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In this paper, analytical and numerical studies of species separation in vertical and horizontal porous, cylindrical annular cells were presented. The binary fluid, saturating the porous medium, is a water-ethanol mixture. The thickness e of the horizontal and vertical columns is equal to [Formula: see text], where [Formula: see text] and [Formula: see text] are, respectively, the internal and the external radii. H is the height of the vertical cell, and the length of the horizontal cell. Constant temperatures, [Formula: see text] and [Formula: see text], were imposed on the inner and outer cylinders. Since an important species separation, in thermogravitational column (TGC), is obtained for [Formula: see text], the same assumption was made for the two configurations. The analytical solution was obtained using the parallel flow approximation for both configurations. The governing equations were solved numerically for 2D and 3D configurations using two different software (Comsol Multiphysics and a spectral collocation method with Gauss-Lobatto-Chebyshev points). Velocity, temperature, mass fraction fields and time to reach steady state were compared for the two configurations. The amount of species separated at the top or the bottom of each cell was also compared for each configuration.

International audience; The coupling between shear-driven convection and thermo-diffusion is a complex phenomenon due to the interactions between the different forces inside the fluid mixture. This paper studies the species separation that may appear in a parallelepipedic cell filled by a binary mixture. Dirichlet boundary conditions are imposed with differentially heated horizontal walls. Each wall can be animated with a uniform velocity. More precisely, two velocities.

An analytical and numerical study of the action of ultrasonic waves on species separation within a rectangular cavity filled with a binary fluid (water-ethanol mixture) is presented. An ultrasonic wave was emitted on a portion of one of its vertical walls, while the opposite wall was perfectly absorbent. The two horizontal walls were differentially heated. A progressive acoustic wave was used to generate, at a large scale, a stationary flow of the viscous binary fluid (Eckart Streaming) within the cavity. The authors analytically determined the temperature T, mass fraction C and velocity fields under the parallel flow hypothesis used for cells with high aspect ratio [Formula: see text], in the presence of the gravitational field. From the analysis of the velocity fields the authors concluded that the associated flow is either unicellular or consists of three counter-rotating cells superimposed in the horizontal direction of the cavity. They also found the variation domains of the physical parameters leading to one or the other of these two types of flows. The algebraic equation allowing the calculation of the mass fraction gradient and hence the species separation between the vertical walls of the cavity was determined. The variation of the dimensional mass fraction gradient, for the water-ethanol mixture, as a function of the two control parameters of the problem, namely the acoustic parameter A and the temperature difference [Formula: see text] imposed on the two horizontal walls was studied.

In this paper, an analytical and numerical study of species separation in binary mixtures taking account of the presence of bounding plates for the cell is presented. A rectangular horizontal porous cavity saturated by a binary mixture and heated from below is considered. This cavity is bounded by thin plates of uniform thickness, the outer surfaces of which are subjected to a constant heat flux. The transition from the equilibrium solution to the convective one, either stationary or oscillatory, was previously studied by Ouattara et al. (2012). Thus in the first part of this paper, the critical parameters associated with the onset of long wavelength disturbances, obtained analytically, are recalled. Then the hypothesis of parallel flow is used to determine an analytical solution which describes the unicellular flow which may appear in the case of a large aspect ratio cell for a given range of separation ratio, ψ , Rayleigh number, Ra , Lewis number, Le, the ratio of the plate to the porous layer thickness, δ, and their thermal conductivity ratio, d. The analytical results are corroborated by direct numerical simulations. We verify that if d goes to infinity, the walls become infinitely conductive and we find the results obtained by Charrier-Mojtabi et al. (2007). When d tends to 0, the walls become infinitely thin the results obtained by Yacine et al. (2016) are recovered. A linear stability analysis of the unicellular flow is also presented. The eigenvalue problem resulting from the temporal stability analysis is solved by a Tau spectral method. The optimal Rayleigh number leading to an optimal value of the separation horizontal gradient is determined for different values of physical parameters. We show that the species separation depends sensitively on the ratio of the plate to the porous layer thickness, and the ratio of their thermal conductivities. Furthermore, we have shown that in the stationary state and for a given value of the thermal conductivity ratio (d = 29), the maximum separation is almost equal for walls of the same thickness than the one of porous cavity or for the case of porous cell delimited by the infinitely thin walls.

International audience; The onset of Soret-driven convection in a shallow porous horizontal layer filled with a binary mixture and submitted to a vertical heat flux on the two horizontal walls is investigated. We study the species separation between the two vertical ends of the cell, which appears for low values of the Rayleigh number and for positive separation ratio. An analytical solution is performed and the separation is expressed in terms of thermal Rayleigh number, Lewis number, separation ratio and aspect ratio of the cell. These analytical results are corroborated by direct non-linear numerical simulations. Then, the results are compared with those obtained in a differentially heated vertical cavity.

Resume On etudie l'action des vibrations sur les ecoulements de convection thermique et thermosolutale dans les fluides incompressibles, fortement compressibles (critiques) et dans les milieux poreux satures. Ces ecoulements convectifs sont observes dans de nombreux processus naturels ou industriels (croissance cristalline, … ) . L'objectif est de controler, par le biais des vibrations, ces ecoulements en vue d'optimiser les echanges de chaleur et de masse induits par ces ecoulements. On montre les resultats suivants : – Les vibrations peuvent, dans certaines conditions, conduire soit a l'augmentation soit a la reduction des transferts de chaleur et de masse. – Les vibrations peuvent retarder ou avancer la naissance de la convection. – Les vibrations peuvent permettre de selectionner la structure de l'ecoulement.

This paper reports an analytical and numerical study of double-diffusive naturalconvection through a fluid-saturated, vertical and homogeneous porous annulus subjected touniform fluxes of heat and mass from the side. The influence of each leading parameter andespecially curvature, a major parameter of this geometry, has been numerically investigated.Solutions are presented for 1 ⩽ A ⩽ 10, 1 ⩽ RaT ⩽ 200, 1 ⩽ Le ⩽ 20, 0.1 ⩽ N ⩽ 10 and 0 ⩽ γ ⩽ 10 where A, RaT, Le, N and γ denote the aspect ratio, thermal Rayleigh number, Lewis number, buoyancy ratio andcurvature parameter, respectively. For the steady state, an analytical solution valid for stratifiedflow in slender enclosures is presented. A good agreement is observed between the analyticalpredictions and the numerical simulation for sufficiently high aspect ratios.

We studied the onset of double-diffusive convection in a rectangular cell filled with a porous medium saturated with a binary fluid. Uniform heat and mass fluxes are applied to the vertical walls while the horizontal walls are impermeable and adiabatic. When the ratio of resulting solutal to thermal buoyancy forces is equal to −1, we obtain a purely diffusive regime (motionless). We demonstrate that this regime is linearly stable while the thermal Rayleigh number is lower than a critical value, Ra c , depending of the aspect ratio of the cell, A , and the Lewis number, Le . For a square cavity, we obtained Ra c | Le −1| = 209.84 and for an infinite layer, the critical parameters are found to satisfy Ra c | Le −1| = 105.33 with a wavenumber k c = 2.51. These analytical results are in good agreement with direct numerical simulations. These numerical simulations show that subcritical solutions disappear for a thermal Rayleigh number R 0 Ra c , which depends on the aspect ratio of the cell and the Lewis number. For thermal Rayleigh numbers lower than R 0 , only the purely diffusive solution is stable.

Resume Cette note constitue une etude numerique du phenomene de thermogravitation existant dans un milieu poreux sature par un fluide binaire et soumis a un gradient horizontal de temperature. Elle vise a analyser l'influence des differents parametres adimensionnels caracteristiques sur la separation des constituants. Nous proposons une resolution numerique par methode spectrale du probleme bidimensionnel. Nous retrouvons l'existence d'un nombre de Rayleigh optimal, correspondant a une permeabilite optimale, pour laquelle la separation des constituants est maximale. Les resultats de cette etude sont ensuite compares a des resultats analytiques et experimentaux anterieurs.

This work describes a numerical study of double-diffusive natural convection in a square cavity filled with a porous medium. The flow is driven by a combined buoyancy effect due to both temperature and concentration variations. Several different flow models for porous media, such as Darcy flow, Forchheimer's extension, Brinkman's extension, and the generalized flow are considered. The coupled equations are solved using a finite volume approach with a projection algorithm for the momentum equation. Non-Darcian effects are analyzed through investigating the average heat and mass transfer rates. This study consists of a global analysis of each model and the comparison between them when the Darcy number varies. This work also focuses on the influence of the Lewis number on the inertial and boundary effects. It is shown that the inertial and boundary effects have a profound effect on the double-diffusive convection.

A numerical investigation of two-dimensional and three-dimensional free convection flows in a saturated porous horizontal annulus heated from the inner surface is carried out, using a Fourier-Galerkin approximation for the periodic azimuthal and axial directions and a collocation-Chebyshev approximation in the confined radial direction. The numerical algorithm integrates the Darcy-Boussinesq's equations formulated in terms of pressure and temperature. This method gives an accurate description of the 2-D multicellular structures for a large range of Rayleigh number and radii ratio. Some considerations about the existence of the various 2-D solutions previously described in the literature are reported. The 3-D spiral flows are described in the vicinity of the transition from the 2-D unicellular flows. Bifurcation points between 2-D unicellular flows and either 2-D multicellular or 3-D flows are also determined numerically.

In this paper, an analytical and numerical study to determine the species separation process in a binary fluid mixture by decoupling the thermal gradient from the convective velocity was performed. The configuration considered is a horizontal rectangular cavity of large aspect ratio, filled with a binary fluid. A constant tangential velocity is applied to the upper horizontal wall. The two horizontal impermeable walls are maintained at different and uniform temperatures T1 and T2 with ΔT = T2 − T1. Species separation is governed by two control parameters, the temperature difference and the velocity of the upper plate U e x → . The intensity of the thermodiffusion is controlled by the temperature gradient, while the velocity U e x → controls the convective flow. This problem depends on six dimensionless parameters, namely, the separation ratio ψ , the Lewis number Le, the Prandlt number Pr, the aspect ratio of the cell A and two control parameters: the thermal Rayleigh number, Ra and the Peclet number Pe. In this study, the separation (mass fraction difference between the two ends of the cell) is obtained analytically as a function of mass Peclet number (Pem = PeLe) and mass Rayleigh number ( Ra m = ψ R a L e ) . The optimal separation m = 42 / 15 ≈ 0.432 is obtained for Pe m = 42 and Ram = 540. The numerical results, obtained using the full governing equations, are in good agreement with the analytical results based on a parallel flow approximation.

International audience; In this paper, a numerical and analytical analysis is performed in order to improve the species separation process in a binary fluid mixture by decoupling the thermal gradient from the convective velocity. The configuration considered is a horizontal rectangular cavity, of large aspect ratio, filled with a binary fluid. A constant tangential velocity is applied to the upper horizontal wall. The two horizontal impermeable walls are maintained at different and uniform temperatures T1 and T2 with ΔT = T1 − T2 Species separation is governed by two control parameters, the temperature difference ΔT and the velocity of the upper plate Uex. The intensity of the thermodiffusion is controlled by the temperature, while the velocity Uex controls the convective flow. This problem depends on six dimensionless parameters, namely, the separation ratio, ψ, the Lewis number, Le, the Prandtl number Pr, the aspect ratio of the cell, A and two control parameters: the thermal Rayleigh number, Ra and the Péclet number Pe. In this study, the formulation of the separation (mass fraction difference between the two ends of the cell) as a function of the Péclet number and the Rayleigh number is obtained analytically. For a cell heated from below, the optimal separation m = √42/15 is obtained for Pe = √42/Le and Ra = 540/(Leψ). 2D numerical results, obtained by solving the full governing equations, are in good agreement with the analytical results based on a parallel flow approach.

International audience; This paper reports a theoretical and numerical study of species separation in a binary liquid mixture saturating a shallow porous layer heated from below or from above and inclined with respect to the vertical axis. It is shown that the separation can be increased using this configuration and the stability of the unicellular flow obtained in this case is investigated. The critical Rayleigh number obtained is much higher than the one leading to the maximum separation. Experiments performed with a solution of CuSO4 give results which are almost in good agreement with the analytical and the numerical results.

International audience; Heat transfer in Rayleigh-Bénard convection is investigated for three types of nanofluid. Instead of using the expressions commonly found in the literature for specific heat capacity and thermal expansion coefficient, we used two relations that are in agreement with the laws of thermodynamics. The influence of nanoparticles on conductive and convective heat transfer is studied. It is shown that adding nanoparticles in a fluid delays the onset of convection. Contrary to what is argued by many authors, we prove by direct numerical simulations that the use of nanofluids can reduce heat transfer instead of increasing it.

We study the influence of vertical high-frequency and small-amplitude vibrations on the separation of a binary mixture saturating a shallow horizontal porous layer heated from below. The monocellular flow obtained for a separation ratio ψ > ψ mono > 0 leads to a migration of the species towards the two vertical boundaries of the cell. The 2D direct numerical simulations and the linear stability analysis of the averaged governing equations show that the vertical vibrations delay the transition from monocellular flow to multicellular flow. The vibrations also decrease the value of ψ mono , which allows species separation for a wide variety of binary mixtures.

A numerical and experimental study of free convection two-dimensional (2D) flows in a saturated porous horizontal annulus is reported. The fluid motion, described by 2D Darey-Boussinesq equations, is obtained from two different numerical methods, namely Fourier-Galerkin and collocation-Chebyshev methods, for high approximations. A comparison of the spectral accuracy of these two methods is carried out. Numerical results indicate that the collocation-Chebyshev method gives a better accuracy especially for the description of the boundary layers developed near the inner and outer cylinders. An experimental study using the Christiansen effect for the visualization of the thermal fields shows the existence of bicellular 2D structures. These structures, which have never been previously observed for concentric cylinders and by means of the Christiansen effect, are in good agreement with numerical results.

Thermogravitational separation has, until now, been used in differentially heated vertical cells, called thermogravitational columns (TGCs). The cell can be either a slit or an annular cavity whose two isothermal faces are maintained at different temperatures T1 and T2. In this study, we show contrary to what has been done until now, that it is possible to carry out the separation of the species of a binary mixture in the classical configuration of Rayleigh-Bénard (horizontal cell heated from below with the separation ratio psipsi(mono)0 or from above with psi0); we obtain a monocellular flow for psipsi(mono) at the onset of convection. The species separation, in a horizontal cell, is obtained without fear of the remixing observed in vertical thermogravitational cells for fluids with negative Soret coefficients. In this situation, the heaviest component concentrates in the upper part of the cell creating an unstable physical situation. We improve the efficiency of separation for the development of an industrial process of separation. We study the thermogravitational separation of the components of a binary fluid mixture saturating a horizontal porous layer, in the presence of the Soret effect. The horizontal walls of the cavity are impermeable and maintained at constant and different temperatures. The system of equations governing the problem has an equilibrium solution with a vertical stratification of temperature and concentration. This solution loses its stability via a stationary bifurcation for values of the separation ratio psipsi(0) while, for psipsi(0), stability is lost via a Hopf bifurcation. For a cavity heated from below, the flow resulting from the stationary bifurcation becomes monocellular beyond a certain value of psipsi(mono)0 and leads to a separation of the species between the two ends of the cell. We propose to determine whether the monocellular flow remains stable until the optimum of separation and if we can obtain high separation in the case of a horizontal cavity. We verify that the critical Rayleigh number Ra(c2), associated with the transition between monocellular flow and multicellular flow, is higher than the optimum Rayleigh number leading to maximum separation of species. Thus this study reveals that it is possible to obtain optimal separation before the monocellular flow loses its stability. We show also that the separation inside the horizontal cell in a Rayleigh-Bénard configuration permits us to produce the same degree of separation but with a greater quantity of each species compared to a thermogravitational vertical column.

What is Thermal Vibration? The importance of mechanical vibration, as a source of pattern generating mech-anism on the surface of a container filled with liquid, was recognized as early as the beginning of the 19th century by Faraday (1831). However, its importance as a mechanism controlling the convective motion has only been recognized during the 20th century. Originally, mechanical vibration was used in mathematical modeling aimed at increasing the stability threshold of thermo-fluid system (Gershuni et al. 1970, Gresho and Sani 1970). The space exploration and especially the benefits expected from material production in space stations accelerated its development (Alexander 1994). Formally, the thermo-vibrational convection studies concern the form of a mean flowin a confined cavity filled with a fluid presenting temperature non-homogeneities. Compared to the gravity-induced convection, this type of convection presents the advantage that it may exist under weightlessness condition. Under micro-gravity conditions, the gravitational force is reduced drastically. However this situation may cause other forces, which under earth conditions are of minor importance, to be more significant.

International audience; We present an analytical and numerical stability analysis of Soret-driven convection in a porous cavity saturated by a binary fluid. Both the mechanical equilibrium solution and the monocellular flow obtained for particular ranges of the physical parameters of the problem are considered. The porous cavity, bounded by horizontal infinite or finite boundaries, is heated from below or from above. The two horizontal plates are maintained at different constant temperatures while no mass flux is imposed. The influence of the governing parameters and more particularly the role of the separation ratio, characterizing the Soret effect and the normalized porosity, are investigated theoretically and numerically. From the linear stability analysis, we find that the equilibrium solution loses its stability via a stationary bifurcation or a Hopf bifurcation depending on the separation ratio and the normalized porosity of the medium. The role of the porosity is important, when it decreases, the stability of the equilibrium solution is reinforced. For a cell heated from below, the equilibrium solution loses its stability via a stationary bifurcation when the separation ratio >0(Le,), while for 0, while a stationary or an oscillatory bifurcation occurs if mono the monocellular flow loses stability via a Hopf bifurcation. As the Rayleigh number increases, the resulting oscillatory solution evolves to a stationary multicellular flow. For a cell heated from above and

Resume Nous presentons une etude analytique et numerique de l'action conjuguee de la thermodiffusion et de la convection, dite diffusion thermogravitationnelle, dans les melanges binaires. Il s'agit de quantifier l'importance de la separation dans le cas d'une cavite parallelepipedique horizontale remplie d'un melange binaire et soumise a des gradients horizontaux de temperature constants sur les deux parois horizontales, les autres parois sont impermeables a la matiere et thermiquement isolees. Cette nouvelle procedure differe de celle habituellement utilisee et qui consiste a obtenir la separation thermogravitationnelle dans une cellule verticale dont deux parois verticales sont maintenues a des temperatures constantes et differentes. Pour citer cet article : B. Elhajjar et al., C. R. Mecanique 334 (2006).

CONTENTS 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3228.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 8.1.2 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 8.1.3 Other Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3268.2 Influence of Vibration on a Porous Layer Saturated by a Pure Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 8.2.1 Infinite Porous Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3318.2.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 8.2.1.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 8.2.1.3 Time-averaged formulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 8.2.1.4 Scale analysis method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 8.2.1.5 Time-averaged system of equations . . . . . . . . . . . . . . . . . . . 335 8.2.1.6 Stability analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 8.2.1.7 Comparison of the two methods. . . . . . . . . . . . . . . . . . . . . . . 345 8.2.1.8 Effect of the direction of vibration . . . . . . . . . . . . . . . . . . . . . 3488.2.2 Confined Cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 8.2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 8.2.2.2 Stability analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 8.2.2.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 8.2.2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3538.3 Influence of Vibration on a Porous Layer Saturated by a Binary Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 8.3.1 Infinite Horizontal Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3548.3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 8.3.1.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 8.3.1.3 The time-averaged formulation . . . . . . . . . . . . . . . . . . . . . . . . 356 8.3.1.4 Stability analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357KAMB: “dk3120_c008” — 2005/2/17 — 18:08 — page 322 — #28.3.1.5 Limiting case of the long-wave mode . . . . . . . . . . . . . . . . . 360 8.3.2 Confined Cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3618.3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 8.3.2.2 Governing equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 8.3.2.3 Stability analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 8.3.2.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3648.4 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3688.1.1 DefinitionNatural convection is a fluid flowmechanism inwhich the convectivemotion is produced by the density difference in a fluid subjected to a body force. This difference is usually caused by thermal and/or chemical species diffusion. Consequently, to obtain natural convection, two necessary conditions should be satisfied; the existenceof adensityvariationwithin afluidand the existence of a body force. Some common examples of body forces include gravitational, centrifugal, and electromagnetic forces, which may be constant, like gravitational force or may exhibit spatial variation as in centrifugal force. It should be noted that the existence of the body force and the density variation do not guarantee the appearance of convective motion. The relative orientation of the density gradient to the body force provides the sufficient condition for the onset of convection. The possibility of controling the hydrodynamic stability of flows by mod-ulation has attracted the attention of researchers for many years [1]. Two types of modulations have been extensively studied; the temperature modulation and the gravity modulation. It is shown that by proper selection of the modulation parameters, dramatic modification in the stability behavior of the dynamic system can be observed [2]. In some applications, it may be desirable to operate at Rayleigh num-bers higher than the critical one at which the convection occurs and yet have no convection. Also it is advantageous to suppress undesired chaotic motions in order to remove temperature oscillations which may exceed safe operational conditions. In the context of the temperature or heat flux modulation in porous media, we may mention the study of Caltagirone [3] and of Rees [4], Rudariah and Malashetty [5] in the Rayleigh-Benard configuration by temperature modulation and Antohe and Lage [6] in a square cavity heated laterally by flux modulation. Thermo-vibrational convection belongs to a special class of periodic flows in which the buoyancy force is time dependent. In this class, which is different from the problems concerning spatial variations of body forces [7-9], the action of external force fieldKAMB: “dk3120_c008” — 2005/2/17 — 18:08 — page 323 — #3(namely, mechanical vibration) in the presence of a nonhomogeneous scalar field (e.g., temperature or concentration) may be used to control the onset of convective motion. Under microgravity conditions, the gravitational force will be reduced drastically and consequently the buoyancy induced convection. However this situation may cause other forces, which under normal conditions are of secondary importance, to become significant. Therefore, residual vibration, which naturally exists in a spacecraft, maybe used to increase the rate of heat or mass transfer. In its simplest form, the imposed vibration can be considered as a harmonic oscillation having zero average over a vibration period. Aswith any subject concerning thermofluid science, the study of the effects of a vibration mechanism on convective motion has been motivated by practical considerations. It is a known fact that, in the presence of gravitational field, the temperature and concentration gradients may produce natural flows. This, in turn, drastically affects material processing; for example, the rate of crystal growth, etc. With the progress of the space industry, there is an opportunity to grow perfect crystals aboard a spacecraft where there exists a highly reduced gravitational environment. Further, it was thought that the unfavorable effects of natural convectionwouldbeeliminated. Therefore,manycrystal growthexperiments were conducted aboard Skylab and theMir space station. However the results were surprisingly much less interesting than expected [10]. It was confirmed experimentally that the space station did not represent an acceleration-free environment; there are transientdisturbancesdue to space stationmaneuvers, impulsive crew movement, and operation of life supporting systems. These residual accelerations are referred to as g-jitter, which can be modeled as harmonic oscillations [11-14]. The theory of thermo-vibrational convection in the fluid medium is summarized in the book written by Gershuni and Lyubimov [15] which reports the Russian studies in this field. In contrast to the thermo-vibrational problem in fluid media, work on thevibrational problem in porous media is quite recent. We can classify these studies according to geometry, direction of vibration, range of frequency, the number of saturating fluids (mono-component or multi-component), type of boundary conditions, and transport modeling.

International audience; Dans cette Note on étudie la convection mixte en fluide binaire avec effet Soret en canal rectangulaire chauffé par le bas. On s'intéresse plus particulièrement à la transition vers les rouleaux transversaux 2D qui apparaissent pour de faibles nombres de Reynolds et de Rayleigh. L'analyse de stabilité linéaire de l'écoulement de Poiseuille, stratifié linéairement en température et concentration, montre l'influence du facteur de séparation sur le nombre de Rayleigh critique de la transition vers les rouleaux transversaux. A faible nombre de Reynolds, on met en évidence la présence de rouleaux se propageant dans le même sens ou en sens inverse de l'écoulement moyen. On montre également que dans ces conditions, la fréquence de propagation des rouleaux correspond à la somme des fréquences obtenues en convection mixte thermique et en convection naturelle thermosolutale.

This paper concerns the thermal stability analysis of porous layer saturated by a binary fluid under the influence of mechanical vibration. The linear stability analysis of this thermal system leads us to study the following damped coupled Mathieu equations: B H + B(π2 + k2) + 1 H +(π2 + k2) − k2k2 + π2 RaT(1 + R sin ω* t*) H =k2k2 + π2 (NRaT)(1 + R sin ω* t*) Fe* B F + B π2 + k2Le + e* F +π2 + k2Le − k2k2 + π2 NRaT(1 + R sin ω* t*) F =k2k2 + π2 RaT(1 + R sin ω* t*) H where RaT is thermal Rayleigh number, R is acceleration ratio (bω2 /g), Le is the Lewis number, k is the dimensionless wave-number, e* is normalized porosity and N is the buoyancy ratio (H and F are perturbations of temperature and concentration fields). In the follow up, the non-linear behavior of the problem is studied via a generalization of the Lorenz model (five coupled non-linear differential equations with periodic coefficients). In the presence or absence of gravity, the stability limit for the onset of stationary as well as Hopf bifurcations is determined.Copyright © 2005 by ASME

Theoretical studies of Rayleigh-Benard convection subjected to sinusoidal acceleration modulations have been conducted by several workers. Linear and weakly nonlinear stability analyses have been developed by Gresho and Sani [4], Clever et al. [2] and by the Russian workers Gershuni and Lyubimov [3]. As shown by these studies, the Rayleigh number for the onset of synchronous convection increases with the frequency of vibration for a layer oscillating with constant vertical amplitude until a certain frequency of modulation is reached and at which the onset is in the form of subharmonic motions.

International audience; It is not unusual for a Soret coefficient to change sign with temperature. We develop the theory for the onset of convection in such systems, heated from below or from above, provided that the mean temperature is precisely that at which the change of sign occurs. We also consider the realistic case of rigid, conducting, impervious boundaries for later comparison with laboratory experiments.

International audience; Nous étudions la naissance de la convection dans une couche poreuse infinie, saturée par un fluide binaire, en présence d'effet Soret. Les parois horizontales, imperméables à la matière, sont maintenues à des températures constantes et différentes. Pour une cellule chauffée par le bas, la solution d'équilibre perd sa stabilité via une bifurcation stationnaire lorsque le facteur de séparation ψ est supérieur à ψ0, fonction du nombre de Lewis et de la porosité normalisée ; pour ψ0, alors qu'elle perd sa stabilité via une bifurcation stationnaire pour ψ0, while a stationary bifurcation occurs if ψ

International audience; Convective oscillations in porous and fluid media are studied numerically. A two-dimensional, square, differentially heated cavity, filled with a porous medium saturated by a binary fluid or simply by a binary fluid, is considered. This cavity is subjected to linear harmonic oscillations in the vertical direction. The formulation is based on the Darcy-Brinkman-Forchheimer-Boussinesq model. The time dependent Darcy-Brinkman-Forchheimer-Boussinesq equations are solved using a pseudo-spectral Legendre collocation method. The instantaneous and mean characteristics of the flows are studied and discussed. An intensification of the heat and mass transfers is observed at low frequency for sufficiently high vibration intensity. A comparison between the response to the imposed vibrations is made for Darcy numbers varying from Da = 10-7 to Da = 10.

International audience; In the present work, we study the onset of double-diffusive convective regimes in a tilted rectangular cavity, filled with a porous medium, saturated by a binary fluid. Two opposite walls are maintained at different but uniform temperatures and concentrations while the two other walls are impermeable and adiabatic. When the thermal and solutal buoyancy forces are comparable in intensity but have opposite signs, the motionless double-diffusive regime with linear temperature and concentration profiles is a solution of the problem. The first part of the study consists of a linear stability analysis of the motionless regime. We determine the critical thermal Rayleigh number for the onset of stationary and oscillatory convection. Indeed, we point out that there exist primary Hopf bifurcations for the studied problem in porous medium, while in the same configuration with a fluid medium only primary stationary bifurcations exist. When the first primary bifurcation creates a steady state branch of solutions, the bifurcation is either transcritical or pitchfork depending on the aspect ratio, A and the tilt, φ of the cavity. The onset of oscillatory convection (Hopf bifurcation) depends not only on A and φ but also on the Lewis number, Le and the normalized porosity, ε. Then, we determine the parts of the (Le, ε) parameter space for which the first primary bifurcation is stationary or oscillatory. In particular, it is found that in the case Le⩾1 and for ε Le21 only stationary primary bifurcations exist. In the case Le

International audience; The onset of thermosolutal convection in a rectangular cell filled with a porous medium saturated by a binary fluid is studied. The vertical walls are maintained at different, but uniform, temperatures and concentrations. The horizontal walls are impermeable and adiabatic. When the ratio of resulting solutal and thermal buoyancy forces is equal to −1, we obtain an equilibrium solution corresponding to a purely diffusive regime. We demonstrate that this regime is linearly stable up to a critical thermal Rayleigh number, Rac1, depending on the aspect ratio of the cell and the Lewis number. These analytical results are in good agreement with numerical direct simulations which allow us to describe the double-diffusive convective regimes taking place when the equilibrium regime loses its stability.

The aim of this study was to investigate the effect of conducting boundaries on the onset of convection in a binary fluid-saturated porous layer. The isotropic saturated porous layer is bounded by two impermeable but thermally conducting plates, subjected to a constant heat flux. These plates have identical conductivity. Moreover, the conductivity of the plates is generally different from the porous layer conductivity. The overall layer is of large extent in both horizontal directions. The problem is governed by seven dimensionless parameters, namely the normalized porosity of the medium ", the ratio of plates over the porous layer thickness and their relative thermal conductivities ratio d, the separation ratio , the Lewis number Le and thermal Rayleigh number Ra. In this work, an analytical and numerical stability analysis is performed. The equilibrium solution is found to lose its stability via a stationary bifurcation or a Hopf bifurcation depending on the values of the dimensionless parameters. For the long-wavelength mode, the critical Rayleigh number is obtained as Racs = 12(1+2d)/ [1+ (2d Le+ Le+1)] and kcs = 0 for > uni> 0. This work extends an earlier paper by Mojtabi and Rees (2011 Int. J. Heat Mass Transfer 54 293‐301) who considered a configuration where the porous layer is saturated by a pure fluid. (Some figures may appear in colour only in the online journal)

The evolution of an initially homogeneous solution consisting of two species subjected to a thermal gradient which generates a mass transfer, which can lead to the separation species from the binary fluid, is studied. The configuration chosen to study the separation is a rectangular (or parallelepipedic) cell, horizontal and placed in the gravitational field. The presence of a piezoelectric source on one of the vertical walls of the cavity makes it possible to generate a stationary flow on a large scale. The flow is induced by the propagation of ultra-sonic waves within the viscous fluid : the dissipation of the acoustic energy of the wave within the fluid is called Eckart streaming. We seek to optimize the separation by combining thermal gradient and acoustic source. The first part consists of the study of the isothermal flow generated by the ultrasonic wave in a monoconstituent fluid. After calculating the acoustic intensity field with the Rayleigh integral, the profile is implemented in a Comsol Multiphysics finite element code. The numerical results are compared with previous experimental results. In a second part, we consider a cavity containing a binary fluid. A configuration heated from the top is privilegied to allow the insertion of an acoustic source. The separation (difference of mass fraction between the two ends of the cell) is determined analytically using the Maple software as a function of the control parameters of the problem. 2D and 3D numerical simulations showed a good agreement between the analytical and numerical results, for a constant acoustic parameter. The problem considered depends on eight dimensionless parameters. Three of them are specific to the nature of the binary fluid : the Lewis number Le, the Prandtl number Pr and the separation factor ψ. There are then two control parameters, the thermal Rayleigh number Ra and the adimensional acoustic force A. Finally, the other dimensionless parameters are the two aspect ratios of the cavity, as well as the relative thickness of the beam.; On étudie l'évolution d'une solution initialement homogène constituée de deux espèces soumises à un gradient thermique qui génère un transfert de matière, ce qui peut conduire à la séparation des espèces du fluide binaire. La configuration choisie pour étudier la séparation est une cellule rectangulaire (ou parallélépipédique), horizontale et placée dans le champ de pesanteur. La présence d'une source piézo-électrique, sur l'une des parois verticales de la cavité, permet de générer un écoulement stationnaire à grande échelle. L'écoulement est induit par la propagation d'ondes ultrasonores au sein du fluide visqueux : la dissipation de l'énergie acoustique de l'onde au sein du fluide porte le nom d'Eckart streaming. On cherche à optimiser la séparation en combinant gradient thermique et source acoustique. La première partie consiste en l'étude de l'écoulement isotherme généré par l'onde ultrasonore dans un fluide mono-constituant. Après avoir calculé le champ d'intensité acoustique avec l'intégrale de Rayleigh, le profil est implémenté dans un code aux éléments finis Comsol Multiphysics. Les résultats numériques sont comparés avec des résultats expérimentaux antérieurs. Dans une seconde partie, on considère une cavité contenant un fluide binaire. On détermine analytiquement, à l'aide du logiciel Maple, la séparation (différence de fraction massique entre les deux extrémités de la cellule) en fonction des paramètres de contrôle du problème. Des simulations numériques 2D et 3D ont montré un bon accord entre les résultats analytiques et numériques, pour un paramètre acoustique constant et un chauffage par le bas ou par le haut de la cellule. Le problème considéré dépend alors de huit paramètres adimensionnels. Trois d'entre eux sont propres à la nature du fluide binaire : le nombre de Lewis Le, de Prandtl Pr et le facteur de séparation ψ. Il y a ensuite deux paramètres de contrôle, le nombre de Rayleigh thermique Ra et la force acoustique adimensionnelle A. Enfin, les autres paramètres adimensionnels sont les deux rapports d'aspect de la cavité, ainsi que l'épaisseur relative du faisceau acoustique

The evolution of an initially homogeneous solution consisting of two species subjected to a thermal gradient which generates a mass transfer, which can lead to the separation species from the binary fluid, is studied. The configuration chosen to study the separation is a rectangular (or parallelepipedic) cell, horizontal and placed in the gravitational field. The presence of a piezoelectric source on one of the vertical walls of the cavity makes it possible to generate a stationary flow on a large scale. The flow is induced by the propagation of ultra-sonic waves within the viscous fluid : the dissipation of the acoustic energy of the wave within the fluid is called Eckart streaming. We seek to optimize the separation by combining thermal gradient and acoustic source. The first part consists of the study of the isothermal flow generated by the ultrasonic wave in a monoconstituent fluid. After calculating the acoustic intensity field with the Rayleigh integral, the profile is implemented in a Comsol Multiphysics finite element code. The numerical results are compared with previous experimental results. In a second part, we consider a cavity containing a binary fluid. A configuration heated from the top is privilegied to allow the insertion of an acoustic source. The separation (difference of mass fraction between the two ends of the cell) is determined analytically using the Maple software as a function of the control parameters of the problem. 2D and 3D numerical simulations showed a good agreement between the analytical and numerical results, for a constant acoustic parameter. The problem considered depends on eight dimensionless parameters. Three of them are specific to the nature of the binary fluid : the Lewis number Le, the Prandtl number Pr and the separation factor ψ. There are then two control parameters, the thermal Rayleigh number Ra and the adimensional acoustic force A. Finally, the other dimensionless parameters are the two aspect ratios of the cavity, as well as the relative thickness of the beam., On étudie l'évolution d'une solution initialement homogène constituée de deux espèces soumises à un gradient thermique qui génère un transfert de matière, ce qui peut conduire à la séparation des espèces du fluide binaire. La configuration choisie pour étudier la séparation est une cellule rectangulaire (ou parallélépipédique), horizontale et placée dans le champ de pesanteur. La présence d'une source piézo-électrique, sur l'une des parois verticales de la cavité, permet de générer un écoulement stationnaire à grande échelle. L'écoulement est induit par la propagation d'ondes ultrasonores au sein du fluide visqueux : la dissipation de l'énergie acoustique de l'onde au sein du fluide porte le nom d'Eckart streaming. On cherche à optimiser la séparation en combinant gradient thermique et source acoustique. La première partie consiste en l'étude de l'écoulement isotherme généré par l'onde ultrasonore dans un fluide mono-constituant. Après avoir calculé le champ d'intensité acoustique avec l'intégrale de Rayleigh, le profil est implémenté dans un code aux éléments finis Comsol Multiphysics. Les résultats numériques sont comparés avec des résultats expérimentaux antérieurs. Dans une seconde partie, on considère une cavité contenant un fluide binaire. On détermine analytiquement, à l'aide du logiciel Maple, la séparation (différence de fraction massique entre les deux extrémités de la cellule) en fonction des paramètres de contrôle du problème. Des simulations numériques 2D et 3D ont montré un bon accord entre les résultats analytiques et numériques, pour un paramètre acoustique constant et un chauffage par le bas ou par le haut de la cellule. Le problème considéré dépend alors de huit paramètres adimensionnels. Trois d'entre eux sont propres à la nature du fluide binaire : le nombre de Lewis Le, de Prandtl Pr et le facteur de séparation ψ. Il y a ensuite deux paramètres de contrôle, le nombre de Rayleigh thermique Ra et la force acoustique adimensionnelle A. Enfin, les autres paramètres adimensionnels sont les deux rapports d'aspect de la cavité, ainsi que l'épaisseur relative du faisceau acoustique