title: Optimal propulsive flapping in Stokes flows.
abstract: Swimming fish and flying insects use the flapping of fins and wings to generate thrust. In contrast, microscopic organisms typically deform their appendages in a wavelike fashion. Since a flapping motion with two degrees of freedom is able, in theory, to produce net forces from a time-periodic actuation at all Reynolds numbers, we compute in this paper the optimal flapping kinematics of a rigid spheroid in a Stokes flow. The hydrodynamics for the force generation and energetics of the flapping motion is solved exactly. We then compute analytically the gradient of a flapping efficiency in the space of all flapping gaits and employ it to derive numerically the optimal flapping kinematics as a function of the shape of the flapper and the amplitude of the motion. The kinematics of optimal flapping are observed to depend weakly on the flapper shape and are very similar to the figure-eight motion observed in the motion of insect wings. Our results suggest that flapping could be a exploited experimentally as a propulsion mechanism valid across the whole range of Reynolds numbers.
authors.affiliation: Department of Mechanical and Aerospace Engineering, University of California San Diego, 9500 Gilman Drive, La Jolla CA 92093-0411, USA.
authors.forename: Loïc
authors.initials: L
authors.lastname: Was
authors.forename: Eric
authors.initials: E
authors.lastname: Lauga
mesh_terms: D000818:Animals; D001683:Biological Clocks / Q000502:physiology*; D032701:Biomimetics / Q000379:methods*; D003198:Computer Simulation; D004735:Energy Transfer / Q000502:physiology; D005426:Flight, Animal / Q000502:physiology*; D008954:Models, Biological; D005082:Physical Exertion / Q000502:physiology; D012212:Rheology / Q000379:methods*; D013550:Swimming / Q000502:physiology*; D014921:Wings, Animal / Q000502:physiology*