On a quasi-conformal joukowski type transformation of second order in RM+1

Abstract

The classical Joukowski transformation plays an important role in different applications of conformal mappings, in particular in the study of flows around airfoils. Generalizations of this transformation where used in the 1920s by J. L. Walsh in order to approximate a continuous function on the boundary of its domain. Later, in the 1980s, H. Haruki and M. Barran studied generalized Joukowski transformations of higher order in the complex plane but from the perspective of functional equations. The aim of our contribution is to present a second order Joukowski type transformation in Rm+1, but the construction also shows how to proceed in the case of higher orders. Like in the complex plane it still preserves some of the main properties of the ordinary Joukowski transformation (thereby justifying to be called a Joukowski type transformation), but also reveals some new and less expected properties. We deal in some detail only with the 3D-case corresponding to m = 2 and discuss its properties and visualizations for different geometric configurations.