Propulsion and maneuvering of an artificial microswimmer by two closely spaced waving elastic filaments

This work examines the effect of hydrodynamic interaction between two closely spaced waving elastic filaments on the propulsion and maneuvering of an artificial microswimmer. The filaments are actuated by a forced oscillation of the slope at their clamped end and are free at the opposite end. We obtain an expression for the interaction force and apply an asymptotic expansion based on a small parameter representing the ratio between the elastic deflections and the distance between the filaments. The leading-order interaction forces yield asymmetric oscillation patterns at the two frequencies $\left({\omega }_1,{\omega }_2\right)$ in which the filaments are actuated. Higher orders oscillate at frequencies which are combinations of the actuation frequencies, where the first order includes the $2{\omega }_1,2{\omega }_2,{\omega }_1+{\omega }_2$, and ${\omega }_1-{\omega }_2$ harmonics. For configurations with ${\omega }_1\approx {\omega }_2$, the ${\omega }_1-{\omega }_2$ mode represents the dominant first-order interaction effect due to significantly smaller effective Sperm number. For in-phase actuation with ${\omega }_1={\omega }_2$, the deflection dynamics are identical to an isolated filament with a modified Sperm number. Phase difference between the filaments is shown to have significant effect on the time-averaged forces. Optimal Sperm numbers for in-phase and antiphase actuation are calculated. Turning moments due to phase difference between the filaments are presented, yielding optimal maneuvering for phase of ${90}^{\circ }$. Calculation of the effect of hydrodynamic interaction on the propulsive forces yielded that antiphase beating is more efficient than the in-phase scenario, in contrast with the commonly used assumption of maximal efficiency of the synchronized state. Experiments are conducted to verify and illustrate some of the theoretical predictions.

Figure 1

Illustration of the examined configuration consisting of two oscillating elastic filaments immersed in a viscous fluid. The filaments, at rest, are parallel to the $x$ axis, and their centers oscillate within the $x\text{−}y$ plane. The distance between the bases of the filaments is $d_0$, and the deflections are denoted by $v_1$ and $v_2$ for filament 1 and 2, respectively. The length of both filaments is $l$.

Figure 2

Axial distributions of the amplitude and phase of the frequencies comprising $v_1$ for various configurations. In all cases filament 1 is actuated with amplitude ${\varphi }_1={15}^{\circ }$ and frequency ${\mathrm{\Omega }}_1=0.5$, the amplitude of filament 2 is ${\varphi }_2={\varphi }_1,\varepsilon =0.1$, and $S_p=2.1$. Panels (a)–(d) examine the effect of phase $\gamma$ difference for ${\mathrm{\Omega }}_2={\mathrm{\Omega }}_1$. The modes presented in panels (a), (b) oscillate at frequency ${\mathrm{\Omega }}_1$, and the modes presented in panels (c), (d) oscillate at frequency $2{\mathrm{\Omega }}_1$. Panels (e)–(h) examine the effect of an adjacent oscillating filament with ${\mathrm{\Omega }}_2=0.2{\mathrm{\Omega }}_1$ and $\gamma =0$. Panels (i)–(l) examine the effect of ${\mathrm{\Omega }}_2=5{\mathrm{\Omega }}_1$ and $\gamma =0$.

Figure 3

Schematic description of an artificial swimmer with two waving elastic filaments actuated at the base. Turning moment may be generated by difference in the average propulsion forces of the filaments.

Figure 4

Average force in the $x$ direction (a) and the ratio $\langle {\mathcal{F}}_i\rangle /\langle {\mathcal{M}}_i\rangle$ (b), which is proportional to energetic efficiency, vs the Sperm number $S_p=l/l_0$. In all cases, ${\varphi }_1={\varphi }_2={25}^{\circ },l_0=\sqrt[4]{s/\left(2\pi \omega {\xi }_{\perp }\right)}=70\text{mm}$ (and ${\omega }_1={\omega }_2)$, $d_0=14\text{mm}$, and $r_c=0.5\text{mm}$. Isolated filament is marked by smooth lines. Beating filament with an adjacent filament beating in-phase in marked by dashed lines. Beating filament with an adjacent filament beating in antiphase is marked by dotted line.

Figure 5

Average propulsion force vs Sperm number for filaments 1 (dotted lines) and 2 (dashed lines) with phase difference of ${45}^{\circ }$ (a), ${90}^{\circ }$ (b), and ${135}^{\circ }$ (c). The steering moment is marked by the red solid line in panels (a)–(c). Panels (d), (e), and (f) present deflection patterns of filament 1 for ${45}^{\circ },{90}^{\circ }$, and ${135}^{\circ }$, respectively. Panels (g), (h), and (i) present deflection patterns for $S_p=2.45$ of filament 2 for ${45}^{\circ },{90}^{\circ }$, and ${135}^{\circ }$, respectively. The tags at panels (d)–(i) mark six equally spaced times along the full oscillation period of the first filament. With the exception of phase, all parameters are identical to Fig. 4.

Figure 6

(a) Schematic description of the experimental setup consisting of two elastic filaments deforming due to a prescribed oscillation of the slope at their bases. (b) An illustrative frame obtained with the Canon EOS 60D DSLR camera during an experiment.

Figure 7

Deflection $V_1$ of filament 1 for ${\mathrm{\Omega }}_1=5$ and amplitude ${\varphi }_1={\varphi }_2={15}^{\circ }$ at six equally spaced times along the full oscillation period $P$. Dimensional values are $v_1=V_1\times 1.4\text{mm},\omega =\mathrm{\Omega }\times 0.1\text{Hz},x=X\times 150\text{mm,}$ and $t=T\times 0.62\text{s}$. Smooth lines denote theoretical results, dots denote experimental data, and dashed lines denote experimental data polynomial fit. The examine cases are (a) without an adjacent filament, (b) an adjacent filament with ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2$, (c) ${\mathrm{\Omega }}_2=0.8{\mathrm{\Omega }}_1$, and (d) ${\mathrm{\Omega }}_2=1.2{\mathrm{\Omega }}_1$. Inserts present the deflection of the filament at its free end $V_1(X=1)$ vs time. See movies 1–4 in Ref. [26].

Figure 8

Fourier decomposition of the experimental deflection $V_1$ at $X=1$, presenting amplitude $A_{\mathrm{\Omega }}$ vs frequency $\mathrm{\Omega }$. Panels (a)–(d) present the frequency decomposition of inserts of panels (a)–(d) in Fig. 7, respectively. Filled circle markers denote experimental data, and smooth blue lines denote the theoretical predictions. Red circles denote harmonics expected from the theoretical results. Panels (e)–(h) are magnifications of (a)–(d) in order to present the amplitudes of the $2{\mathrm{\Omega }}_1,2{\mathrm{\Omega }}_2,\text{and}{\mathrm{\Omega }}_1+{\mathrm{\Omega }}_2$ first-order frequencies.