The study reported in this paper is based on computational fluid dynamics simulations of steady viscoelastic flows through a planar two-dimensional T-junction. The purpose of this study is to analyse the influence of constitutive model and fluid elasticity upon the main recirculating flow characteristics formed at the junction and the shear stress fields. The fluid here considered has some characteristics similar to blood which makes the study relevant to hemodynamics. By taking into account some fluid complexities, such as inherent viscoelasticity, which is shown to affect flow recirculation and in particular the shear stress distribution, the study gives a contribution to fundamental issues in the genesis of vascular diseases. We restrict attention in this study to steady state flows and analyse the influence of elastic effects, either by varying the Deborah number or by increasing the dispersed phase elasticity-inducing concentration parameter and extensibility. In these steady simulations we apply either the FENE-CR or the FENE-MCR constitutive models, which are based on the non-linear finite extensibility dumbbell model but assume different simplifications in the derivation of the respective differential model equations. Those simplifications were assumed to be only important in unsteady flows but the present results show that Lagrangian acceleration in steady-state flow do also lead to significant differences. (C) 2014 Elsevier B.V. All rights reserved.