The derived analytical expressions and developed numerical algorithms were tested for certain limiting cases (namely the flow of Newtonian fluids in circular tubes, the flow of Newtonian fluids in elliptical tubes and the flow of power law fluids in circular tubes) and expected trends and correlations (such as the expected shear thinning and shear thickening behaviors and their effects on the variation of flow velocity profile and volumetric flow rate) where these analytical expressions and numerical algorithms passed all these tests.