The time evolution of the mother body of a planar uniform vortex moving in an inviscid fluid

The time evolution of the mother body of a planar, uniform vortex that moves in an incompressible, inviscid fluid is investigated. The vortex is isolated, so that its motion is just due to self-induced velocities. Its mother body is defined as the part internal to the vortex of the singular set of the Schwarz function of its boundary. In the present analysis, it is an arc of curve (branch cut), starting and ending in the two internal branch points of this function, across any point of which the Schwarz function experiences a finite jump. By looking at the mother body from the outside of the vortex, it behaves as a vortex sheet having density of circulation given by the jump of the Schwarz function. Its name (mother body) is taken from Geophysics, and it is here used due to its property of generating, outside the vortex and on its boundary, the same velocity as the vortex itself. The shape of the branch cut and the jump of the Schwarz function across any point of it change in time, by following the motion of the vortex boundary. As it happens for a physical vortex sheet, the mother body is not a material line, so that it does not move according to the velocities induced by the vortex. In the present paper, the cut shape, the above jumps, as well as the cut velocities are deduced from the time evolution equation of the Schwarz function. Numerical experiments, carried out by building the branch cut and calculating the limit values of the Schwarz function on its sides during the vortex motion, confirm the analytical calculations. Some global quantities (circulation, first and second order moments) are here rewritten as integrals on the cut, and their conservation during the vortex motion is analytically and numerically verified. Indeed, the numerical simulations show that they behave in the same way as their classical contour dynamics forms, written in terms of integrals on the vortex boundary. This proves that the shape of the cut, as well as the limit values of the Schwarz function on its sides, are correctly calculated during the motion.