Carpet cleaning: Lagrangian transport by cilia*

Cilia are hair-like appendages that protrude from the surface of a range of eukaryotic cells. Their deformation in a wavelike manner generates a flow in the surrounding fluid. This flow is important in a myriad of biological contexts, from mucus transport in the sinus to microorganism locomotion. The classical theoretical approach for fluid transport by cilia is Taylor's waving sheet model, where an asymptotic calculation in the sheet amplitude leads to a nonzero time-averaged Eulerian flow. We revisit this classical problem and calculate explicitly the Lagrangian transport of suspended tracers induced by small-amplitude deformations of the sheet. Using a multiple-scale analysis, we show that the Lagrangian drift transports particles on trajectories characterised by the superposition of a periodic orbit and a slower time horizontal drift, in a manner that, in contrast with the Eulerian flow, varies with the distance to the sheet. Our theoretical results are validated using numerical computations.