Surfing on metachronal waves: ciliary transport by inertial coasting

Abstract

Motile cilia drive biological fluid transport through whip-like beating motions that synchronize into metachronal waves. The lengths of these cilia span three orders of magnitude, from microns in human airways to millimeters in ctenophores. While recent studies have considered ciliary flows at intermediate Reynolds numbers, the effect of inertia on coordinated particle transport remains unexplored. Here, we address this gap using "Pufflets," the inertial counterparts of Stokeslets. These Pufflets describe rapidly accelerating flows generated by short-lived impulses, encoded by spatiotemporally singular momentum injections. To produce such rapid impulses experimentally, we designed an Atwood machine that generates long-lived Pufflet flows, which we capture with high-speed PIV measurements that agree well with analytical theory and simulations. Moreover, we find that pairs of equal and opposite Pufflets can drive net particle displacements and mixing due to time reversal symmetry breaking, which would be impossible in Stokes flow. Finally, we consider metachronal waves of Pufflets. Remarkably, we discover that particles can surf on these waves by coasting inertially from one cilium to the next, leading to highly efficient particle transport. This work paves the way toward understanding rapidly accelerating flows and collective transport driven by biological and artificial cilia.

Figures (7)

• Figure 1: Ciliary transport across the scales.(A) Organisms using cilia at intermediate transient Reynolds numbers: Stylonichia spp. (left), Spirostomum ambiguum (center) and the ctenophore Mnemiopsis leidyi (right). Image by Bruno Vellutini, CC3.0. (B) Tracer particle transported by a metachronal wave on the cell surface of Spirostomum ambiguum. (C) Characteristic length and time scales determining the Reynolds number $\text{Re}$ and transient Reynolds number $\text{Re}_t$. (D) Map of $\text{Re}$ and $\text{Re}_t$ for ciliated organisms across the scales. From databases velho_rodrigues_bank_2021 and heimbichner_goebel_scaling_2020. (E) Flow relaxation for non-inertial Stokeslets and inertial Pufflets. (F) Diffusion of vorticity generated by a Pufflet. Contour plots present the magnitude of vorticity. The point of maximal vorticity (red dot) does not coincide with the vortex center (green dot) surrounded by streamlines (white).
• Figure 2: Measuring Pufflet flows. (A) Experimental set-up with Atwood machine, laser sheet, and high-speed camera. (B) Cartoons describing the strong but short-lived force actuation. (C) Velocity of the sphere as a function of time, compared to the velocity of the fluid at a point located at $x = 2.5 r$ and $y=z=0$. (D) Comparison of experimental and theoretically modeled streamlines at $t=8$ ms (after the beginning of ball movement). (E) Experimental and theoretical velocity profile along the red dashed line shown in panel D, at the same $t=8$ ms. The vortex position is found as the root of the velocity profile. (F) Experimental vortex positions against time, compared with theory. (G) Simulation of Lagrangian grid displaced, distorted by a single Pufflet. Characteristic trajectories are plotted in color.
• Figure 3: Irreversible mixing by cyclic forcing. (A) Experimental setup and schematic cartoon for a two-way forcing Cyclet experiment. (B) Comparison of experimentally obtained and theoretically modeled trajectories starting at the same location. (C) Histogram of normalized displacement after the whole cycle (two opposite impulses), for experimental Stokeslet and Pufflet cases. (D) Lagrangian grid distortion and representative trajectories illustrating the time-irreversibility of Pufflets. (E) Mixing by multiple iterations of Cyclets. (F) Mixing number as a function of the number of iterations $k$, obtained from simulations presented in panel E.
• Figure 4: Long-ranged transport by surfing on a wave of Pufflets.(A) Deformation of a square grid of fluid parcels by a wave of Pufflets with different spacings (see sketch), $\delta = 1, \, 0.5, \, \text{and} \, 0.25$. Scalebars in all panels are $0.25$. (B) Example trajectories for the same waves as in panel A. Yellow trajectories show transport along the waves, and blue ones display only transient motion. (C) Poincaré maps in the steady state for two different waves, plotted in the wave reference frame. Each trajectory shows the motion of a point transported by a wave, with the origin being the last active Pufflet. (D) Transported volume by a Pufflet wave, which is non-zero for sufficiently small spacing $\delta$ and scales as $\delta^{-3}$.
• Figure S1: (A) Smoothed sphere velocity for a single Pufflet with $T_{acc}$ as the time between the sphere's half-max and max velocities. (B) Smoothed sphere acceleration for a single Pufflet with $T_{acc}$ as the time between the max acceleration and $a=0$. A double-Gaussian fit is performed to calculate $U_0$ and $T_{acc}$.