Multidimensional fluid motions with planar waves

TL;DR

This paper generalizes classical fluid solutions to three dimensions by allowing fluid particles in a plane to move in different directions, capturing complex nonlinear processes and singularities in plasma, gas, or liquids.

Contribution

It introduces a symmetry-based generalization of 1D fluid solutions to 3D, incorporating variable directions of particle motion dependent on all spatial coordinates.

Findings

Describes 3D nonlinear fluid motions with superposed trajectories.

Models singularities in plasma, gas, and liquids.

Provides a framework for analyzing complex multidimensional fluid behaviors.

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.