Numerical simulation of incompressible laminar flows of Newtonian and non–Newtonian Fluids is studied.For non–Newtonian fluids, a shear thinning Power law model is employed. The problem attempted in the study is a two–dimensional 1:4sudden expansions as well as 4:1 contraction channels in Cartesian coordinates.Axisymmetric pipe geometry in cylindrical polar coordinates is moreover considered. All domains may be filled with porous matrix or in absence of Porous Media.For two–dimensional model,the basic governing system of equations covers the equation of continuity and Darcy–Brinkman momentum equation.For well posed problems suitable initial and boundary conditions are adopted. A numerical formulationapplied in the existing investigation is based upon the time–dependent finite element scheme.Applied numerical method is so called Taylor–Galerkin/pressure–correction algorithm. Algorithm is design in such a way that, time derivative is discretised in two steps. At first step, forward difference is used at half time step. While, for full time step, a central difference scheme is adopted, to make the scheme second order accurate. Pressure is dealt with projection method introducedCrank–Nicolson choice (Ɵ = 0.5) and on diffusion term implicitness is applied to form a semi–implicit scheme. Adopting time marching scheme steady solutionsare sought to investigate various flow features. In both expansion and contraction flows, main interesting phenomena is development of vortices at different places, vortex enhancement and excess pressure drop. Other motivated features of the problem are the effects of fluid inertia on recirculation flow rate, size in terms of length and intensity, and location of vortices. For excess pressure drop,Couette–Correction is also inspired phenomena.For non–Newtonian fluids, investigation of shear–thinning effects is besides very important flow feature. In expansion flows in absence of porous material, at two different locations vortices develops, such as, the silent corner and the lip of expansion channel.For Newtonian flow, these vortices grow with increasing inertia. While, further increasing inertia progressively these vortices enriched and merge a centre of both eddies is observed. Furthermore, at high value of inertia vortex augmentation in the downstream is perceived. Regarding vortex size in terms of vortex intensity and length correspondingly growth with increase in inertial values is witnessed. Regarding pressure drop, the Couette–Correction decline linearly up to Re = 05, while, after this stage the Couette–Correction enhances linearly with increasing inertia. Whereas, in the presence of porous space in expansion flows, all flow features diminish such as, vortex enhancement, vortex length and intensity. No evidence of vortex development has been detected even at high inertial values and changing permeability of porous material. The Couette–Correction remains static with increasing inertia in porous matrix. In contraction channel, initially, at low inertial value in salient corner a small vortex is observed. With increasing inertial values this initial small vortex diminishes. Fluid inertia pushes the fluid towards contraction wall and reduces the size of eddy. The Couette–Correction enhances linearly with increasing inertia. While, with the introduction of porous material no evidence of vortex is admitted even with changing permeability/porosity. However, the Couette–Correction persist static with increasing inertia. For the simulation of axisymmetric 1:4 expansion tube, the flow of Newtonian fluids has been investigated. Initially, a small vortex develops in the silent corner at low value of inertia (Re=1). Subsequently, with increasing fluid inertia the vortex enhancement is observed. Both vortex intensity and length, initially, increase linearly. However, at high level of fluid inertia these quantities increase in non–linear fashion and Couette–Correction decrease in asymptoticshape. With the introduction of porous space, no vortex activity is detected and Couette–Correction remains stationary. For shear–thinning non–Newtonian fluids, adopting the Power law model at various indices, flow through axisymmetric pipe has been analysed. At high value of Power law index close to Newtonian fluid, with increasing inertia recirculation flow rate of fluid is observed in non–linear tendency. While, at low value of Power law index tendency of vortex development has been changed and eddy size remain steady. And Couette–Correction decreases asymptotically. In the presence of porous material, no movement of recirculation is found and Couette–Correction remains stationary. The excellent agreement of numerical predictions is attended based upon the employed finite element algorithm against other experimental as well as numerical solution.