Swimming by switching

Abstract

In this paper we investigate different strategies to overcome the scallop theorem. We will show how to obtain a net motion exploiting the fluid's type change during a periodic deformation. We are interested in two different models: in the first one that change is linked to the magnitude of the opening and closing velocity. Instead, in the second one it is related to the sign of the above velocity. An interesting feature of the latter model is the introduction of a delay-switching rule through a thermostat. We remark that the latter is fundamental in order to get both forward and backward motion.

Optimal Motion of a Scallop: Some Case Studies

“…Note that in (7), since the control u is in L ∞ , we do not consider the initial and final condition on the control u because now we are interesting on its value on positive measure intervals instead of on some points. Moreover, even if the control is no more continuous the system remains controllable (as it is shown in the final numerical simulations in [16]…”

“…where ξ and η are respectively the drag coefficients in the directions parallel, e i , and perpendicular, e ⊥ i , to each link. Integrating the density of force and neglecting inertia, (sum of forces equal zero), we obtain the following dynamics (see [16] for details)…”

“…Therefore we can split (7) in two simpler sub-problems. Indeed, fixed ∆x = F 2 (θ(t 1 )) − F 1 (θ(t 1 )) according to [16] there exists θ 1 such that θ(t 1 ) = θ 1 , with t 1 the first switching instant, such that the corresponding ∆x is exactly x f − x 0 . Then we have…”

“…Others, instead, supposed the scallop immersed in a non Newtonian fluid, in which the viscosity is not constant, ending up with a non reversible dynamics [7]- [15]. Here, inspired by an idea of A. Bressan, we use the approach proposed in [16] where a change in the fluid's regime between the opening and closing of the valves is supposed. Moreover an hysteresis operator is introduced through a thermostat, see Fig.…”

Special issue on ‘active behavior in soft matter and mechanobiology’

“…Finally, Bagagiolo, Maggistro, and Zoppello deal with the topic of biological motility and in particular explore the possibility of achieving locomotion by switching between a slow and a fast actuation strategy [16].…”