The influence of the non-Newtonian stress-strain relation of blood on the oscillatory shear index (OSI) and mean wall shear stress (WSS) are described. The unsteady linear 1D momentum equation is solved for a viscoelastic fluid, with six elements of the Maxwell type and one dashpot element connected in parallel. A novel numerical algorithm is described which uses the upwind finite difference method to solve the equation of momentum. Results obtained by using a finite difference approach show significantly higher values of OSI when blood is assumed to be a viscoelastic fluid compared with those of simplified Newtonian fluid model. The calculation of OSI in human normal conditions for the Newtonian fluid differs in 12% (if alpha=0.02) from the results obtained from using the viscoelastic model.