Initial stages of the interaction between uniform and pointwise vortices in an inviscid fluid

The motion of a uniform vortex in presence of a pointwise one in an isochoric, inviscid fluid is analytically investigated. The uniform vortex is initially circular and the point vortex lies inside or outside this circle. At successive times, the shape of the uniform vortex is accounted for by means of the Lagrangian form of the Schwarz function of its boundary. A novel mathematical approach is adopted, based on the time evolution equation of this function. It leads to a non-linear singular integral system, the analytical solution of which is addressed by means of successive approximations. The 0th order one neglects the non-linear terms, while the kth (k≥1) approximation accounts these terms as forcing ones, once they are evaluated in correspondence to the (k−1)th approximation. However, due to the increasing algebraic difficulties in handling these approximations, the present analysis is limited to the 1st order one. In several sample cases its description of the motion is compared to the fully non-linear numerical simulation and a satisfactory agreement is found, at least for small times.