Navier–Stokes Equation (NSE) in the Transport Phenomena for Momentum Transport in Pipe Flow

The flow of fluids through pipes is a fundamental problem in engineering, governing the performance of systems ranging from water distribution networks to chemical reactors. This work presents a comprehensive analysis of internal pipe flow using the Navier–Stokes equations as the foundational framework. Under assumptions of steady, incompressible, Newtonian, and fully developed flow, the governing equations are simplified and solved analytically for laminar conditions, yielding the classical parabolic velocity profile, the Hagen–Poiseuille solution. The derivation is complemented by dimensional analysis, which reveals the universal nature of the laminar velocity distribution when expressed in dimensionless form. Numerical validation using realistic fluid properties confirms the theoretical predictions and illustrates the sensitivity of flow rate to pipe radius. A comparison with turbulent flow regimes highlights the transition in momentum-transport mechanisms as Reynolds number increases, with turbulent profiles described by empirical correlations such as the one-seventh power law and the logarithmic law of the wall. The results provide not only a rigorous theoretical foundation but also practical insights for the design and optimization of piping systems in engineering applications.