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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.

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.

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.

While it is well known that the phenomenon of Faraday instability is a manifestation of vibrational acceleration, its behaviour when both the capillary and gravitational effects vanish, remains unexplored theoretically and experimentally. A detailed experimental and theoretical study is performed to understand the physics of this phenomenon in small rectangular geometry where the proximity of wall results in considerable sidewall stresses. A novel binary liquids system is utilized with low interfacial tension for a near flat interface. Thermal control of fluid system is utilized for achieving reduction in capillary force with study of miscible fluids where interfacial tension reduces to almost zero. In order to discriminate between gravity and capillarity effects, experiments were performed in parabolic flight campaigns. . For the theoretical approach a linear stability analysis is performed through Navier-Stokes equations in a Newtonian incompressible viscous fluid system. This is achieved through a Fourier Floquet method resulting into an eigenvalue problem. Equations are solved by including viscous damping effects for compensating sidewall stresses. Experimentally binary fluids provided a convenient option of changing the coefficient of interfacial tension by temperature control and going through immiscible to miscible system without change of liquid charge. Viscous damping rate is determined experimentally by measuring the linear damping rate. The correction in the theoretical calculations with the viscous damping rate helped in achieving a better understanding of the prediction of the experimental critical amplitude for sub-harmonic and harmonic modes.; Alors qu'il est bien connu que le phénomène d'instabilité de Faraday est une manifestation d'ondes de gravité capillaire, son comportement lorsque les effets capillaires et gravitationnels disparaissent reste inexploré théoriquement et expérimentalement. Une étude expérimentale et théorique détaillée est réalisée pour comprendre la physique de ce phénomène dans une petite cavité rectangulaire où la proximité des murs entraîne des contraintes considérables sur les parois latérales. Un couple de liquides binaires est utilisé avec une faible tension interfaciale pour une interface presque plate. Le contrôle thermique de ce système de fluide est utilisé pour diminuer la force capillaire et d’étudier ainsi les instabilités de Faraday dans les fluides miscibles où la tension interfaciale s’annule. Afin de prendre en compte les effets gravitationnels, l'expérience a été réalisée dans des campagnes de vols paraboliques. Pour l'approche théorique, une analyse de stabilité linéaire est effectuée à l'aide d'équations de Navier-Stokes dans un système de fluide visqueux incompressible et newtonien. Ceci est réalisé grâce à une méthode de Fourier-Floquet résultant en un problème aux valeurs propres. Les comparaisons montrent des différences non négligeables. Les équations sont ensuite résolues en incluant des effets d'amortissement visqueux pour compenser les contraintes des parois latérales. Les fluides binaires ont fourni une option commode pourchanger le coefficient de tension interfaciale en augmentant la température jusqu’à la température critique, ce qui a permis de passer d’un système de fluides non miscibles à celui des fluides miscibles tout en restant au-dessous de la température d’ébullition. Le taux d'amortissement visqueux linéaire est mesuré expérimentalement. La correction des calculs théoriques en prenant en compte le taux d'amortissement visqueux a permis une amélioration nette des résultats et donc de mieux comprendre la prédiction de l'amplitude critique expérimentale pour les modes sous-harmonique et harmonique.

In this study, the influence of vertical vibrations on species thermo-gravitational separation of a binary fluid, saturating a porous medium, is presented. The cell is horizontal of large aspect ratio and situated in the gravity field. A formulation using time average equations is used. Two configurations have been considered and compared. In the first one, a constant heat flux is imposed on the horizontal walls and in the second case, a constant heat flux is imposed on the bottom wall while a constant temperature is imposed on the top wall. For each configuration, stable unicellular solutions leading to species separation are obtained, depending on the dimensionless parameters of the problem. The analytical results are in good agreement with those obtained by direct numerical simulations. In both cases, a linear stability analysis of the equilibrium solution and the unicellular one is presented using a spectral method.; Dans cette étude, nous proposons une étude théorique et numérique de l'influence des vibrations de hautes fréquences et de faibles amplitudes sur la séparation thermo-gravitationnelle des constituants d'un mélange binaire saturant un milieu poreux. La cellule considérée est horizontale, de grand rapport d'aspect et placée dans le champ de pesanteur. La formulation mathématique est obtenue en utilisant le formalisme des équations moyennées. Les conditions aux limites imposées au niveau des parois horizontales diffèrent de celles du problème de Rayleigh Bénard. On considère respectivement le cas où un flux constant est imposé sur ces parois puis le cas où un flux thermique constant est imposé sur la paroi inférieure alors qu'une température constante est imposée sur la paroi supérieure. Dans les deux configurations, onmet en évidence de solutions stables monocellulairesconduisant à la séparation des constituants du mélange et ce pour une large gamme des paramètres adimensionnels régissant le problème. Les résultats analytiques et de simulations numériques directes sont en très bon accord. Dans les deux cas, une étude de stabilité linéaire de la solution d'équilibre et de la solution monocellulaireest réalisée par méthode spectrale.