First, we obtain the and components of the velocity vector.
Second, we make the change of variables
We substitute the expressions for and in terms of and in the equations for and . Then, we set , and we define and . Now we write
Third, evaluate the radial velocity at the circles surface. The radial velocity can be written in terms of the velocities and according to the equation
With this equation and the equations for and at the surface of the circle, we have