1304.5046 (Fangxu Jing et al.)

Stability of Periodic Locomotion in Potential Flow    [PDF]

Fangxu Jing, Eva Kanso

Most aquatic vertebrates swim by lateral flapping of their bodies and caudal fins. While much effort has been devoted to understanding the flapping kinematics and its influence on the swimming efficiency, little is known about the stability (or lack of) of periodic swimming. In this paper, we examine the stability of periodic locomotion due to sideways flapping in unbounded potential flow. It is believed that stability limits maneuverability and body designs/flapping motions that are adapted for stable swimming are not suitable for high maneuverability and vice versa. Here, we consider a simplified model where the swimmer is a planar elliptic body undergoing prescribed periodic heaving and pitching. We show that periodic locomotion can be achieved due to the resulting hydrodynamic forces, and its value depends on several parameters including the aspect ratio of the body, the amplitudes and phases of the prescribed flapping. We obtain closed-form solutions for the locomotion and efficiency for small flapping amplitudes, and numerical results for finite flapping amplitudes. We then study the stability of the (finite amplitude flapping) periodic locomotion using Floquet theory. We find that stability depends nonlinearly on all parameters. Interesting trends of switching between stable and unstable motions emerge and evolve as we continuously vary the parameter values. This suggests that, when it comes to live organisms, maneuverability and stability need not be thought of as disjoint properties, rather the organism may manipulate its motion in favor of one or the other depending on the task at hand.

View original: http://arxiv.org/abs/1304.5046