Locomotion of Deformable Bodies in an Ideal Fluid: Newtonian versus Lagrangian Formalisms

Abstract

This paper is concerned with comparing Newtonian and Lagrangian methods in Mechanics for determining the governing equations of motion (usually called Euler–Lagrange equations) for a collection of deformable bodies immersed in an incompressible, inviscid fluid whose flow is irrotational. The bodies can modify their shapes under the action of inner forces and torques and are endowed with thrusters, which means that they can generate fluid jets by sucking and blowing out fluid through some localized parts of their boundaries. These capabilities may allow them to propel and steer themselves. Our first contribution is to prove that under smoothness assumptions on the fluid-bodies interface, Newtonian and Lagrangian formalisms yield the same equations of motion. However, and rather surprisingly, this is no longer true for nonsmooth shaped bodies. The second novelty brought in this paper is treating for the first time a broad spectrum of problems in which several bodies undergoing any kind of shape changes can be involved and to display the Euler–Lagrange equations under a form convenient to study locomotion. These equations have been used to develop a Matlab toolbox (Biohydrodynamics Toolbox) that allows one to study animal locomotion in a fluid or merely the motion of submerged rigid solids. Examples of such simulations are given in this paper.