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 waveleng th of the two walls; it allows for arbitrary phase relations and arbit rary (not too small) spacing, and it includes an externally imposed pr essure gradient. General results of a second-order perturbation analys is of creeping flow are presented. The time-averaged steady mean veloc ity may be viewed as the sum of two contributions: that of the pressur e gradient (Poiseuille flow), and that of ciliary-driven motion which, owing to nonlinearities, also depends on the pressure gradient and re duces to pure streaming in the absence of a pressure gradient. For zer o pressure gradient, the ratio of the streaming velocity of the channe l and that of a single sheet shows the degree to which streaming is au gmented or impeded by flow interaction. This ratio increases for the s ymplectic and peristaltic cases, but decreases for the antiplectic cas e, as the width of the channel decreases for fixed values of phase rel ation and amplitudes. The net flow arising from streaming and pressure gradient is shown as pump characteristics, and associated efficiencie s are given. The results indicate that propulsion (pumping) is greates t and most effective for symplectic metachronism in ciliated channels with predominantly transverse waves, that it is nearly as good for per istaltic motion, but that it is considerably inferior for antiplectic metachronism in channels with predominantly longitudinal waves.