The rheological modeling of a polymer melt in the course of its extrusion through a circular cylindrical channel inside a shaping die equipped with a rotating mandrel and a rotating or stationary nozzle is investigated. This paper attempts to provide a detailed rheological description of the physical processes of a polymer melt based on Leonov model inside the channel of an extruder under its kinematic deformation upon two-dimensional shear flow. Based on the obtained rheological equations of state, the flow process can be described in a form of dependencies reflecting kinematics of complex polymer deformation in the die channel. These expressions allow estimation of the velocity distribution for the flow elements within the channel as well as of the flow-pressure characteristics of the process. A quantitative relation between flow characteristics, rheological properties of polymer, and technical parameters of the channel of an extruder was determined. Comparison of this theoretical model with experimental results validates the suggested model.