Section: New Results

Analysis and control of fluids and of fluid-structure interactions

Participants : Thomas Chambrion, Antoine Henrot, Alexandre Munnier, Yu Ning Liu, Jean-François Scheid, Erica Schwindt, Mario Sigalotti, Takéo Takahashi, Marius Tucsnak, Jean-Claude Vivalda, Jérôme Lohéac.

The study of a fluid-structure system depends on the nature of the fluid considered and in particular on the Reynolds number. We have split the new results of this section according to the viscosity of the fluid. The first part is devoted to the case of a viscous fluid. This is the case that has received more attention from mathematicians in the recent years. In the second part, we have put the results concerning an inviscid fluid. This case is more classical in Fluid Mechanics and could be more interesting to understand self-propelled motions which is one of the main goal of our work. In the last part, we have given some numerical results.

Incompressible viscous fluids

In [31] , García and Takahashi present some abstract results giving a general connection between null-controllability and several inverse problems for a class of parabolic equations. They obtain some conditional stability estimates for the inverse problems consisting of determining the initial condition and the source term, from interior or boundary measurements. They apply this framework for Stokes system with interior and boundary observations, for a coupling of two Stokes system and a linear fluid-structure system.

Nečasová, Takahashi and Tucsnak consider in [43] the three-dimensional motion of a self-propelled deformable structure into a viscous incompressible fluid. The deformation of the solid is given whereas its position is unknown. Such a system could model the propulsion of fish-like swimmers. The equations of motion of the fluid are the Navier-Stokes equations and the equations for the structure are deduced from Newton's laws. The corresponding system is a free-boundary problem and the main result they obtain is the existence of weak solutions for this problem.

In [29] we give a controllability result for a simplified 1D fluid-structure system.

In [39] we give a detailed analysis of a phase field type model describing the motions of vesicles in a viscous incompressible fluid.

In [40] we study a controllability problem for a simplified one dimensional model for the motion of a rigid body in a viscous fluid. One of the novelties brought in with respect to the existing literature consist in the fact that we use a single scalar control. Moreover, we introduce a new methodology, which can be used for other nonlinear parabolic systems, independently of the techniques previously used for the linearized problem. This methodology is based on an abstract argument for the null controllability of parabolic equations in the presence of source terms and it avoids tackling linearized problems with time dependent coefficients.

Ideal fluids

In [42] , the author studies the motion of an hyperelastic body immersed in a perfect fluid. The recourse to a strain energy density function in the modeling allows many different constitutive equations for the hyperelastic material to be considered. Numerical simulation are performed, aiming to study passive locomotion (i.e. locomotion at zero energy cost).

In [27] , we study the approximate controllability of 2D swimmer in an ideal fluid. The result includes an approximate tracking result of both the shape and the position of the swimmer.