Laminar channel flow over long and moderate waves

Laminar Newtonian flow through a 2-dimensional channel with one sinusoidal wall is studied for wavelengths comparable to, or much larger than, the channel height. In the long wave limit which is solved analytically, the pressure is in phase with the wave slope and the shear stress is in phase with the wave height. As the wavelength is reduced, the shear stress maximum moves upstream of the peak and the pressure minimum moves toward the wave crest. Weakly-nonlinear effects increase the shear on the upstream side of the crest and cause a net increase in the average shear and pressure drop over the length of the wave. For moderate wavelengths the pressure drop and drag are found, from finite element calculations, to increase as amplitude to approximately the third power. The maximum shear stress moves upstream of the crest as Reynolds number is increased and the phase angle decreases as the amplitude is increased. The minimum pressure moves downstream, away from the inviscid value, as the amplitude is increased.