Locomotion of deformable bodies in an ideal fluid: Newtonian versus Lagrangian formalisms

Alexandre Munnier (1, 2)

1 CORIDA - Robust control of infinite dimensional systems and applications
2 IECN - Institut Élie Cartan de Nancy

It is commonly admitted in Classical Mechanics that as long as the system under consideration is monogenic, Newton's laws and Hamilton's (or D'Alembert's) principle both are equivalent in the sense that each one suffices in providing the same governing equations of motion. For more general configurations, in particular when other kind of forces (which do not derive from a scalar potential) entered the scene, the equivalence between Newtonian and Lagrangian formalisms has not been proved so far. This paper is concerned with comparing Newtonian and Lagrangian approaches for determining the governing equations of motion (usually called Euler-Lagrange's equations) for a collection of deformable bodies (also called {\it swimmers}) immersed in an incompressible fluid whose flow is inviscid and irrotational. The swimmers are assumed to be able to change their shapes under the action of inner forces and are endowed with {\it thrusters}, what means that they can generate fluid jets by sucking and blowing out fluid through some localized parts of their boundaries. These capabilities (called generically {\it controls}) 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 quit surprisingly this is no longer true for nonsmooth shaped swimmers. The second novelty brought in in this paper is to treat a broad spectrum of problems (which may involve one or seve\-ral swimmers being able to undergo any kind of deformation) and to display the Euler-Lagrange's equations as a {\it fully} explicit system of ODE's, involving only the value of the fluid potential function on the boundaries of the swimmers and geometric data of these surfaces. Thus, the resulting equations are very convenient for being used in a numerical scheme for simulations.

Mots clés: Biohydrodynamics; ideal fluid; Lagrangian and Newtonian mechanics; PDE-ODE coupled system; shape sensitivity analysis

Domaines: Equations aux dérivées partielles [math.AP]; Biomécanique [physics.med-ph]; Biomécanique [physics.med-ph]

Alexandre Munnier. Locomotion of deformable bodies in an ideal fluid: Newtonian versus Lagrangian formalisms. 2008. ⟨hal-00276252v1⟩