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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.

The purpose of this paper is to present an analytical model for the thermogravitational separation process in a porous rectangular cavity, saturated by a multi-component mixture. In the first part, an analytical and a numerical study of the onset of the Soret driven convection of an n-component mixture saturating a homogeneous porous horizontal layer are presented. This study is based on the classical Darcy-Boussinesq equations which admit a mechanical equilibrium solution associated with the pure double diffusive regime. This solution is linearly stable up to a critical value of the stability parameters. The dispersion relation is obtained for an n-component mixture and the associated critical parameters are calculated analytically and numerically using the Galerkin spectral method for the stationary transition. Concerning the oscillatory transition, only the Galerkin spectral method is used. The second part of the study presents an analytical model based on the parallel flow approximation. The analytical solution for the unicellular flow is obtained and the separation is expressed in terms of the Lewis numbers ( Le ), the separation ratio (ψ), the cross diffusion coefficients ( Cr ) and the thermal Rayleigh number (Ra). The analytical results are validated using direct numerical simulations. The numerical and the analytical results obtained are in good agreement.

On designe par milieu poreux un solide de forme complexe renfermant des cavites appelees pores. Les milieux poreux sont le siege de multiples phenomenes physico-chimiques et de transport. L'etude de ces phenomenes necessite la connaissance des proprietes de stockage des fluides, des proprietes de transferts et eventuellement des proprietes mecaniques. Les transferts de chaleur en milieux poreux sont souvent complexes. Cet article introduit les notions indispensables concernant la caracterisation des milieux poreux, puis etablit une synthese et une initiation a des ouvrages plus specialises.

Ce dossier, consacre au transfert de chaleur avec changement de phase en milieu poreux, fait suite au dossier « Transfert de chaleur en milieu poreux. Conduction, convection, rayonnement » [BE 8 250] 1 auquel le lecteur pourra utilement se referer pour de nombreuses notions non presentees a nouveau ici par souci de concision. Le lecteur peu familier avec les phenomenes de transport en milieu poreux aura en fait tout interet a n’aborder le present dossier qu’apres lecture de la reference 1 . Les phenomenes de changement de phase en milieux poreux occupent une place importante dans de nombreux domaines : l’exploitation des gisements d’hydrocarbures, l’isolation thermique, la geothermie, les transferts entre le sol et l’atmosphere, les echangeurs thermiques, les piles a combustible, le genie chimique... Comme tous les processus polyphasiques, ils sont d’une grande complexite et sur de nombreux points, leur connaissance n’est encore que partielle. En raison de son importance pratique particuliere, on ne s’interessera dans ce qui suit qu’au seul cas du changement de phase liquide – vapeur (evaporation-condensation), le changement de phase liquide – solide comparativement moins etudie (bien qu’il intervienne dans de nombreuses applications : fonte des neiges, metallurgie/fonderie, etc.) n’etant pas aborde. Signalons simplement que la solidification d’un melange multiconstituant fait intervenir deux mecanismes, domines par les instabilites eventuelles de l’interface fluide/solide 2 : croissance colonnaire a partir de la surface de refroidissement, qui conduit a la presence d’un milieu poreux (« mushy zone »), croissance de grains equiaxes dans la solution liquide en refroidissement. Des modeles du type « milieux poreux » ont donc ete developpes. Le lecteur interesse par ces questions trouvera quelques references bibliographiques dans 3 4 ainsi que dans 5 pour une discussion sur les modelisations non-equilibre local. Le changement de phase liquide – vapeur en milieux poreux a notamment ete etudie en relation avec les problemes de sechage 6 7 8 de geothermie 9 , d’ evaporateurs 10 11 12 , d’environnement (câbles electriques de forte puissance enterres, stockage des dechets radioactifs...) d'ameliorations des echanges par revetement poreux 13 ou encore de surete nucleaire 14 15 16 17 . Ce phenomene, qui intervient comme suite a l'evolution des conditions thermodynamiques du ou des fluides saturant l'espace des pores, peut etre provoque par des modifications des conditions aux limites sur les frontieres du domaine limitant le milieu poreux (sechage convectif...) ou par des processus internes concernant le volume poreux dans son ensemble (chauffage micro-ondes...). (d'apres 18 ) (d'apres 18 ) Evaporation a travers une surface limite permeable pour T

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 effect of both magnitude and orientation of a uniform magnetic field on the critical transition occurring within an electrically conducting binary fluid layer, stratified in temperature and concentration, taking into account the Soret effect, is investigated numerically. For such a configuration, the results show that the critical thresholds corresponding to an arbitrary orientated magnetic field can be derived from those obtained for a vertical magnetic field and that the axes of the marginal cells are aligned with the horizontal component of the magnetic field. Moreover, an analytical study is conducted to investigate the impact of the magnetic field on long-wavelength instabilities. The effect of the magnetic field on such instabilities reveals a new phenomenon consisting in major changes of the unstable modes that lose their unicellular nature to regain their multi-roll characteristic, as it is the case without magnetic field fo rψψ0, la valeur du nombre de Hartmann Ha(ψ, Le)correspondant à cette transition responsable d’une modification significative des transferts de masse et de chaleur peut être déterminée à partir des relations analytiques établies dans le cadre de ce travail.

We present an analytical and numerical stability analysis of Soret-driven convection in a porous cavity saturated by a binary fluid mixture and subjected to vertical high-frequency and small-amplitude vibrations. Two configurations have been considered and compared: an infinite horizontal layer and a bounded domain with a large aspect ratio. In both cases, the initial temperature gradient is produced by a constant uniform heat flux applied on the horizontal boundaries. A formulation using time-averaged equations is used. The linear stability of the equilibrium solution is carried out for various Soret separation ratios $$\varphi $$ , vibrational Rayleigh numbers Rv, Lewis numbers Le and normalized porosity. For an infinite horizontal layer, the critical Rayleigh number $$\mathrm{Ra}_c$$ is determined analytically. For a steady bifurcation to a one-cell solution (the critical wavenumber is zero), we obtain $$\mathrm{Ra}_{c}={12}/{({\varphi }(\mathrm{Le}+1)+1)}$$ for all Rv. When the bifurcation is a Hopf bifurcation or when the critical wavenumber is not zero, we use a Galerkin method to compute the critical values. Our study is completed by a nonlinear analysis of the bifurcation to one-cell solutions in an infinite horizontal layer that is compared to numerical simulations in bounded horizontal domains with large aspect ratio.

International audience; In this paper, a time dependent one-dimensional linear advection–diffusion equation with Dirichlet homogeneous boundary conditions and an initial sine function is solved analytically by separation of variables and numerically by the finite element method. It is observed that when the advection becomes dominant, the analytical solution becomes ill-behaved and harder to evaluate. Therefore another approach is designed where the solution is decomposed in a simple wave solution and a viscous perturbation. It is shown that an exponential layer builds up close to the downstream boundary. Discussion and comparison of both solutions are carried out extensively offering the numericist a new test model for the numerical integration of the Navier–Stokes equation

The authors studied the effect of vertical high-frequency and small-amplitude vibrations on the separation of a binary mixture saturating a porous cavity. The horizontal bottom plate was submitted to constant uniform heat flux and the top one was maintained at constant temperature while no mass flux was imposed. The influence of vertical vibrations on the onset of convection and on the stability of the unicellular flow was investigated for positive separation ratio $\psi$ . The case of high-frequency and small-amplitude vibrations was considered so that a formulation using time averaged equations could be used. For an infinite horizontal porous layer, the equilibrium solution was found to lose its stability via a stationary bifurcation leading to unicellular flow or multicellular one depending on the value of $\psi$ . The analytical solution of the unicellular flow was obtained and separation was expressed in terms of Lewis number, separation ratio and thermal Rayleigh number. The direct numerical simulations using the averaged governing equations and analytical stability analysis showed that the unicellular flow loses its stability via oscillatory bifurcation. The vibrations decrease the value of $\psi_{uni}$ , which allows species separation for a wide variety of binary mixtures. The vibrations can be used to maintain the unicellular flow and allow species separation over a wider range of Rayleigh numbers. The results of direct numerical simulations and analytical model are in good agreement.

International audience; The onset of convection in a porous layer heated from below is considered, and we determine how the presence of two solid but heat-conducting bounding plates of finite thickness alters the manner in which convection ensues. Heat is generated by the lower plate (with an insulating lower boundary), but the upper one is passive with a fixed upper boundary temperature. It is shown that this composite layer may mimic in turn one of the three different types of classical single-layer onset problems which are well-known in the literature. The type which is selected (or indeed whether it corresponds to a transitional case) depends quite critically on the precise values of the relative thickness of the solid layers and their conductivity ratio. It is also shown that care needs to be taken over declaring that the solid plates are thin: extreme values of the conductivity ratio can yield a stability criterion which appears to be different from that suggested by the imposed boundary conditions.

National audience; Thermogravitational separation has until now, been used in different heated vertical cells called thermogravitational columns. The cell can be an annular cavity with two isothermal faces maintained at different temperatures. The main objective of this paper is to study the two dimensional coupled convection with thermodiffusion process. It concerns a theoretical and numerical investigation of species separation in a binary liquid mixture saturating a horizontal porous annulus space where the inner cylinder is heated isothermally. This kind of geometry is used instead of the annular vertical cell, hence the novelty of this technique. Analytical resolution is performed using the perturbation method function of time versus the corresponding physics (Raleigh and Lewis numbers...). Results reveal that the separation can be increased with an optimum for small values of Rayleigh number. Further, these values are less important than the critical value of Raleigh leading to the loss of unicellular flow stability found in literature.

International audience; The aim of this work is to present a theoretical analysis of the separation of an N-component mixture. In this study, two analytical models explaining the thermo-gravitational separation of components in N-component mixtures for vertical cavity filled by a porous medium are presented and assessed. The basic state and the separation are expressed in terms of the separation ratio, and the Lewis, cross-diffusion and Rayleigh numbers. Our computational analysis confirms that, for the given values of the mass fractions, thermodiffusion can be measured with a thermo-gravitational column, strongly supporting the experimentally determined transport coefficients.

In this paper we study by numerical simulation, the cooling of a solar cell by forced convection in the presence of a nanofluid. The inclined walls of the cavity are adiabatic but the silicon solar cells are subjected to a constant heat temperature. The nanofluid is introduced into the cavity with a constant vertical speed and subjected to room temperature. The equations governing the flow hydrodynamics and heat transfer are described by the Navier-Stockes and energy equations. For the physical parameters of Al2O3-Water nanofluid, we use the model of Brinkman and Wasp. The finite elements method is used to solve the system of differential equations that is based on the Galerkin method. We consider the effect of solid volume fraction for different values of Reynolds number on the results in the form of isotherms and modified local and average Nusselt number.

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.

Two-dimensional, double diffusion, natural convection in a partially porous cavity satured with a binary fluid is investigated numerically. Multiple motions are driven by the external temperature and concentration differences imposed across vertical walls. The wavy interface between fluid and porous layer is horizontal. The equations which describe the fluid flow and heat and mass transfer are described by the Navier-Stokes equations (fluid region), Darcy-Brinkman equation (porous region) and energy and mass equations. The finite element method was applied to solve the governing equations. The fluid flow and heat and mass transfer has been investigated for different values of the amplitude and the wave number of the interface and the buoyancy ratio. The results obtained in the form of isotherms, stream lines, isoconcentrations and the Nusselt and Sherwood numbers; show that the wavy interface has a significant effect on the flow and heat and mass transfer.

PurposeThe purpose of this paper is to present a numerical and an analytical study of the thermohaline convection with Soret effect in a square enclosure filled with a binary fluid mixture.Design/methodology/approachThe horizontal boundaries of the enclosure are impermeable and heated from below while its vertical walls are assumed to be adiabatic and impermeable. The Navier‐Stokes equations under the Boussinesq‐Oberbeck approximation are solved numerically. The results are given for different values of the separation ratio. The critical Rayleigh number at the onset of convection is determined analytically and numerically. The Hopf frequency at the onset of convection is obtained.FindingsThe existence of two stable stationary bifurcation branches is illustrated. Furthermore, it is shown that the existence of stable traveling waves in the transition from one branch to the other depends on the value of the separation ratio. For some values of Rayleigh number, asymmetric flows are observed. A good agreement is found between the numerical solution and analytical analysis.Originality/valueThe present work is the first to consider thermosolutal convection with Soret effect in a square enclosure.

The objective of this study is the comparison between the model of Darcy-Brinkman and the Navier-Stokes equations modified, in the case of the natural convection. The study is made in a porous vertical square cavity saturated by a Newtonian fluid. A cylindrical heat source maintained at a uniform heat flux is introduced into porous medium. The equations which describe the thermal transfer and the hydrodynamic flow of the two models are solved numerically by means of the software package Femlab 3.2 based on the finite element method. The results obtained are in the form of average kinetic energy per unit mass, the local and the average Nusselt numbers, the pressure and the viscous force per unit area.

International audience; An analytical and numerical study of Soret-driven convection in a horizontal porous layer saturated by a binary fluid and subjected to uniform cross heat fluxes is presented. The flow is driven by the combined buoyancy effect due to temperature and induced mass fraction variations through a binary water ethanolmixture. In the first part of the study, a closed-form analytical solution in the limit of a large aspect ratio of the cell (A >> 1) is developed. We are mainly concerned with the determination of the mass fraction gradient of the component of interest along the horizontal direction, which determines the species eparation. In the second part, numerous numerical simulations are carried out in order to validate the analytical results and extend heat and mass transfer to an area not covered by the analytical study. Good agreement is found between analytical and numerical results concerning the species separation obtained for a unicellular flow. In this configuration, the Soret separation process is improved by two control parameters: the heat flux density imposed on the horizontal walls of the cell and the ratio, a, of heat flux density imposed on vertical walls to that on horizontal walls. The influence of the heat flux density ratio, a, on the transient regime (relaxation time) is also investigated numerically. We observe that an increase in the parameter a leads to a decrease in the relaxation time. Thus, for a cell heated from below withoutlateral heating, the onset of convection from the mechanical equilibrium state is analyzed. The linear stability analysis shows 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 linear stability results are widely corroborated by direct 2D numerical simulations. The thresholds of various multicellular solutions are determined in terms of the governing parameters of the problem using nonlinear direct numerical simulations

International audience; Our present work focuses on the coupling between thermal diffusion and convection in order toimprove the thermal gravitational separation of mixture components. The separation phenomenonwas studied in a porous medium contained in vertical columns. We performed analytical andnumerical simulations to corroborate the experimental measurements of the thermal diffusion coefficientsof ternary mixture n-dodecane, isobutylbenzene, and tetralin obtained in microgravity in theinternational space station. Our approach corroborates the existing data published in the literature.The authors show that it is possible to quantify and to optimize the species separation for ternarymixtures. The authors checked, for ternary mixtures, the validity of the “forgotten effect hypothesis”established for binary mixtures by Furry, Jones, and Onsager. Two complete and different analyticalresolution methods were used in order to describe the separation in terms of Lewis numbers, theseparation ratios, the cross-diffusion coefficients, and the Rayleigh number. The analytical model isbased on the parallel flow approximation. In order to validate this model, a numerical simulation wasperformed using the finite element method. From our new approach to vertical separation columns,new relations for mass fraction gradients and the optimal Rayleigh number for each component of theternary mixture were obtained.

We examine the linear stability of a liquid layer heated from below (the classical Rayleigh–Benard problem) but laterally confined between four vertical rigid and adiabatic boundaries. The main feature of the present study is that the height of the layer is much greater than the two other horizontal dimensions. The Soret effect is also taken into account. The ultimate objective of the study is a better knowledge of the operation of thermogravitational columns, and the search for a possible new way to measure positive Soret coefficients based on the variation of the critical Rayleigh number. To cite this article: J.K. Platten et al., C. R. Mecanique 335 (2007).

International audience; Double diffusion natural convection in an enclosure filled with a liquid and subjected to differential heating and differential species concentration is investigated. Four models were developed to address the hydraulic effect of the upper lid on the rate of heat and mass transfer and on the flow structures. It is found that free surface yields higher rate of heat and mass transfer. Also, there is noticeable difference between the results of twoandthree-dimensional results. Hence, the effect of upper lid condition cannot be underestimated. The effect of boundary condition on the rate of heat transfer is more profound for natural convection without double diffusion effects.

International audience; The dynamic behavior of convective motion in a confined porous medium saturated by a pure fluid under the action of mechanical vibration is studied. A redefinition of vibrational Rayleigh number is proposed from which we distinguish the domain of validity of the mean flow. The weakly nonlinear stability analysis performed demonstrates, contrary to published results, that the bifurcation is of supercritical nature and the subcritical branch does not exist. It is emphasized that, in order to find the thermal behavior of the system for the onset of convection, we should separate the vibrational effect from the thermal effect involving the temperature difference.

The generation of two-dimensional thermal convection induced simultaneously by gravity and high-frequency vibration in a bounded rectangular enclosure or in a layer is investigated theoretically and numerically. The horizontal walls of the container are maintained at constant temperatures while the vertical boundaries are thermally insulated, impermeable and adiabatic. General equations for the description of the time-averaged convective flow and, within this framework, the generalized Boussinesq approximation are formulated. These equations are solved using a spectral collocation method to study the influence of vibrations (angle and intensity). Hence, a theoretical study shows that mechanical quasi-equilibrium (i.e., state in which the averaged velocity is zero but the oscillatory component is in general non-zero) is impossible when the direction of vibration is not parallel to the temperature gradient. In the other case, it is proved that the mechanical equilibrium is linearly stable up to a critical value of the unique stability parameter, which depends on the vibrational field. In this paper, it is shown that high-frequency vertical oscillations can delay convective instabilities and, in this way, reduce the convective flow. The isotherms are oriented perpendicular to the axis of vibration. In the case where the direction of vibration is perpendicular to the temperature gradient, small values of the Grashof number, the stability parameter, induce the generation of an average convective flow. When the aspect ratio is large enough, the character of the bifurcation is practically the same as in the limiting case of an infinitely long layer.

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.

International audience; The effect of natural convection on solute segregation in the horizontal Bridgman configuration is studied. The objective is to check whether a single non-dimensional number, based on the fluid flow induced interface shear stress, is able to capture the physics of the mass transport phenomena. A number of heat and mass transfer numerical simulations are carried out in the laminar convection regime, and the segregation results are found to be in good agreement with the predictions of the scaling analysis. At the higher convective levels relevant for the comparison with existing experimental data, a direct computation of the segregation phenomena is not possible, but numerical simulations accounting for turbulence modeling can provide the interface shear stress. With this procedure, a good agreement between the experimentally measured segregation and the predictions of the scaling analysis is again observed, thus validating the choice of the interface shear stress as a key parameter for the segregation studies.

This work brings new insight to the question of the piston effect, which has been found to be the main cause of temperature equilibration in the vicinity of the liquid–vapor critical point under weightlessness conditions. The thermalization process of a near-critical fluid confined in a cavity and submitted to local heating is modeled with special emphasis on the role of gravity and boundary conditions. The solution of the unsteady Navier–Stokes equations written for a hypercom-pressible low-heat-diffusing van der Waals gas is obtained in a 2-D configuration by means of a finite-volume numerical code. Under Earth gravity conditions, the results show that the thermal plume rising from a heat source strongly decreases and rapidly cancels bulk fluid heating when it strikes the top thermo-stated wall. It is proved that convection does not prevent heat transfer by the piston effect but that it causes a sudden enhancement of the cooling piston effect generated at the thermostated top boundary, which leads to an early equilibrium between the cooling and heating piston effects.

National audience; Lorsqu'on soumet une solution initialement homogène, constituée d'au moins deux espèces chimiques, à un gradient thermique, celui-ci induit un transfert des constituants au sein du mélange : ce phénomène est appelé thermodiffusion ou effet Soret. Le couplage entre la convection et la thermodiffusion, appelé diffusion thermogravitationnelle conduit, sous certaines conditions, à une séparation des espèces plus importante qu'en thermodiffusion. Dans les colonnes thermogravitationnelles verticales le gradient thermique imposé induit non seulement la thermodiffusion mais également le mouvement convectif. Nous proposons dans ce travail une nouvelle technique permettant d'améliorer le procédé de séparation en découplant le gradient thermique de la vitesse convective. On considère pour cela, une cavité rectangulaire, horizontale, remplie d'un fluide binaire et soumise à un flux de chaleur vertical. La paroi supérieure est animée d'une vitesse uniforme, ce qui permet de disposer de deux paramètres de contrôle indépendants. Les résultats analytiques et numériques obtenus sont en parfait accord.

International audience; We investigate the onset of convection in a uniform, constant-thickness, horizontal porous layer which is heated from below. The layer is bounded above and below by thermally conducting but impermeable layers. Our aim is to determine the effect on the onset of convection of the interaction between the presence of these outer conducting layers and a horizontal background flow. A linear stability analysis is performed and a dispersion relation is derived from which the stability characteristics of the layer are computed. Convection cells are found move along the layer at a speed which is lower than that of the imposed flow due to a thermal drag caused by the presence of the bounding solid layers. Neutral curves and streamline/ isotherm patterns are presented in order to understand the physical role played by the governing nondimensional parameters. When the diffusivity of the solid layers is much lower than the diffusivity of the porous layer there exists a regime where the neutral curve can exhibit two minima, and at one point in parameter space there exists a neutral curve with a quartic minimum.

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; Two-dimensional thermovibrational convection in rectangular cavities under the condition of weightlessness is studied. The problem is based on the system of equations of the mean fields of velocity, pressure, and temperature. A pseudospeetral Chebyshev collocation method is used. The case of rectangular cavities (a layer of finite length) is considered subject to high-frequency transversal vibrations and a longitudinal temperature gradient. In the case of a square cavity the instability of the main flow exists, and the bifurcation to other symmetry takes place. The same behavior is observed when the cavity is elongated in the direction of the temperature gradient. It is shown that the intensity of the thermovibrational convective flow decreases, in general, while the aspect ratio increases in accordance with linear stability theory, in which it was proven that, in the limiting case of an infinitely long layer subject to a longitudinal temperature gradient and a transversal axis of vibrations, the absolute stability of the quasi-equilibrium state lakes place.

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.

The aim of this article is to present briefly a summary of the state of art in theoretical, experimental and numerical approaches in thermodiffusion. The concepts and equations giving the mass flux of constituents in binary, ternary and multicomponent mixtures are presented.

International audience; We consider convection in a uniform fluid-saturated porous layer which is bounded by conducting plates and heated from below. The primary aim is to determine the identity of the postcritical convection planform as a function of the thicknesses and conductivities of the bounding plates relative to that of the porous layer. This work complements and extends an early paper by Riahi (J Fluid Mech 129:153–171, 1983) who considered a situation where the porous layer is bounded by infinitely thick conducting media.We present regions in parameter space wherein convection in the form of rolls is unstable and within which cells with square planform form the preferred pattern.

International audience; We present an analytical and numerical stability analysis of the onset of natural convection in a horizontal fluid-saturated porous cavity. The cavity is bounded by thin horizontal plates with uniform thickness whose outer surfaces are subject to a constant heat flux. The main aim is to determine the effect of the presence of the bounding plates on the onset of convection. The onset criterion is found to be sensitively dependent on the relative thickness of the plates and the porous layer, delta, and their relative conductivities, d. For the long wavelength mode it is precisely Ra-c = 12(1 + 2 delta d).

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.

Natural convection is investigated numerically and experimentally in a cylindrical annulus. The governing equations based on primitive variables are solved using Chorin’s method. In addition to the unicellular flows reported in the literature, depending on initial conditions, bicellular flows are observed for high Rayleigh numbers. The bifurcation point is determined numerically. The velocity field for unicellular flows is measured by laser-Doppler anemometry in an air-filled annulus. A perturbation solution is also presented. The experimental results are in good agreement with numerical predictions and the perturbation solution.

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.

Temporal and spatio-temporal instabilities of binary liquid films flowing down an inclined uniformly heated plate with Soret effect are investigated by using the Chebyshev collocation method to solve the full system of linear stability equations. Seven dimensionless parameters, i.e. the Kapitza, Galileo, Prandtl, Lewis, Soret, Marangoni, and Biot numbers (Ka,G,Pr,L, χ,M,B), as well as the inclination angle (β) are used to control the flow system. In the case of pure spanwise perturbations, thermocapillary S- and P-modes are obtained. It is found that the most dangerous modes are stationary for positive Soret numbers (χ≥0), and oscillatory for χ0 and even merges with the long-wave S-mode. In the case of streamwise perturbations, a long-wave surface mode (H-mode) is also obtained. From the neutral curves, it is found that larger Soret numbers make the film flow more unstable as do larger Marangoni numbers. The increase of these parameters leads to the merging of the long-wave H- and S-modes, making the situation long-wave unstable for any Galileo number. It also strongly influences the short-wave P-mode which becomes the most critical for large enough Galileo numbers. Furthermore, from the boundary curves between absolute and convective instabilities (AI/CI) calculated for both the long-wave instability (S- and H-modes) and the short-wave instability (P-mode), it is shown that for small Galileo numbers the AI/CI boundary curves are determined by the long-wave instability, while for large Galileo numbers they are determined by the short-wave instability.

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.