A finite-volume (FV) procedure is applied to the prediction of two-dimensional (2-D) laminar flow through a 4:1 planar contraction of upper convected Maxwell (UCM) and simplified Phan-Thien-Tanner (SPTT) fluids. The method incorporates general coordinates, indirect addressing for easy mapping of complex domains, and is based on the collocated mesh arrangement. Calculations with the UCM model at a Reynolds number of 0.01 were carried out with three consecutively refined meshes which enabled the estimation of the accuracy of the predictions of the main vortex characteristics through Richardson＇s extrapolation. Converged solutions with the first-order upwind differencing scheme for the convective terms were obtained up to at least De = 8 in the finest mesh, but were limited to De less than or equal to 1, De less than or equal to 3 and De less than or equal to 5 for the fine, medium and coarse meshes, respectively, when using the second-order linear upwind scheme. The predicted flow patterns for increasing Deborah numbers with the UCM model resemble the well known lip vortex enhancement mechanism reported in the literature for constant-viscosity fluids in axisymmetric contractions and shear-thinning fluids in planar contraction, but very fine meshes were required in order to capture the described vortex activity. Predictions with the SPTT model also compared well with the behaviour reported in the literature.