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In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from (Lasiecka et al. in Nonlinear Anal 44:54–85, 2018). An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generalizing the results of (Wollner and Wick in J Math Fluid Mech 21:34, 2019) to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners.

We construct gradient structures for free boundary problems with moving capillary interfaces with nonlinear (hyper)elasticity and study the impact of moving contact lines. In this context, we numerically analyze how phase-field models converge to certain sharp-interface models when the interface thickness tends to zero. In particular, we study the scaling of the Cahn--Hilliard mobility with certain powers of the interfacial thickness. In the presence of interfaces, it is known that the intended sharp-interface limit holds only for a particular range of powers However, in the presence of moving contact lines we show that some scalings that are valid for interfaces produce significant errors and the effective range of valid powers of the interfacial thickness in the mobility reduces.

We consider a non-linear system modelling the dynamics of a linearly elastic body immersed in an incompressible viscous fluid. We prove local existence of strong solutions and global existence and uniqueness for small data. The main result is the characterization of long-time behaviour of the elastic displacement. We show convergence either to a rest state or rigid motion, or to a time-periodic pressure wave that may occur only in specific geometric settings., Comment: 39 pages, 1 figure

We propose a numerical methodology for the simultaneous numerical simulation of four states of matter: gas, liquid, elastoplastic solids, and plasma. The distinct, interacting physical processes are described by a combination of compressible, inert, and reactive forms of the Euler equations, multi-phase equations, elastoplastic equations, and resistive MHD equations. Combinations of systems of equations are usually solved by coupling finite element for solid modelling and CFD models for fluid modelling or including material effects through boundary conditions rather than full material discretisation. Our simultaneous solution methodology lies on the recasting of all the equations in the same, hyperbolic form allowing their solution on the same grid with the same finite volume numerical schemes. We use a combination of sharp- and diffuse-interface methods to track or capture material interfaces, depending on the application. The communication between the distinct systems of equations (i.e., materials separated by sharp interfaces) is facilitated by means of mixed-material Riemann solvers at the boundaries of the systems, which represent physical material boundaries. To this end, we derive approximate mixed-material Riemann solvers for each pair of the above models based on characteristic equations. To demonstrate the applicability of the new methodology, we consider a case study, where we investigate the possibility of ignition of a combustible gas that lies over a liquid in a metal container that is struck by a plasma arc akin to a lightning strike. We study the effect of the metal container material and its conductivity on the ignition of the combustible gas, as well as the effects of an additional dielectric coating, the sensitivity of the gas, and differences between scenarios with sealed and pre-damaged metal surfaces.

For many biological systems that involve elastic structures immersed in fluid, small length scales mean that inertial effects are also small, and the fluid obeys the Stokes equations. One way to solve the model equations representing such systems is through the Stokeslet, the fundamental solution to the Stokes equations, and its regularized counterpart, which treats the singularity of the velocity at points where force is applied. In two dimensions, an additional complication arises from Stokes' paradox, whereby the velocity from the Stokeslet is unbounded at infinity when the net hydrodynamic force within the domain is nonzero, invalidating the solutions. A straightforward computationally inexpensive method is presented for obtaining valid solutions to the Stokes equations for net nonzero forcing. The approach is based on imposing a mean zero velocity condition on a large curve that surrounds the domain of interest. The condition is shown to be equivalent to a net-zero force condition, where the opposite forces are applied on the large curve. The numerical method is applied to models of cellular motility and blebbing., Comment: 33 pages, 9 figures, submitted to Communications in Applied Mathematics and Computational Science

In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from [Lasiecka, Szulc, and Zochoswki, Nonl. Anal.: Real World Appl., 44, 2018]. An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generaziling the results of [Wollner and Wick, J. Math. Fluid Mech., 21, 2019] to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners.

We present a computational multiscale model for the efficient simulation of vascularized tissues, composed of an elastic three-dimensional matrix and a vascular network. The effect of blood vessel pressure on the elastic tissue is surrogated via hyper-singular forcing terms in the elasticity equations, which depend on the fluid pressure. In turn, the blood flow in vessels is treated as a one-dimensional network. Intravascular pressure and velocity are simulated using a high-order finite volume scheme, while the elasticity equations for the tissue are solved using a finite element method. This work addresses the feasibility and the potential of the proposed coupled multiscale model. In particular, we assess whether the multiscale model is able to reproduce the tissue response at the effective scale (of the order of millimeters) while modeling the vasculature at the microscale. We validate the multiscale method against a full scale (three-dimensional) model, where the fluid/tissue interface is fully discretized and treated as a Neumann boundary for the elasticity equation. Next, we present simulation results obtained with the proposed approach in a realistic scenario, demonstrating that the method can robustly and efficiently handle the one-way coupling between complex fluid microstructures and the elastic matrix. Supplementary Information The online version contains supplementary material available at 10.1007/s10439-021-02804-0.

We present a framework to systematically derive variational formulations for fluid-structure interaction problems based on thermodynamical driving functionals and geometric structures in different coordinate systems by suitable transformations within this formulation. Our approach provides a promising basis to construct structure-preserving discretization strategies.

We prove existence of a weak solution for a nonlinear, multi-physics, multi-scale problem of magnetorheological suspensions introduced in Nika & Vernescu (Z. Angew. Math. Phys., 71(1):1--19, '20). The hybrid model couples the Stokes' equation with the quasi-static Maxwell's equations through the Lorentz force and the Maxwell stress tensor. The proof of existence is based on: i) the augmented variational formulation of Maxwell's equations, ii) the definition of a new function space for the magnetic induction and the proof of a Poincaré type inequality, iii) the Altman--Shinbrot fixed point theorem when the magnetic Reynold's number, Rm, is small.

International audience; We study control properties of a linearized fluid-structure interaction system, where the structure is a rigid body and where the fluid is a viscoelastic material. We establish the approximate controllability and the exponential stabilizability for the velocities of the fluid and of the rigid body and for the position of the rigid body. In order to prove this, we prove a general result for this kind of systems that generalizes in particular the case without structure. The exponential stabilization of the system is obtained with a finite-dimensional feedback control acting only on the momentum equation on a subset of the fluid domain and up to some rate that depends on the coefficients of the system. We also show that, as in the case without structure, the system is not exactly null-controllable in finite time.

A derivation of the model for a poroelastic elliptic membrane shell is undertaken. The flow and deformation in a three-dimensional shell domain is described by the quasi-static Biot equations of linear poroelasticity. We consider the limit when the shell thickness goes to zero and look for the limit equations. Using the technique developed in the seminal articles by Ciarlet, Lods, Miara et al.and the recent results on the rigorous derivation of the equations for poroelastic plates and flexural poroelastic shells by Marciniak-Czochra, Mikelić, and Tambača, we present a rigorous derivation of the linear poroelastic elliptic membrane shell model. After rescaling, the corresponding velocity and the pressure field are close in the C([0, T] ; (H1x)2×(L2x)2) norm and the stresses in C([0, T] ; (L2x)9) norm. We note the major difference with respect to the flexural case: (i) it is not anymore the rescaled total stress divided by the scaling parameter, but the rescaled total stress itself which converges ; (ii) the same comment applies to the pore fluid pressure ; and (iii) there is a deterioration of the convergence for the vertical component of the rescaled displacement. Consequence of the above differences is that the effective model remains of the 2nd order in space. In the case of a spherical membrane shell, we confirm the results by Taber from the literature.

Flexible filaments and fibres are essential components of important complex fluids that appear in many biological and industrial settings. Direct simulations of these systems that capture the motion and deformation of many immersed filaments in suspension remain a formidable computational challenge due to the complex, coupled fluid--structure interactions of all filaments, the numerical stiffness associated with filament bending, and the various constraints that must be maintained as the filaments deform. In this paper, we address these challenges by first describing filament kinematics using quaternions to resolve both bending and twisting, applying implicit time-integration to alleviate numerical stiffness, and using quasi-Newton methods to obtain solutions to the resulting system of nonlinear equations. In particular, we employ geometric time integration to ensure that the quaternions remain unit as the filaments move. We also show that our framework can be used with a variety of models and methods, including matrix-free fast methods, that resolve low Reynolds number hydrodynamic interactions. We provide a series of tests and example simulations to demonstrate the performance and possible applications of our method. Finally, we provide a link to a MATLAB/Octave implementation of our framework that can be used to learn more about our approach and as a tool for filament simulation.

In rough elastohydrodynamic lubricated contacts the geometry often exhibits two clearly separated scales: a macroscopic scale –the one of the bearing– and a microscopic scale, that of the surface roughness. In numerical simulation of lubricated contacts, this difference in scales leads to large systems of equations to solve. Assuming periodicity or pseudo-periodicity of the small scale, several methods to decouple the macro scale from the micro scale have been proposed, the formal approach being the homogenization theory. However, the approximation errors due to the classical asymptotic assumptions can be considerable. In this work we introduce a homogenized model which takes into account the non-negligible pressures and deformations of the micro scale, thus extending the applicability of the classical asymptotic homogenized approaches.

We study a recently introduced formulation for fluid-structure interaction problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. The time discretization of the problem leads to a mixed problem for which a rigorous stability analysis is provided. The finite element space discretization is discussed and optimal convergence estimates are proved.

We consider a multiscale approach based on immersed methods for the efficient computational modeling of tissues composed of an elastic matrix (in two or three-dimensions) and a thin vascular structure (treated as a co-dimension two manifold) at a given pressure. We derive different variational formulations of the coupled problem, in which the effect of the vasculature can be surrogated in the elasticity equations via singular or hyper-singular forcing terms. These terms only depend on information defined on co-dimension two manifolds (such as vessel center line, cross sectional area, and mean pressure over cross section), thus drastically reducing the complexity of the computational model. We perform several numerical tests, ranging from simple cases with known exact solutions to the modeling of materials with random distributions of vessels. In the latter case, we use our immersed method to perform an in silico characterization of the mechanical properties of the effective biphasic material tissue via statistical simulations., 37 pages, 23 figures, 7 tables

Flexible filaments and fibres are essential components of important complex fluids that appear in many biological and industrial settings. Direct simulations of these systems that capture the motion and deformation of many immersed filaments in suspension remain a formidable computational challenge due to the complex, coupled fluid–structure interactions of all filaments, the numerical stiffness associated with filament bending, and the various constraints that must be maintained as the filaments deform. In this paper, we address these challenges by describing filament kinematics using quaternions to resolve both bending and twisting, applying implicit time-integration to alleviate numerical stiffness, and using quasi-Newton methods to obtain solutions to the resulting system of nonlinear equations. In particular, we employ geometric time integration to ensure that the quaternions remain unit as the filaments move. We also show that our framework can be used with a variety of models and methods, including matrix-free fast methods, that resolve low Reynolds number hydrodynamic interactions. We provide a series of tests and example simulations to demonstrate the performance and possible applications of our method. Finally, we provide a link to a MATLAB/Octave implementation of our framework that can be used to learn more about our approach and as a tool for filament simulation.

International audience; In this paper, we consider an elastic structure immersed in a compressible viscous fluid. The motion of the fluid is described by the compressible Navier-Stokes equations whereas the motion of the structure is given by the nonlinear Saint-Venant Kirchhoff model. For this model, we prove the existence and uniqueness of regular solutions defined locally in time. To do so, we first rewrite the nonlinearity in the elasticity equation in an adequate way. Then, we introduce a linearized problem and prove that this problem admits a unique regular solution. To obtain time regularity on the solution, we use energy estimates on the unknowns and their successive derivatives in time and to obtain spatial regularity, we use elliptic estimates. At last, to come back to the nonlinear problem, we use a fixed point theorem.

We consider periodically heterogeneous fluid-saturated poroelastic media described by the Biot model with inertia effects. The weak and semistrong formulations for displacement, seepage and pressure fields involve three equations expressing the momentum and mass balance and the Darcy law. Using the two-scale homogenization method, we obtain the limit two-scale problem and prove the existence and uniqueness of its weak solutions. The Laplace transformation in time is used to decouple the macroscopic and microscopic scales. It is shown that the seepage velocity is eliminated from the macroscopic equations involving strain and pressure fields only. The plane harmonic wave propagation is studied using an example of layered medium. Illustrations show some influence of the orthotropy on the dispersion phenomena.

This paper presents a mathematical model for an industry-inspired problem of vibration isolation applied to elastic fluid-filled containers. A fundamental problem of suppression of vibrations within a finite-width frequency interval for a multi-scale fluid-solid system has been solved. We have developed a systematic approach employing full fluid-solid interaction and dispersion analysis, which can be applied to finite and periodic multi-scale systems. The analytical findings are accompanied by numerical simulations, including frequency response analyses and transient regime computations., 33 pages, 20 figures

We study a coupled system of equations describing the movement of a rigid body which is immersed in a viscoelastic fluid. It is shown that under natural assumptions on the data and for general goemetries of the rigid body, excluding contact scenarios, a unique local-in-time strong solution exists.

International audience; Thanks to the Cagniard-de Hoop we derive the solution to the problem of wave propagation in an infinite bilayered acoustic/poroelastic media, where the poroelastic layer is modelled by the biphasic Biot's model. This first part is dedicated to solution to the two dimensional problem. We illustrate the interest of the solution by using it to validate a numerical code.

The problem of feedback controlled Rayleigh-Bénard convection is considered. For this problem with the simple flow structure in the vertical direction, a Galerkin method that uses only a few basis functions in this direction is presented. This approximation yields considerable simplification of the problem, explicitly incorporates the non-classical boundary conditions at the horizontal boundaries of the fluid layer resulting from feedback control and reduces the dimension of the original problem by one. This method is in spirit very similar to lubrication theory, where the simple laminar flow in the vertical direction is integrated out across the height of the fluid layer. Using a minimal set of appropriate basis functions to capture the nonlinear behaviour of the flow, we investigate the effects of feedback control on amplitude, wavelength and selection of patterns via weakly nonlinear analysis and numerical simulations of the resulting dimension-reduced problems in two and three dimensions. In the second part of this study we discuss the derivation of the appropriate basis functions and prove convergence of the Galerkin scheme. © 2008 IOP Publishing Ltd and London Mathematical Society.

We study the existence of weak solution for unsteady fluid-structure interaction problem for shear-thickening flow. The time dependent domain has at one part a flexible elastic wall. The evolution of fluid domain is governed by the generalized string equation with action of the fluid forces. The power-law viscosity model is applied to describe shear-dependent non-Newtonian fluids.

We detail and theoretically analyse the so-called fast vector (or velocity) penalty-projection methods (VPP ε) of which the main ideas and features are briefly introduced in [8,9,10]. This family of numerical schemes proves to efficiently compute the solution of unsteady Navier-Stokes/Brinkman problems governing incompressible or low Mach multi-phase viscous flows with variable mass density and/or viscosity or anisotropic permeability. In this paper, we describe in detail the connections and essential differences with usual methods to solve the Navier-Stokes equations. The key idea of the basic (VPP ε) method is to compute at each time step an accurate and curl-free approximation of the pressure gradient increment in time. This is obtained by proposing new Helmholtz-Hodge decomposition solutions of L 2-vector fields in bounded domains to get fast methods with suitable adapted right-hand sides; see [11]. This procedure only requires a few iterations of preconditioned conjugate gradients whatever the spatial mesh step. Then, the splitting (VPP ε) method performs a two-step approximate divergence-free vector projection yielding a velocity divergence vanishing as O(ε δt), δt being the time step, with a penalty parameter ε as small as desired until the machine precision, e.g. ε = 10 −14 , whereas the solution algorithm can be extremely fast and cheap. Indeed, the proposed velocity correction step typically requires only one or two iterations of a suitable pre-conditioned Krylov solver whatever the spatial mesh step [10]. Moreover, the robustness of our method is not sensitive to large mass density ratios since the velocity penalty-projection step does not include any spatial derivative of the density. 2 In the present work, we also prove the theoretical foundations as well as global sol-vability and optimal unconditional stability results of the (VPP ε) method for Navier-Stokes problems in the case of homogeneous flows, which are the main new results. Keywords Vector penalty-projection method · divergence-free penalty-projection · penalty method · splitting prediction-correction scheme · fast Helmholtz-Hodge decompositions · Navier-Stokes/Brinkman equations · stability analysis · incompressible homogeneous flows · dilatable flows · low Mach number flows · incompressible non-homogeneous or multiphase flows

We consider the problem of phase separation in eutectic alloy such as e.g. SnPb. For this we derive a phase field model from an atomistic point of view. We find the surface energy to be anisotropic, having in general a nonlinear dependence on concentration. We use matched asymptotic analysis to obtain a corresponding sharp-interface model. The resulting expression for the surface tension agrees with that found on the basis of classical thermodynamics for jump conditions at singular interfaces. A boundary integral formulation of the sharp-interface model enables us to numerically describe the motion and deformation of the binary alloy.

Spiders' webs and gossamer threads are often paraded as paradigms for lightweight structures and outstanding polymers. Probably the most intriguing of all spider silks is the araneid capture thread, covered with tiny glycoprotein glue droplets. Even if compressed, this thread remains surprisingly taut, a property shared with pure liquid films, allowing both thread and web to be in a constant state of tension. Vollrath and Edmonds proposed that the glue droplets would act as small windlasses and be responsible for the tension, but other explanations have also been suggested, involving for example the macromolecular properties of the flagelliform silk core filaments. Here we show that the nanolitre glue droplets of the capture thread indeed induce buckling and coiling of the core filaments: microscopic in-vivo observations reveal that the slack fibre is spooled into and within the droplets. We model windlass activation as a structural phase transition, and show that fibre spooling essentially results from the interplay between elasticity and capillarity. This is demonstrated by reproducing artificially the mechanism on a synthetic polyurethane thread/silicone oil droplet system. Fibre size is the key in natural and artificial setups which both require micrometer-sized fibres to function. The spools and coils inside the drops are further shown to directly affect the mechanical response of the thread, evidencing the central role played by geometry in spider silk mechanics. Beside shedding light on araneid capture thread functionality, we argue that the properties of this biological system provide novel insights for bioinspired synthetic actuators., 7 pages, 6 figures

Most biological flyers undergo orderly deformation in flight, and the deformations of wings lead to complex fluid-structure interactions. In this paper, an aerodynamic-structural coupling method of flapping wing is developed based on ANSYS to simulate the flapping of flexible wing. Fluent module and Transient Structural module are connected through the System Coupling module to make a two-way fluid-structure Coupling computational framework. Comparing with the rigid wing of a cicada, the coupling results of the flexible wing shows that the flexible deformation can increase the aerodynamic performances of flapping flight., 9 pages, 6 figures, International Journal of Recent advances in Mechanical Engineering (IJMECH) Vol.3, No.4, November 2014

We consider the feedback stabilization of a simplified 1d model for a fluid–structure interaction system. The fluid equation is the viscous Burgers equation whereas the motion of the particle is given by the Newton's laws. We stabilize this system around a stationary state by using feedbacks located at the exterior boundary of the fluid domain. With one input, we obtain a local stabilizability of the system with an exponential decay rate of order σ σ 0 . An arbitrary order for the exponential decay rate can be proved if a unique continuation result holds true or if two inputs are used to stabilize the system. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domains of the stationary state and of the stabilized solution are different.

Consider a time-harmonic acoustic plane wave incident onto a doubly periodic (biperiodic) surface from above. The medium above the surface is supposed to be filled with a homogeneous compressible inviscid fluid of constant mass density, whereas the region below is occupied by an isotropic and linearly elastic solid body characterized by its Lamé constants. This paper is concerned with a variational approach to the fluid-solid interaction problems with unbounded biperiodic Lipschitz interfaces between the domains of the acoustic and elastic waves. The existence of quasi-periodic solutions in Sobolev spaces is established at arbitrary frequency of incidence, while uniqueness is proved only for small frequencies or for all frequencies excluding a discrete set. A finite element scheme coupled with Dirichlet-to-Neumann mappings is proposed. The Dirichlet-to-Neumann mappings are approximated by truncated Rayleigh series expansions, and, finally, numerical tests in 2D are performed.

We consider the system of equations modeling the free motion of a rigid body with a cavity filled by a viscous (Navier-Stokes) liquid. We give a rigorous proof of Zhukovskiy's Theorem, which states that in the limit of time going to infinity, the relative fluid velocity tends to zero and the rigid velocity of the full structure tends to a steady rotation around one of the principle axes of inertia. The existence of global weak solutions for this system was established previously. In particular, we prove that every weak solution of this type is subject to Zhukovskiy's Theorem. Independently of the geometry and of parameters, this shows that the presence of fluid prevents precession of the body in the limit. In general, we cannot predict which axis will be attained, but we show stability of the largest axis and provide criteria on the initial data which are decisive in special cases., Comment: 18 pages, 0 figures

International audience; We study a coupled fluid-structure system. The structure corresponds to a part of the boundary of a domain containing an incompressible viscous fluid. The structure displacement is modeled by an ordinary differential equation. We prove the local null controllability of the system when the control acts on a fixed subset of the fluid domain.

8 pages; International audience; We present a new {\em fast vector penalty-projection method (VPP$_{\eps}$)} to efficiently compute the solution of unsteady Navier-Stokes problems governing incompressible multiphase viscous flows with variable density and/or viscosity. The key idea of the method is to compute at each time step an accurate and curl-free approximation of the pressure gradient increment in time. This method performs a {\em two-step approximate divergence-free vector projection} yielding a velocity divergence vanishing as $\cO(\eps\,\dt)$, $\dt$ being the time step, with a penalty parameter $\eps$ as small as desired until the machine precision, {\em e.g.} $\eps=10^{-14}$, whereas the solution algorithm can be extremely fast and cheap. Indeed, the proposed {\em vector correction step} typically requires only a few iterations of a suitable preconditioned Krylov solver whatever the spatial mesh step. The method is numerically validated on three benchmark problems for non-homogeneous or multiphase flows where we compare it to the Uzawa augmented Lagrangian (UAL) and scalar incremental projection (SIP) methods. Moreover, a new test case for fluid-structure interaction problems is also investigated. That results in a very robust method running faster than usual methods and being able to efficiently and accurately compute sharp test cases whatever the density, viscosity or anisotropic permeability jumps, whereas other methods crash.

International audience; In this note we consider the motion of a solid body in a two dimensional incompressible perfect fluid. We prove the global existence of solutions in the case where the initial vorticity belongs to $L^p$ with $p>1$ and is compactly supported. We do not assume that the energy is finite.

The motion of a rigid body immersed in an incompressible perfect fluid which occupies a three-dimensional bounded domain have been recently studied under its PDE formulation. In particularclassical solutions have been shown to exist locally in time. In this note, following the celebratedresult of Arnold [1] concerning the case of a perfect incompressible fluid alone, we prove that theseclassical solutions are the geodesics of a Riemannian manifold of infinite dimension, in the sense thatthey are the critical points of an action, which is the integral over time of the total kinetic energy ofthe fluid-rigid body system.Keywords. Perfect incompressible fluid, fluid-rigid body interaction, least action principle.AMS Subject Classification. 76B99, 74F10. 1 Introduction We consider the motion of a rigid body immersed in an incompressible homogeneous perfect fluid, so thatthe system fluid-rigid body occupies a smooth open and bounded domain Ω ⊂ R 3 . The solid is supposedto occupy at each instant t>0 a smooth closed connected subset S(t) ⊂ Ω which is surrounded by aperfect incompressible fluid filling the domain F(t) := Ω\S(t).For the point of view of PDEs, this system have been recently studied in [8], [9], [10], [6], [5] whichhave set a Cauchy theory for classical solutions.The aim of this note is to provide a rigorous proof that the classical solutions can be equivalentlythought as geodesics of a Riemannian manifold of infinite dimension, in the sense that they are thecritical points of an action, which is the integral over time of the total kinetic energy of the fluid-rigidbody system. It was pointed out in a famous paper by Arnold [1] that both the Euler equations fora rigid body as well as the Euler equations for a perfect fluid can be derived with this approach. Themotion of a rigid body in a frame attached to its center of mass can be considered as a geodesic on thespecial orthogonal group SO(3). On the other hand the motion of a perfect fluid filling a container Ω(without any immersed rigid body in it) can be considered as a geodesic equation on the space Sdiff

The skin is made of three main layers which are, from the top to the bottom: the epidermis, the dermis and the hypodermis. We consider the dermis as made of a Stokes fluid interacting with a periodic network of elastic fibers, assumed to obey the linearized elasticity law of behaviour. Above and below, the epidermis and the hypodermis are elastic solids. As the dimension of the thickness is very small compared to the two others, we assume periodic boundary conditions in those two planar directions. We study the 3d fluid-structure interaction system in a first part, and in a second part, we make the characteric size of the periodic element of the network go to zero in order to find an homogenized law for the whole skin. Starting from linear elastic materials, we find a viscoelastic law at the limit.

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International audience; We are interested in the modeling of wave propagation in an infinite bilayered acoustic/poroelastic media. We consider the biphasic Biot's model in the poroelastic layer. The first part was devoted to the calculation of analytical solution in two dimensions, thanks to Cagniard de Hoop method. In this second part we consider the 3D case.

We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. The main peculiarity of the model is the assumption that the transversal displacements of the plate are negligible relative to in-plane displacements. This kind of models arises in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. We also show that the corresponding linearized system generates exponentially stable $C_0$-semigroup. We do not assume any kind of mechanical damping in the plate component. Thus our results means that dissipation of the energy in the fluid due to viscosity is sufficient to stabilize the system., Comment: 18 pages

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International audience; This paper is devoted to the study of a coupled system which consists of a wave equation and a heat equation coupled through a transmission condition along a steady interface. This system is a linearized model for fluid-structure interaction introduced by Rauch, Zhang and Zuazua for a simple transmission condition and by Zhang and Zuazua for a natural transmission condition. Using an abstract theorem of Burq and a new Carleman estimate proved near the interface, we complete the results obtained by Zhang and Zuazua and by Duyckaerts. We prove, without a Geometric Control Condition, a logarithmic decay of the energy.

International audience; In this paper we investigate the motion of a rigid ball surrounded by an incompressible perfect fluid occupying RN. We prove the existence, uniqueness, and persistence of the regularity for the solutions of this fluid-structure interaction problem.

A two-dimensional body, exhibiting a slight rotational movement, moves in a rarefied medium of particles which collide with it in a perfectly elastic way. In previously realized investigations by the first two authors, Plakhov & Gouveia (2007, Nonlinearity, 20), shapes of nonconvex bodies were sought which would maximize the braking force of the medium on their movement. Giving continuity to this study, new investigations have been undertaken which culminate in an outcome which represents a large qualitative advance relative to that which was achieved earlier. This result, now presented, consists of a two-dimensional shape which confers on the body a resistance which is very close to its theoretical supremum value. But its interest does not lie solely in the maximization of Newtonian resistance; on regarding its characteristics, other areas of application are seen to begin to appear which are thought to be capable of having great utility. The optimal shape which has been encountered resulted from numerical studies, thus it is the object of additional study of an analytical nature, where it proves some important properties which explain in great part its effectiveness., Accepted (April 16, 2009) for publication in the journal "Applied Mathematics and Computation"

International audience; Dynamical behavior of the head during an impact is important for analyzing the nduced local damage or diffuse damage in the brain tissue. We determine in the present tudy the natural frequencies and the modal shapes of the system of brain, cerebro-spinal luid and skull. Two models are presented in this work: an elastic-acoustic model assuming a igid skull and an elastic-acoustic-elastic model assuming a deformable skull. It is shown that atural frequencies and more significantly the modal shapes are strongly influenced by the nteraction between solid phases (brain and skull) and the cerebro-spinal fluid.

We introduce a framework that is capable to model the appearance of mechanical stresses due to inelastic deformations. Among these we consider in particular thermal expansions, diffusion and phase transitions. Among the quantities of central importance are the eigenstrain and the misfit strain. They describe the phenomenon that different material volumes of a compact body may not be compatible to each other in a stress-free reference configuration, so that here a compact body may not exist. We shall show that it is possible to find a further reference configuration, where the body is compact but not free of stress. A typical example where misfit appears concerns a body whose local parts differently transform their phase. This might be a change of the crystal lattice from the ferrite to the austenite symmetry in steel, or the formation of liquid droplets in crystalline gallium arsenide. In both cases the new interior phase has with respect to the parent phase different volume or shape in its state that is free of stress. In this study we consider the eigenstrain problem for pure substances as well as for mixtures. In the latter case subtle arguments are needed for an appropriate description. Special focus is given to the equivalence of interface boundaries with discontinues and continues displacement vectors.

International audience; Understanding fish-like locomotion as a result of internal shape changes may result in improved underwater propulsion mechanism. In this ar- ticle, we study a coupled system of partial differential equations and ordinary differential equations which models the motion of self-propelled deformable bodies (called swimmers) in an potential fluid flow. The deformations being prescribed, we apply the least action principle of Lagrangian mechanics to de- termine the equations of the inferred motion. We prove that the swimmers degrees of freedom solve a second order system of nonlinear ordinary differen- tial equations. Under suitable smoothness assumptions on the fluid's domain boundary and on the given deformations, we prove the existence and regularity of the bodies rigid motions, up to a collision between two swimmers or between a swimmer with the boundary of the fluid. Then we compute explicitly the Euler-Lagrange equations in terms of the geometric data of the bodies and of the value of the fluid's harmonic potential on the boundary of the fluid.

We consider the flow around a rigid fishing net. We describe the model and show an existence result. Numerical results are shown and compared with experimental data., L'objet de cette Note est la modélisation et la simulation d'un écoulement turbulent au voisinage d'un filet de pêche rigide. Nous décrivons la modélisation, un résultat d'existence au système d'équations puis quelques résultats numériques comparés avec des résultats expérimentaux pour valider le code.

This work deals with a class of evolution problems consisting of a pair of coupled equations for modelling propagation of elastic waves in fluid-saturated porous media. The type of the first equation depends on two physical parameters (density and secondary consolidation) which can vanish while the second one is always parabolic. In case the density never vanishes, the first equation is second-order hyperbolic type and a weak solution to the problem is constructed using a variational method in a Sobolev framework. Next, the proof of uniqueness involves Ladyzenskaja's test-functions used to compensate a lack of regularity that would be required in a standard energy method. This approach gives rise to a priori estimates which are useful to prove that the linearized thermoelasticity and the quasi-static systems are defined as asymptotic models of the Biot problem when the secondary consolidation coefficient or the density is small.

The paper is devoted to the thermodynamic construction of a two-component model of poroelastic media undergoing, in contrast to earlier works on this subject, nonisothermal processes. Under the constitutive dependence on partial mass densities, deformation gradient of skeleton, relative velocity, temperature, temperature gradient and porosity (simple poroelastic material) as well as the assumption of small deviations from the thermodynamic equilibrium we construct explicit relations for fluxes, prove the splitting of the free energy into partial contributions without mechanical couplings and construct a chemical potential for the fluid component important for the formulation of boundary conditions on permeable boundaries. We discuss as well a modification of the porosity balance equation in which we account for time changes of equilibrium porosity. This modification yields the behavior of the model characteristic for granular materials.

In this paper we consider two established structural acoustic models: Mathematically, the first model couples a hyperbolic (wave) equation, defined within a two- or three-dimensional acoustic chamber, with an elastic plate (or beam) equation, possibly with structural damping (parabolic type), defined on its elastic (flat) wall. Instead, the second model couples the same hyperbolic equation this time with a thermoelastic plate (or beam), either of parabolic type or else of {\em hyperbolic-dominated} type, defined on its flexible (flat) wall. The thermoelastic component may be supplemented by any canonical boundary conditions (B.C.), including the coupled free B.C. Moreover, its differential operators may have variable coefficients (in space). In either of the two models, coupling takes place on the elastic wall. This coupled PDE system (possibly with hyperbolic/parabolic interaction) generates a strongly continuous contraction semigroup $e^{At}$ on a natural energy space $Y$. The main result of the present paper is a backward uniqueness theorem for such structural acoustic semigroups: $e^{AT} y_0 = 0$ for some $T > 0$ and $y_0 \in Y$ implies $y_0 =0$.