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Here presented a study of finite element Taylor–Galerkin pressure correction scheme to predict the transport fluid phenomena of shear thinning Non–Newtonian fluid through the ratio 1:4 expansion pipe. The fluid flow problem is non–linear when shear thinning non–Newtonian fluid occurs due to change in viscosity in terms of shear rate. The main important model power law employed to investigate the shear thinning non-Newtonian fluid behave with different index rates 0.95, 0.90 and 0.80.The numerical finite element scheme study will affect the flow structure, streamlines patterns of Shear thinning Non-Newtonian fluids with enhancing the value of Reynolds number and employing power law Computations on various index rates of the recirculation flow rate (Ψv), vortex length(X are plotted. The numerical results are compared and excellent results are achieved.