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 $ B\gg1$ , 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 $ \Delta T$ imposed on the two horizontal walls was studied.

La convection thermogravitationnelle, qui génère de s variations locales de la masse volumique du fluide en fonction de la température, conduit final ement à une séparation des espèces dans la cavité. On s’intéresse dans ce travail à la séparation des espèces d’une solution binaire remplissant une cavité parallélépipédique poreuse, placée dans le c hamp de la pesanteur, et soumise au niveau de ses parois horizontales et verticales à des densités de flux uniformes et croisés. Nous déterminons aussi les paramètres de contrôle conduisant à la séparati on optimale. En utilisant l’étude de la stabilité linéaire, on a vérifié que le nombre de Rayleigh cr itique linéaire, , associé à la perte de la stabilité de la solution unicellulaire est supérieur au nombr e de Rayleigh optimal, , conduisant à la séparation des constituants du mélange.

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.

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 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.

In this paper, we consider the effect of mechanical vibration on the onset of convection in porous media. The porous medium is saturated either by a pure fluid or by a binary mixture. The importance of a transport model on stability diagrams is presented and discussed. The stability threshold for the Darcy–Brinkman case in the Ra Tc - R and k c - R diagrams is presented (where Ra Tc , k c and R are the critical Rayleigh number, the critical wave number and the vibration parameters, respectively). It is shown that there is a significant deviation from the Darcy model. In the thermo-solutal case with the Soret effect, the influence of vibration on the reduction of multi-cellular convection is emphasized. A new analytical relation for obtaining the threshold of mono-cellular convection is derived. This relation shows how the separation factor Ψ is related to the controlling parameters of the problem, Ψ = f ( R , ε *, Le ), when the wave number k → 0. The importance of vibrational parameter definition is highlighted and it is shown how, by using a proper definition for vibrational parameter, we may obtain compact relationship. It is also shown how this result may be used to increase component separation.

In this paper, we study the effect of a homogeneous longitudinal through flow on the onset of convection in a horizontal porous layer saturated by a binary fluid and heated from below or above. The layer boundaries are subjected to a constant heat flux. The investigation is made by taking the Soret effect into account. It is found that in the case of positive separation ratio when the denser component moves toward the cooler wall, through flow has no effect on the stability threshold but exerts an orientating effect on the convective patterns. For negative separation ratio, a strong destabilization occurs of the spatially homogeneous state with respect to long-wave disturbances. The stability range for long-wavelength convective rolls is defined.

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.

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 two- and three-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.

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, ψ, characterizing the Soret effect and the normalized porosity, ε, 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 ψ>ψ0(Le,ε), while for ψ<ψ0(Le,ε), it loses stability via a Hopf subcritical bifurcation. The oscillatory solution is unstable and becomes stationary. For a cell heated from above, the equilibrium solution is linearly stable if ψ>0, while a stationary or an oscillatory bifurcation occurs if ψ<0. The results obtained from the linear stability analysis are widely corroborated by direct 2D numerical simulations. In the case of long-wave disturbances, for ψ<0 and for ψ higher than a particular value called ψmono, we observe that the monocellular flow leads to a separation of the species between the two ends of the cell. First, we determined the velocity, temperature, and concentration fields analytically for monocellular flow. Then we studied the stability of this flow. For a cell heated from below and for ψ>ψmono 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 ψ<0, the monocellular flow remains linearly stable. We verified numerically that this problem admits other stable multicellular stationary solutions for this range of parameters.

The linear stability of a plane layer with horizontal temperature and concentration stratification corresponding to gradient zone of a solar pond is investigated. The problem is described by Navier-Stokes equations with Boussinesq-Oberbeck approximation. Two source terms are introduced in the energy equations: the absorption of solar energy characterized by the extinction radiative coefficient μe and by the parameter f defined as the ratio of extracted heat flux to absorbed heat flux in the lower convective zone. The influence of the parameters μe and f on the onset of thermosolutal convection in the case of confined and infinite layers is analyzed. It is found that convection starts in an oscillatory state, independently of the RaS value. Different convection solutions were found for marginal stability and steady state.

A theoretical model describing the response of a single-phase near-critical fluid to small-amplitude, high-frequency translational vibrations is developed on the basis of the multiple-scale and averaging methods. Additionally to the usual terms of thermal convection, the equations and boundary conditions contain new terms responsible for the generation of pulsating and average flows due to vibrational forcing and fluid compressibility. The effect of compressibility is taken into account both in the boundary layers and in the bulk. A number of classical problems of convection and thermoacoustics are considered on the basis of the new model.

This work presents a numerical validation of a developed analytical model. For the assembly, the total force-displacement behavior is well defined. Thus the analytical model makes it possible to determine the rigidity of the assembly and to obtain a simple formulation very rapidly which gives the total behavior of the assembly.

The governing equations and effective boundary conditions to describe thermal vibrational convection in a near-critical fluid are derived with the help of the multiple-scale method and averaging procedure. In contrast to Part 1, this paper focuses on the effects of density non-homogeneities caused not by external heating but by vibrational and gravity stratifications due to the divergent mechanical compressibility of near-critical media. It is shown that vibrations generate non-homogeneities in the average temperature, which result in the onset of thermal convection even under isothermal boundary conditions. An agreement with the results of previous numerical and asymptotical analyses and with experiments is found.

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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: BH¨+B(π2+k2)+1H˙+(π2+k2)−k2k2+π2RaT(1+Rsinω*t*)H=k2k2+π2(NRaT)(1+Rsinω*t*)Fε*BF¨+Bπ2+k2Le+ε*F˙+π2+k2Le−k2k2+π2NRaT(1+Rsinω*t*)F=k2k2+π2RaT(1+Rsinω*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, ε* 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.

In this chapter we consider the instability of double-diffusive convection in porous media under the effect of mechanical vibration. The so-called time-averaged formulation has been adopted. This formulation can be effectively applied to study the vibrational induced thermo-solutal convection problem. The influence of high-frequency and small-amplitude vibration on the onset of thermo-solutal convection, in a confined porous cavity with various aspect ratios and saturated by a binary mixture has been presented. Linear stability analysis of the mechanical equilibrium or quasi-equilibrium solution is performed. A theoretical examination of the limiting case of the long-wave mode in the case of Soret driven convection under the action of vibration has been carried out. The 2D numerical simulations of the problem are presented which allow us to corroborate the results obtained from the linear stability analysis for both stationary and Hopf bifurcations.

This paper presents a study of the stress analysis in cylindrical assemblies. For the present study we use a cylindrical assembly of two tubes. We write all the components of the stress field function of the σzz(1)(z) stress in the first tube and then we introduce these components into the potential energy formulation. Our method is a variational method applied on the potential energy of deformation. The model can predict the intensity and the distributions of stresses in the assembly. We can also analyse the influence of some geometrical or material parameters on the stress field.

Mixed convection in binary fluid with Soret effect: study of the transition towards transverse 2D rolls. This Note deals with mixed convection in binary fluid with Soret effect in a rectangular duct heated from below. In particular, we study the transition towards transverse 2D rolls appearing at low Reynolds and Rayleigh numbers. The linear stability analysis of Poiseuille flow, with linearly stratified temperature and concentration fields, shows the influence of the separation ratio on the critical Rayleigh number at the transition towards the transversal 2D convective patterns. It highlights the presence, at low Reynolds numbers, of propagating transverse rolls in the downwards as well as in the upwards direction. Finally, we point out that, under these conditions, the propagating frequency of the rolls is the sum of two well defined frequencies: the first related to the Reynolds, the second to the separation ratio. (C) 2004 Academie des sciences. Publie par Elsevier SAS. Tous droits reserves.