Consistent section-averaged equations of quasi-one-dimensional laminar flow

Abstract

Section-averaged equations of motion, widely adopted for slowly varying flows in pipes, channels and thin films, are usually derived from the momentum integral on a heuristic basis, although this formulation is affected by known inconsistencies. We show that starting from the energy rather than the momentum equation makes it become consistent to first order in the slowness parameter, giving the same results that have been provided until today only by a much more laborious two-dimensional solution. The kinetic-energy equation correctly provides the pressure gradient because with a suitable normalization the first-order correction to the dissipation function is identically zero. The momentum equation then correctly provides the wall shear stress. As an example, the classical stability result for a free falling liquid film is recovered straightforwardly.