Uniqueness and partial identification in a geometric inverse problem for the Boussinesq system

We analyze the inverse problem of the identification of a rigid body immersed in a fluid governed by the stationary Boussinesq system. First, we establish a uniqueness result. Then, we present a new method for the partial identification of the body. The proofs use local Carleman estimates, differentiation with respect to domains, data assimilation techniques and controllability results for PDEs. To cite this article: A. Doubova et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Two-dimensional local null controllability of a rigid structure in a Navier–Stokes fluid

We consider the two-dimensional motion of a rigid structure immersed in an incompressible fluid governed by Navier–Stokes equations. The control force acts on a fixed subset of the fluid domain. We prove that our system is null controllable; that is, for small initial data, the system can be driven at rest and the structure can be driven to the origin at a given $T>0$. The result holds for a structure symmetric with respect to the center of mass and for initial conditions satisfying strong compatibility conditions. To cite this article: M. Boulakia, A. Osses, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Rigid Binary Relations on a 4-Element Domain

An n-ary relation ρ on a set U is strongly rigid if it is preserved only by trivial operations. It is projective if the only idempotent operations in P o l ρ are projections. Rosenberg, (Rocky Mt. J. Math. 3, 631–639, 1973) characterized all strongly rigid relations on a set with two elements and found a strongly rigid binary relation on every domain U of at least 3 elements. Larose and Tardif (Mult.-Valued Log. 7(5-6), 339–362, 2001) studied the projective and strongly rigid graphs and constructed large families of strongly rigid graphs. ?uczak and Ne?et?il (J. Graph Theory. 47, 81–86, 2004) settled in the affirmative a conjecture of Larose and Tardif that most graphs on a large set are projective, and characterized all homogenous graphs that are projective. ?uczak and Ne?et?il (SIAM J. Comput. 36(3), 835–843, 2006) confirmed a conjecture of Rosenberg that most relations on a big set are strongly rigid. In this paper, we characterize all strongly rigid relations on a set with at least three elements to answer an open question by Rosenberg, (Rocky Mt. J. Math. 3, 631–639, 1973) and we classify the binary relations on the 4-element domain by rigidity and demonstrate that there are merely 40 pairwise nonisomorphic rigid binary relations on the same domain (among them 25 are pairwise nonisomorphic strongly rigid).     

Peripheral layer viscosity effects on periodic blood flow through rigid tube of thin diameter

A mathematical model has been presented for periodic blood flow in a rigid circular tube of thin diameter. Blood is presented as a 3-layered fluid by considering core fluid as a casson fluid which is covered by a thin layer of Newtonian fluid (plasma). The energy integral method has been used to obtain the unsteady pressure gradients as suggested by Elkouh [2]. The results obtained for velocity profiles have been compared with the experimental results of Bugliarello and Sevilla (Biorheology 7 (1970), 85). The effects of various parameters on wall shearing stress has also been brought out and discussed.     

A new transfer principle in the geometry of numbers

In my paper (Proc. Roy. Soc. Edinburgh Sect. A 64 (1956), 223–238), I gave a general transfer principle in the geometry of numbers which consisted of inequalities linking the successive minima of a convex body in n dimensions with those of a convex body in N dimensions where in general N is greater than n. This result contained in particular my earlier theorem on compound convex bodies (Proc. London Math. Soc. (3) 5 (1955), 358–379). In the present paper I apply essentially the same method to prove a new transfer principle which connects the successive minima of a convex body in m dimensions and those of a convex body in n dimensions with the successive minima of a convex body in mn dimensions.     

A stochastic differential equation from friction mechanics

The existence and uniqueness of solutions to multivalued stochastic differential equations of the second order on Riemannian manifolds are proved. The class of problem is motivated by rigid body and multibody dynamics with friction and an application to the spherical pendulum with friction is presented. To cite this article: F. Bernardin et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Mouvements rigides associés à des masses positives et négatives

The Note deals with rigid solutions of the N-Body Problem, i.e. solutions with constant mutual distances between the bodies. It is shown that for these motions, the configuration is balanced in the sense of Albouy and Chenciner [Invent. Math. 131 (1998) 151–184] even when the masses are of different signs. This fact was proved only for positive masses, using the scalar product they define. A consequence of the result is the constancy of the rotation velocity. It is also shown that any configuration can generate non-planar rigid motions for certain masses. Such motions do not exist with positive masses. All the results can be generalized to systems with N charged particles. To cite this article: M. Celli, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

An exact Block–Newton algorithm for solving fluid–structure interaction problems

In this Note, we introduce a partitioned Newton based method for solving nonlinear coupled systems arising in the numerical approximation of fluid–structure interaction problems. The originality of this Schur–Newton algorithm lies in the exact Jacobians evaluation involving the fluid–structure linearized subsystems which are here fully developed. To cite this article: M.Á. Fernández, M. Moubachir, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On period spaces for p-divisible groups

In their book Rapoport and Zink constructed rigid analytic period spaces for Fontaine's filtered isocrystals, and period morphisms from moduli spaces of p-divisible groups to some of these period spaces. We determine the image of these period morphisms, thereby contributing to a question of Grothendieck. We give examples showing that only in rare cases the image is all of the Rapoport–Zink period space. To cite this article: U. Hartl, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A modified Lagrange–Galerkin method for a fluid-rigid system with discontinuous density

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 (k?+?1)-th discrete time level into the approximate rigid body at k-th time. Convergence results are proved for both semi-discrete and fully-discrete schemes.     

The Rayleigh–Stokes problem for an edge in an Oldroyd-B fluid

The velocity fields corresponding to an incompressible fluid of Oldroyd-B type subject to a linear flow within an infinite edge are determined for all values of the relaxation and retardation times. The well known solution for a Navier–Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appears as a limiting case of our solutions. To cite this article: C. Fetecau, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 979–984.     

A particle model for fluid–structure interactionUne méthode particulaire pour le couplage fluide–structure

We propose a particle method to handle fluid–structure interactions on a 1D model problem. Interactions between fluid and solid particles implicitly enforce the continuity of stresses on the interface. Comparisons with results obtained by ALE methods allow one to evaluate the robustness and accuracy of the method. To cite this article: G.-H. Cottet, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 833–838.     

A three-dimensional phase transition model in ferromagnetism: Existence and uniqueness

We scrutinize both from the physical and the analytical viewpoint the equations ruling the paramagnetic-ferromagnetic phase transition in a rigid three-dimensional body. Starting from the order structure balance, we propose a non-isothermal phase-field model which is thermodynamically consistent and accounts for variations in space and time of all fields (the temperature θ, the magnetic field vector H and the magnetization vector M). In particular, we are able to establish a well-posedness result for the resulting coupled system.     

Free vibration analysis of a rigid bar supported by arbitrary elastic beams

For convenience, a two-node conventional elastic beam element (C beam element) with the displacements of its 2nd node replaced by those of center of gravity (c.g.) of the joined rigid bar is called the modified beam element (M beam element). The objective of this paper is to present a modified finite element method (modified FEM) such that the free vibration characteristics of a rigid bar supported by a number of elastic beams can be easily determined. First of all, the displacements for the 2nd node of a C beam element joined with the rigid bar are determined in terms of those for the c.g. of the joined rigid bar to establish the M beam element. Next, the mass and stiffness matrices for the M beam element are derived based on the displacements for the 1st node of the C beam element and those for the c.g. of the joined rigid bar. Then, the overall property matrices of the entire unconstrained vibrating system (i.e. a rigid bar supported by a number of elastic beams) can be determined by using the assembly technique of the conventional FEM and considering the effects of lumped mass and rotary inertia of the rigid bar. Finally, the boundary (supporting) conditions are imposed to produce the effective property matrices of the constrained vibrating system and then the free vibration characteristics are determined with the standard approach. In order to confirm the presented theory and the developed computer program, the rigid bar is modeled by a number of C beam elements with bigger Young’s modulus (E_R) and the conventional FEM is used to determine the natural frequencies and associated mode shapes of the vibrating system. It is found that the latter will converge to the corresponding ones obtained from the presented modified FEM when the magnitude of E_R increases to certain values.     

Dynamical shape gradient for the Navier–Stokes system

This Note deals with the sensitivity analysis of a newtonian incompressible fluid driven by the Navier–Stokes equations with respect to the dynamic of the fluid domain boundary. The structure of the gradient with respect to the velocity of the domain for a given cost function is established. This result is obtained using new shape derivation tools for Eulerian functionals and the Min–Max derivation principle. To cite this article: R. Dziri et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

An added-mass free semi-implicit coupling scheme for fluid–structure interaction

In this Note we propose a semi-implicit coupling scheme for the numerical simulation of fluid–structure interaction systems involving a viscous incompressible fluid. The scheme is stable irrespectively of the so-called added-mass effect and allows for conservative time-stepping within the structure. The efficiency of the scheme is based on the explicit splitting of the viscous effects and geometrical/convective non-linearities, through the use of the Chorin–Temam projection scheme within the fluid. Stability relies on the implicit treatment of the pressure stresses and on the Nitsche's treatment of the viscous coupling. To cite this article: M. Astorino et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A projection algorithm for fluid–structure interaction problems with strong added-mass effect

This Note aims at introducing a semi-implicit coupling scheme for fluid–structure interaction problems with a strong added-mass effect. Our main idea relies on the splitting of added-mass, viscous effects and geometrical/convective non-linearities, through a Chorin–Temam projection scheme within the fluid. We state some theoretical stability results, in the linear case, and provide some numerical experiments. The main interest of the proposed scheme is its efficiency compared to the implicit approach. To cite this article: M.A. Fernández et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the motion of a rigid body in a two-dimensional ideal flow with vortex sheet initial data

A famous result by Delort about the two-dimensional incompressible Euler equations is the existence of weak solutions when the initial vorticity is a bounded Radon measure with distinguished sign and lies in the Sobolev space H⁻¹$H^{-1}$. In this paper we are interested in the case where there is a rigid body immersed in the fluid moving under the action of the fluid pressure. We succeed to prove the existence of solutions à la Delort in a particular case with a mirror symmetry assumption already considered by Lopes Filho et al. (2006) [11], where it was assumed in addition that the rigid body is a fixed obstacle. The solutions built here satisfy the energy inequality and the body acceleration is bounded. When the mass of the body becomes infinite, the body does not move anymore and one recovers a solution in the sense of Lopes Filho et al. (2006) [11].     

Vibrations of elastic systems with a large number of tiny heavy inclusionsVibrations de systèmes élastiques avec un grand nombre de petites inclusions à forte densité

We consider a spectral problem modeling natural vibrations of a complex medium that consists of an elastic medium and tiny rigid inclusions. We study the asymptotic behaviour of solutions of this problem when the total number of inclusions and their density tend to infinity. We obtain a limit problem being a spectral problem for a linear fractional operator pencil that describes the macroscopic behaviour of the system (global vibrations). To cite this article: V. Rybalko, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 245–250.     

Weak convergence results for inhomogeneous rotating fluid equations

We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vector B(x): this is a generalization of the usual rotating fluid model (where B is constant). We prove the weak convergence of Leray-type solutions towards a vector field which satisfies the usual 2D Navier–Stokes equation in the regions of space where B is constant, with Dirichlet boundary conditions, and a heat–type equation elsewhere. The method of proof uses weak compactness arguments. To cite this article: I. Gallagher, L. Saint-Raymond, C. R. Acad. Sci. Paris, Ser. I 336 (2003).