Fig. 4.

Numerical calculations of hydrodynamic torque on cilia arrays of varying density. (A) The beat pattern of a single cilium from ependymal culture, data taken from Hamilton et al. (2020), is taken as an input. The cilium centre line is characterised by its tangent profile ψ(s,t) as a function of the arclength along the cilium s and at the time t. (B) We model each cilium by a rigid rod attached to a surface, tilted to an angle, calibrated from the cilium beat pattern. In the figure we also mark the angle that the cilium reaches during the transient period between power and recovery stroke. (C) An external shear flow u_∞ is applied along the y-axis; cilia beating direction is identified with the unit vector , in the x–y plane. The angle between the cilia beating direction and the external flow is θ. (D) Cilia carpets are then represented by regular arrays of N_r cilia, all tilted with an angle α along the beating direction . (E) The average hydrodynamic torque on an individual cilium, as a function of the angle with the external flow, θ, and cilia density ρ. The torque experienced by an individual cilium in a dense array of cilia in a shear flow decreases as the cilia density increases. The reported torques are normalised with T₀(θ=π/2)=0.55, which is the torque experienced by an isolated single cilium.