The numerical simulation of viscoelastic flows was studied in this work. In particular the effect of mesh refinement on the quality and the convergence of the finite element method was examined, as well as the differences that may be found by using several rheological models to describe the behaviour of the non-Newtonian fluids. The finite element simulation of viscoelastic fluid flows results in non-linear simultaneous equation systems that have to be solved iteratively. The iterations for all the viscoelastic models and most flow geometries have been found to diverge when the stress level or the elasticity of the flow increases above some certain level. The limit of convergence depends both on the mesh used for the discretization of the flow domain and on the rheological model. The limit usually decreases with mesh refinement. The effect of the mesh refinement on the convergence and the accuracy of the solution was studied here in two flow geometries: flow into an abrupt contraction (4/1 contraction ratio) and slit flow over a transverse slot. The penalty formulation of the finite element method (FEM) was used to numerically calculate the stress and the velocity fields in the flow domain using a number of coarse and fine meshes. Several rheological models were used, with their parameters chosen so that they would best fit a certain polystyrene melt. The solutions obtained were compared to results of g flow birefringence measurements and streamline photographs of the same material flowing under the same conditions that were simulated. The range of conditions that were covered by the calculations was shear stress at the die wall of 0-43 kPa, flow rates of 0-17 (mm³/ sec mm-width) and elasticity of 0-11 Deborah number. Even though oscillations in all numerical solutions were observed around the corners of the flow domain, it was found that the overall agreement of the numerical results with the experimental data was reasonable. The coarse meshes showed lower oscillations near the comers, but the accuracy of their predictions were poor. The limit of convergence for such meshes was the highest. Finer meshes on the other hand, showed higher oscillations near the comers and lower limit of convergence, but more accurate results away from the corner. It seems that the optimum mesh for an engineering calculation is an intermediate fine mesh that will give relatively high limits and reasonable accuracy. On the effect of the rheological model, it was found that the lower limit of convergence was given by the upper convected Maxwell model (UCM). The Leonov-like model also gave low limits. The Phan-Thien Tanner (P-T T) and the White-Metzner (W-M) models, on the other hand, showed quite higher limits in terms of the maximum stress levels and flow rates that they could handle. In terms of the quality of the solution inside the convergence range of each model, there is very little difference between the results of the models. In general, the Phan-Thien Tanner and the White-Metzner models show slightly better solutions. A possible reason for the better behaviour of these two models is believed to be the shear thinning viscosity and primary normal stress difference coefficient that these models are able to predict in simple flows. A few other characteristics of the two flows that were studied included the hole pressure, the entrance pressure loss and the presence of extensional fields around the contraction. It was found that the numerical method gave lower results for the hole pressure than the experimental data. Two models (W-M and UCM) gave a maximum in the entrance pressure loss and then a decrease towards negative values as the wall shear stress in the die increased. The P-T T and the Leonov-like models predicted a monotonic increase with the wall shear stress. Finally, there are two areas with strong elongational flow field in the contraction flow. One extends along the centerline of the die and the other lies along a line that starts from the reentrant comer and extends towards the upstream wall at an angle of around 45° (but depending on the flow rate). It is believed that this area is related to the natural entry angle, at which the viscoelastic fluid enters the contraction. / Ph. D. / incomplete_metadata