Solution of the Ovsyannikov Problem of Two-Dimensional Isothermal Motion of a Polytropic Gas

We study an overdetermined system of partial differential equations which describes the two-dimensional isothermal motion of a polytropic gas. The system is reduced to a passive form and is completely integrated. The resulting solutions are treated as ideal incompressible fluid flows bounded by a free surface or a moving solid wall.     

Solitary wave packets beneath a compressed ice cover

A family of plane solitary wave packets of a small (but finite) amplitude on the surface of an ideal incompressible fluid of finite depth beneath an ice cover is described. The solitary wave trains correspond to solutions of the two-dimensional system of Euler’s equations of an ideal incompressible fluid of the type of a traveling wave which decreases at infinity and has identical phase and group velocities. The ice cover is simulated by an elastic Kirchhoff-Love plate freely floating on the fluid surface in the compressed state.     

A two-dimensional effective model describing fluid–structure interaction in blood flow: analysis, simulation and experimental validation

We derive a closed system of effective equations describing a time-dependent flow of a viscous incompressible Newtonian fluid through a long and narrow elastic tube. The 3D axially symmetric incompressible Navier–Stokes equations are used to model the flow. Two models are used to describe the tube wall: the linear membrane shell model and the linearly elastic membrane and the curved, linearly elastic Koiter shell model. We study the behavior of the coupled fluid–structure interaction problem in the limit when the ratio between the radius and the length of the tube, , tends to zero. We obtain the reduced equations that are of Biot type with memory. An interesting feature of the reduced equations is that the memory term explicitly captures the viscoelastic nature of the coupled problem. Our model provides significant improvement over the standard 1D approximations of the fluid–structure interaction problem, all of which assume an ad hoc closure assumption for the velocity profile. We performed experimental validation of the reduced model using a mock circulatory flow loop assembled at the Cardiovascular Research Laboratory at the Texas Heart Institute. Experimental results show excellent agreement with the numerically calculated solution. Major applications include blood flow through large human arteries. To cite this article: S. Čanić et al., C. R. Mecanique 333 (2005).     

Simultaneous solution of the Navier-Stokes and elastic membrane equations by a finite element method

Fluid flow through a significantly compressed elastic tube occurs in a variety of physiological situations. Laboratory experiments investigating such flows through finite lengths of tube mounted between rigid supports have demonstrated that the system is one of great dynamical complexity, displaying a rich variety of self-excited oscillations. The physical mechanisms responsible for the onset of such oscillations are not yet fully understood, but simplified models indicate that energy loss by flow separation, variation in longitudinal wall tension and propagation of fluid elastic pressure waves may all be important. Direct numerical solution of the highly non-linear equations governing even the most simplified two-dimensional models aimed at capturing these basic features requires that both the flow field and the domain shape be determined as part of the solution, since neither is known a priori. To accomplish this, previous algorithms have decoupled the solid and fluid mechanics, solving for each separately and converging iteratively on a solution which satisfies both. This paper describes a finite element technique which solves the incompressible Navier-Stokes equatikons simultaneously with the elastic membrane equations on the flexible boundary. The elastic boundary position is parametized in terms of distances along spines in a manner similar to that which has been used successfully in studies of viscous free surface flows, but here the membrane curvature equation rather than the kinematic boundary condition of vanishing normal velocity is used to determine these diatances and the membrane tension varies with the shear stresses exerted on it by the fluid motions. Bothy the grid and the spine positions adjust in response to membrane deformation, and the coupled fluid and elastic equations are solved by a Newton-Raphson scheme which displays quadratic convergence down to low membrane tensions and extreme states of collapse. Solutions to the steady problem are discussed, along with an indication of how the time-dependent problem might be approached.     

Nonlinear waves in incompressible viscoelastic Maxwell medium

In this paper we study two-dimensional flows of incompressible viscoelastic Maxwell media with Jaumann corotational derivative in the rheological constitutive law. In the general case, due to the incompressibility condition, the equations of motion have both real and complex characteristics. Group properties of this system are studied. On this basis, two submodels of the Maxwell model are selected, which can be reduced to hyperbolic ones. More precisely, we consider plane shear flow between two parallel planes and Couette type flow caused by the inertial cylinder rotation. As a result, we obtain the closed systems of three equations of mixed type, which describe nonlinear transverse waves in an incompressible Maxwell fluid. It is demonstrated that discontinuities can develop in elastic media even from smooth initial data. Stability of shocks in the Maxwell fluid with and without retardation time is discussed.     

On a Fluid-Structure Model in Which the Dynamics of the Structure Involves the Shear Stress Due to the Fluid

In this paper we consider a model for fluid-structure interaction. The hybrid system describes the interaction between an incompressible fluid in a three-dimensional container with interior a fixed domain and a thin elastic plate, the interface, which coincides with a flexible flat part of the surface of the vessel containing the fluid. The motion of the fluid is described by the linearized Navier–Stokes equations and the deformation of the plate by the classical plate equations for in-plane motions, modified to include the viscous shear stress which the fluid exerts on the plate as well as damping of Kelvin–Voigt type. We establish the existence of a unique weak solution of the interactive system of partial differential equations by considering an appropriate variational formulation. Uniform stability of the energy associated with the model is shown under the assumption that the potential plate energy is dominated by the dissipation induced by the viscosity of the fluid. The retention of the physical parameters in the problem is an a priori requirement in this physical condition.      

Behavior of a Semi-Infinite Ice Cover Under a Uniformly Moving Load

This paper consideres the behavior of a semi-infinite ice cover on the surface of an ideal incompressible fluid of finite depth under the action of a load moving with constant velocity along the edge of the cover at some distance from it. The ice cover is modeled by a thin elastic plate of constant thickness. In a moving coordinate system, the deflection of the plate is assumed to be steady. An analytic solution of the problem is obtained using the Wiener–Hopf technique. The wave forces, the deflection of the plate, and the elevation of the free surface of the fluid at different velocities of the load are investigated.     

Effective equations describing the flow of a viscous incompressible fluid through a long elastic tubeEquations efficaces décrivant l'écoulement d'un fluide visqueux incompressible dans un tuyau long et élastique

We study the flow of a viscous incompressible fluid through a long and narrow elastic tube whose walls are modeled by the Navier equations for a curved, linearly elastic membrane. The flow is governed by a given small time dependent pressure drop between the inlet and the outlet boundary, giving rise to creeping flow modeled by the Stokes equations. By employing asymptotic analysis in thin, elastic, domains we obtain the reduced equations which correspond to a Biot type viscoelastic equation for the effective pressure and the effective displacement. The approximation is rigorously justified by obtaining the error estimates for the velocity, pressure and displacement. Applications of the model problem include blood flow in small arteries. We recover the well-known Law of Laplace and provide a new, improved model when shear modulus of the vessel wall is not negligible. To cite this article: S. ?ani?, A. Mikeli?, C. R. Mecanique 330 (2002) 661–666.     

Parallel finite element simulation of mooring forces on floating objects

The coupling between the equations governing the free‐surface flows, the six degrees of freedom non‐linear rigid body dynamics, the linear elasticity equations for mesh‐moving and the cables has resulted in a fluid‐structure interaction technology capable of simulating mooring forces on floating objects. The finite element solution strategy is based on a combination approach derived from fixed‐mesh and moving‐mesh techniques. Here, the free‐surface flow simulations are based on the Navier–Stokes equations written for two incompressible fluids where the impact of one fluid on the other one is extremely small. An interface function with two distinct values is used to locate the position of the free‐surface. The stabilized finite element formulations are written and integrated in an arbitrary Lagrangian–Eulerian domain. This allows us to handle the motion of the time dependent geometries. Forces and momentums exerted on the floating object by both water and hawsers are calculated and used to update the position of the floating object in time. In the mesh moving scheme, we assume that the computational domain is made of elastic materials. The linear elasticity equations are solved to obtain the displacements for each computational node. The non‐linear rigid body dynamics equations are coupled with the governing equations of fluid flow and are solved simultaneously to update the position of the floating object. The numerical examples includes a 3D simulation of water waves impacting on a moored floating box and a model boat and simulation of floating object under water constrained with a cable. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical simulation of viscous flow interaction with an elastic membrane

A numerical fluid–structure interaction model is developed for the analysis of viscous flow over elastic membrane structures. The Navier–Stokes equations are discretized on a moving body‐fitted unstructured triangular grid using the finite volume method, taking into account grid non‐orthogonality, and implementing the SIMPLE algorithm for pressure solution, power law implicit differencing and Rhie–Chow explicit mass flux interpolations. The membrane is discretized as a set of links that coincide with a subset of the fluid mesh edges. A new model is introduced to distribute local and global elastic effects to aid stability of the structure model and damping effects are also included. A pseudo‐structural approach using a balance of mesh edge spring tensions and cell internal pressures controls the motion of fluid mesh nodes based on the displacements of the membrane. Following initial validation, the model is applied to the case of a two‐dimensional membrane pinned at both ends at an angle of attack of 4° to the oncoming flow, at a Reynolds number based on the chord length of 4 × 10³. A series of tests on membranes of different elastic stiffness investigates their unsteady movements over time. The membranes of higher elastic stiffness adopt a stable equilibrium shape, while the membrane of lowest elastic stiffness demonstrates unstable interactions between its inflated shape and the resulting unsteady wake. These unstable effects are shown to be significantly magnified by the flexible nature of the membrane compared with a rigid surface of the same average shape. Copyright © 2007 John Wiley & Sons, Ltd.     

Unsteady behavior of an elastic articulated beam floating on shallow water

The unsteady behavior of an elastic beam composed of hinged homogeneous sections, which freely floats on the surface of an ideal incompressible fluid, is studied within the framework of the linear shallow water theory. The unsteady behavior of the beam is due to incidence of a localized surface wave or initial deformation. Beam deflection is sought in the form of an expansion with respect to eigenfunctions of oscillations in vacuum with time-dependent amplitudes. The problem is reduced to solving an infinite system of ordinary differential equations for unknown amplitudes. The beam behavior with different actions of the medium and hinge positions is studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 54–65, July–August, 2009.     

Finite‐element simulation of incompressible fluid flow in an elastic vessel

Finite‐element simulation was performed to predict the incompressible Navier–Stokes flow in a domain, partly bounded by an elastic vessel, which is allowed to vary with time. Besides satisfying the physical conservation laws, both surface and the volume conservation laws are satisfied at the discrete level for ensuring the balance between physical and geometrical variables. Several problems which are amenable to analytical solutions were tested for validating the method. The simulated results are observed to agree favourably with analytical solutions. Having verified the applicability of the finite‐element code to problems involving moving grids, we consider an incompressible fluid flow bounded by rigid and elastic vessel walls. Our emphasis was placed on the validation of the formulation developed within the moving‐grid framework. Copyright © 2003 John Wiley & Sons, Ltd.     

Fluid flow of incompressible viscous fluid through a non-linear elastic tube

The study of viscous flow in tubes with deformable walls is of specific interest in industry and biomedical technology and in understanding various phenomena in medicine and biology (atherosclerosis, artery replacement by a graft, etc) as well. The present work describes numerically the behavior of a viscous incompressible fluid through a tube with a non-linear elastic membrane insertion. The membrane insertion in the solid tube is composed by non-linear elastic material, following Fung’s (Biomechanics: mechanical properties of living tissue, 2nd edn. Springer, New York, 1993) type strain–energy density function. The fluid is described through a Navier–Stokes code coupled with a system of non linear equations, governing the interaction with the membrane deformation. The objective of this work is the study of the deformation of a non-linear elastic membrane insertion interacting with the fluid flow. The case of the linear elastic material of the membrane is also considered. These two cases are compared and the results are evaluated. The advantages of considering membrane nonlinear elastic material are well established. Finally, the case of an axisymmetric elastic tube with variable stiffness along the tube and membrane sections is studied, trying to substitute the solid tube with a membrane of high stiffness, exhibiting more realistic response.     

WAVE PROPAGATION IN A SYSTEM OF COAXIAL TUBES FILLED WITH INCOMPRESSIBLE MEDIA: A MODEL OF PULSE TRANSMISSION IN THE INTRACRANIAL ARTERIES

The propagation of harmonic waves through a system formed of coaxial tubes filled with incompressible continua is considered as a model of arterial pulse propagation in the craniospinal cavity. The inner tube represents a blood vessel and is modelled as a thin-walled membrane shell. The outer tube is assumed to be rigid to account for the constraint imposed on the vessels by the skull and the vertebrae. We consider two models: in the first model the annulus between the tubes is filled with fluid; in the second model the annulus is filled with a viscoelastic solid separated from the tubes by thin layers of fluid. In both models, the elastic tube is filled with fluid. The motion of the fluid is described by the linearized form of the Navier–Stokes equations, and the motion of the solid by classical elasticity theory. The results show that the wave speed in the system is lower than that for a fluid-filled elastic tube free of any constraint. This is due to the stresses generated to satisfy the condition that the volume in the system has to be conserved. However, the effect of the constraint weakens as the radius of the outer tube is increased, and it should be insignificant for the typical physiological parameter range.     

Investigation of the interaction between a viscous incompressible fluid layer and walls of a channel formed by coaxial vibrating discs

The interaction between a viscous incompressible fluid layer and walls of a channel formed by two concentric discs moving perpendicularly to their planes due to vibration of the base on which the channel is mounted is investigated. The case of two absolutely rigid discs with elastic suspension and the case in which one of the discs is an elastic plate with the rigid restrain on the edges are considered. The velocity and pressure distributions over the fluid and the laws of motion of the walls and their frequency characteristics which make it possible to determine the resonance vibration frequencies of the mechanical system considered are found.     

Steady magnetic waves

Steady simple waves are investigated in an incompressible conducting ideal inhomogeneously and isotropically magnetizable fluid moving along the lines of force of a magnetic field. The integration of the system of equations describing such waves is reduced to the calculation of quadrature expressions in the case of an arbitrary magnetization law. It is shown that, depending on the magnetic properties of the medium, different types of steady waves are possible: magnetizing waves in a diamagnetic fluid and demagnetizing waves in a paramagnetic fluid. The results are given of calculations of demagnetizing waves in a conducting ferromagnetic fluid. An analysis is made of the various possible flow regimes of a conducting magnetizable fluid at the point of a perfectly conducting corner.     

Kinetic model for the motion of compressible bubbles in a perfect fluid

Collective behavior of compressible gas bubbles moving in an inviscid incompressible fluid is studied. A kinetic approach is employed, based on an approximate calculation of the fluid flow potential and formulation of Hamilton's equations for generalized coordinates and momenta of bubbles. Kinetic equations governing the evolution of a distribution function of bubbles are derived. These equations are similar to Vlasov's equations. Conservation laws which are direct consequences of the kinetic system are found. It is shown that for a narrowly peaked distribution function they form a closed system of hydrodynamical equations for the mean flow parameters. The system yields the analogue of Rayleigh–Lamb's equation governing oscillations of bubbles. A variational principle for the hydrodynamical system is established and the linear stability analysis is performed.     

The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations

The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations. Unlike our approach in [5] for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in strong norms of our sequence of regularized problems.     

Existence of Weak Solutions for the Three-Dimensional Motion of an Elastic Structure in an Incompressible Fluid

We study here the three-dimensional motion of an elastic structure immersed in an incompressible viscous fluid. The structure and the fluid are contained in a fixed bounded connected set Ω. We show the existence of a weak solution for regularized elastic deformations as long as elastic deformations are not too important (in order to avoid interpenetration and preserve orientation on the structure) and no collisions between the structure and the boundary occur. As the structure moves freely in the fluid, it seems natural (and it corresponds to many physical applications) to consider that its rigid motion (translation and rotation) may be large. The existence result presented here has been announced in [4]. Some improvements have been provided on the model: the model considered in [4] is a simplified model where the structure motion is modelled by decoupled and linear equations for the translation, the rotation and the purely elastic displacement. In what follows, we consider on the structure a model which represents the motion of a structure with large rigid displacements and small elastic perturbations. This model, introduced by [15] for a structure alone, leads to coupled and nonlinear equations for the translation, the rotation and the elastic displacement.     

Microstructure-dependent nonlinear viscoelasticity due to extracellular flow within cellular structures

A constitutive model that highlights a special viscoelastic property of materials with cellular microstructures is developed. We model the microstructure as a regularly arranged system of the same elastic cells that are mutually interconnected by elastic linkages. The space between cells is filled by a fluid that may flow freely within this extracellular space. The macroscopic behavior of the whole structure is studied by means of continuum mechanics using a differential scheme with internal variables. Here, the internal variables are chosen as the distances that separate neighboring cells. The evolution equations are derived from the Clausius–Planck inequality, which considers the internal dissipation to be exclusively due to the extracellular fluid movement. Special attention is paid to incompressible materials in the context of uniaxial load. In this context, the importance of the fluid viscosity on material behavior is related to microstructural parameters like the cells’ dimensions and the relative stiffness between the cells and matrix elastic reinforcement.