# Stability of Confined Vortex Sheets

## TL;DR

This paper introduces a model for the evolution of inviscid vortex sheets in confined channels, revealing that solid boundaries do not alter the growth rates of instabilities but can accelerate their nonlinear development.

## Contribution

The study extends the Birkhoff-Rott equation to include boundary effects and analyzes the stability of vortex sheets in confined geometries, showing unchanged linear growth rates but faster nonlinear instability growth.

## Key findings

- Unstable modes' growth rates remain unchanged with confinement.
- Solid boundaries do not stabilize the vortex sheet.
- Confinement accelerates nonlinear instability growth.

## Abstract

We propose a simple model for the evolution of an inviscid vortex sheet in a potential flow in a channel with parallel walls. This model is obtained by augmenting the Birkhoff-Rott equation with a potential field representing the effect of the solid boundaries. Analysis of the stability of equilibria corresponding to flat sheets demonstrates that in this new model the growth rates of the unstable modes remain unchanged as compared to the case with no confinement. Thus, in the presence of solid boundaries the equilibrium solution of the Birkhoff-Rott equation retains its extreme form of instability with the growth rates of the unstable modes increasing in proportion to their wavenumbers. This linear stability analysis is complemented with numerical computations performed for the nonlinear problem which show that confinement tends to accelerate the growth of instabilities in the nonlinear regime.