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

Figure 2

Sketch of the invariant circle 𝒞_𝑖(𝑡) in the planes of the variables 𝐳 and 𝜻 and its image 𝐶_𝑖(𝑡) in the physical plane. The points 𝐳_𝑖𝑜(±1) and their images 𝐱^±_𝑖𝑜 are represented with green dots. The same colours (blue and red) are used for the arcs that are images through the maps 𝐳↦𝜻^⋆ and 𝐳↦𝐱 of the two arcs in which 𝒞_𝑖(𝑡) is divided by the points 𝐳=𝐳_𝑖𝑜(±1). For 𝐳 running on 𝒞_𝑖(𝑡), the point 𝐱(𝐳) moves on the closed curve 𝐶_𝑖(𝑡), first in a direction and then in the reverse one. 𝐶_𝑖(𝑡) is composed by two superimposed arcs of the circle 𝐶_𝑖(𝑡) (yellow line) between the points 𝐱⁻_𝑖𝑜 and 𝐱⁺_𝑖𝑜: its inside has a vanishing area.