Hemodynamics is a complex problem with several distinct characteristics; fluid is non-Newtonian, flow is pulsatile in nature, flow is three-dimensional due to cholesterol/plague built up, and blood vessel wall is elastic. In order to simulate this type of flows accurately, any proposed numerical scheme has to be able to replicate these characteristics correctly, efficiently, as well as individually and collectively. Since the equations of the finite difference lattice Boltzmann method (FDLBM) are hyperbolic, and can be solved using Cartesian grids locally, explicitly and efficiently on parallel computers, a program of study to develop a viable FDLBM numerical scheme that can mimic these characteristics individually in any model blood flow problem was initiated. The present objective is to first develop a steady FDLBM with an immersed boundary (IB) method to model blood flow in stenoic artery over a range of Reynolds numbers. The resulting equations in the FDLBM/IB numerical scheme can still be solved using Cartesian grids; thus, changing complex artery geometry can be treated without resorting to grid generation. The FDLBM/IB numerical scheme is validated against known data and is then used to study Newtonian and non-Newtonian fluid flow through constricted tubes. The investigation aims to gain insight into the constricted flow behavior and the non-Newtonian fluid effect on this behavior.