|Feb1-13, 02:18 PM||#1|
learn about the proof of Lamb-Chaplygin dipole
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-t...=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!
|Mar6-13, 07:30 AM||#2|
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.
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!
|chaplygin, dipole, hydrodinamics, physics|
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