On the Self-displacement of Deformable Bodies in a Potential Fluid Flow

Understanding fish-like locomotion as a result of internal shape changes may result in improved underwater propulsion mechanism. In this article, we study a coupled system of partial differential equations and ordinary differential equations which models the motion of self-propelled deformable bodies (called swimmers) in an potential fluid flow. The deformations being prescribed, we apply the least action principle of Lagrangian mechanics to determine the equations of the inferred motion. We prove that the swimmers degrees of freedom solve a second order system of nonlinear ordinary differential equations. Under suitable smoothness assumptions on the fluid’s domain boundary and on the given deformations, we prove the existence and regularity of the bodies rigid motions, up to a collision between two swimmers or between a swimmer with the boundary of the fluid. Then we compute explicitly the Euler-Lagrange equations in terms of the geometric data of the bodies and of the value of the fluid’s harmonic potential on the boundary of the fluid.