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Explanation for the next steps of chaplygin dipole

this post is the Chaplygin dipole, it's an interesting issue. Can someone explain me these steps in other words please? any Explanation of any step will help me, I hope that together I will ...
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Explanation for the next steps of lamb-chaplygin dipole

Can someone explain me the steps please (I mean steps 2 and 3)? I didn't understand it. 1) Any flow field represented by a stream function $\psi$(x,r), defined as: v = - $\frac{\partial ...
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Lamb-Chaplygin dipole

There is an interesting issue of hydrodynamics: whirlpools. I want to learn about the Lamb-Chaplygin dipole. Lamb-Chaplygin dipole corresponds to a steady solution of the two dimensional Euler ...