Theoretical solution for laminar flow in partially-filled pipes

Partially-filled pipe flow as occurs in subsurface drains and sewers is computed by Manning's resistance equation or using the cross-sectional velocity distribution. Yet, Manning's equation is valid only for turbulent flow and no theoretical solutions and experiments are available for laminar, partially-filled pipe flow, although fully-filled pipe flow is well understood. This research solves the Navier-Stokes equations, using bipolar coordinates and the Fourier transform, for partially-filled pipe flow under steady uniform conditions, resulting in theoretical solutions for the cross-sectional velocity distribution, discharge, boundary shear stress and friction coefficient. Although the solutions are not tested with laminar flow data (a research need), they satisfy all boundary conditions and special cases. Particularly, their graphical interpretations agree qualitatively with related turbulent flow data, providing a benchmark for formulating analytical or empirical solutions for turbulent flow in the future. The proposed stage-discharge relationship is also useful for discharge measurements in drainage and sewerage systems.