A 3D vortex formed by trajectories of the Navier–Stokes equation

Abstract

.The 3D Navier–Stokes equation for incompressible fluid is resolved by using the Lie group analysis method. The solution shows a spiral vortex structure and some strange related aspects. There are two running modes of the vortex, up-spiral or down-spiral, and the modes are determined by the viscosity of fluid. The calculated z-directional curl of the velocity presents a zero-piercing, which indicates that the variation from the up-spiral structure to the down-spiral may correspond to the state change of the fluid from gas to liquid. In the corresponding parameter region, there is likely a so-called two-phase-coexistence of gas and liquid. Within the vortex, the dependence of the angle velocity of a mass particle on its radial position is dominated by a Gaussian-like curve in the plane of angle velocity versus radial position. This work provides a vivid description of the vortex that is in accordance with a real vortex structure.