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 understand all! please explain me the steps in other words because I didn't understand what is written
In 1903 Chaplygin published another remarkable paper (Chaplygin 1903) devoted to the motion associated with a compact vorticity distribution in a two-dimensional unbounded inviscid flow. In the introduction of that paper he gave a precise formulation of the problem: Consider an unbounded mass of incompressible fluid in which the motion is parallel to the OXY plane; let the motion outside some circular cylinder be irrotational, the velocity being equal to zero at infinity. The question is to find a distribution of vortex lines inside the cylinder that gives rise to a uniformly translating vortex column with a continuous velocity distribution and with a positive pressure all around.
As a first example of the solution Chaplygin considered in detail a case of rectilinear motion of a circular vortex of radius $a$ with a constant translation velocity $v_0$. By superimposing on the whole fluid fluid a uniform velocity $-v_0$ he obtained a stationary problem of a steady vortex cylinder placed in a potential flow with uniform velocity at infinity. By choosing the polar coordinate system $(r,\theta)$ , with the origin at the centre of the cylinder, the stream function $\psi_1$ for the potential flow around the cylinder is written as: