The onset of unsteadiness of two-dimensional bodies falling or rising freely in a viscous fluid: a linear study

We consider the transition between the steady vertical path and the oscillatory path of two-dimensional bodies moving under the effect of buoyancy in a viscous fluid. Linearization of the Navier–Stokes equations governing the flow past the body and of Newton’s equations governing the body dynamics leads to an eigenvalue problem, which is solved numerically. Three different body geometries are then examined in detail, namely a quasi-infinitely thin plate, a plate of rectangular cross-section with an aspect ratio of 8, and a rod with a square cross-section. Two kinds of eigenmodes are observed in the limit of large body-to-fluid mass ratios, namely ‘fluid’ modes identical to those found in the wake of a fixed body, which are responsible for the onset of vortex shedding, and four additional ‘aerodynamic’ modes associated with much longer time scales, which are also predicted using a quasi-static model introduced in a companion paper. The stability thresholds are computed and the nature of the corresponding eigenmodes is investigated throughout the whole possible range of mass ratios. For thin bodies such as a flat plate, the Reynolds number characterizing the threshold of the first instability and the associated Strouhal number are observed to be comparable with those of the corresponding fixed body. Other modes are found to become unstable at larger Reynolds numbers, and complicated branch crossings leading to mode switching are observed. On the other hand, for bluff bodies such as a square rod, two unstable modes are detected in the range of Reynolds number corresponding to wake destabilization. For large enough mass ratios, the leading mode is similar to the vortex shedding mode past a fixed body, while for smaller mass ratios it is of a different nature, with a Strouhal number about half that of the vortex shedding mode and a stronger coupling with the body dynamics

A quasi-static approach to the stability of the path of heavy bodies falling within a viscous fluid

We consider the gravity-driven motion of a heavy two-dimensional rigid body freely falling in a viscous fluid.We introduce a quasi-static linear model of the forces and torques induced by the possible changes in the body velocity,or by the occurrence of a nonzero incidence angle or a spanwise rotation of the body. The coefficients involved in this model are accurately computed over a full range of Reynolds number by numerically resolving the Navier 13Stokes equations, considering three elementary situations where the motion of the body is prescribed. The falling body is found to exhibit three distinct eigenmodes which are always damped in the case of a thin plate with uniform mass loading or a circular cylinder,but may be amplified for other geometries,such as in the case of a square cylinder

Evolution and rupture of vulnerable plaques: a review of mechanical effects

Atherosclerosis occurs as a result of the buildup and infiltration of lipid streaks in artery walls, leading to plaques. Understanding the development of atherosclerosis and plaque vulnerability is of critical importance, since plaque rupture can result in heart attack or stroke. Plaques can be divided into two distinct types: those that rupture (vulnerable) and those that are less likely to rupture (stable). In the last few decades, researchers have been interested in studying the influence of the mechanical effects (blood shear stress, pressure forces, and structural stress) on the plaque formation and rupture processes. In the literature, physiological experimental studies are limited by the complexity of in vivo experiments to study such effects, whereas the numerical approach often uses simplified models compared with realistic conditions, so that no general agreement of the mechanisms responsible for plaque formation has yet been reached. In addition, in a large number of cases, the presence of plaques in arteries is asymptomatic. The prediction of plaque rupture remains a complex question to elucidate, not only because of the interaction of numerous phenomena involved in this process (biological, chemical, and mechanical) but also because of the large time scale on which plaques develop. The purpose of the present article is to review the current mechanical models used to describe the blood flow in arteries in the presence of plaques, as well as reviewing the literature treating the influence of mechanical effects on plaque formation, development, and rupture. Finally, some directions of research, including those being undertaken by the authors, are described

Dynamique non-linéaire des écoulements confinés : application à l'instabilité de Marangoni-Bénard et aux écoulements entre surfaces texturées

Le travail porte sur deux problématiques scientifiques : la formation de structures convectives induites par l'instabilité de Marangoni-Bénard et les propriétés de transport des écoulements entre surfaces texturées. Bien que physiquement distincts, ces deux systèmes présentent les points communs d'être assujettis à de fortes contraintes spatiales. Il sont analysés par le biais de la théorie des bifurcations. L'étude de la convection de Marangoni-Bénard a été menée dans des géométries cylindriques à section transverse circulaire et faiblement elliptique. La comparaison des deux situations dans le régime non-linéaire a été menée par l'étude des changements induits sur les diagrammes de bifurcation eux mêmes interprétés par la théorie des bifurcations en présence de symétries. Nous avons ensuite mené l'étude de cette instabilité en présence de mélanges fluides binaires sujets à l'effet Soret et dans des couches fluides bidimensionnelles. Ce travail a révélé la formation de structures convectives spatialement localisées appelées convectons dont nous avons révélé la formation sur un fond d'ondes de plus faible amplitude. Enfin, nous avons étudié les propriétés de transport des écoulements entre surfaces texturées. Le système étudié est confiné transversalement à la direction de l'écoulement ce qui place cette étude dans le contexte de la microfluidique et de l'élaboration de micro-mélangeurs passifs. La simulation numérique et l'analyse des propriétés de transport de traceurs passifs est menée sur les équations issues d'un développement asymptotique faiblement inertiel dans un canal formé d'une succession périodique de cellules texturées.The work focuses on two different physical situations: the convective structures resulting from the Marangoni-Bénard instability and the flow between patterned surfaces. The two systems are spatially constrained and are analysed using dynamical systems theories. Marangoni-Bénard convection has been studied in cylindrical geometries with either a circular or a weakly elliptical cross-section. The comparison of the two situations is carried out in the non-linear regime and the corresponding bifurcation diagrams are analysed using bifurcation theory with symmetries. Two-dimensional Marangoni convection in binary mixtures with Soret effect has also been studied in large periodic domains. The results show the formation of steady convective structures localized in space called convectons and the onset of stable convectons embedded in a background of small amplitude standing waves. Finally, the transport properties of flows in between patterned surfaces under weak inertia influence is studied. The flow is induced by a constant applied pressure gradient and the velocity field is calculated using an extension of the lubrication approximation taking into account the first order inertial corrections. Trajectories of tracers are obtained by integrating numerically the quasi-analytic velocity field. The transport properties are analysed by the study of Poincaré sections and their invariants

Spatially localized states in Marangoni convection in binary mixtures

Two-dimensional Marangoni convection in binary mixtures is studied in periodic domains with large spatial period in the horizontal. For negative Soret coefficients convection may set in via growing oscillations which evolve into standing waves. With increasing amplitude these waves undergo a transition to traveling waves, and then to more complex waveforms. Out of this state emerge stable stationary spatially localized structures embedded in a background of small amplitude standing waves. The relation of these states to the time-independent spatially localized states that characterize the so-called pinning region is investigated by exploring the stability properties of the latter, and the associated instabilities are studied using direct numerical simulation in time

Nonlinear Marangoni convection in circular and elliptical cylinders

The spatial organization of single-fluid Marangoni convection in vertical cylinders with circular or elliptical horizontal cross section is described. The convection is driven by an imposed heat flux from above through Marangoni stresses at a free but undeformed surface due to temperature-dependent surface tension. The solutions and their stability characteristics are obtained using branch-following techniques together with direct numerical simulations. The changes in the observed patterns with increasing ellipticity are emphasized. In some cases, the deformation of the cylinder results in the presence of oscillations

Path instabilities of heavy bodies in free fall in a viscous fluid: wake dynamics vs. aerodynamic effects

Solid bodies in free fall in a viscous fluid generally fall along a non-straight path, and a variety of periodic (fluttering, tumbling) and non-periodic regimes can be observed. We analyze the structure of the couplings between the fluid and the body, restricting to a linear stability framework. Introducing a simple toy model consisting of a infinitely long plate sliding along a vertical wall, we show that in the limit of large solid-to-fluid masses a decoupling takes place, allowing us to distinguish two kinds of modes: ``wake'' modes in which the body motion has virtually no influence, and ``body'' modes for which the intrinsic wake dynamics can be neglected. Turning to more realistic objects, we show that the ``body'' modes can be described through a rationally derived aerodynamic model (based on quasi-static assumptions), yielding either a static instability, or a dynamic, low-frequency, instability. Considering 2D rectangular rods and 3D disks, we explore the competition between the three kinds of instabilities. For objects elongated in the spanwise direction, it is found that wake instability dominates in case of 2D rectangles and low-frequency instability dominates in case of disks. For objects elongated in the streamwise direction, static instability always dominate

Stabilité globale linéaire et faiblement non-linéaire du sillage d'objets axisymétriques

Des études récentes ont montré que les premières étapes du scénario de transition à la turbulence d'un sillage de disque observé par simulation numérique directe (Fabre et al. 2008) peuvent être prédites par un modèle analytique basé sur l'interaction faiblement non linéaire des modes d'instabilité dominants (Meliga et al. 2009). Nous étendons ce type d'analyse globale à d'autres objets axisymétriques au repos (disques d'épaisseur finie, ellipsoïdes, sphères). Pour chacun, nous déterminons le système d'équations d'amplitude ainsi que le diagramme de bifurcation correspondant

Identification of effective elastic modulus using modal analysis : application to canine cancellous bone

Mechanical properties of cancellous bone play a role in osteoporosis and fracture induction, bone tumor microenvironment, fracture healing and implant fixation. Most characterization methods used to identify cancellous bone Young modulus are compressive tests, which are known to comprise significant limitations especially when they are performed on small size specimens. We hypothesized that modal analysis of straight beams could be proposed as an alternative methodology to obtain effective elastic properties. Theoretical key-points were provided to determine the elastic modulus from natural frequencies and mode shapes. In a first step, the methodology was validated using a synthetic bone model as control. Then, water-jet cutting allowed collecting fourteen regular beam-like specimens in specific zones of canine distal femurs. X-ray microtomography confirmed the preservation of tissue microarchitecture and homogeneity. The first natural frequency in clamped-free boundary conditions was used to obtain mean values of Young modulus, which ranged from 210 MPa to 280 MPa depending on the specimen’s collection site. This was in good agreement with literature data obtained with uniaxial compressive tests. Experimental tests were rapid and reproducible, non-destructive and did not depend on scale factor. Therefore, beam modal analysis can be a compelling methodology for exploring mechanical properties of fragile and scarce biological tissues

Numerical and in vitro experimental study of arterial deformation and buckling under hypertension and atherosclerotic conditions

Cardiovascular diseases remain the major cause of mortality worldwide. Pathologies of the vasculature such as atherosclerosis are often related to biochemical and genetic factors as well as mechanical effects that strongly change the function and shape of arteries. The present work is part of a general research project which aims to better understand the mechanical mechanisms responsible for atherosclerotic plaque formation and rupture. The chosen approach is to use numerical fluidstructure interaction (FSI) methods to study the relative influence of hemodynamic parameters on the structural stresses generated on plaques. To this aim, a numerical study of a simplified straight vessel exposed to lumen pressure was investigated under quiescent and steady flow conditions. As the internal pressure or the steady velocity increases, the vessel buckles lead-ing to a non-linear large deformation behaviour. The results have been validated using theoretical predictions for the buckling thresholds. Further studies on idealised cardiovascular conditions such as stenosis (i.e., lumen constriction) or aneurysm like (i.e., arterial wall expansion) formation have also been performed