49 research outputs found
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 between the solid and the
cavity's bottom tends to zero. Denoting by 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 (real shock case), and with
null velocity for (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 ,
we transform the Laplace Neumann problem into a generalized Neumann problem set
on a domain containing a horizontal strip ,
where
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 to the
vorticity of the solution in the punctured domain (i.e. the whole domain with
the point 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