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Compared with the growing applications of peritectic alloy, none research on the freckle formation during peritectic solidification has been reported before. Observation on the dendritic mushy zone of Sn-36 at.%Ni peritectic alloy during directional solidification at different growth velocities shows that the freckles are formed in two different regions: region I before peritectic reaction and region II after peritectic reaction. In addition, more freckles can be observed at lower growth velocities. Examination on the experimental results demonstrates that both the temperature gradient zone melting (TGZM) and Gibbs-Thomson (G–T) effects have obvious influences on the morphology of dendritic network during directional solidification. The current theories onKI Rayleigh number Ra characterizing the thermosolutal convection of dendritic mushy zone to predict freckle formation through the maximum of Ra can only explain the existence of region I while the appearance of region II after peritectic reaction cannot be predicted. Thus, a new Rayleigh number RaP is proposed in consideration of evolution of dendritic mushy zone by both effects and peritectic reaction. Theoretical prediction of RaP also shows a maximum after peritectic reaction in addition to that before peritectic reaction, thus, agreeing well with the freckle formation in region II. In addition, more severe thermosolutal convection can be predicted by the new Rayleigh number RaP at lower growth velocities, which further demonstrates the reliability of RaP in describing the dependence of freckle formation on growth velocity.

Different from other alloys, the observation in this work on the dendritic mushy zone shows that the freckles are formed in two different regions before and after peritectic reaction in directional solidification of Sn–Ni peritectic alloys. In addition, the experimental results demonstrate that the dendritic morphology is influenced by the temperature gradient zone melting and Gibbs–Thomson effects. A new Rayleigh number (RaP) is proposed in consideration of both effects and peritectic reaction. The prediction of RaP confirms the freckle formation in two regions during peritectic solidification. Besides, heavier thermosolutal convection in samples with larger diameter is also demonstrated.

International audience; Experiments of directional solidification of TiAl cylindrical samples were conducted within the frame of the ESA GRADECET project. The experiments were performed in the ESA "Large Diameter Centrifuge" using a furnace with a well defined thermal protocol. The furnace was mounted in the centrifuge and free to tilt in such a way that the total apparent gravity (sum of terrestrial gravity and centrifugal acceleration) was aligned to the cylinder centerline. Several centrifugation levels were investigated besides to one reference case out of the centrifuge. In this work, we present 3D numerical simulations of these experiments paying special attention in the liquid thermosolutal buoyancy convection and aluminum macrosegregation. The numerical model accounts for the non-inertial accelerations that appear in the rotating reference frame (centrifugal and Coriolis), motionless solid, the-mosolutal Boussinesq approximation and an infinitely fast microscopic diffusion model between the phases to depict the solid growth. The results showed that the Coriolis acceleration entirely modifies the liquid flow regime during solidification leading to a 3D aluminum segregation pattern with respect to the case solidified under normal terrestrial gravity conditions. Additionally the magnitude of aluminum segregation increases with the level of centrifu-* Corresponding author

In this paper, three-dimensional thermosolutal natural convection and entropy generation in a cubic cavity filled with a non-Newtonian Carreau-Yasuda fluid has been simulated by Lattice Boltzmann Method (LBM). This study has been conducted for certain pertinent parameters of Rayleigh number (Ra = 104 and 105), the Buoyancy ratio (N = -1, 0.1, 1), the Lewis number (Le = 0.5, 2.5, 10, 100, and 1000), power-law indexes (n = 0.5, 1, and 1.5), Carreau number (Cu = 0.01, 0.1, 1, 10 and 100), and Carreau-Yasuda parameter (m = 0.1, 0.5, and 1). Results indicate that the rise of Rayleigh number enhances heat and mass transfer for various studied parameters. The increase in power-law index provokes heat and mass transfer to drop gradually. However, the effect of power-law index on heat and mass transfer rises steadily as Rayleigh number rises. The enhancement of Carreau number decreases heat and mass transfer, but; the increases in the Carreau-Yasuda parameter augments heat and mass transfer significantly. The augmentation of the buoyancy ratio number enhances heat and mass transfer. It was found that mass transfer increases as Lewis number augments. The augmentation of Rayleigh number enhances different entropy generations and declines the average Bejan number. The increase in the power-law index provokes various irreversibilities to drop significantly. The enhancement of the buoyancy ratio causes the summation entropy generations to increase considerably. The rise of Carreau number and Carreau-Yasuda parameter decreases and increases the total irreversibilities; respectively.

The effects of magnetic field dependent (MFD) thermosolutal convection and MFD viscosity of the fluid dynamics are investigated between squeezing discs rotating with different velocities. The unsteady constitutive expressions of mass conservation, modified Navier-Stokes, Maxwell and MFD thermosolutal convection are coupled as a system of ordinary differential equations. The corresponding solutions for the transformed radial and azimuthal momentum as well as solutions for the azimuthal and axial induced magnetic field equations are determined, also the MHD pressure and torque which the fluid exerts on the upper disc is derived and discussed in details. In the case of smooth discs the self-similar equations are solved using Homotopy Analysis Method (HAM) with appropriate initial guesses and auxiliary parameters to produce an algorithm with an accelerated and assured convergence. The validity and accuracy of HAM results is proved by comparison of the HAM solutions with numerical solver package BVP4c. It has been shown that magnetic Reynolds number causes to decrease magnetic field distributions, fluid temperature, axial and tangential velocity. Also azimuthal and axial components of magnetic field have opposite behavior with increase in MFD viscosity. Applications of the study include automotive magneto-rheological shock absorbers, novel aircraft landing gear systems, heating up or cooling processes, biological sensor systems and biological prosthetic etc. Keywords: MFD viscosity, Thermosolutal convection, Magnetic field, Reynolds numbers, HAM

During the transmission of heat and mass inside an annulus, the impacts of the thermal diffusion & diffusion-thermo should be considered. Then the target of this paper is solving the thermosolutal convection of a nanofluid inside an annulus amongst a square cavity and an inner prismatic shape with three circular cylinders. The ranges of the considered parameters are the solid volume fraction (0≤φ≤0.1), circular cylinder's radius (0.01≤Rc≤0.075), Hartmann number (0≤Ha≤40), Dufour number (0.03≤Du≤0.6), Rayleigh number (103≤Ra≤106), and Soret number (0.1≤Sr≤2). The results indicated that the radius of the circular cylinders is improving the nanofluid moments and strengths of convection flow in an annulus. The maximum of the streamlines is reducing according to an augmentation in Hartmann number or solid volume fraction. The extra buoyancy forces at higher Rayleigh numbers are improving the strengths of the temperature, concentration, and streamlines within an annulus. The mean Nusselt/Sherwood numbers are enhanced along with an expansion in the solid volume fraction or radius of the circular cylinders.

We propose a new phase-field model formulated within the system of lattice Boltzmann (LB) equation for simulating solidification and dendritic growth with fully coupled melt flow and thermosolutal convection–diffusion. With the evolution of the phase field and the transport phenomena all modeled and integrated within the same LB framework, this method preserves and combines the intrinsic advantages of the phase-field method (PFM) and the lattice Boltzmann method (LBM). Particularly, the present PFM/LBM model has several improved features compared to the existing phase-field models including: (1) a novel multiple-relaxation-time (MRT) LB scheme for the phase-field evolution is proposed to effectively model solidification coupled with melt flow and thermosolutal convection–diffusion with improved numerical stability and accuracy, (2) convenient diffuse interface treatments are implemented for the melt flow and thermosolutal transport which can be applied to the entire domain without tracking the interface, and (3) the evolution of the phase field, flow, concentration, and temperature fields on the level of microscopic distribution functions in the LB schemes is decoupled with a multiple-time-scaling strategy (despite their full physical coupling), thus solidification at high Lewis numbers (ratios of the liquid thermal to solutal diffusivities) can be conveniently modeled. The applicability and accuracy of the present PFM/LBM model are verified with four numerical tests including isothermal , iso-solutal and thermosolutal convection–diffusion problems, where excellent agreement in terms of phase-field and thermosolutal distributions and dendritic tip growth velocity and radius with those reported in the literature is demonstrated. The proposed PFM/LBM model can be an attractive and powerful tool for large-scale dendritic growth simulations given the high scalability of the LBM.

Embedded the thermal and solutal buoyancy into the momentum conservation equation as an additional force term using the Boussinesq approximation, a 2D CA-FVM model is extended to simulate the dendritic growth with thermosolutal convection. The model is firstly validated by comparison of numerical predictions with the benchmark test of Rayleigh – Benard convection and the analytical solutions of the stagnant film model for the free dendritic growth with thermosolutal convection, and good agreements between the numerical results with analytical solutions are obtained. Later, numerical simulations for both the equiaxed and columnar dendritic growth of Fe-0.82wt%C binary alloy with thermosolutal convection are performed. The results show that, for the equiaxed dendritic growth in an undercooled melt, the dendrite tip growth rapidly decreases from the high velocity to a relative low steady-state value. With the further growth of dendrite, the thermosolutal convection induced by the solute rejection and latent heat release is enhanced and four vortexes are developed between the dendrite arms. Thus, the asymmetries of the dendrite morphology, temperature and solute profiles are intensified. For the columnar dendritic growth with thermosolutal convection under the unidirectional solidification process, the thermosolutal convection transports the rejected solute downward and makes the solute enrich at the interdendritic region. The thermosolutal convection facilitates the upstream dendritic growth, but inhibits the downstream dendritic growth. Moreover, with the increase of deflection angle of gravity, the advection on the top region and the clock-wise vortex flow at the interdendritic region intensified, and finally the columnar dendrite morphology becomes more asymmetrical.

Using a perturbation method, we solve asymptotically the nonlinear partial differential equations that govern double-diffusive convection (with heat and solute diffusing) in a two-dimensional rectangular domain near a critical point in parameter space where the linearized operator has a quadruple-zero eigenvalue. The asymptotic solution near this codimension-four point is found to depend on two slow-time-dependent amplitudes governed by two nonlinearly-coupled Van der Pol–Duffing equations. Through numerical approximation of the 3-dimensional Poincare map in the four-dimensional state space of the amplitude equations, we detect and analyze the bifurcations of the amplitude equations as the thermal Rayleigh number, R T , is increased (for R S ≪ R T , the solute Rayleigh number) with all other parameters fixed. The bifurcations observed include: Hopf, pitchfork and Neimark–Sacker bifurcations of limit cycles, symmetric and asymmetric saddle–node bifurcations of 2-tori, and reverse torus-doubling cascades. In addition, chaotic solutions are found numerically to emerge via two different types of routes: (1) a route involving a homoclinic explosion in the Poincare map and; (2) type-I intermittency routes near saddle–node bifurcations of 2-tori. The homoclinic explosion occurs when two unstable 2-tori form homoclinic connections with a saddle limit cycle, thereby creating a homoclinic butterfly in the Poincare map that leads to a discrete Lorenz-like attractor.

Peritectic alloys have attracted more and more attention during past decades due to their growing applications. However, different from the current analysis on other alloys, freckle formation is witnessed in two regions in Sn–Ni peritectic alloy during directional solidification: the former (latter) one before (after) peritectic reaction. Besides, compared with the steady growth condition, fewer freckles can be observed in deceleration growth while more freckles are formed in acceleration growth. In addition, it has been demonstrated that the dendritic morphology is greatly influenced by the Gibbs-Thomson (G-T) and temperature gradient zone melting (TGZM) effects. A Rayleigh number RaP in consideration of both effects was proposed. The prediction of RaP in the mushy zone shows two maximums during peritectic solidification, confirming the freckle formation in two regions in this work. Furthermore, the influence of unsteady growth condition on the freckle formation can also be successfully predicted by RaP.

International Conference on Materials and Energy (ICOME), La Rochelle, FRANCE, MAY 17-20, 2016; International audience; An hybrid Lattice-Boltzman finite-difference method is proposed to assess the impact of the inclination angle of the cavity on thermosolutal natural convection of a binary fluid confined in a square cavity differentially heated and salted from two opposite walls with constant temperatures and concentrations. This method seems more flexible in implementing additional effects and apparently more efficient than the classical CFD methods. The study is conducted in the case of aiding buoyancy forces and many other controlling parameters. Results obtained showed important qualitative and quantitative effects of the governing parameters on fluid flow and heat and mass transfer characteristics. The heatlines and masslines concepts are also implemented to obtain additional details on heat and mass transfer processes. (C) 2017 The Authors. Published by Elsevier Ltd.

A numerical investigation of Soret and Dufour effects on heat and mass transfer in a Salt Gradient Solar Pond (SGSP) and its storage efficiency under meteorological conditions of Tangier (Morocco) is presented. This SGSP consists of an open square cavity filled with a saline solution. Its vertical and bottom walls are adiabatic and impermeable. Transfers are described by the Navier-Stokes, thermal energy and diffusion equations which are solved with a finite volume scheme and the Gauss method. The link between the pressure and velocity fields is carried out by the SIMPLE algorithm. The numerical model, developed in Fortran 95 language, is satisfactorily validated by comparisons with previously published numerical and experimental studies. Results show that the Dufour effect on thermosolutal convection in a SGSP is more relevant than the Soret effect. Nevertheless, increasing the value of the Dufour coefficient from 0 to 0.8 improves heat and mass transfer in the SGSP. Therefore, the amount of the thermal energy accumulated in the storage zone is reduced by about 5.56 %. In addition, the heat losses through the SGSP free surface are improved by around 8.53 %.

In this study, the bifurcation phenomenon and the existence of dual asymmetry solutions of double-diffusive convection driven by an internal thermal and solutal source in a horizontal cavity have been investigated systematically. The problem is solved by the SIMPLE algorithm with the QUICK scheme in a non-uniform staggered grid system. The Rayleigh numbers, 200 ≤ Ra ≤ 106, buoyancy ratios, −20 ≤ Nc ≤ 20, Soret numbers, 0 ≤ Sr ≤ 0.8, and Dufour numbers, 0 ≤ Df ≤ 0.8, are considered in this study. Different streamlines for the onset and evolution of static bifurcation are tracked. The paper aims to determine the critical Rayleigh numbers and buoyancy ratios for the onset of symmetry-breaking through bifurcation diagrams based on the vortex length. The results indicate that increasing buoyancy can destabilize the symmetric system. The strong couple diffusion effect delays the onset of bifurcation flow. A pair of asymmetric modes can be obtained under different initial conditions.

The present study simulates the 3D unsteady double-diffusive natural convection subject to opposing thermal and solutal buoyancy forces (N

The unsteady double-diffusive natural convection flow in a rectangular enclosure filled with soft rocks which are subjected to humid air, including water vapor absorption and thermoelasticity, is studied numerically. The high order compact finite difference schemes are adopted for better simulation of this problem, and a Darcy–Brinkman–Forchheimer extended model combined with the absorption and thermoelasticity effects is introduced. The effects of the thermal and solutal Rayleigh numbers, hygroscopicity, Darcy numbers and thermoelasticity on the flow are investigated in detail. Results are presented in the form of streamlines and isolines of temperature and mass fraction as well as the plots of average Nusselt and Sherwood numbers over the walls under different conditions. The flow structure develops from conduction-dominated to steady convection-dominated as increasing Rayleigh number or increasing buoyancy ratio as well as increasing Darcy numbers. The thermoelasticity effects of soft rocks play an important role in the analysis of thermal convection flow by the change of temperature and permeability, while the water vapor absorption effects may have a slight effect on the flow in the beginning and vanish with the absorption effects becoming weak. Nevertheless, the effect may significant reduce the mechanical strength properties of soft rocks.

Thermosolutal convection with reaction in a porous media of Darcy type is studied using the energy method. The effect of the reaction terms on the temperature and salt Rayleigh numbers are presented graphically. Furthermore, nonlinear energy stability boundaries for different values of the reaction terms are compared with the linear instability boundaries obtained by Pritchard and Richardson (2007).

A stability analysis is carried out to investigate the onset of thermosolutal convection in a horizontal porous layer when the solid and fluid phases are not in a local thermal equilibrium, and the solubility of the dissolved component depends on temperature. To study how the reaction and thermal non-equilibrium affect the double-diffusive convection, the effects of scaled inter-phase heat transfer coefficient H and dimensionless reaction rate k on thermosolutal convection are discussed . The critical Rayleigh number and the corresponding wave number for the stability and overstability convections are obtained. Specially, asymptotic analysis for both small and large values of H and k is presented, and the corresponding asymptotic solutions are compared with numerical results. At last, a nonlinear stability analysis is presented to study how H and k affect the Nusselt number.

The mechanism for thermosolutal convection and the coupling effects of flow, temperature and solute fields during the vertical Bridgman growth of Hg1−xCdxTe single crystals have been numerically analyzed. The calculations take into account the thermophysical properties and their dependence on temperature and composition. The results show that there are two main thermal convection cells in the melt caused by two temperature gradients during the growth of Hg1−xCdxTe single crystals. The stabilizing axial solute gradient in the melt will significantly damp the thermal convection cells when the absolute value of the solute Rayleigh number is close to the value of the thermal Rayleigh number. When there is a large solute gradient in the melt, the thermosolutal convection will become unstable, and the upper flow cell will evolve into a two-cell flow pattern. During the growth of Hg1−xCdxTe single crystals, the radial solute segregation in the melt develops non-monotonically with two minima values. Thus, the methods that solely aim at either damping or enhancing the thermosolutal convection are not always able to improve the radial solute segregation.

On the basis of Brinkman model, the onset of double-diffusive (thermosolutal) convection with a reaction term in a horizontal sparsely packed porous media is studied using the normal mode analysis. The effects of Brinkman term, the reaction term, the normalized porosity e and Lewis number Le on the Rayleigh number for stationary and oscillatory case are presented graphically. The Darcy number destabilize the system in case of both stationary and oscillatory mode. The effects of Lewis number and reaction term depend on the value of the values of solutal Rayleigh number. Furthermore, some results of Darcy model can be recovered in limit cases.

The effect of magnetic field dependent (MFD) viscosity on thermosolutal convection in ferromagnetic fluid is considered for a ferromagnetic fluid layer heated and soluted from below in the presence of a uniform vertical magnetic field. Using the linearized stability theory and the normal mode analysis method, an exact solution is obtained for the case of two free boundaries. For the case of stationary convection, stable solute gradient and MFD viscosity have stabilizing effects on the onset of instability. The magnetization may have destabilizing or stabilizing effect in the presence of MFD viscosity, whereas magnetization has always destabilizing effect in the absence of MFD viscosity. The critical wave number and critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of magnetic parameter M"1 and results are depicted graphically. The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid heated from below in the absence of stable solute gradient. The oscillatory modes are introduced due to the presence of the stable solute gradient, which were non-existent in its absence. A sufficient condition for the non-existence of overstability is also obtained.

Combined effect of thermodiffusion and lateral heating on double diffusive natural convection in a horizontal porous layer, filled with a binary fluid and subjected to uniform fluxes of heat and mass on its long sides, is studied analytically and numerically. The short sides of the layer are impermeable to mass transfer and exposed to a perturbating constant heat flux. The governing parameters of problem under study are the Rayleigh number R T , the Lewis number Le , the buoyancy ratio N , the separation parameter φ , the ratio of the horizontal to vertical heat flux a and the aspect ratio A r of the layer. The thermodiffusion effect on the multiplicity of solutions is studied. It is demonstrated that the φ – N plane can be divided into four regions presenting different behaviors. The heat transfer is found to be considerably affected by the Soret effect.

Double diffusive natural convection within a vertical porous enclosure with localized heating and salting from one side is numerically studied by the finite element based finite volume method. In the formulation of the problem, use is made of the Darcy model, which allows the slip boundary condition on a solid wall to be satisfied. Comparisons with benchmark solutions for natural convection in fluid saturated porous enclosures are first presented to validate the code. Following that, an extensive series of numerical simulations is conducted in the range of −55 ⩽ N ⩽ + 55 and 0.125 ⩽ L ⩽ 0.875, where N and L are the buoyancy ratio and the element location, respectively. Streamlines, heatlines, masslines, isotherms and iso-concentrations in the system are produced to illustrate the flow structure transition from solutal dominated opposing to thermal dominated and solutal dominated aiding flows, respectively. The computed average Nusselt and Sherwood numbers provide guidance for locating the heating and salting element.

Thermosolutal convection during vertical Bridgman directional solidification of Ga 1-x In x Sb alloys has been studied by numerical simulation. The transient analysis of heat, momentum and species transport has been performed by using the finite element code FIDAP R . In the case of vertical Bridgman configuration, the thermal convection is driven by the radial temperature gradients. The solute (InSb) rejected at the solid-liquid interface, which is heavier than the GaSb component, damps the thermally driven convection. The solutal effect on the melt convection has been analyzed for low (x = 0.01) and high (x = 0.1) doped Ga 1-x In x Sb alloys. It is found that the damping effect is negligible for Ga 0.99 In 0.01 Sb alloy grown at low pulling rates (V = 1 μm/s), but cannot be neglected if the pulling rate is increased. In the case of concentrated alloys, the low level of convection intensity leads to an increase of radial segregation and interface curvature during the whole growth process as also shown by experiments. The effect of solutal buoyancy force on the melt convection is analyzed for the horizontal Bridgman configuration under microgravity conditions. An inverse but lower solutal effect on the melt convection, as compared with vertical Bridgman arrangement, is observed. The results are in good agreement with the experimental data, and show that convective transport can be observed even for low (2 x 10 -6 g 0 ) residual gravity levels.

The onset of thermosolutal convection due to thermal diffusion in a fluid saturated isotropic porous layer, subject to a gravity gradient is investigated using the Galerkin technique. Results reveal that the Soret parameter affects the pattern of convection only when its magnitude is large, both in the presence and absence of variable gravity field.

This paper reports the results obtained by means of scale analysis of two dimensional double diffusive natural convection in a square porous cavity isothermally heated from below and cooled from the upper surface, whereas its vertical boundaries are subjected to a horizontal solutal gradient. In the problem formulation, the Darcy model is considered with the Boussinesq approximation. The analysis is performed on the basis of the scaling law approach to predict the order of magnitude heat and mass transfer characteristics in limit cases corresponding to thermal or solutal driven flows. The results of the scale analysis are validated numerically and useful correlations of Nu and Sh are proposed for wide ranges of the governing parameters, which are the thermal Rayleigh number, Ra H , the Lewis number, Le and the buoyancy ratio, N .

This article investigates the effect of thermosolutal convection in liquid phase electroepitaxial (LPEE) growth of GaAs through a two-dimensional numerical simulation model. The model accounts for heat transfer and electric current distribution with Peltier and Joule effects, diffusive and convective mass transport including the effect of electromigration, and fluid flow coupled with temperature and concentration fields. Simulations are performed for two growth cell configurations and the results are analyzed to determine growth rates, substrate shape evolution, and relative contributions of Peltier cooling and electromigration. The simulations predict and help explain a number of experimentally observed features which previous diffusion-based models fail to reproduce. In general, electromigration is found to be the dominant growth mechanism, but the contribution of Peltier cooling to the overall growth rate is found to be significantly enhanced by thermosolutal convection in the solution, and Peltier cooling can in fact become the dominant growth mechanism for certain growth conditions and growth cell configurations. The overall growth rate is found to increase with increasing furnace temperature and applied electric current density. The thermosolutal convection model predicts increased non-uniformity of the grown layers compared with the pure-diffusion model. The shape of the grown layers is also shown to be very sensitive to changes in growth cell configurations.

The present paper mollifies the nastily behaving governing equations of Dufour-driven thermosolutal convection of the Veronis type by the construction of an appropriate linear transformation and extends the results of M. B. Banerjeeet al.(Proc. Roy. Soc. London Ser. A378, 1981, 301;J. Math. Anal. Appl.179, 1993, 327) concerning the linear growth rate and the behaviour of oscillatory motion.

The purpose of this paper is to investigate the influence of the Soret effect on natural convection in a vertical cavity filled with a binary liquid. The two vertical walls of the cavity are subject to constant fluxes of heat while the two horizontal ones are adiabatic. The analysis deals with the particular situation where the buoyancy forces induced by the thermal and solutal effects are opposing each other and of equal intensity. In the limit of tall enclosures an analytical model is derived on the basis of a parallel flow approximation. The study is completed by a numerical solution of the full governing equations.

A numerical study is carried out on double-diffusive natural convection within a vertical circular porous annulus. Motions are driven by the externally applied constant temperature and concentration differences imposed across the vertical walls of the enclosure. In the formulation of the problem, use is made of the Brinkman extended Darcy model which allows the no-slip boundary condition on a solid wall, to be satisfied. The flow is assumed to be laminar and two dimensional. The density variation is taken into account by the Boussinesq approximation. The control-volume approach is used for solving the governing equations. Solutions for the flow fields, temperature and concentration distributions and Nusselt, Nui and Sherwood, Shi numbers are obtained in terms of the governing parameters of the problem. The effect of both the Darcy number, Da, and the radius ratio, κ, on Nui and Shi is found to be significant. Results for a pure viscous fluid and a Darcy (densely packed) porous medium emerge from the present model as limiting cases.

The onset of double diffusive convective flows in an inclined fluid layer, when constant fluxes of heat and mass are applied on the two oposing boundaries of the layer, is investigated. The case of equal and opposing buoyancy forces is considered. A numerical linear stability theory is used to determine the critical Rayleigh number for the onset of convection. The existence of a subcritical Rayleigh number, for the onset of finite amplitude convection, is demonstrated on the basis of the parallel flow approximation.

Two dimensional, thermosolutal convection in molten Pb-Sn housed within a square enclosure, is simulated using a spectral method. The predictions illustrate two regimes of liquid convection which develop in response to simultaneous horizontal temperature and species concentration gradients applied to the system. Sherwood numbers for the high Schmidt number fluid are quite high, and change markedly depending upon the regime of operation. Heat transfer rates are only moderately affected by convection due to the low Prandtl number of the melt. The predictions at relatively large Rayleigh number and buoyancy ratio display highly oscillatory behavior in a manner consistent with previous experimental observations.

The rate of layer growth during thermosolutal convection in a square enclosure is studied both experimentally and analytically. The thermosolutal convection is induced by the combined thermal and solutal buoyancies which either augment or oppose each other. The layer growth process is attributable to the vertical solutal boundary flow which accumulates and stratifies along the horizontal wall. A mathematical model, based on the filling box process used to predict the layer growth rate, is developed. The solutal boundary flow that enters or exits the stratified layer is calculated by using an integral solution for natural convection due to combined thermal and solutal buoyancies along a vertical plate. Comparison of layer growth rate between the prediction and the data is made, and the agreement is excellent if the actual thickness of the solutal boundary layer flow at the inlet and the outlet of the stratified layer can be accurately determined. The actual thickness of the solutal boundary layer flow at the inlet and the outlet is found to increase with increasing buoyancy ratio.

Numerical solutions are presented for the problem of thermosolutal convection in a rectangular enclosure subjected to simultaneous heating of the bottom and sidewalls. A steady model based on primitive variables and an unsteady model solved using a high-accuracy numerical scheme are used to study the problem. Effects of the Lewis number, buoyancy ratio, aspect ratio and the Rayleigh number on flow and heat and mass transfer rates are studied. Details of oscillatory solutions and flow bifurcations are presented.

This paper reports the numerical simulations of double diffusive natural convection flows in a binary fluid contained in a two-dimensional enclosure, with imposed horizontal temperature and concentration differences between the vertical walls. The analysis concerns the influence of the different parameters governing the problem in the heat transfer characteristics and on the flow structure. The study is focused on steady-state solutions in the cooperating situation. At high Lewis numbers, numerical simulations show that heat transfer decreases with increasing buoyancy ratios. This phenemenon is analysed on the basis of the flow structure, and a scale analysis is provided to estimate the decrease in heat transfer. Then, the influence of the parameters on the formation of a multicellular flow structure is discussed.

This paper reports the analytical and numerical results on double diffusive natural convection in a binary fluid contained in a two-dimensional enclosure where horizontal temperature and concentration differences are specified. A numerical code, based on a finite volume procedure, and a scaling law approach are used to analyse the influence of the different parameters which characterize those thermosolutal flows: in this first paper, the mass transfer problem is studied in the steady-state. A general mass transfer correlation is proposed, which is valid over a wide range of parameters.

This paper focuses on the effect of natural convection on the accuracy of diffusion coefficient measurements in concentrated liquid alloys. This convective effect is modelled using the concept of effective diffusivity, introduced in a former work for the case of dilute systems. Our computer simulations support the validity of this approach and show that it is very difficult in practice to reach conditions of negligible convective solute transport. The possibility of realizing the measurements of diffusion coefficients in microgravity is also discussed.

We consider the dynamical system associated with two-dimensional thermosolutal convection and show the existence of a maximal attractor. We derive estimates on the Lyapunov exponents, the Hausdorff and fractal dimension of the attractor.

The thermosolutal instability in porous medium in a viscous, conducting, incompressible and heterogeneous fluid layer confined between two free boundaries is studied under linear theory, when the whole system is immersed in a uniform vertical magnetic field. The existence of stationary, oscillatory and non-oscillatory modes are discussed. A variational formulation of the problem is also established and numerical computations are done to study the influence of various parameters. It is found that heterogeneity has a destabilizing effect and the Chandrasekhar number and porosity parameter have stabilizing effects on the system.

The effect of relaxing imposed symmetries is investigated in two-dimensional numerical experiments. For rolls constrained to have point symmetry about their axes bifurcations leading to chaos persist as the mesh is refined. When the constraints is relaxed symmetry-breaking gives rise to mixed-mode periodic solutions with more complicated spatial structure. As the aspect ratio is increased further bifurcations lead to spatiotemporal chaos.

Observations of the macroscopic shape of the solid-liquid interface and radial solute segregation have been carried out on quenched Pb-30 wt%TI alloys which were grown upwards, that is, for the case for which thermosolutal convection may take place. The solidification front exhibits a structure which is composed of solid plateaux separated by liquid channels and whose amplitude presents a maximum when plotted versus growth rate, which suggests a concomitant maximum in the level of thermosolutal convection. The convective flow adjacent to the solidification front has then been deduced.

The influence of laminar thermosolutal convection on solute segregation and heat transfer in the floating zone crystal growth of a dilute binary alloy is considered. The solute gradient is assumed to be destabilizing, and the thermal gradient is assumed perpendicular to it. The evolution of non-linear states from the critical points predicted by linear theory is studied. For vanishing thermal Rayleigh numbers, two solutions, corresponding to upflow and downflow at the meniscus, are obtained. The presence of side heating establishes upflow as the preferential direction; the downflow solution is found to persist for low thermal Rayleigh numbers. Flow and heat transfer for high thermal Rayleigh numbers are also computed.

Lead-tin-telluride has been grown in a thermally stable mode (solutally unstable) and in a solutally stable (thermally unstable) mode in a Bridgman configuration. Significant differences in the crystal morphology and the compositional homogeneity have been found between the two configurations. In addition, for the solutally stable configuration, evidence has been found that the flow characteristics in the melt change drastically during the course of the run.

Natural convection of salt water solution due to combined horizontal temperature and concentration gradients in studies experimentally in a rectangular enclosure of aspect ratio 0.2. Both gradients are imposed in such a way that their effects are either cooperating or opposing. Flow visualization shows two types of global flow patterns in the enclosure—a unicell flow pattern and a multilayer flow pattern—and the latter is observed to appear in a certain range of Ra T -Ra S plane. Formation and growth of a layered flow structure with time is observed and described. More layers appear in the case of opposing thermal and solutal buoyancy forces than the cooperating case. Due to the double-diffusive nature of heat and salt, interesting temperature and concentration profiles are obtained in the multilayer flow regime.

When a solution containing a non-linear profile of solute concentration is heated from below, convection can occur in layers. Observations of this phenomenon are presented, and these are compared with calculations based on a simple model. It is concluded that the model contains the essential features of the process.

Summary The conservation equations for binary nanofluid system are solved to get eigenvalue equation. Valid approximations are used to simplify the complex expressions for analytical results. Oscillatory motions are not possible and hence mode of convection is invariably through stationary mode. Numerical computations are carried out for water based nanofluids to analyze solutal effects on the stability of the system using the software Mathematica. Higher conductivity and density of copper makes it more stable than alumina.

Corrigendum to this publication: http://hal.archives-ouvertes.fr/hal-00528029; International audience; This call describes a numerical comparison exercise for the simulation of ingot solidification of binary metallic alloys. Two main steps are proposed, which may be treated independently: 1. The simulation of the full solidification process. First a specified 'minimal' solidification model is used and the contributors are provided with the corresponding sets of equations. The objective is to verify the agreement of the numerical solutions obtained by different contributors. Then different physical solidification models may be compared to check the features that allow for the best possible prediction of the physical phenomena. 2. A separate preliminary exercise is also proposed to the contributors, only concerned with the convective problem in the absence of solidification, in conditions close to those met in solidification processes. Two problems are considered for the case of laminar natural convection: transient thermal convection for a pure liquid metal with a Prandtl number on the order of 10(-2), and double-diffusive convection in an enclosure for a liquid binary metallic mixture with a Prandtl number on the order of 10(-2) and a Lewis number on the order of 10(4).

An extension of a slightly compressible flow model to double-diffusive convection of a binary mixture of ideal gases enclosed in a cavity is presented. The problem formulation is based on a low Mach number approximation and the impermeable surface assumption is not invoked. The main objectives of this paper are the statement of a new problem formulation, and the analysis of some significant results showing the influence of density variations on transient solutions for pure thermal or pure solutal convection. At steady-state, it is shown that the heat and mass transfer analogy may be applied for non-dilute mixtures at parameter ranges larger than those usually considered. To cite this article: H. Sun, G. Lauriat, C. R. Mecanique 337 (2009).

Experiments were carried out to test the existence of solutal or thermosolutal convection and its effect on the solute distribution in the solid. Dilute Al-Mg binary alloys were solidified unidirectionally upward in the planar growth range and the temperature variations in the melt ahead of the solidification front were measured by differential thermal analysis (DTA). The Solidification interface was revealed by quenching and the solute distribution in the solid was analyzed. The almost flat solidification interface at the macroscopic scale indicated that thermal convection due to a radial temperature gradient was minimal. Temperature fluctuations in the melt were very small. However, the concentration profile along the axis of the crystal showed macroscopic segregation due to convection, which was more pronounced at slower growth rates and at higher concentration. The character of the convection is discussed in terms of solutal and thermosolutal convection, and a convection criterion is proposed which combines the solutal Rayleigh number, Ra s , and the thermal Rayleigh number, Ra T .