = -(k^2)*ψ. where k is a constant, and transform to polar coordinates r, θ, we get:

(d^2)ψ/ dr^2 + (1/r) * (dψ/dr) + (1/r^2) * (d^2)ψ/ dθ^2 + (k^2)*ψ = 0

which is satisfied by:

where Js is a bassel's function.


any help appreciated! thanks!