Utilizing passive elements to break time reversibility at low Reynolds number: a swimmer with one activated element

Abstract

In the realm of low Reynolds number, the shape-changing biological and artificial matters need to break time reversibility in the course of their strokes to achieve motility. This necessity is well described in the so-called scallop theorem. In this work, considering low Reynolds number, a novel and versatile swimmer is proposed as an example of a new scheme to break time reversibility kinematically and, in turn, produce net motion. The swimmer consists of one sphere as a cargo or carried body, joined by one activated link with time-varying length, to another perpendicular rigid link, as the support of two passively flapping disks, at its end. The disks are free to rotate between their fixed minimum and maximum angles. The system’s motion in two dimensions is simulated, and the maneuverability of the swimmer is discussed. The minimal operating parameters for steering of the swimmer are studied, and the limits of the swimmer are identified. The introduced swimming mechanism can be employed as a simple model system for biological living matters as well as artificial microswimmers.

Graphic abstract

Appendices

Appendix A ODEs for the case of straight motion

For the case of symmetric constraining angles, the governing ODEs can be derived upon substituting from Eqs. 2 and 3 into Eqs. 12–14 and solving the subsequent linear system of the equations. Considering each phase separately, the ODEs are calculated as:

For the first and third phases:

$$\begin{aligned}{} & {} \varvec{{\dot{\theta }}_1}= -{\dot{l}} \left( \frac{27b\pi \text {cos}(\theta )}{a(160a + 45\pi b - 64a\text {cos}^2(\theta ))}\right) {\hat{i}}_3 \end{aligned}$$

(A1)

$$\begin{aligned}{} & {} \varvec{v_O}={\dot{l}}\left( \frac{45b\pi }{96a + 45\pi b+ 64a\text {sin}^2(\theta ) }\right) {\hat{i}}_2 \end{aligned}$$

(A2)

For the second phase:

$$\begin{aligned} \varvec{v_O}={\dot{l}} \left( \frac{9b\pi }{32a + 9\pi b + 16a\text {cos}^2 (\theta _{1,\text {min}})}\right) {\hat{i}}_2 \end{aligned}$$

(A3)

For the fourth phase:

$$\begin{aligned} \varvec{v_O}={\dot{l}} \left( \frac{9b\pi }{32a + 9\pi b + 16a}\right) {\hat{i}}_2 \end{aligned}$$

(A4)

By integrating Eq. A1, expansion (contraction) length of the body link in the phase 1 (phase 3) can be found, which is denoted by $l_{\text {sym}}$ and given in Eq. 16. From the above equations, it is evident that only the second and fourth phases contribute to the swimmer’s displacement over each cycle, as also explained in Sect. 3.

Appendix B General description of the trajectory for an arbitrary cyclic motion of a swimmer in 2D

Here we consider the motion of a cyclic swimmer in the XY plane of the inertial frame XYZ, in which each cycle of motion leads to the resultant translation $\delta$ and rotation $\gamma$ as illustrated in Fig. 9.

Fig. 9

The trajectory of an arbitrary swimmer (green line) over two cycles (starting from point A and passing point B, which is the swimmer’s final position in the first cycle and the initial position for the second cycle, the swimmer reaches the eventual position at point C in the second cycle). The body-fixed coordinate system $E_{1}^AE_{2}^AE_{3}^A$ initially coincides with the laboratory frame XYZ

We will show that all initial and final positions of the swimmer in each cycle are concyclic. We consider an arbitrary cycle during the motion of the swimmer, initially starting from point A and reaching point B at the end of this cycle. We assume $\Theta$ to be the angle between $\overrightarrow{AB}$, the vector joining point A to point B, and the positive X axis. Without loss of generality, we assume that $E_{1}^AE_{2}^AE_{3}^A$ is the body-fixed frame initially coinciding with XYZ frame, and $E_{1}^BE_{2}^BE_{3}^B$ is the same body-fixed frame at the end of the first cycle at point B (Fig. 9). It can be shown that there is a point ${\varvec{o}}_c=(o_{c,1},o_{c,2})$ such that its position (mathematical representation) is the same in both the former and the latter frames (i.e., ${\varvec{o}}_c=(o_{c,1},o_{c,2})_{E_{1}^AE_{2}^AE_{3}^A}=(o_{c,1},o_{c,2})_{E_{1}^BE_{2}^BE_{3}^B}$). This point can be calculated as

$$\begin{aligned} {\varvec{o}}_c=\left( \frac{\delta \sin {(\Theta -\gamma /2)}}{2\sin {\gamma /2}}, \frac{\delta \cos {(\Theta -\gamma /2)}}{2\sin {\gamma /2}}\right) \end{aligned}$$

(B5)

As we obtained the point ${\varvec{o}}_c$ in an arbitrary cycle of the motion, this point has the same property for all the proceeding cycles of the swimmer; thus, it is the center of the circle passing through all the initial and final positions of the swimmer. Each cycle gives rise to the rotation of the body frame about point $\varvec{o_c}$ with angle $\gamma$. Thus, the positions of the swimmer at the end of each cycle are concyclic, with ${\varvec{o}}_c$ being the center of the circle. The radius of this circle $\rho$ can be derived as

$$\begin{aligned} \rho =\frac{\delta }{2\sin {\gamma /2}} \end{aligned}$$

(B6)

The system closes the circle after n cycles of motion, simply calculated as, $n=\lceil \frac{2\pi }{\gamma }\rceil$, where $\lceil x\rceil$ is the ceiling function mapping the least integer greater than or equal to x [40]. For small values of $\gamma$, the noise-free trajectory, analogous to Reference [33], is a circle.