A bi-directional low-Reynolds-number swimmer with passive elastic arms

Abstract

It has been recently shown that it is possible to design simple artificial swimmers at low Reynoldsnumber that possess only one degree of freedom and, nevertheless, can overcome Purcell's celebratedscallop theorem. One of the few examples is given by Montino and DeSimone, Eur. Phys. J. E, vol.38, 2015, who consider the three-sphere Swimmer of Najafi and Golestanian, replacing one active armwith a passive elastic spring. We further generalize this idea by increasing the number of springs andshow that it is possible to invert the swimming direction using the frequency of the single actuatedarm.

Figures (7)

• Figure 1: Three-sphere swimmer as proposed by Montino and DeSimone in montino_three-sphere_2015. Two spheres are connected by a spring with rest length $l_1$ and spring constant $k$. The third sphere is connected by an actuated arm. The distances between the spheres are given by $L_1(t)$ and $L_2(t)$.
• Figure 2: Numerical results for Montino and DeSimone's three-sphere swimmer. (A) Curves in configuration space resulting from numerical solution of equations (\ref{['equation:L2_Montino_deSimone']} - \ref{['equation:L2dot_v2_v3_Montino_DeSimone']}). The different colors correspond to different frequencies $\omega$ (in $\unit[]{rad ~s^{-1}}$) of the actuated arm. Note that the unit of length is $\unit[10^{-4}]{m}$. (B) Displacement $\Delta x$ (in $\unit[10^{-4}]{m}$) of the three-sphere swimmer with respect to the normalized time $t/T=t\cdot \omega/(2\pi)$ for different actuating frequencies $\omega$. montino_three-sphere_2015
• Figure 3: Four-sphere swimmer where the middle arm connecting spheres 2 and 3 is the activated part. Springs have rest lengths $l_i$ and spring constants $k_i$. Distances between the spheres are given by $L_1(t)$, $L_2(t)$ and $L_3(t)$. In this case, the motion of the middle arm is prescribed as: $L_2 = l_2(1 + \varepsilon \sin(\omega t))$.
• Figure 4: Three dimensional trajectories of the four-sphere swimmer depicting limit cycles. (A) Effect of various frequencies $\omega$ (in $\unit[]{rad~ s^{-1}}$) for a fixed $\kappa = 10$. (B) Effect of stiffness ratio $\kappa = k_3/k_1$ for a fixed $\omega = \unit[1]{rad~ s^{-1}}$.
• Figure 5: Average displacement per period ($\Delta X$, in $m$) of the swimmer computed directly by integration (in red) and using the circulation of the first-order approximation \ref{['eq:curlxi']} of $\text{curl}(\xi)$ (in blue), plotted as a function of $\omega$ (in $rad \, s^{-1}$), for $\varepsilon = 0.3$.