Laminar flow of a viscoelastic fluid obeying the linear simplified Phan-Thien/Tanner model (LPTT) is numerically studied in a planar channel partially obstructed by a cosinusoidal constriction. Based on published data (Tammadon-Jahromi et al., 2011) there is no excess pressure drop for this particular fluid when flowing through an orifice-plate. Numerical results obtained using OpenFoam software at a typically low Reynolds number suggest that there exists a strong competition between the fluid’s strain-hardening/shearthinning behavior on the one side with its first normal-stress difference in extension, on the other side, in controlling the pressure drop caused by the presence of the constriction. It is shown that, an excess-pressuredrop (epd) can correctly be predicted provided that use is made of a proper (inelastic) baseline in the definition of the “epd”. At moderate Reynolds numbers a flow-reversal is predicted to occur at the lee side of the constriction ruling out this technique as an extensional rheometer. It is argued that such vortices can be very useful in high-throughput microfluidic systems for mixing enhancement. To reduce the excessive pressure drop experienced by the fluid when working at high Reynolds numbers, it is shown that the Deborah number of the flow should be increased.