A continuum model for flow induced by metachronal coordination between beating cilia

In this numerical study we investigate the flow induced by metachronal coordination between beating cilia arranged in a densely packed layer by means of a continuum model. The continuum approach allows us to treat the problem as two-dimensional as well as stationary, in a reference frame moving with the speed of the metachronal wave. The model is used as a computationally efficient design tool to investigate cilia-induced transport of a Newtonian fluid in a plane channel. Contrary to prior continuum models, the present approach accounts for spatial variations in the porosity along the metachronal wave and thus ensures conservation of mass within the cilia layer. Using porous-media theory the governing volume-averaged Navier–Stokes (VANS) equations are derived and closure formulations are given explicitly for the model. This makes it possible to investigate cilia-induced flow with a continuum model in both the viscous regime and the inertial regime. The results show that metachronal coordination can act as a transport mechanism in both regimes. Porosity variations appear to be the key mechanism for correct prediction of the fluid transport in the viscous flow regime. The reason is that spatial variations in the porosity break the symmetry of the drag distribution along the metachronal wave. A new insight that has been gained is that the fluid transport reverses, thus switches from plectic to antiplectic metachronism, for the same cilia beat cycle when the wavespeed is increased such that inertial effects occur. Based on a parameter study, the net transport in the channel is described by a power-law relation of the amplitude, length and speed of the metachronal wave. It is found that the wavelength has the strongest effect on the viscosity-dominated fluid transport.