A note on ciliated plane channel flow with a pressure gradient

An envelope model is applied to the case of a two-dimensional channel with ciliated parallel walls. The formulation assumes identical values of the longitudinal and transverse amplitudes, frequency and wavelength of the two walls; it allows for arbitrary phase relations and arbitrary (not too small) spacing, and it includes an externally imposed pressure gradient. General results of a second-order perturbation analysis of creeping flow are presented. The time-averaged steady mean velocity may be viewed as the sum of two contributions: that of the pressure gradient (Poiseuille flow), and that of ciliary-driven motion which, owing to nonlinearities, also depends on the pressure gradient and reduces to pure streaming in the absence of a pressure gradient. For zero pressure gradient, the ratio of the streaming velocity of the channel and that of a single sheet shows the degree to which streaming is augmented or impeded by flow interaction. This ratio increases for the symplectic and peristaltic cases, but decreases for the antiplectic case, as the width of the channel decreases for fixed values of phase relation and amplitudes. The net flow arising from streaming and pressure gradient is shown as pump characteristics, and associated efficiencies are given. The results indicate that propulsion (pumping) is greatest and most effective for symplectic metachronism in ciliated channels with predominantly transverse waves, that it is nearly as good for peristaltic motion, but that it is considerably inferior for antiplectic metachronism in channels with predominantly longitudinal waves.