The present investigation deals with the effect of the shape of a stenosis on the flow characteristics of blood, having shear-thinning viscoelastic rheological properties by using a suitable mathematical model. Keeping the relevance of the physiological situation, the mathematical model is developed by treating blood as a non-Newtonian shear-thinning viscoelastic fluid characterised by unsteady Oldroyd-3-constant model through an axisymmetric irregular arterial stenosis obtained from casting of a mildly stenosed artery (cf. Back et al., 1984). Comparison with the well-known cosine-shaped stenosis, in order to estimate the effect of surface roughness on the flow characteristics of blood, has however not been ruled out from the present study. Numerical illustrations are presented for a physiological flow, as well as for an equivalent simple pulsatile flow with equal stroke volume to that of the physiological flow, and the differences in their flow behaviour are recorded and discussed. The Marker and Cell method is developed in cylindrical co-ordinate system in order to tackle the highly nonlinear governing equations of motion. The effects of the quantities of significance such as Reynolds number, Deborah number, blood viscoelasticity and flow pulsatility, as well on the velocity components, pressure drop, wall shear stress and patterns of streamlines are quantitatively investigated graphically. Comparison of the results reveals that although the behaviour of two different pulses are similar at the same instant of time, there exist some important deviations in the flow pattern, pressure drop and wall shear stress as well. The present results also predict that the excess pressure drop across the cosine stenosis compared with the irregular one is consistent with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration.