Exact Solutions of the Euler Equations for Rotationally-Symmetrical Motions of Incompressible
Liquid

Abstract:
Rotationally-symmetrical motions of ideal incompressible liquid present a great interest in connection with the possibility of their description of catastrophical effects in the environment such as whirlpools, waterspouts, tornado and cyclonic vortices.

For plane and axial-symmetrical motions of ideal incompressible liquid there were proved global existence and uniqueness theorems, while in the case of rotational symmetry there are still only local results. Therefore it is important to have a wide set of exact solutions to these equations.

In the current paper we present three classes of such solutions: invariant solutions, obtained with the help of Andreyev-Rodionov transformation, partially invariant solutions and solutions, built on the base of partially invariant ones. Finding most of these solutions is reduced to the integration of systems of equations with two independent variables.

The built solutions describe motions with a localized swirl domain, flows in curvilinear channels, motions with peculiarities on the axis of symmetry such as sources or vortex lines.