Confinement-induced motion of ciliates

TL;DR

Addressing how ciliates move in corrugated channels where beating dynamics matter, the paper coarse-grains metachronal ciliary activity into a time- and space-varying slip velocity and uses lubrication theory to derive an Adler-equation form for the ciliate position. The method combines analytical reduction to $\dot{\xi} = a - b \sin \xi$ with Fourier decomposition and supports results with lattice-Boltzmann simulations. The key finding is a resonance between the wall corrugation and the metachronal wave that yields net propulsion even when bulk fluid motion is blocked, producing oscillatory and ballistic regimes and even reversals of swimming direction relative to the bulk. The work reveals nontrivial flow morphologies, a dip in mean speed near a critical corrugation $R_1^*/R_0$, and extends to higher harmonics and finite-length ciliates, with implications for microfluidic filtering and design of artificial microswimmers.

Abstract

The time dynamics of flagellar and ciliary beating is often neglected in theories of microswimmers, with the most common models prescribing a time-constant actuation of the surrounding fluid. By explicitly introducing a metachronal wave, coarse-grained to a sinusoidal surface slip velocity, we show that a spatial resonance between the metachronal wave and the corrugation of a confining cylindrical channel enables a ciliate to swim even when it cannot move forward in a bulk fluid. Using lubrication theory, we reduce the problem to the Adler equation that reveals an oscillatory and ballistic swimming regime. Interestingly, a ciliate can even reverse its swimming direction in a corrugated channel compared to the bulk fluid.

Figures (5)

• Figure 1: (a) Schematic of a long ciliate (green cylinder) that swims in a cylindrical channel with corrugated walls, which are depicted in black while the surrounding fluid is drawn in blue. The cilia are coarse-grained to an effective slip velocity $v_c$, represented by purple arrows on the surface of the ciliate, while the ciliate moves with velocity $v_p$. (b) Schematics of a reciprocal (above) and non-reciprocal (below) ciliary beat pattern. Pink arrows indicate the direction of motion of the single cilia. The metachronal wave travels from left to right with speed $v_m >0$. For a sufficiently sparse ciliary carpet, the net fluid flow of the reciprocal beat pattern is negligible.
• Figure 2: (a) Position $x_p$ of the ciliate plotted vs. time for different values of $v_0/v_m$ and $v_1/v_m$. (b) Snapshot of the fluid flow for $v_0=0$, $R_0/R_c = 3/5$, $R_1/R_0=1/3$, $R_0/\lambda_c=3/20$, $v_1/v_m=3.6$, taken at $t/T_c = 3.78$. The fluid streamlines are shown in black and the color indicates the flow-velocity component $v_x$. In this moment, the ciliate moves to the left. (c) Time-averaged ciliate velocity $\langle v_p \rangle$ and (d) period of oscillatory motion $T$ plotted vs. the amplitude of ciliary slip velocity, $v_1/v_m$, times the amplitude of channel corrugation, $R_1/R_0$, for several values of $v_0/v_m$. In all panels $\lambda_c = \lambda_w$.
• Figure 3: Time-averaged ciliate velocity $\langle v_p \rangle$ as function of (a) ciliary wave amplitude $v_1/v_m$, (b) corrugation amplitude $R_1/R_0$, and (c) ciliate curvature $R_0/R_c$ for $\lambda_c = \lambda_w$ and $v_0 = 0$. The two varied parameters of the curves are indicated in the legends by color and line style, respectively. Colored symbols show results from lattice-Boltzmann simulations. Inset in (a): master curve $\langle v_p \rangle / v_m = \sqrt{1-v_1^2/v_{*}^{2}} - 1$ with $v_* = 2v_m/\mathcal{G}_1$. In (b), (c) the magnitude $|\langle v_p \rangle|$ is plotted.
• Figure 4: Time-averaged ciliate velocity $|\langle v_p \rangle|$vs. the ratio $\lambda_w/\lambda_c$ for various $v_1/v_m$. (a) Long ciliate with length $L_c/\lambda_w \gg 1$. (b) Results from lattice-Boltzmann simulations for a finite ciliate with length $L_c / \lambda_c = 2.4$ and $d/L_c = 0.25$, where $d$ is the surface-to-surface distance between the finite ciliate and its periodic image. For both panels, $R_0/R_c = 3/5$, $R_0/\lambda_c=1/4$, and $R_1/R_0=1/3$.
• Figure S1: (a) Fourier components $\mathcal{G}_n$ as a function of corrugation amplitude $R_1/R_0$ for various $n$. (b) Mobility function $\mathcal{G}(x)$ as a function of position $x/\lambda_c$ for various $R_1/R_0$. (c) Flow rate $Q$ and (d) vorticity $\Omega$ as a function of time for various $R_1/R_0$. (e) Time-averaged flow rate $\langle Q \rangle$ and (f) vorticity $\langle \Omega \rangle$ as a function of $R_1/R_0$. For all panels, $R_0/R_c=1$, $v_1/v_m=5$, $v_0=0$, $\lambda_c/\lambda_w=1$.