Dynamics and stability of a class of low Reynolds number swimmers near a wall

We study the dynamic stability of low Reynolds number swimming near a plane wall from a control-theoretic viewpoint. We consider a special class of swimmers having a constant shape, focus on steady motion parallel to the wall, and derive conditions under which it is passively stable without sensing or feedback. We study the geometric structure of the swimming equation and highlight the relation between stability and reversing symmetry of the dynamical system. Finally, our numerical simulations reveal the existence of stable periodic motion. The results have implications for design of miniature robotic swimmers, as well as for explaining the attraction of micro-organisms to surfaces.

Figure 1

(Color online) Drawing of the swimmers. (a) Two-sphere swimmer. (b) $2+1$-sphere swimmer. (c) Three-sphere swimmer.

Figure 2

(Color online) The two-sphere swimmer near wall. (a) Plot of $bc$ as a function of $h$. (b) Phase plot in $\left(y,\theta \right)$ plane for ${\mathbf{u}}_e=\left(1,-1.0068\right)$. (c) Snapshots of wave-like motion along the wall. (d) Snapshots of spiral-like motion along the wall.

Figure 3

(Color online) Simulation of the $2+1$-sphere swimmer. (a) Plot of $y\left(t\right)$. (b) Plot of $\theta \left(t\right)$. (c) Snapshots of swimmer’s motion.

Figure 4

(Color online) The three-sphere swimmer. (a) Real part of eigenvalues as a function of ${\theta }_e$. (b) Phase plot for ${\theta }_e=60°$. (c) Snapshots of the periodic motion along the wall.