Alexandre Munnier

Locomotion and Control of a Self-Propelled Shape-Changing Body in a Fluid

In this paper we study the locomotion of a shape-changing body swimming in a two-dimensional perfect fluid of infinite extent. The shape changes are prescribed as functions of time satisfying constraints ensuring that they result from the work of internal forces only: conditions necessary for the locomotion to be termed self-propelled. The net rigid motion of the body results from the exchange of momentum between these shape changes and the surrounding fluid.The aim of this paper is three-fold.First, it describes a rigorous framework for the study of animal locomotion in fluid. Our model differs from previous ones mostly in that the number of degrees of freedom related to the shape changes is infinite. The Euler–Lagrange equation is obtained by applying the least action principle to the system body fluid. The formalism of Analytic Mechanics provides a simple way to handle the strong coupling between the internal dynamics of the body causing the shape changes and the dynamics of the fluid. The Euler–Lagrange equation takes the form of a coupled system of ordinary differential equations (ODEs) and partial differential equations (PDEs). The existence and uniqueness of solutions for this system are rigorously proved.Second, we are interested in making clear the connection between shape changes and internal forces. Although classical, it can be quite surprising to select the shape changes to play the role of control because the internal forces they are due to seem to be a more natural and realistic choice. We prove that, when the number of degrees of freedom relating to the shape changes is finite, both choices are actually equivalent in the sense that there is a one-to-one relation between shape changes and internal forces.Third, we show how the control problem, consisting in associating with each shape change the resulting trajectory of the swimming body, can be analysed within the framework of geometric control theory. This allows us to take advantage of the powerful tools of differential geometry, such as the notion of Lie brackets or the Orbit Theorem and to obtain the first theoretical result (to our knowledge) of control for a swimming body in an ideal fluid. We derive some interesting and surprising tracking properties: For instance, for any given shape changes producing a net displacement in the fluid (say, moving forward), we prove that other shape changes arbitrarily close to the previous ones exist, which lead to a completely different motion (for instance, moving backward): This phenomenon will be called Moonwalking. Most of our results are illustrated by numerical examples.

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

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.

Generic Controllability of 3D Swimmers in a Perfect Fluid

We address the problem of controlling a dynamical system governing the motion of a 3D weighted shape changing body swimming in a perfect fluid. The rigid displacement of the swimmer results from the exchange of momentum between prescribed shape changes and the flow, the total impulse of the fluid-swimmer system being constant for all times. We prove the following tracking results: (i) Synchronized swimming: Maybe up to an arbitrarily small change of its density, any swimmer can approximately follow any given trajectory while, in addition, undergoing approximately any given shape changes. In this statement, the control consists in arbitrarily small superimposed deformations; (ii) Freestyle swimming: Maybe up to an arbitrarily small change of its density, any swimmer can approximately tracks any given trajectory by combining suitably at most five basic movements that can be generically chosen (no macro shape changes are prescribed in this statement).

Locomotion of articulated bodies in an ideal fluid: 2d model with buoyancy, circulation and collisions

Articulated solid bodies (ASB) is a basic model for the study of shape-changing underwater vehicles made of rigid parts linked together by pivoting joints. In this paper we study the locomotion of such swimming mechanisms in an ideal fluid. Our study ranges over a wide class of problems: several ASBs can be involved (without being hydrodynamically decoupled), the fluid-bodies system can be partially or totally confined and fluid circulation, buoyancy force and possible collisions between bodies are taken into account. We derive the Euler–Lagrange equation governing the dynamics of the system, study its well-posedness and describe a numerical scheme implemented in a Matlab toolbox (Biohydrodynamics Toolbox).

Detection of a moving rigid solid in a perfect fluid

In this paper, we consider a moving rigid solid immersed in a potential fluid. The fluid–solid system fills the whole two-dimensional space and the fluid is assumed to be at rest at infinity. Our aim is to study the inverse problem, initially introduced in Conca et al (2008 Inverse Problems 24 045001), that consists in recovering the position and the velocity of the solid assuming that the potential function is known at a given time. We show that this problem is in general ill-posed by providing counterexamples for which the same potential corresponds to different positions and velocities of a same solid. However, it is also possible to find solids having a specific shape, such as ellipses for instance, for which the problem of detection admits a unique solution. Using complex analysis, we prove that the well-posedness of the inverse problem is equivalent to the solvability of an infinite set of nonlinear equations. This result allows us to show that when the solid enjoys some symmetry properties, it can be partially detected. Besides, for any solid, the velocity can always be recovered when both the potential function and the position are supposed to be known. Finally, we prove that by performing continuous measurements of the fluid potential over a time interval, we can always track the position of the solid.

Large Time Behavior for a Simplified N-Dimensional Model of Fluid–Solid Interaction

ABSTRACT In this paper, we study the large time behavior of solutions of a parabolic equation coupled with an ordinary differential equation (ODE). This system can be seen as a simplified N-dimensional model for the interactive motion of a rigid body (a ball) immersed in a viscous fluid in which the pressure of the fluid is neglected. Consequently, the motion of the fluid is governed by the heat equation, and the standard conservation law of linear momentum determines the dynamics of the rigid body. In addition, the velocity of the fluid and that of the rigid body coincide on its boundary. The time variation of the ball position, and consequently of the domain occupied by the fluid, are not known a priori, so we deal with a free boundary problem. After proving the existence and uniqueness of a strong global in time solution, we get its decay rate in L p (1 ≤ p ≤ ∞), assuming the initial data to be integrable. Then, working in suitable weighted Sobolev spaces, and using the so-called similarity variables and scaling arguments, we compute the first term in the asymptotic development of solutions. We prove that the asymptotic profile of the fluid is the heat kernel with an appropriate total mass. The L ∞ estimates we get allow us to describe the asymptotic trajectory of the center of mass of the rigid body as well. We compute also the second term in the asymptotic development in L 2 under further regularity assumptions on the initial data.

Asymptotic Analysis of a Neumann Problem in a Domain with Cusp. Application to the Collision Problem of Rigid Bodies in a Perfect Fluid

We study a two dimensional collision problem for a rigid solid immersed in a cavity filled with a perfect fluid. We are led to investigate the asymptotic behavior of the Dirichlet energy associated with the solution of a Laplace--Neumann problem as the distance

Generic 3D swimmers in a perfect fluid are controllable

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

between the solid and the cavitys bottom tends to zero. Denoting by

On the detection of small moving disks in a fluid

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