The mechanics of the blood flow in a flexible tapered artery with stenosis is studied from the viewpoint of a mathematical model. The flowing blood is represented by a two-fluid model, consisting of a core region of suspension of all erythrocytes assumed to be characterized by a Casson fluid and a peripheral plasma layer free from cells of any kind as a Newtonian fluid. The moving wall of the artery is treated as an anisotropic, linear viscoelastic incompressible circular cylindrical membrane cell. The effect of the surrounding connective tissues on the motion of the artery wall is also given due attention. The unsteady flow mechanism, subjected to a pulsatile pressure gradient has been solved using the finite difference scheme by exploiting the appropriate physically realistic prescribed conditions. The present model is also employed to study the effect of taper angle, the wall deformation, the severity of the stenonis, the viscosity of the peripheral layer, and the non-Newtonian rheology of streaming blood on the dynamic flow field. Finally, the numerical illustration presented at the end of the paper provides an effective quantitative measure of the flux, the resistive impedance and the wall shear stress through their graphical representations and also a few comparisons with the existing results have been made in order to validate the applicability of the present model.