# A fluid mechanic's analysis of the teacup singularity

## TL;DR

This paper investigates the formation of singularities in inviscid, wall-bounded fluid flows using numerical simulations of the Euler equations, emphasizing the role of pressure and flow confinement in the blowup process.

## Contribution

It introduces a pressure decomposition method and a primitive-variables model to analyze the singularity mechanism in axisymmetric swirling flows.

## Key findings

- Numerical simulations support the existence of finite-time singularities.
- Pressure confinement within the cylinder is crucial for velocity gradient blowup.
- The proposed model captures key mechanics of the singularity formation.

## Abstract

The mechanism for singularity formation in an inviscid wall-bounded fluid flow is investigated. The incompressible Euler equations are numerically simulated in a cylindrical container. The flow is axisymmetric with swirl. The simulations reproduce and corroborate aspects of prior studies reporting strong evidence for a finite-time singularity. The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity. Linearity of the pressure Poisson equation is exploited to decompose the pressure field into independent contributions arising from the meridional flow and from the swirl, and enforcing incompressibility and enforcing flow confinement. The key pressure field driving the blowup of velocity gradients is that confining the fluid within the cylinder walls. A model is presented based on a primitive-variables formulation of the Euler equations on the cylinder wall, with closure coming from how pressure is determined from velocity. The model captures key features in the mechanics of the blowup scenario.