In this paper, the unsteady pulsatile magneto-hydrodynamic blood flows through porous arteries concerning the influence of externally imposed periodic body acceleration and a periodic pressure gradient are numerically simulated. Blood is taken into account as the third-grade non-Newtonian fluid. Besides the numerical solution, for small Womersley parameter (such as blood flow through arterioles and capillaries), the analytical perturbation method is used to solve the nonlinear governing equations. Consequently, analytical expressions for the velocity profile, wall shear stress, and blood flow rate are obtained. Excellent agreement between the analytical and numerical predictions is evident. Also, the effects of body acceleration, magnetic field, third-grade non-Newtonian parameter, pressure gradient, and porosity on the flow behaviors are examined. Some important conclusions are that, when the Womersley parameter is low, viscous forces tend to dominate the flow, velocity profiles are parabolic in shape, and the center-line velocity oscillates in phase with the driving pressure gradient. In addition, by increasing the pressure gradient, the mean value of the velocity profile increases and the amplitude of the velocity remains constant. Also, when non-Newtonian effect increases, the amplitude of the velocity profile. 
Pulsatile magneto-hydrodynamic blood flows through porous blood vessels using a third grade non-Newtonian fluids model