Analytical Results on the Role of Flexibility in Flapping Propulsion

Wing or fin flexibility can dramatically a↵ect the performance of flying and swimming animals. Both laboratory experiments and numerical simulations have been used to study these e↵ects, but analytical results are notably lacking. Here, we develop small-amplitude theory to model a flapping wing that pitches passively due to a combination of wing compliance, inertia, and fluid forces. Remarkably, we obtain a class of exact solutions describing the wing’s emergent pitching motions, along with expressions for how thrust and e ciency are modified by compliance. The solutions recover a range of realistic behaviors and shed new light on how flexibility can aid performance, the importance of resonance, and the separate roles played by wing and fluid inertia. The simple, robust estimates a↵orded by our theory will likely be valuable even in situations where details of the flapping motion and wing geometry di↵er.