Ovsyannikov plane vortex: The equations of the submodel

Abstract

In the classical one-dimensional solution of fluid dynamics equations all unknown functions depend only on time t and Cartesian coordinate x. Although fluid spreads in all directions (velocity vector has three components) the whole picture of motion is relatively simple: trajectory of one fluid particle from plane x=const completely determines motion of the whole plane. Basing on the symmetry analysis of differential equations we propose generalization of this solution allowing movements in different directions of fluid particles belonging to plane x=const. At that, all functions but an angle determining the direction of particle's motion depend on t and x only, whereas the angle depends on all coordinates. In this solution the whole picture of motion superposes from identical trajectories placed under different angles in 3D space. Orientations of the trajectories are restricted by a finite relation possessing functional arbitrariness. The solution describes three-dimensional nonlinear processes and singularities in infinitely conducting plasma, gas or incompressible liquid.