Under consideration for publication in J. Fluid Mech. 1 On the best design for undulatory swimming

(Received?; revised?; accepted?.- To be entered by editorial office) Most aquatic vertebrates swim by passing a bending wave down their bodies, a swimming mode known as undulatory propulsion. Except for very elongated swimmers like eels and lampreys, these animals have evolved to a similar shape: an anterior streamlined region of large volume separated from a caudal fin by a caudal peduncle of reduced cross-section. However, the link between this particular shape and the hydrodynamical constraints re-mains to be explored. Here, this question is addressed by seeking the optimal design for undulatory swimmers with an evolutionary algorithm. Animals of varying elliptic cross-section are considered whose motions are prescribed by arbitrary periodic curvature laws. In the elongated-body limit, reactive and resistive forces can be formulated at any cross-section, allowing to calculate the recoil motion and the mean swimming speed of a given animal. A bi-objective optimisation problem then consists in finding body shapes and corresponding motions associated with the highest swimming velocities, the lowest ener-getic costs, or any trade-offs between the two. For biologically relevant parameters, this optimisation calculation yields two distinct ‘species’: one specialised in economical swim-ming and the other in fast swimming. By comparing, the attributes and performances of these numerically-obtained swimmers with data on undulatory-swimming animals, it is argued that evolution is consistent with low energetic costs. 1