## A simplified model of aquatic locomotion

We have developed a simple model of aquatic locomotion. Using the theory of complex variables, we have estimated the hydrodynamic forces acting on an infinite thin rigid plate of length L, following the seminal Work of Theordorsen [1].

By considering the different possible motions of the swimmer, we calculate the velocity potential to derive the pressure by means of the generalised Bernoulli relation. We show that the effect of flow unsteadiness is the principal mechanism for locomotion [2].

We impose a periodic rotation of the tail in order to approximate the undulatory motion of the swimmer. We show the linear dependence of longitudinal velocity on the angular frequency predicted by Gazzola et al [3] . We also predict that the transverse motion presents the same frequency as the forcing whereas the longitudinal motion is a linear function of time plus a periodic term with double frequency.

Finally, by taking the angle of the tail as a small parameter we perform a perturbative expansion to obtain an equation linking swimming velocity to the different parameters involved in swimming. The results arised from this perturbative method are in high accordance with the numerical results.

[1] Theodorsen, T., General theory of aerodynamic instability and the mechanism of flutter, NACA TR No. 496, 1934

[2] Garrick, I. E., Propulsion of a flapping and oscillating airfoil, NACA TR No. 567, 1936

[3] Gazzola, M., Argentina, M., & Mahadevan, L , Scaling macroscopic aquatic locomotion, Nature Physics 10 (10), 758-761, 2014