Fig. 5: Comparison of advective and diffusive transport.

Here the active carpet is a square lattice of perpendicular Stokeslets. a, b Flows (Eq. (88)) produced by a carpet of λ = 10 (sparse) and λ = 2.5 (dense), respectively, in the plane y = 0. Colours shows the flow magnitude and black arrows are streamlines. c The total advection flow, v_adv,z (x, 0, 2.5), normalised by the flow of a single actuator, u_⊥,z (0, 0, 2.5), for different lattice spacings: λ = 10 (blue), λ = 2.5 (green), λ = 1 (red). The flows vanish as λ decreases. d Comparison of advective and active diffusive transport. Black points show the normalised advection flow, Φ(ζ) (see Eq. (89)). The dashed lines are the decaying functions e^−ζ (dashed green) and \({e}^{-{\zeta }^{2}}\) (dashed blue), so the advection vanishes for dense carpets and large distances from the surface. However, the normalised diffusive transport (Eq. (90)) increases with ζ (red lines). e Mean square displacement (MSD) of a particle near a carpet with very weak active fluctuations compared to a strong mean force, Var(f) = 10⁻⁶ and \(\bar{f}=-1\). The other parameters used are density n = 1, h = 1, τ = 0.1. The red dashed lines show the prediction of Eq. (3). f Corresponding space-dependent diffusivity. The solid lines show the prediction of Eq. (4). Despite the strong advection currents near the carpet, the theory still holds beyond a certain distance from the surface.