The mathematical and computer modeling of axisymmetric flows is considered. We constructed a mathematical model of motion of a viscous incompressible axisymmetric flow of fluid in channels of complex section. We developed a software for the numerical realization of this model by the method of R -functions in the POLE system. Computational experiments results are presented here. Real devices developed by using the proposed approach are shown. The development of many devices and technologies is based on the study of characteristics of hydrodynamic flows. At present, mathematical modeling and computational experiments are more and more preferred in the research of various processes. However, in order that result of modeling correspond to a real process, correct mathematical models and efficient methods of solution of problems stated based on them are required. In the solution of problems of hydrodynamics, taking into account adequately geometric information on channels, which is realized in various computational methods with different degrees of efficiency, is of importance. The R-function method (RFM) developed by V. L. Rvachev [7, 9, 11], an academician of the National Academy of Sciences of Ukraine, enables one to take into account geometric information on channels on the analytical level and satisfy exactly the boundary conditions. Axially symmetric flows form a broad class. Among them are flows through various nozzles of circular section, confusers, diffusers, flows around bodies of revolution, etc. In the computation of these flows, methods for arbitrary three-dimensional domains are predominantly used [14, 17]. In this case, modeling is performed on the basis of a system of Navier–Stokes equations for physical variables (the velocity components V 1 , V 2 , and V 3 , and pressure P ). Taking into account symmetry enables one to reduce the order of the system and, thus, substantially decrease the time of computation. In the two-dimensional case, the computation of flows can be performed on the basis of equations for the stream and vortex functions and the Poisson equation for static pressure [16]. In the two-dimensional case, one can also remove the equation for vortex from the system and reduce the system of Navier–Stokes equations to a sequence of equations for the stream function and static pressure [2]. In the case of a three-dimensional space, it is impossible to introduce the stream function in a general case, but, in the axisymmetric case, the stream function exists, and modeling can be performed on the basis of equations for this stream function [12]. In the present paper, we consider a method for the solution of hydrodynamic problems in axisymmetric channels by the R -function method on the basis of equations for the stream function and static pressure. The use of this method for plane channels was considered in [1, 2, 10, 13]. The axisymmetric case was considered in [4, 12], in which the statement of the axisymmetric problem was realized in the form of equations for the stream function and static pressure. In the present article, we describe in detail and deduce the boundary conditions for the Navier–Stokes equation for the stream function and the Poisson equation for the function of static pressure, and present a sequence of linearized equations for the stream function and the structures of the solution of the