In this investigation, shear-thinning and shear-thickening inelastic fluids through a contraction channel are presented based on a power-law inelastic model. In this regard, Navier–Stokes partial differential equations are used to describe the motion of fluids. These equations include a time-dependent continuity equation for the conservation of mass and time-dependent equations for the conservation of momentum. Numerically, a time-stepping Taylor Galerkin-pressure correction finite element method is used to treat the governing equations. A start-up of Poiseuille flow through axisymmetric 4:1 contraction channel for inelastic fluid are taken into consideration as instances to satisfy the method analysis. Here, the impacts of different parameters, such as Reynolds number (Re), the consistency parameter (k), and the power-law index (n), are examined. Mainly, the effect of these parameters on the convergence levels of solution components considering it the most important point of view. The findings demonstrate that the inelastic parameters have a significant influence on the rates of velocity and pressure temporal convergence, and this effect is observed significantly. Fundamentally, the rate of convergence for shear-thickening flow is found to be greater than the convergence for shear-thinning flow. In addition, the critical level of Reynolds number is also determined for shear-thinning and shear-thickening situations. In this context, we captured that the critical level of Re for a shear-thickening case is much higher than that found for the shear-thinning case.