Influence of the aspect ratio on boiling flows in rectangular mini-channels

This article presents experiments conducted with two single rectangular mini-channels of same hydraulic diameter (1.4 mm) and different aspect ratios for conditions of horizontal boiling flow. The Forane® 365 HX used was subcooled (ΔT_sub = 15 °C) for all the boiling curves presented in the paper. Local heat transfer coefficients were measured for heat flux ranging from 25 to 62 kW m⁻² and mass flux from 200 kg m⁻² s⁻¹ to 400 kg m⁻² s⁻¹. The boiling flows were observed with two different cameras (depending on the flow velocity) through a visualization window. The flow patterns in the two channels were compared for similar conditions. The results show that the boiling heat transfer coefficient and the pressure drop values are different for the two single mini-channels. For low heat flux condition, the channel with lowest aspect ratio (H/W = 0.143) has a higher heat transfer coefficient. On the other hand, for high heat flux condition, the opposite situation occurs, namely the heat transfer coefficient becomes higher for the channel with highest aspect ratio (H/W = 0.43). This is probably due to the earlier onset of dryout in the channel with lowest aspect ratio. For the two cases of heating, the pressure drop for the two-phase flow remains lower for the channel with lowest aspect ratio. These results show that the aspect ratio plays a substantial role for boiling flows in rectangular channels. As for single-phase flows, the heat transfer characteristics are significantly influenced (even though the hydraulic diameter remains the same) by this parameter.     

An elastoplastic model for granular materials taking into account grain breakage

Granular materials are constituted of an assembly of particles. In spite of the simplicity of this assembly, its mechanical behaviour is complex. In the first stage we propose a framework to establish correlations between parameters of the supposedly continuous medium and grain properties which are assumed to be constant. However, this hypothesis is no longer valid in the case where physical (shape, size…) or mechanical properties (Young modulus E_g, Poisson's ratio ν_g…) of grains evolve during loading, causing the behaviour of the assembly to modify. We study the influence of the physical and mechanical parameters on grain breakage. We subsequently propose a way to model the influence of the grain breakage on granular materials and we introduce this influence in an elastoplastic constitutive model. Validations are made on two kinds of sands under isotropic and triaxial loading. Since the results of numerical computations corresponded well with the experimental data, we believe that the new model is capable of accurately simulating the behaviour of granular materials under a wide range of stresses and of taking into account, through new parameters, the individual strength of grains.     

Simulations of Three-Dimensional Turbulent Mixing for Schmidt Numbers of the Order 1000

We report basic results from new numerical simulations of passive scalar mixing at Schmidt numbers (Sc) of the order of 1000 in isotropic turbulence. The required high grid-resolution is made possible by simulating turbulence at very low Reynolds numbers, which nevertheless possesses universality in dissipative scales of motion. The results obtained are qualitatively consistent with those based on another study (Yeung et al., Phys. Fluids 14 (2002) 4178-4191) with a less extended Schmidt number range and a higher Reynolds number. In the stationary state maintained by a uniform mean scalar gradient, the scalar variance increases slightly with Sc but scalar dissipation is nearly constant. As the Schmidt number increases, there is an increasing trend towards k ^?1 scaling predicted by Batchelor (Batchelor, J. Fluid Mech. 5 (1959) 113-133) for the viscous-convective range of the scalar spectrum; the scalar gradient skewness approaches zero; and the intermittency measured by the scalar gradient flatness approaches its asymptotic state. However, the value of Sc needed for the asymptotic behavior to emerge appears to increase with decreasing Reynolds number of the turbulence. In the viscous-diffusive range, the scalar spectrum is in better agreement with Kraichnan's (Kraichnan., Phys. Fluids 11 (1968) 945-953) result than with Batchelor's.     

Effet de l'anisotropie élastique cristalline sur la distribution des facteurs de Schmid à la surface des polycristauxInfluence of crystalline elasticity anisotropy on Schmid factor distribution at the free surface of polycrystals

Experimental studies of the plasticity mechanisms of polycrystals are usually based on the Schmid factor distribution supposing crystalline elasticity isotropy. A numerical evaluation of the effect of crystalline elasticity anisotropy on the apparent Schmid factor distribution at the free surface of polycrystals is presented. Cubic elasticity is considered. Order II stresses (averaged on all grains with the same crystallographic orientation) as well as variations between averages computed on grains with the same crystallographic orientation but with different neighbour grains are computed. The Finite Element Method is used. Commonly studied metals presenting an increasing anisotropy degree are considered (aluminium, nickel, austenite, copper). Concerning order II stresses in strongly anisotropic metals, the apparent Schmid factor distribution is drifted towards small Schmid factor values (the maximum Schmid factor is equal to 0.43 instead of 0.5) and the slip activation order between characteristic orientations of the crystallographic standard triangle is modified. The computed square deviations of the stresses averaged on grains with the same crystallographic orientation but with different neighbour grains are a bit higher than the second order ones (inter-orientation scatter). Our numerical evaluations agree quantitatively with several observations and measures of the literature concerning stress and strain distribution in copper and austenite polycrystals submitted to low amplitude loadings. Hopefully, the given apparent Schmid factor distributions could help to better understand the observations of the plasticity mechanisms taking place at the free surface of polycrystals. To cite this article: M. Sauzay, C. R. Mecanique 334 (2006).     

Effect of the grain elongation on the behaviour of granular materials in biaxial compression

In this article we study the effect of the grain elongation on the shearing behaviour of dense granular materials by means of 2D numerical simulations of biaxial compression tests, using the Contact Dynamics method. The case of grains with hexagonal shapes (and four possible different elongations) is studied in comparison to the case of a sample with circular grains. For the shapes studied, samples with polygonal grains exhibit initial densities higher than the sample with discs, and the initial density decreases when the grain elongation increases. The friction angle at the residual state increases linearly with the particle elongation ratio. The cumulative rotations of discs are higher than those of polygons. Finally, in the case of elongated hexagons, the grains with highest rotations are located along thin bands because the shear bands are thinner and more persistent for these shapes.     

Analytic calculation of the stresses in an ensiled granular mediumCalcul analytique des contraintes dans un matériau granulaire ensilé

We present in this Note a new analytic approach, of continuous medium type, which improves the Janssen theory and enables us to calculate the stresses in an ensiled granular medium. This approach is based on the two dimensional equilibrium equations, coupled with the Mohr–Coulomb criterion and a slip condition at the walls of the silo. An analytic resolution is developed to compute the stresses for cohesive and non cohesive materials in the whole silo. To cite this article: O. Millet et al., C. R. Mecanique 334 (2006).     

Compatibility Equations of Nonlinear Elasticity for Non-Simply-Connected Bodies

Compatibility equations of elasticity are almost 150 years old. Interestingly, they do not seem to have been rigorously studied, to date, for non-simply-connected bodies. In this paper we derive necessary and sufficient compatibility equations of nonlinear elasticity for arbitrary non-simply-connected bodies when the ambient space is Euclidean. For a non-simply-connected body, a measure of strain may not be compatible, even if the standard compatibility equations (“bulk” compatibility equations) are satisfied. It turns out that there may be topological obstructions to compatibility; this paper aims to understand them for both deformation gradient F and the right Cauchy-Green strain C = F ^T F. We show that the necessary and sufficient conditions for compatibility of deformation gradient F are the vanishing of its exterior derivative and all its periods, that is, its integral over generators of the first homology group of the material manifold. We will show that not every non-null-homotopic path requires supplementary compatibility equations for F and linearized strain e. We then find both necessary and sufficient compatibility conditions for the right Cauchy-Green strain tensor C for arbitrary non-simply-connected bodies when the material and ambient space manifolds have the same dimensions. We discuss the well-known necessary compatibility equations in the linearized setting and the Cesàro-Volterra path integral. We then obtain the sufficient conditions of compatibility for the linearized strain when the body is not simply-connected. To summarize, the question of compatibility reduces to two issues: i) an integrability condition, which is d(F dX) = 0 for the deformation gradient and a curvature vanishing condition for C, and ii) a topological condition. For F dx this is a homological condition because the equation one is trying to solve takes the form dφ = F dX. For C, however, parallel transport is involved, which means that one needs to solve an equation of the form dR/ ds = RK, where R takes values in the orthogonal group. This is, therefore, a question about an orthogonal representation of the fundamental group, which, as the orthogonal group is not commutative, cannot, in general, be reduced to a homological question.     

Quantifying the extent of crushing in granular materials: A probability-based predictive method

Grain crushing is one of the micromechanisms that governs the stress-strain behaviour of a granular material, and also its permeability by altering the grain size distribution. It is therefore advantageous to be able to predict the point of onset of crushing and to quantify the subsequent evolution of crushing. This paper uses the data of Discrete Element Method (DEM) simulations to inform a statistical model of granular crushing. Distributions of normalised contact forces are first obtained. If the statistical distribution of the crushing strength of the grains is then known, the onset of crushing within an assembly of grains should be predictable. Two different cases, one in which grain strength was statistically independent of grain size and one showing an arbitrary trend, were used to compare with DEM results and so confirm the validity of the statistical method.     

Investigation of internal pressure gradients generated in electrokinetic flows with axial conductivity gradients

Field amplified sample stacking (FASS) is used to increase sample concentrations in electrokinetic flows. The technique uses conductivity gradients to establish a non-uniform electric field that accumulates ions within a conductivity gradient, and can be readily integrated with capillary electrophoresis. Conductivity gradients also cause gradients in near-wall electroosmotic flow velocities. These velocity gradients generate internal pressure gradients that drive secondary, dispersive flows. This dispersion leads to a significant reduction in the efficiency of sample stacking. This paper presents an experimental investigation of internally generated pressure gradients in FASS using micron-resolution particle image velocimetry (μPIV). We measure velocity fields of particles seeded into an electrokinetic FASS flow field in a glass microchannel with a single buffer–buffer interface. μPIV allows for the direct quantification of local, instantaneous pressure gradients by analyzing the curvature of velocity profiles. Measurements show internally generated pressure-driven velocities on the order of 1mm/s for a typical applied electric field of 100 V/cm and a conductivity ratio of 10. A one-dimensional (1D) analytical model for the temporal development of the internal pressure gradient generation is proposed which is useful in estimating general trends in flow dynamics.

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Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations

The higher-order stress work-conjugate to slip gradient in single crystals at small strains is derived based on the self-energy of geometrically necessary dislocations (GNDs). It is shown that this higher-order stress changes stepwise as a function of in-plane slip gradient and therefore significantly influences the onset of initial yielding in polycrystals. The higher-order stress based on the self-energy of GNDs is then incorporated into the strain gradient plasticity theory of Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5-32] and applied to single-slip-oriented 2D and 3D model crystal grains of size D. It is thus found that the self-energy of GNDs gives a D^-1-dependent term for the averaged resolved shear stress in such a model grain under yielding. Using published experimental data for several polycrystalline metals, it is demonstrated that the D^-1-dependent term successfully explains the grain size dependence of initial yield stress and the dislocation cell size dependence of flow stress in the submicron to several-micron range of grain and cell sizes.     

Analysis of local behaviour in granular materials

A local scale, called the meso-scale, has recently been introduced to the multi-scale approach for 2D granular materials. This local scale is defined at the level of meso-domains enclosed by particles in contact. Stress and strain have been defined at this local scale, and their relation with the local structure has been studied. The purpose of this paper is to analyse the behaviour of granular materials at the meso-scale, i.e. the stress–strain–structure relationship at this scale. Analyses are performed on a 2D numerical granular sample subjected to a biaxial compression test and simulated with the Discrete Element Method (DEM). The sample is quite dense and it is loaded at a relatively low strain rate so that the state of the sample can be considered as being quasi-static. The size of sub-domains in the sample varies largely from 3 to 12 particles. It is shown that the evolution of the internal state of the sample corresponds, at the meso-scale, to a clear evolution of the quantity of meso-domains oriented in different directions. In addition, the behaviour of meso-domains is highly governed by their orientation rather than their density, especially for the strongly elongated meso-domains: the meso-domains oriented in the compression (resp. extension) direction behave like a dense (resp. loose) granular material.     

Résolution des équations de dispersion–diffusion en présence de forts gradientsResolution of advection–diffusion equation in presence of sharp gradient

In order to resolve the convection terms in the simulation of coastal flows, we propose a numerical method based on the Total Variation Diminishing scheme completed with the Artificial Compression Method to reduce the artificial diffusion and to sharpen the tracer profile in strong gradient regions. This method was applied successfully at discontinuous regions, but out of these regions a limitation of its application is necessary. An application is made to estimate the width of the fresh water band in the Eastern English Channel. To cite this article: C. Sart et al., C. R. Mecanique 330 (2002) 159–165.     

Derivation of a well-posed and multidimensional drift-flux model for boiling flows

In this Note, we derive a multidimensional drift-flux model for boiling flows. Within this framework, the distribution parameter is no longer a scalar but a tensor that might account for the medium anisotropy and the flow regime. A new model for the drift-velocity vector is also derived. It intrinsically takes into account the effect of the friction pressure loss on the buoyancy force. On the other hand, we show that most drift-flux models might exhibit a singularity for large void fraction. In order to avoid this singularity, a remedy based on a simplified three field approach is proposed. To cite this article: O. Grégoire, M. Martin, C. R. Mecanique 333 (2005).     

Some Indentities for the Gradient of the Principal Invariants, Traces and Determinants via Grassmann Calculus

Let A be a second order tensor in a finite dimensional space. In this work we determine the gradient of the principal invariants of A and obtain some trace and determinant identities using only some standard rigorous statements concerning Grassmann calculus. We recover some of the results of Dui et al. (J. Elast. 75:193–196, 2004) and of Truesdell and Noll (The Non-linear Field Theories of Mechanics, Springer, Berlin, 2002) and solve an old problem proposed in SIAM Review concerning a determinant identity from a new perspective in a concise and simple manner.     

Stabilité de la forme de la goutte d'eau formée entre deux grains solides : méthode de résolutionStability of the shape of a liquid drop placed between two solid grains: method of resolution

We discuss the positiveness of the second variation of a Lagrangian Action subjected to an integral constraint when the end-points of the range of integration of the integrals can be displaced along prescribed curves. The Lagrangian does not depend explicitly on the variable of integration. The results are illustrated by the problem of a liquid drop placed between two solid grains. To cite this article: P. Bérest et al., C. R. Mecanique 332 (2004).     

Comportement mécanique et rupture de milieux granulaires cohésifsMechanical behaviour and rupture of cohesive granular media

The modelling of mechanical behaviour and rupture of cohesive granular media is carried out on a model medium made of aluminium cylinders and cohesive bonds playing the role of cohesion points. The mechanical behaviour of the cohesion point is studied under different loading situations (compression, traction, shearing, torque). The results are introduced into a code, which is based on a discrete element method extended to the cohesive case. The validation is achieved with compression tests of macroscopic samples. We notice a good correlation between simulation and experiment on the mechanical behaviour, on the appearance of fractures and their propagation. To cite this article: J.-Y. Delenne et al., C. R. Mecanique 330 (2002) 475–482.     

On localization modes in coupled thermo-hydro-mechanical problems

A perturbation approach is used to study localization phenomena in saturated porous media when thermo-mechanical loadings and thermo-hydro-mechanical couplings are fully taken into account. We show that various types of localization modes are possible depending on the constitutive behavior and loading conditions. Examination of the associated conditions in the light of the classical band approach reveals that the differences between these modes lie in their structure which may involve jumps in different variables (beside the velocity gradient) such as the gradients of heat and fluid fluxes, the temperature and the pressure rates. To cite this article: A. Benallal, C. R. Mecanique 333 (2005).     

Approche globale,minima relatifs et Critère d'Amorçage en Mécanique de la RuptureGlobal approach,relative minima and yield criterion in Fracture Mechanics

We show that it is possible to predict the onset of cracking by searching relative minima of the total energy of a body, the yield criterion depending on the choice of the energies. In particular in the case where the bulk energy is elastic and the surface energy depends on the jump of the displacement, we prove that the yield criterion is formulated in terms of the stress tensor and of intrinsic curve type. To cite this article: J. Laverne, J.-J. Marigo, C. R. Mecanique 332 (2004).     

Steady Flow of a Navier-Stokes Fluid Around a Rotating Obstacle

Let ? be a body immersed in a Navier-Stokes liquid ? that fills the whole space. Assume that ? rotates with prescribed constant angular velocity ω. We show that if the magnitude of ω is not “too large”, there exists one and only one corresponding steady motion of ? such that the velocity field v(x) and its gradient grad?v(x) decay like |x|^?1 and |x|^?2, respectively. Moreover, the pressure field p(x) and its gradient grad?p(x) decay like |x|^?2 and |x|^?3, respectively. These solutions are “physically reasonable” in the sense of Finn. In particular, they are unique and satisfy the energy equation. This result is relevant to several applications, including sedimentation of heavy particles in a viscous liquid.     

Thermocapillary convection in a three-layer system with the temperature gradient directed along the interfaces

The paper presents numerical simulations of the Marangoni–Bénard convection in a real symmetric three-layer system. The temperature gradient is directed along the interfaces. Nonlinear regimes of steady and oscillatory convective flows are investigated by means of the finite-difference method. Transitions between the motions with different spatial structures are studied. To cite this article: V. Shevtsova et al., C. R. Mecanique 333 (2005).