3-D Quasi-Conformal Mappings and Generalization of Axisymmetric Case

Abstract

In this chapter, we consider quasi-conformal mappings of axisymmetric regions as a special case of 3-D transformations. For a steady irrotational flow of an incompressible inviscid fluid, two streamline functions are introduced. Any solenoid vector can be presented as a vector product of the gradients of two streamline functions. In order to determine the potential of the velocities, we obtain the relation between the velocity components and the streamline functions. These transformations are the basis for the harmonic mappings according to M.A. Lavrent’ev. On the other hand, these conditions can be considered as a generalization of the Cauchy-Riemann conditions in the 3-D case. As shown in our previous work, generalized 3-D Cauchy-Riemann conditions for harmonic mappings are reduced to the Cauchy-Riemann conditions for two functions of a usual complex variable. An analog of 3-D quasi-conformal mappings of axisymmetric regions is obtained as combination one functions of a polar complex variable and usual 2-D function. The application of the M.A. Lavrent’ev’s type of harmonic mappings allows to construct the analogs of quasi-conformal mappings of axisymmetric regions. The quasi-conformal mappings generalize the application of 2-D conformal mappings to the 3-D case in a natural way. The examples of visualization of quasi-conformal mappings of axisymmetric domains and their generalizations are given. The best proof of obtained results is their visualization.