Two problems are solved in the work. Formulation of rheological model for polymer flows on the base of mesoscopic approach and mathematical modelling of three-dimensional flow in a converging channel with a rectangular cross section. The Vinogradov-Pokrovskii rheological model was generalised to a multi-mode case for a description of rheological behaviour of branched polymer melts. The contribution of each independent mode to a stress tensor corresponds to the individual polymer fractions differing in relaxation time and viscosity. Simultaneously a determination of an internal friction is expressed by means of the parameters depending on the first invariant of the anisotropy tensor. The theoretical predictions of the generalised model compared with the measured steady and transient rheological characteristics of two branched low density polyethylenes provide good agreement. Single mode rheological model is used for solving the problem of mathematical modeling of three-dimensional flow of a nonlinear viscoelastic fluid in a 14:1 planar contraction. Discrete analogs for partial differencial equations were obtained with control volume method separating physical processes. The numerical implementation is carried out using GPU-based parallel computing technology. Velocity and pressure fields have been calculated for two samples of polyethylene melts and we note circulating flow in the entrance of the slit channel. It is shown that the size of the vortex zone depends significantly on the melt rheology. The calculations imply that the increase in speed in the slit of the channel due to the three-dimensional nature of the flow field and the stress caused by the increase in the flow of LDPE compared with a sample of LLDPE. The three-dimensional nature of the flow LDPE confirmed the presence of the neutral component of the velocity in the flow direction, which was not observed for the flow of LLDPE.