<abstract xmlns="http://www.w3.org/1999/xhtml"><p>The planar interactions between pair of vortices in an inviscid fluid are analytically investigated, by assuming one of the two vortices pointwise and the other one uniform. A novel approach using the Schwarz function of the boundary of the uniform vortex is adopted. It is based on a new integral relation between the (complex) velocity induced by the uniform vortex and its Schwarz function and on the time evolution equation of this function. They lead to a singular integrodifferential problem. Even if this problem is strongly nonlinear, its nonlinearities are confined inside two terms, only. As a consequence, its solution can be analytically approached by means of successive approximations. The ones at 0th (nonlinear terms neglected) and 1st (nonlinear terms evaluated on the 0-order solution) orders are calculated and compared with contour dynamics simulations of the vortex motion. A satisfactory agreement is keept for times which are small with respect to the turn-over time of the vortex pair.</p></abstract>

The planar interactions between pair of vortices in an inviscid fluid are analytically investigated, by assuming one of the two vortices pointwise and the other one uniform. A novel approach using the Schwarz function of the boundary of the uniform vortex is adopted. It is based on a new integral relation between the (complex) velocity induced by the uniform vortex and its Schwarz function and on the time evolution equation of this function. They lead to a singular integrodifferential problem. Even if this problem is strongly nonlinear, its nonlinearities are confined inside two terms, only. As a consequence, its solution can be analytically approached by means of successive approximations. The ones at 0th (nonlinear terms neglected) and 1st (nonlinear terms evaluated on the 0-order solution) orders are calculated and compared with contour dynamics simulations of the vortex motion. A satisfactory agreement is keept for times which are small with respect to the turn-over time of the vortex pair.

A study of the interactions between uniform and pointwise vortices in an inviscid fluid

Department of Industrial and Information Engineering, Second University of Naples, Italy, and CNR-INSEAN, Marine Technology Research Institute, Italy

Communications in Applied and Industrial Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

The planar interactions between pair of vortices in an inviscid fluid are analytically investigated, by assuming one of the two vortices pointwise and the...

A study of the interactions between uniform and pointwise vortices in an inviscid fluid

Published Online: May 20, 2016

Page range: 4 - 22

Received: Nov 06, 2014

Accepted: May 18, 2015

DOI: https://doi.org/10.1515/caim-2016-0016

Keywords
inviscid two-dimensional vortex dynamics, uniform vortex, point vortex, contour dynamics, Schwarz function, nonlinear singular integral equation

© 2016 Giorgio Riccardi, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

A study of the interactions between uniform and pointwise vortices in an inviscid fluid

Published Online: May 20, 2016

Page range: 4 - 22

Received: Nov 06, 2014

Accepted: May 18, 2015

DOI: https://doi.org/10.1515/caim-2016-0016

Keywordsinviscid two-dimensional vortex dynamics, uniform vortex, point vortex, contour dynamics, Schwarz function, nonlinear singular integral equation

© 2016 Giorgio Riccardi, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
inviscid two-dimensional vortex dynamics, uniform vortex, point vortex, contour dynamics, Schwarz function, nonlinear singular integral equation