Fluid transport by cilia between parallel plates

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Abstract
The problem of fluid transport by cilia is investigated using the Green's function for a Stokeslet between two parallel plates. The discrete-cilia approach is used in building the model, and a readily usable expression for the velocities is obtained. Dependence on the direction of the metachronal wave and on time is not averaged out. Velocity fields, pressure fields and fluxes due to a single Stokeslet and to an infinite line of Stokeslets are discussed. It is found that the flux associated with Stokeslets in between two parallel plates is always zero, in contrast to a Stokeslet parallel to, and above, one plate. In the model one also has to add a plane Poiseuille flow, which incorporates non-zero flux. The flow due to the Stokeslet solution imposes a positive pressure gradient downstream, and the Poiseuille flow a negative pressure gradient. Calculated velocity profiles, in the pumping range, are seen to be time-independent in the centre of the channel and vary between a negative parabolic profile and a plug flow. The reason for these profiles and some possible biological applications are discussed.

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