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International audience; We study the interaction of surface water waves with a floating solid constraint to move only in the vertical direction. The first novelty we bring in is that we propose a new model for this interaction, taking into consideration the viscosity of the fluid. This is done supposing that the flow obeys a shallow water regime (modeled by the viscous Saint-Venant equations in one space dimension) and using a Hamiltonian formalism. Another contribution of this work is establishing the well-posedness of the obtained PDEs/ODEs system in function spaces similar to the standard ones for strong solutions of viscous shallow water equations. Our well-posedness results are local in time for every initial data and global in time if the initial data are close (in appropriate norms) to an equilibrium state. Moreover, we show that the linearization of our system around an equilibrium state can be described, at least for some initial data, by an integro-fractional differential equation related to the classical Cummins equation and which reduces to the Cummins equation when the viscosity vanishes and the fluid is supposed to fill the whole space. Finally, we describe some numerical tests, performed on the original nonlinear system, which illustrate the return to equilibrium and the influence of the viscosity coefficient.

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.

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.

International audience; We propose a new model for the motion of a viscous incompressible fluid. More precisely, we consider the Navier–Stokes system with a boundary condition governed by the Coulomb friction law. With this boundary condition, the fluid can slip on the boundary if the tangential component of the stress tensor is too large. We prove the existence and uniqueness of weak solution in the two–dimensional problem and the existence of at least one solution in the three–dimensional case, together with regularity properties and an energy estimate. We also propose a fully discrete scheme of our problem using the characteristic method and we present numerical simulations in two physical examples.

International audience; We propose a new model in a fluid-structure system composed by a rigid body and a viscous incompress-ible fluid using a boundary condition based on Coulomb's 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.

International audience; In this paper, we propose a new characteristics method for the discretization of the two dimensional fluid-rigid body problem in the case where the densities of the fluid and the solid are different. The method is based on a global weak formulation involving only terms defined on the whole fluid-rigid domain. To take into account the material derivative, we construct a special characteristic function which maps the approximate rigid body at the discrete time level $t_{k+1}$ into the approximate rigid body at time $t_k$. Convergence results are proved for both semi-discrete and fully-discrete schemes.

International audience; We propose a new characteristics method for the time discretization of a fluid-rigid system in the case when the densities of the fluid and the solid are different. This method is based on a global weak formulation involving only terms defined on the whole fluid-rigid domain. The main idea is to construct a characteristic function which preserves the rigidity of the solid at the discrete time levels. A convergence result for this semi-discrete scheme is then given.

This paper considers the periodic spectral problem associated with the Laplace operator written in \({\mathbb{R}^N}\) (N = 3, 4, 5) periodically perforated by balls, and with homogeneous Dirichlet condition on the boundary of holes. We give an asymptotic expansion for all simple eigenvalues as the size of holes goes to zero. As an application of this result, we use Bloch waves to find the classical strange term in homogenization theory, as the size of holes goes to zero faster than the microstructure period.

In this paper, we study the motion of rigid bodies in a perfect incompressible fluid. The rigid-fluid system fils a bounded domain inR 3 . Adapting the strategy from Bourguignon and Brezis [1], we use the stream lines of the fluid and we eliminate the pressure by solving a Neumann problem. In this way, the system is reduced to an ordinary dierential equation on a closed infinite dimensional manifold. Using this formulation, we prove the local in time existence and uniqueness of strong solutions. Notation. Throughout this paper › denotes an open bounded and connected subset of R 3 and S0 is a closed set with nonempty interior and with smooth boundary such that S0 ‰ ›. We denote as usual by SO3(R) the special orthogonal group on R 3 . We will often use functions defined from a time interval to R 3 or to SO3(R). these functions will be denoted using bold characters, such as h : [0;T] ! R 3 or R : [0;T] ! SO3(R). The same kind of notation will be used for three other time dependent vector fields k, !, · and » which will be defined in the sequel. The five time dependent fields mentioned above will define the state z of the fluid-solid system. A vector from R 3 or a matrix from SO3(R) will be denoted by h or by R, respectively. The transposed of a matrix will be denoted by ⁄ so that the column vector of components a and b is denoted either µ a

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.

Papier accepté pour la publication dans Quarterly of Applied Mathematics; International audience; 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.

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.

We consider the two-dimensional motion of several non-homogeneous rigid bodies immersed in an incompressible non-homogeneous viscous fluid. The fluid, and the rigid bodies are contained in a fixed open bounded set of R 2 . The motion of the fluid is governed by the Navier-Stokes equations for incompressible fluids and the standard conservation laws of linear and angular momentum rule the dynamics of the rigid bodies. The time variation of the fluid domain (due to the motion of the rigid bodies) is not known a priori, so we deal with a free boundary value problem. The main novelty here is the demonstration of the global existence of weak solutions for this problem. More precisely, the global character of the solutions we obtain is due to the fact that we do not need any assumption concerning the lack of collisions between several rigid bodies or between a rigid body and the boundary. We give estimates of the velocity of the bodies when their mutual distance or the distance to the boundary tends to zero.

In Heintz et al. (Electron. J. SADIO 1(1) (1998) 37), Castro et al. (Found., Comput. Math. (2003) to appear) and Pardo (Proceedings EACA’2000, 2000, pp. 25–51), the authors have shown that universal solving procedures require exponential running time. Roughly speaking, a universal solving procedure takes as input a system of multivariate polynomial equations and outputs complete symbolic information on the solution variety. Here, we introduce a non-universal solving procedure adapted to Generalised Pham Systems. The aim is to compute partial information of the variety defined by the input system. The Algorithm is based on an homotopic deformation and on a non-Archimedean lifting procedure from a non-singular zero of the homotopy curve. The complexity of the procedure is also stated and it depends on some intrinsic quantity called the deformation degree of the given input system.

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.

This Ph.D. thesis aims to obtain and to develop some mathematical models to understand some aspects of the dynamics of heterogeneous granular fluids. More precisely, the expected result is to develop three models, one where the dynamics of the granular material is modeled using a mixture theory approach, and the other two, where we consider the granular fluid is modeled using a multiphase approach involving rigid structures and fluids. More precisely : • In the first model, we obtained a set of equations based on the mixture theory using homogenization tools and a thermodynamic procedure. These equations reflect two essential properties of granular fluids : the viscous nature of the interstitial fluid and a Coulomb-type of behavior of the granular component. With our equations, we study the problem of a dense granular heterogeneous flow, composed by a Newtonian fluid and a solid component in the setting of the Couette flow between two infinite cylinders. • In the second model, we consider the motion of a rigid body in a viscoplastic material. The 3D Bingham equations model this material, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. • In the third model, we consider the motion of a perfect heat conductor rigid body in a heat conducting Newtonian fluid. The 3D Fourier-Navier-Stokes equations model the fluid, and the Newton laws and the balance of internal energy model the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is composed by the balance of momentum and the balance of total energy equation which includes the pressure of the fluid, and it involves a free boundary (due to the motion of the rigid body). To obtain an integrable pressure, we consider a Navier slip boundary condition for the outer boundary and the mutual interface; Cette thèse vise à obtenir et à développer des modèles mathématiques pour comprendre certains aspects de la dynamique des fluides granulaires hétérogènes. Plus précisément, le résultat attendu consiste à développer trois modèles. Nous supposons dans un premier temps que la dynamique du matériau granulaire est modélisée à l’aide d’une approche fondée sur la théorie du mélange. D’autre part, pour les deux modèles restant, nous considérons que le fluide granulaire est modélisé à l’aide d’une approche multiphase associant des structures et des fluides rigides. Plus exactement : • Dans le premier modèle, nous avons obtenu un ensemble d’équations basées sur la théorie du mélange en utilisant des outils d’homogénéisation et une procédure thermodynamique. Ces équations reflètent deux propriétés essentielles des fluides granulaires : la nature visqueuse du fluide interstitiel et un comportement de type Coulomb de la composante granulaire. Avec nos équations, nous étudions le problème de Couette entre deux cylindres infinis d’un écoulement hétérogène granulaire dense, composé d’un fluide newtonien et d’une composante solide. • Dans le deuxième modèle, nous considérons le mouvement d’un corps rigide dans un matériau viscoplastique. Les équations 3D de Bingham modélisent ce matériau et les lois de Newton régissent le déplacement du corps rigide. Notre résultat principal est d’établir l’existence d’une solution faible pour le système correspondant. • Dans le troisième modèle, nous considérons le mouvement d’un corps rigide conducteur thermique parfait dans un fluide newtonien conducteur de la chaleur. Les équations 3D de Fourier-Navier-Stokes modélisent le fluide, tandis que les lois de Newton et l’équilibre de l’énergie interne modélisent le déplacement du corps rigide. Notre principal objectif dans cette partie est de prouver l’existence d’une solution faible pour le système correspondant. La formulation faible est composée de l’équilibre entre la quantité du mouvement et l’équation de l’énergie totale, qui inclut la pression du fluide, et implique une limite libre due au mouvement du corps rigide. Pour obtenir une pression intégrable, nous considérons une condition au limite de glissement de Navier pour la limite extérieure et l’interface mutuelle

This Ph.D. thesis aims to obtain and to develop some mathematical models to understand some aspects of the dynamics of heterogeneous granular fluids. More precisely, the expected result is to develop three models, one where the dynamics of the granular material is modeled using a mixture theory approach, and the other two, where we consider the granular fluid is modeled using a multiphase approach involving rigid structures and fluids. More precisely : • In the first model, we obtained a set of equations based on the mixture theory using homogenization tools and a thermodynamic procedure. These equations reflect two essential properties of granular fluids : the viscous nature of the interstitial fluid and a Coulomb-type of behavior of the granular component. With our equations, we study the problem of a dense granular heterogeneous flow, composed by a Newtonian fluid and a solid component in the setting of the Couette flow between two infinite cylinders. • In the second model, we consider the motion of a rigid body in a viscoplastic material. The 3D Bingham equations model this material, and the Newton laws govern the displacement of the rigid body. Our main result is the existence of a weak solution for the corresponding system. • In the third model, we consider the motion of a perfect heat conductor rigid body in a heat conducting Newtonian fluid. The 3D Fourier-Navier-Stokes equations model the fluid, and the Newton laws and the balance of internal energy model the rigid body. Our main result is the existence of a weak solution for the corresponding system. The weak formulation is composed by the balance of momentum and the balance of total energy equation which includes the pressure of the fluid, and it involves a free boundary (due to the motion of the rigid body). To obtain an integrable pressure, we consider a Navier slip boundary condition for the outer boundary and the mutual interface; Cette thèse vise à obtenir et à développer des modèles mathématiques pour comprendre certains aspects de la dynamique des fluides granulaires hétérogènes. Plus précisément, le résultat attendu consiste à développer trois modèles. Nous supposons dans un premier temps que la dynamique du matériau granulaire est modélisée à l’aide d’une approche fondée sur la théorie du mélange. D’autre part, pour les deux modèles restant, nous considérons que le fluide granulaire est modélisé à l’aide d’une approche multiphase associant des structures et des fluides rigides. Plus exactement : • Dans le premier modèle, nous avons obtenu un ensemble d’équations basées sur la théorie du mélange en utilisant des outils d’homogénéisation et une procédure thermodynamique. Ces équations reflètent deux propriétés essentielles des fluides granulaires : la nature visqueuse du fluide interstitiel et un comportement de type Coulomb de la composante granulaire. Avec nos équations, nous étudions le problème de Couette entre deux cylindres infinis d’un écoulement hétérogène granulaire dense, composé d’un fluide newtonien et d’une composante solide. • Dans le deuxième modèle, nous considérons le mouvement d’un corps rigide dans un matériau viscoplastique. Les équations 3D de Bingham modélisent ce matériau et les lois de Newton régissent le déplacement du corps rigide. Notre résultat principal est d’établir l’existence d’une solution faible pour le système correspondant. • Dans le troisième modèle, nous considérons le mouvement d’un corps rigide conducteur thermique parfait dans un fluide newtonien conducteur de la chaleur. Les équations 3D de Fourier-Navier-Stokes modélisent le fluide, tandis que les lois de Newton et l’équilibre de l’énergie interne modélisent le déplacement du corps rigide. Notre principal objectif dans cette partie est de prouver l’existence d’une solution faible pour le système correspondant. La formulation faible est composée de l’équilibre entre la quantité du mouvement et l’équation de l’énergie totale, qui inclut la pression du fluide, et implique une limite libre due au mouvement du corps rigide. Pour obtenir une pression intégrable, nous considérons une condition au limite de glissement de Navier pour la limite extérieure et l’interface mutuelle