Generalized Scallop Theorem for Linear Swimmers

In this article, we are interested in studying locomotion strategies for a class of shape-changing bodies swimming in a fluid. This class consists of swimmers subject to a particular linear dynamics, which includes the two most investigated limit models in the literature: swimmers at low and high Reynolds numbers. Our first contribution is to prove that although for these two models the locomotion is based on very different physical principles, their dynamics are similar under symmetry assumptions. Our second contribution is to derive for such swimmers a purely geometric criterion allowing to determine wether a given sequence of shape-changes can result in locomotion. This criterion can be seen as a generalization of Purcell's scallop theorem (stated in Purcell (1977)) in the sense that it deals with a larger class of swimmers and address the complete locomotion strategy, extending the usual formulation in which only periodic strokes for low Reynolds swimmers are considered.Comment: 14 pages, 10 figure

Invertibility criteria for the biharmonic single-layer potential

While the single-layer operator for the Laplacian is well understood, questions remain concerning the single-layer operator for the Bilaplacian, particularly with regard to invertibility issues linked with degenerate scales. In this article, we provide simple sufficient conditions ensuring this invertibility for a wide range of problems

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 to the solution of a Laplace Neumann problem as the distance $\varepsilon>0$ between the solid and the cavity's bottom tends to zero. Denoting by $\alpha>0$ the tangency exponent at the contact point, we prove that the solid always reaches the cavity in finite time, but with a non zero velocity for $\alpha <2$ (real shock case), and with null velocity for $\alpha \geqslant 2$ (smooth landing case). Our proof is based on a suitable change of variables sending to infinity the cusp singularity at the contact. More precisely, for every $\varepsilon\geqslant 0$, we transform the Laplace Neumann problem into a generalized Neumann problem set on a domain containing a horizontal strip $]0,\ell_\varepsilon[\times ]0,1[$, where $\ell_\varepsilon\to +\infty$

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)

Reconstruction of obstacles in a Stokes flow as a shape-from-moments problem

We address the classical inverse problem of recovering the position and shape of obstacles immersed in a planar Stokes flow using boundary measurements. We prove that this problem can be transformed into a shape-from-moments problem to which ad hoc reconstruction methods can be applied. The effectiveness of this approach is confirmed by numerical tests that show significant improvements over those available in the literature to date

Asymptotic limit for the Stokes and Navier-Stokes problems in a planar domain with a vanishing hole

We show that the eigenvalues of the Stokes operator in a domain with a small hole converge to the eigenvalues of the Stokes operator in the whole domain, when the diameter of the hole tends to 0. The convergence of the eigenspaces and the convergence of the Stokes semigroup are also established. Concerning the Navier--Stokes equations, we prove that the vorticity of the solution in the perforated domain converges as the hole shrinks to a point $r$ to the vorticity of the solution in the punctured domain (i.e. the whole domain with the point $r$ removed). The main ingredients of the analysis are a suitable decomposition of the vorticity space, the formalism elaborated in [7] and some basics of potential theory

Optimal Strokes for Driftless Swimmers: A General Geometric Approach

Swimming consists by definition in propelling through a fluid by means of bodily movements. Thus, from a mathematical point of view, swimming turns into a control problem for which the controls are the deformations of the swimmer. The aim of this paper is to present a unified geometric approach for the optimization of the body deformations of so-called driftless swimmers. The class of driftless swimmers includes, among other, swimmers in a 3D Stokes flow (case of micro-swimmers in viscous fluids) or swimmers in a 2D or 3D potential flow. A general framework is introduced, allowing the complete analysis of five usual nonlinear optimization problems to be carried out. The results are illustrated with examples coming from the literature and with an in-depth study of a swimmer in a 2D potential flow. Numerical tests are also provided