3D Mappings by Generalized Joukowski Transformations

Abstract

The classical Joukowski transformation plays an important role in different applications of conformal mappings, in particular in the study of flows around the so-called Joukowski airfoils. In the 1980s H. Haruki and M. Barran studied generalized Joukowski transformations of higher order in the complex plane from the view point of functional equations. The aim of our contribution is to study the analogue of those generalized Joukowski transformations in Euclidean spaces of arbitrary higher dimension by methods of hypercomplex analysis. They reveal new insights in the use of generalized holomorphic functions as tools for quasi-conformal mappings. The computational experiences focus on 3D-mappings of order 2 and their properties and visualizations for different geometric configurations, but our approach is not restricted neither with respect to the dimension nor to the order.

On generalized Vietoris’ number sequences

“…Of course, the factorization of (16) by separating the factor (n) k k! which is nothing else than (k + 1) for n = 2 was the decisive step for recognizing (17) as the adequate generalization of (8) in the case n > 2.…”

A note on a one-parameter family of non-symmetric number triangles

“…We will try to omit as much as possible details from Clifford analysis, but due to the particular role in Clifford analysis we would like to stress in this introduction at least their origin as coefficients in the construction of generalized Appell polynomials in that framework. Those generalized Appell polynomials have recently received a lot of attention from several authors ( [3,8,15,16,20]) due to their important role in theory and applications ([6, 7, 9, 13]), specially in elasticity [3], PDE and special functions ( [5,12]), or 3D-quasiconformal mapping problems ( [10,11]). …”

3D deformations by means of monogenic functions

“…Therefore, for x 0 ¤ 0, the quasiconformality factor k X 0, 2 .x/ can be computed by formula (30). From formula (34), we can see that the quasiconformality constant k X 0,…”

“…First global results have been recently considered by Malonek et al in [27][28][29] (cf. [30]). The authors have studied the global behavior of higher dimensional analogs of the exponential function and the classical Joukowski transform in the context of quaternion and Clifford analyses.…”