Passive and Self-Propelled Locomotion of an Elastic Swimmer in a Perfect Fluid

In this paper we are interested in studying the free motion of a hyperelastic body (also called swimmer) immersed in a perfect fluid. We derive the Euler-Lagrange equations from the Least Action Principle of Lagrangian Mechanics and prove that they are well-posed when the number of elastic modes is finite. The recourse to a strain energy density function in the modeling allows many different constitutive equations for the hyperelastic material to be considered. We perform numerical simulations, aiming to study passive locomotion (i.e. locomotion at zero energy cost). As a first quite surprising result, we observe that the swimmer does not even have to be elastic to experience passive locomotion in its idealized environment. Indeed, we provide an example of deformable (but non elastic) swimmer, for which the fluid-body system behaves as an oscillating mechanical system. The shape changes caused solely by the hydrodynamical forces on the body’s boundary turn out to be periodic strokes resulting in locomotion. This phenomenon can be seen as a generalization, to deformable bodies, of the famous D’Alembert’s paradox (1752) [11], claiming that the drag force is zero on a rigid solid moving with constant velocity. Many other examples of passive locomotion, involving different types of hyperelastic swimmers, are studied. A special interest is devoted to the study of energy and impulse exchanges between the fluid and the body. In the last section, we assume that the swimmer has the ability to modify its shape by means of internal forces. We prove that in this case, the equations of motion are still well-posed and we illustrate again with numerical simulations that, starting from rest, self-propelled locomotion can be achieved.