Jorge San Martín

A control theoretic approach to the swimming of microscopic organisms

In this paper, we give a control theoretic approach to the slow self-propelled motion of a rigid body in a viscous fluid. The control of the system is the relative velocity of the fluid with respect to the solid on the boundary of the rigid body (the thrust). Our main results show that, there exists a large class of finite dimensional input spaces for which the system is exactly controllable, i.e., one can find controls steering the rigid body in any final position with a prescribed velocity field. The equations we use are motivated by models of swimming of micro-organisms like cilia. We give a control theoretic interpretation of the swimming mechanism of these organisms, which takes place at very low Reynolds numbers. Our aim is to give a control theoretic interpretation of the swimming mechanism of micro-organisms (like ciliata) which is one of the fascinating problems in fluid mechanics.

Convergence of the Lagrange--Galerkin Method for the Equations Modelling the Motion of a Fluid-Rigid System

In this paper, we consider a Lagrange--Galerkin scheme to approximate a two-dimensional fluid-rigid body problem. The equations of the system are the Navier--Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the rigid body. In this problem, the equations of the fluid are written in a domain whose variation is one of the unknowns. We introduce a numerical method based on the use of characteristics and on finite elements with a fixed mesh. Our main result asserts the convergence of this scheme.

OPTIMAL BOUNDS ON DISPERSION COEFFICIENT IN ONE-DIMENSIONAL PERIODIC MEDIA

In this paper, we consider the macroscopic quantity, namely the dispersion tensor associated with a periodic structure in one dimension (see Refs. 5 and 7). We describe the set in which this quantity lies, as the microstructure varies preserving the volume fraction.

Collisions Involving Solids and Fluids

Predictive theories of instantaneous collisions involving rigid and deformable solids as well as fluids are described. They are based on the description of interior percussions to the system made of the colliding bodies. The system made of all the elements (solids or fluids) that are colliding is a deformable system: its form changes even if it is made of rigid elements! If the duration of a collision is small compared to the duration of the evolution, we assume that the collision is instantaneous; thus the velocities are discontinuous We describe the collision of a point with a fixed plane and the simultaneous collisions of a collection of rigid bodies. The impact of an hammer with a bar is an example of collisions of deformables bodies. Experiments show the adequation of the theory. The collision of fluids and solids is illustrated by the description of the belly flop of a diver in a swimming pool.

On the reconstruction of obstacles and of rigid bodies immersed in a viscous incompressible fluid

Abstract We consider the geometrical inverse problem consisting in recovering an unknown obstacle in a viscous incompressible fluid by measurements of the Cauchy force on the exterior boundary. We deal with the case where the fluid equations are the nonstationary Stokes system and using the enclosure method, we can recover the convex hull of the obstacle and the distance from a point to the obstacle. With the same method, we can obtain the same result in the case of a linear fluid-structure system composed by a rigid body and a viscous incompressible fluid. We also tackle the corresponding nonlinear systems: the Navier–Stokes system and a fluid-structure system with free boundary. Using complex spherical waves, we obtain some partial information on the distance from a point to the obstacle.

Higher Order Macro Coefficients in Periodic Homogenization

A first set of macro coefficients known as the homogenized coefficients appear in the homogenization of PDE on periodic structures. If energy is increased or scale is decreased, these coefficients do not provide adequate approximation. Using Bloch decomposition, it is first realized that the above coefficients correspond to the lowest energy and the largest scale. This naturally paves the way to introduce other sets of macro coefficients corresponding to higher energies and lower scales which yield better approximation. The next task is to compare their properties with those of the homogenized coefficients. This article reviews these developments along with some new results yet to be published.

The Dispersion Tensor and Its Unique Minimizer in Hashin–Shtrikman Micro-structures

In this paper, we introduce a macroscopic quantity, namely the dispersion tensor or the Burnett coefficients in the class of generalized Hashin–Shtrikman micro-structures (Tartar in The general theory of homogenization, volume 7 of Lecture notes of the Unione Matematica Italiana, Springer, Berlin, p 281, 2009). In the case of two-phase materials associated with the periodic Hashin–Shtrikman structures, we settle the issue that the dispersion tensor has a unique minimizer, which is the so called Apollonian–Hashin–Shtrikman micro-structure.

Fluid-Rigid Structure Interaction System with Coulomb's Law

We propose a new model in a fluid-rigid structure system composed by a rigid body and a viscous incompressible fluid using a boundary condition based on Coulombs law. This boundary condition allows the fluid to slip on the boundary if the tangential component of the stress is too large. In the opposite case, we recover the standard Dirichlet boundary condition. The governing equations are the Navier--Stokes system for the fluid and the Newton laws for the body. The corresponding coupled system can be written as a variational inequality. We prove that there exists a weak solution of this system.

The Lagrange-Galerkin method for fluid-structure interaction problems

In this paper, we consider a Lagrange-Galerkin scheme to approximate a two-dimensional fluid-structure interaction problem. The equations of the system are the Navier-Stokes equations in the fluid part, coupled with ordinary differential equations for the dynamics of the solid. We are interested in studying numerical schemes based on the use of the characteristics method for rigid and deformable solids. The schemes are based on a global weak formulation involving only terms defined on the whole fluid-solid domain. Convergence results are stated for both semi and fully discrete schemes. This article reviews known results for rigid solid along with some new results on deformable structure yet to be published.