Three-dimensional quasiconformal mappings and axisymmetric problems

Quasiconformal mappings of axisymmetric domains are considered as a special case of three-dimensional transformations. For a three-dimensional steady irrotational flow of an inviscid incompressible fluid, two stream functions are introduced along with the velocity potential. Any solenoidal vector can be represented as the cross product of the gradients of two stream functions. As a result, a relationship between the velocity components and the stream functions is obtained for determining the velocity potential. On the one hand, these transformations underlie Lavrentiev-harmonic mappings. On the other hand, these conditions can be treated as a generalization of the Cauchy–Riemann conditions to the three-dimensional case. In this work, the generalized three-dimensional Cauchy–Riemann conditions for harmonic mappings are reduced to the usual Cauchy–Riemann conditions in polar coordinates of complex variable functions. Lavrentiev-harmonic conditions are used to construct an analogue of quasiconformal mapping of axisymmetric domains and to generalize mappings of axisymmetric domains to arbitrary domains. Examples of visualization of quasiconformal mappings of axisymmetric domains and their generalizations are given.