## Differential and Integral Equations

### A monotone iteration for axisymmetric vortices with swirl

#### Abstract

We consider steady, inviscid axisymmetric vortex flows with swirl in a bounded channel, possibly with obstacles. Such flows can be obtained by solving the nonlinear equation $$-\frac{\partial ^2\psi }{\partial z^2}-r\frac \partial {\partial r}(\frac 1r\frac{\partial \psi }{\partial r})=r^2f(\psi )+h(\psi ), \tag*{(0.1)}$$ where $f$ and $h$ are given functions of the stream function $\psi$, with $\psi$ prescribed on the boundary of the flow domain. We use an iterative procedure to prove the existence of solutions to this Dirichlet problem under certain conditions on $f$ and $h$. Solutions are not unique, and relations between different families of solutions are obtained. These families include not only vortex rings with swirl, but also flows with tubular regions of swirling vorticity as occur in models of vortex breakdown.

#### Article information

Source
Differential Integral Equations, Volume 16, Number 8 (2003), 949-968.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060577

Mathematical Reviews number (MathSciNet)
MR1988954

Zentralblatt MATH identifier
1161.76446

Subjects
Primary: 76B47: Vortex flows
Secondary: 35Q35: PDEs in connection with fluid mechanics

#### Citation

Elcrat, Alan R.; Miller, Kenneth G. A monotone iteration for axisymmetric vortices with swirl. Differential Integral Equations 16 (2003), no. 8, 949--968. https://projecteuclid.org/euclid.die/1356060577