Anisotropic finite-time singularity in the three-dimensional axisymmetric Euler equation with a swirl

TL;DR

This paper investigates finite-time singularities in three-dimensional axisymmetric Euler flows with swirl, showing that anisotropic contraction can lead to multi-valued velocities and genuine blow-up in velocity gradients, relevant for understanding fluid singularities.

Contribution

It demonstrates that anisotropic contraction in axisymmetric Euler flows can cause finite-time singularities, providing a simplified hyperbolic model and analyzing the singular behavior using characteristics.

Findings

Velocity flow exhibits multi-valued solutions near a rim

Genuine blow-up occurs in velocity gradients

Singularity is generic for many initial conditions

Abstract

The search of finite-time singularity solutions of Euler equations is considered for the case of an incompressible and inviscid fluid. Under the assumption that a finite-time blow-up solution may be spatially anisotropic as time goes by such that the flow contracts more rapidly into one direction than into the other, it can be shown that the dynamics of an axially symmetric flow with swirl may be approximated to a simpler hyperbolic system. By using the method of characteristics, it can be shown that generically the velocity flow exhibits multi-valued solutions appearing on a rim at a finite distance from the axis of rotation which displays a singular behavior in the radial derivatives of velocities. Moreover, the general solution shows a genuine blow-up which is also discussed. This singularity is generic for a vast number of smooth finite-energy initial conditions and is characterized by a local singular behavior of velocity gradients and accelerations.