On modified quaternionic analysis, irrotational velocity fields and new applications of the Riesz system in R3

Abstract

Fueter’s construction of revolving planar mappings f around the real axis, developed by Leutwiler in 1992, is interpreted as a mathematical model for the meridional velocity fields in an axially symmetric inhomogeneous medium. We characterize new geometric and applied properties of some typical functions of the reduced quaternionic variable in the framework of generalizations of conformal mappings of the second kind in R3. The ñ-th power and new generalizations of the classical Joukowski transformation in the form of functions of the reduced quaternionic variable are studied. New velocity potentials in the form of φ-harmonic potential functions are given explicitly. Some unknown properties of the logarithmic function and the exponential function of the reduced quaternionic variable are demonstrated. In contrast to theory of functions of a complex variable, in modified quaternionic analysis the Hessian matrix of each potential function has variable trace in R3. The Hessian matrix is interpreted as the deformation rate tensor, which can be decomposed into its spherical part and its deviatoric part with zero trace.