## Differential and Integral Equations

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

### A monotone iteration for axisymmetric vortices with swirl

Alan R. Elcrat and Kenneth G. Miller

#### 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 \begin{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)} \end{equation} 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