Velocity Induced by a Plane Uniform Vortex Having the Schwarz Function of Its Boundary with Two Simple Poles

Table 1

Behaviour with respect to the first property, consequences on the position of the circle 𝒞^⋆=𝜻^⋆(𝒞), and corresponding constraints on 𝛼, 𝛽, and 𝛾. The quantities 𝛽_𝑙 and 𝛽_𝑢 are defined in (B.1).

Kind	Relative position of 𝒞^⋆ and 𝒞	Constraints
1	𝒞^⋆ internal to 𝒞	𝛼>𝛽, 𝛾 and 𝛽<𝛽𝑙
2	𝒞^⋆ includes 𝒞	𝛽<𝛼<𝛾
3	𝒞^⋆ external to 𝒞 and not including it	𝛼<𝛽<𝛾 and 𝛽<𝛽𝑢