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This article deals with the effects of thermophoresis on thermosolutal mixed convection in a vertical wavy porous cavity. Three distinct flow features are chosen by changing the directions of the moving horizontal borders. A nonuniform temperature, as well as concentration distributions, are introduced to the bottom wall while keeping cold temperature and low concentration for the other walls. The governing equations are modeled to describe thermosolutal phenomena. These equations are solved by reconstructing a recently developed compact scheme. Furthermore, the significance of well-defined parameters influencing the fluid rotation and thermosolutal transfer, namely Richardson number ( 0.01 ≤ Ri ≤ 100 ), Buoyancy ratio (N = 1), number of undulations ( 0 ≤ d ≤ 2 ) of the wavy walls, Lewis number ( 1 ≤ Le ≤ 10 ), thermophoretic coefficient ( 0.0 ≤ K T ≤ 1.0 ), Darcy number ( 0.001 ≤ Da ≤ 0.1 ), and porosity of the porous medium ( 0.1 ≤ ε ≤ 1.0 ) are performed generously. Results are displayed in terms of streamlines, isotherms, iso-concentrations, Nusselt, and Sherwood numbers to evoke the thermosolutal phenomena for various physical parameters. It is observed that thermophoresis is effective in certain circumstances. In addition, heat and mass transfer enhancement is noted with decreasing Richardson number while heat and mass transfer reduction is noted with increasing undulation number of the wavy walls. We have noticed that for the change in Le from 1 to 10, heat transfer is diminished up to 2.91% in Case-1, 3.03% in Case-2, and 2.20% in Case-3 while mass transfer is enhanced by 162.85% in Case-1, 164.05% in Case-2, and 87.03% in Case-3. [ABSTRACT FROM AUTHOR]

We investigate the fully nonlinear model for convection in a Darcy porous material where the diffusion is of anomalous type as recently proposed by Barletta. The fully nonlinear model is analysed but we allow for variable gravity or penetrative convection effects which result in spatially dependent coefficients. This spatial dependence usually requires numerical solution even in the linearized case. In this work, we demonstrate that regardless of the size of the Rayleigh number, the perturbation solution will decay exponentially in time for the superdiffusion case. In addition, we establish a similar result for convection in a bidisperse porous medium where both macro- and microporosity effects are present. Moreover, we demonstrate a similar result for thermosolutal convection. [ABSTRACT FROM AUTHOR]

This work presents nonlinear and linear analyses of the rotating Navier–Stokes–Voigt fluid layer that is simultaneously heated and soluted from below, considering different boundary surfaces. The energy method is used to form the eigenvalue problem for nonlinear analysis, whereas the normal mode analysis is used for the linear analysis. The Rayleigh number is numerically calculated by employing the Galerkin technique. Both nonlinear and linear analyses yield the same Rayleigh number, indicating the absence of subcritical regions and implying global stability. The Kelvin–Voigt parameter doesn't affect the Rayleigh number for stationary convection. However, the crucial role of this parameter is established through an energy argument. The presence of rotation, Kelvin–Voigt parameter, and solute gradient give rise to oscillatory modes. Also, the effects of rotation and solute gradient are stabilizing on the system, whereas the stabilizing effect of the Kelvin–Voigt parameter becomes evident when convection exhibits an oscillatory behavior. [ABSTRACT FROM AUTHOR]

Lattice Boltzmann simulations of double-diffusive convection in a rectangular cavity differentially heated and salted are conducted in the presence of internal heat generation and Dufour and Soret effects. The study is performed for an aspect ratio A = 2, an external Rayleigh number RaE = 105, a Lewis number Le = 2, a buoyancy ratio N = -1, and a Prandtl number Pr = 0.71: The study focuses mainly on the effect of Dufour number (Du = 0 and 0.5) to measure its impact on the ranges of steady/unsteady solutions reported in a previous study for the Soret parameter ranging from -0.5 to 0.5 and internal to external Rayleigh numbers ratio ranging from 0 to 80. Compared to Du = 0, the results obtained show that the presence of Dufour mechanism with a positive value leads to a destabilizing effect by widening the ranges of instability observed for the considered values of Sr: The combinations (Sr, Du) the most stabilizing/destabilizing are identified and discussed in this study. [ABSTRACT FROM AUTHOR]

This paper presents a numerical study of the linear and non-linear stability of thermosolutal convection within a porous medium saturated by a non-Newtonian binary fluid. The power-law model is utilized for modeling the behavior of the working medium. The given statement implies that the horizontal boundaries experience thermal and solutal flow rates, whereas the vertical walls are impermeable and thermally isolated. The relevant factors that govern the problem being investigated are the Rayleigh number, RT, the power-law index, n, the cavity aspect ratio, A, the Lewis number, Le, and the buoyancy ratio, N. An analytical solution is obtained for shallow enclosures (A >> 1) using the parallel flow approximation and a modified form of the Darcy equation. By solving the entire set of governing equations, a numerical investigation of the same phenomenon was conducted. One of the most intriguing discoveries from this research is that it identifies a bi-stability phenomenon, this particular phenomenon signifies the existence of two stable solutions. The results obtained from both methods demonstrate a good level of agreement across a diverse range of these governing parameters. [ABSTRACT FROM AUTHOR]

AbstractThermosolutal natural convection originates in a fluid when the density changes due to the presence of two distinct components with varying rates of diffusion. This convection is driven by buoyancy resulting from concurrent temperature and concentration gradients. This study focuses on the thermosolutal free convection flow within a parallelogrammic enclosure. The functioning fluid under consideration is non-Newtonian and formulated by the viscosity model (power-law). The top and bottom walls of the chamber are adiabatic and are assumed to be impermeable. The temperature and concentration on the left side are maintained at a high level, while the conditions on the right side are cold and low. The controlling parameters for the present case include Rayleigh number (105≤Ra≤106), Lewis number (Le = 5, 10), Prandtl number (Pr = 6.2), buoyancy ratio (N = 0.8, −0.8) along with power-law index (n = 0.6, 1, 1.4). The results obtained from this numerical study are validated with existing results and indicate the correctness of the code. The output of the study is shown in terms of velocity and temperature profiles, streamlines, isotherms, and iso-concentrations, the average rate of heat and mass transmission in terms of the average Nusselt number (Nu¯) and the average Sherwood number (Sh¯). The correlation equation derived from the RSM approach demonstrates the relationship between the output responses and the input parameters. It concludes that n negatively affects both heat and mass transfer, while Ra affects positively. The findings of this study could be helpful for understanding of thermosolutal natural convection behavior of non-Newtonian (power-law) fluid in an enclosure and therefore accelerate the industrial application of such fluid in the relevant field, such as HVAC (heating, ventilation, and air conditioning) system. [ABSTRACT FROM AUTHOR]

In a composite system possessing rigid‐rigid boundaries, the significance of thermal diffusion on the onset of triple diffusive convection is analyzed. The Darcy–Brinkman–Rayleigh–Benard model is employed to model the porous media. The regular perturbation methodology has been used to solve the governing equations of the composite system with the Boussinesq approximation. The critical thermal Rayleigh number, which determines the stability of the system, is estimated analytically. A graphical assessment of the impact of multiple physical attributes on the stability of the system is made. It is observed that the Soret parameters and solute Rayleigh numbers have a stabilizing effect, whereas the Darcy number exhibits a destabilizing effect upon the onset of triple diffusive convection in the composite system. [ABSTRACT FROM AUTHOR]

This study investigates the stability analysis of Rayleigh-Bénard configuration for a viscoelastic fluid subject to thermorheological effects, using the D 2 -Chebyshev- τ method. The fluid is modeled as a third-order viscoelastic fluid. This study accentuates how salting the fluid layer affects the thresholds for the onset of instability in a fluid of third order encompassing physically realistic rigid boundaries. The dynamic model incorporates advection-diffusion of temperature and solute concentration and a modified Navier–Stokes equation. We determine instability thresholds for the complex non-Newtonian fluid by analyzing the linear stability of the steady-state conduction solution. Our analysis proves the strong form of the principle of exchange of stabilities, demonstrating that convective motions can only occur through stationary motion. Additionally, a nonlinear stability analysis using the energy method is performed, deriving an unconditional nonlinear stability criterion. The results provide a comprehensive understanding of how variable viscosity and viscoelasticity impact system stability. Both the viscosity parameter and the third-grade fluid parameter exhibit stabilizing effects. Notably, we observe a discrepancy between the linear and global nonlinear stability results, indicating the presence of a subcritical instability region. This study contributes to the understanding of complex fluid dynamics in non-linear mechanical systems, with potential applications in various industrial and natural processes. • A system of differential equations models the double-diffusive convection in viscoelastic fluid. • The effect of temperature-dependent viscosity on the onset of convection is examined. • The strong form of principle of exchange of stabilities is proved. • Linear analysis determines thresholds above which steady solutions become unstable. • Nonlinear analysis proves the total perturbed energy of the system decays asymptotically. [ABSTRACT FROM AUTHOR]

The two-temperature model of local thermal nonequilibrium (LTNE) is utilized to investigate a weakly nonlinear stability of thermosolutal convection in an Oldroyd-B fluid-saturated anisotropic porous layer. The anisotropies in permeability, thermal conductivities of the porous medium, and solutal diffusivity are accounted for by second-order tensors with their principal directions coinciding with the horizontal and vertical coordinate axes. A modified Darcy-Oldroyd model is employed to describe the flow in a porous medium bounded by impermeable plane walls with uniform and unequal temperatures as well as solute concentrations. The cubic-Landau equations are derived in the neighborhood of stationary and oscillatory onset using a modified perturbation approach and the stability of bifurcating equilibrium solutions is discussed. The advantage is taken to present some additional results on the linear instability aspects as well. It is manifested that the solutal anisotropy parameter also plays a decisive role on the instability characteristics of the system. It is found that the stationary bifurcating solution transforms from supercritical to subcritical while the oscillatory bifurcating solution transforms from supercritical to subcritical and revert to supercritical. The Nusselt and Sherwood numbers are used to examine the influence of LTNE and viscoelastic parameters on heat and mass transfer, respectively. The results of Maxwell fluid are outlined as a particular case from this study. [ABSTRACT FROM AUTHOR]

• Heat and mass transfer in a SGSP are investigated experimentally and numerically. • 2D numerical model developed can accurately predict and describe the transfers. • Solar flux influence significantly on the SGSP temperature. • Solar flux influence is insignificant on the SGSP saline concentration. In this paper, experimental and numerical investigations of heat and mass transfer in a Salt Gradient Solar Pond (SGSP) are presented for the first time in the SGSP studies. The experimental approach consists of a thermally insulated parallelipedic tank filled with saline water and exposed under a solar simulator. The numerical model is developed in Fortran 95 basing on 2D Navier-Stokes, thermal energy and diffusion equations with an appropriate treatment of the SGSP boundary conditions. These equations are solved using the implicit Finite Volume Method and the SIMPLE algorithm. The study covers a 10-day period and involves two different values of the sun-like average radiation: 150 W/m2 (case 1) and 500 W/m2 (case 2). Comparisons between experimental and numerical results, for case 1, show that the numerical model developed can accurately predict and describe the heat and mass transfer with the development of the thermosolutal convection in the SGSP. Moreover, the time required to form the linear variation of the SGSP temperature is shorter for case 2 than for case 1 with a difference of 89 h, during which the LCZ temperature increases from 20 °C to 28.82 °C for case 1 and to 48.03 °C for case 2. Therefore, the energy storage takes place after a few days of the SGSP operation. In addition, the salt concentration of the LCZ decreases by about 0.83 kg/m3 and increases in the UCZ by about 1 kg/m3 for both cases. Thus, the sun-like average radiation does not have a significant influence on the mass transfer in the SGSP unlike on the heat transfer. [ABSTRACT FROM AUTHOR]

A full three-dimensional (3D) numerical simulation of solidification was carried out for a benchmark experiment on a binary Sn-10 wt. %Pb alloy. The experiment process involves a melting stage, a first holding stage at constant temperature with electromagnetic stirring, setting a mean horizontal temperature difference (second holding stage), and finally solidification stage by decreasing the temperature under a imposed horizontal temperature gradient. The numerical model is applied only to investigate the solidification stage and compared with the measured temperature fields and macrosegregation obtained from the postmortem analysis. A columnar numerical model based on a two-phase volume-averaged approach is used for the numerical simulation, accounting for thermosolutal convection and assuming perfect microscopic mixing (lever rule) in the mushy zone. It demonstrates that such a model is able to predict stratification in the solute from the liquid phase and mushy zone during the solidification. The effect of the sedimentation on macrosegregations and channel segregation or freckles which develop during the solidification stage is also predicted by the model and compared with experimental data. Emphasis is given to the main factors that have a direct effect on the development and morphology of segregated channels, namely, the remelting phenomenon, dendrite fragmentation, and the solidification front instabilities. [ABSTRACT FROM AUTHOR]

This article concerns the behavior of the initial layer appearing at large Prandtl number in Boussinesq equations with the ill initial data. By using the asymptotic expansion methods of singular perturbation theory, we establish an approximate solution and the rate of convergence as the Prandtl number tends to infinity. Our results improve the existing ones concerning thermosolutal convection. [ABSTRACT FROM AUTHOR]

We present a model for convection in a Navier–Stokes–Voigt fluid when the layer is heated from below and simultaneously salted from below, the thermosolutal convection problem. Instability thresholds are calculated for thermal convection with a dissolved salt field in a complex viscoelastic fluid of Navier–Stokes–Voigt type. The Kelvin–Voigt parameter is seen to play a very important role in acting as a stabilizing agent when the convection is of oscillatory type. The quantitative size of this effect is displayed. Nonlinear stability is also discussed, and it is briefly indicated how the global nonlinear stability limit may be increased, although there still remains a region of potential sub-critical instability, especially when the Kelvin–Voigt parameter increases. [ABSTRACT FROM AUTHOR]

In this article, the behavior of the dissolution process of silicon (Si) in molten germanium (Ge) was mathematically modeled and examined numerically. The transport phenomena during this process were modeled using the axisymmetric model (2D) and the equations of the model were solved numerically using the COMSOL multiphysics package. The numerical simulations were carried out exclusively to explain the experimental observations (carried out previously) on the effect of the presence of a free surface on the transport and the mixture of the solute and the shape of the interface of dissolution. The dissolution experimental work used a configuration in which the sample (source Si) was located at the bottom to mimic for instance the process in the melt replenishment Czochralski growth system. For the samples processed in the dissolution experiments, the dissolved heights of silicon were measured. This measurement gives the quantity of silicon dissolved in the experimental times and must be directly linked to the quantity of silicon transported in the melt. Measurement of the silicon composition profiles in the samples was carried out (in the experimental work previously carried out) using the energy dispersive X-ray spectrometer technique. The present numerical results confirm and complement the experimental observations and show that the effect indicates a tendency to more mixing and the presence of several complex convective melt flow regimes leading to rapid chaotic mixing with the presence of a free surface on the melt. In addition, the numerical and the experimental results reveal that it is necessary to take into account the geometry of the crystal growth system when the source Si material is located at the bottom. Indeed, the dissolution of silicon from the bottom of the melt in the presence of a free surface will occur much faster. This however may lead to instability and crystal growth with nonuniform composition. [ABSTRACT FROM AUTHOR]

Analysis of fluid flow in a bidispersive medium with reactions is explored. The Darcy model is used in the momentum equation with the density being a linear function in temperature and salt concentration. Two cases are regarded which are heated below and salted above system and heated and salted below system. It is presumed that the equilibrium solute concentration follows a linear relationship to temperature. A linear instability and nonlinear stability theories are conducted, and from the outputs of the analysis, the Chebyshev collocation approach is used to solve the resulting eigenvalue systems. The effects of the chemical reaction and other parameters on temperature Rayleigh number are described graphically. [ABSTRACT FROM AUTHOR]

The main purpose of our work is to show the impact of some pertinent parameters; such Lewis and solutal Rayleigh numbers as well as the buoyancy and the aspect ratios; on a viscoplastic materials’ double-diffusive convection which occurs within an SiC annulus; contained between a cold “and less concentric” outer circular cylinder and a hot “and concentric” inner one, to come out with innovative correlations what predict the mean transfer rates into such annulus. To do so, the physical model for the momentum conservation equation is made using the Brinkman extension of the classical Darcy equation. The set of coupled equations is solved using the finite volume method and the SIMPLER algorithm. To handle the cylinder shape in Cartesian coordinates; the Cartesian Cut-Cell approach was adopted. Subsequently, inclusive correlations of overall transfers within such viscoplastic porous annulus are set forth as a function of the governing parameters; which foresee with ±2 to ±3% the numerical predictions. Noted that the validity of the computing code was ascertained by comparing its results with experimental data and numerical ones; already available in the literature. [ABSTRACT FROM AUTHOR]

Through our paper, thermosolutal convection of viscoplastic materials 'so called Bingham plastics' which occurs into a porous matrix with an inner pollutant source has been treated numerically, in the aim to light out the impact of some relevant parameters; such Lewis and porous Rayleigh numbers; as well as the buoyancy ratio and the source dimension one; on a such conjugate phenomenon. To do so, the physical model for the momentum conservation equations is made using the Brinkman extension of the classical Darcy equation. The set of coupled equations is solved using the finite volume method and the SIMPLER algorithm. The heat and solute source within the porous space has taken a circular shape. Simply said, our pollutant source is a transport pipe which presented in 2D. To handle the latter in Cartesian Coordinates; the Cartesian Cut-Cell approach was adopted. After a careful treatment of such double-diffusive convection within the Bingham-porous space; powerful expressions that expect the mean transfer rates in such industrial geometry are set forth as a function of the governing parameters. These correlations, which predicted with '3% the numerical results, may count as a complement to previous Newtonian-fluid researches. It is to note that the validity of the computing code was ascertained by comparing our results with experimental data and numerical ones, already available in the literature. [ABSTRACT FROM AUTHOR]

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 – Bénard 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. [ABSTRACT FROM AUTHOR]

Unsteady coupled natural convective heat and mass transfer with liquid to solid phase change is described to analyze the macrosegregation of a Pb–Sn alloy in a rectangular two-dimensional cavity. The effect of cooling direction is investigated, by comparing the numerical simulation of solidification in rectangular molds with one vertical side cooled and with bottom wall cooling, both with the cold wall temperature changing with time. Fluid mechanics, heat and mass transfer coupled equations with unsteady, convective, diffusion and sources terms are solved with the Control Volume Finite Element Method (CVFEM) in cavities with aspect ratios A = H / B = 1 / 3 , 0.6 and 3. Results for the evolution of the distribution of streamlines, solidification front, isotherms and iso-concentrations are obtained with a code developed to characterize coupled fluid, heat and mass transport phenomena. Complex physical processes such as thermo-solutal convection during solidification, freckle formation and the final pattern of macrosegregation are described in terms of the cavity aspect ratio by the mathematical model. A good description of freckle formation and its relation to direction of cooling and aspect ratio has been obtained by the transient-convective–diffusive mathematical model solved by CVFEM. [ABSTRACT FROM AUTHOR]

• LBM simulation of thermosolutal convection in a SGSP enclosure with porous layer. • Cross-diffusive effects on thermosolutal convection in this SGSP enclosure are analyzed. • Attaching a porous layer at the bottom inhibits the thermal and solutal transfer processes. • Du enhances both thermal and solutal transfer processes, while Sr boosts the solutal one. • Internal heat generation reduces the positive influence of Soret and Dufour effect. Salt gradient solar pond (SGSP) is a promising alternative for low-temperature industrial applications. Thermal and solutal transfer processes occur simultaneously at SGSP, which induce the cross-diffusive effects and make the internal flow more complicated. In the present work, thermosolutal convection under cross-diffusive effects inside a simplified SGSP enclosure attached with a porous layer are investigated through Lattice Boltzmann Method (LBM). Influences of internal heat generation, buoyancy ratio, thickness and permeability of porous layer combined with Soret and Dufour parameters are considered on the thermal and solutal transport characteristics. Average Nusselt and Sherwood numbers as well as mean temperature and concentration are presented to analyze the thermal and solutal transport processes in the SGSP enclosure. Results indicate that Soret effect mainly boosts the solutal transport process, while Dufour effect can greatly enhance both the thermal and solutal transport processes. Internal heat generation can inhibit the thermal and solutal transport processes and reduce the positive influence of Soret and Dufour effect. Buoyancy ratio can boost thermal and solutal transport processes. In the opposing situations (N < 0), Soret effect weakens the thermal transport process, while Dufour effect is reverse. Furthermore, attaching a porous layer at the bottom wall can suppress the thermal and solutal transfer processes, and then increases the thermal storage capacity. Permeability and thickness of porous layer have a significant influence on thermal and solutal transport processes. Present study is beneficial to the thermal storage enhancement and the future development of SGSP. [Display omitted] [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]

The thermosolutal convection in Rivlin-Ericksen elastic-viscous fluid in porous medium is considered to include the effect of suspended particles in the presence of uniform magnetic field, uniform rotation and variable gravity field. It is found that for stationary convection, the stable solute gradient has stabilizing effect on the system. Rotation has stabilizing effect as gravity increases upward and destabilizing effect as gravity decreases upward whereas suspended particles have destabilizing effect as gravity increases upward and stabilizing effect as gravity decreases upward. The medium permeability has stabilizing/destabilizing effect depending on the rotation parameter. The magnetic field has a stabilizing effect in the absence of rotation whereas in the presence of rotation it has a stabilizing/destabilizing effect under certain conditions. The principle of exchange of stabilities is satisfied in the absence of magnetic field, rotation and stable solute gradient. The presence of magnetic field, rotation and stable solute gradient introduces oscillatory modes into the system. [ABSTRACT FROM AUTHOR]

We examine the initial layer problem and the infinite Prandtl number limit of the thermosolutal convection, which is applicable to magma chambers. We derive the effective approximating system of the Boussinesq system at large Prandtl number using two time scale approach [M. Holmes,Introduction to Perturbation Methods, Springer, New York, 1995, A. Majda,Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, Vol. 9, New York, American Mathematical Society, Providence, RI, 2003]. We show that the effective approximating system is nothing but the infinite Prandtl number system with initial layer terms. We also show that the solutions of the Boussinesq system converge to solutions of the effective approximating system with the convergence rate ofO(ϵ). [ABSTRACT FROM AUTHOR]

Abstract: This paper presents the results of a numerical study of double-diffusive convection in an inclined rectangular collector filled with two parallel porous layers. Each porous layer is considered homogeneous, isotropic and saturated with the same fluid. The vertical walls of porous cavity are subjected to uniform temperature and concentration whereas the other surfaces are assumed to be adiabatic and impermeable. The set of equations governing the heat and mass coupled problem within the enclosure are solved numerically. The numerical results are presented and analyzed in terms of streamlines, isotherms, isoconcentrations lines and average Nusselt and Sherwood numbers. A scale analysis is used to characterize the effect of the permeability ratio on the heat and mass transfer. It is found that the numerical solutions of the full governing equations are in good agreement with the scaling results for some specific conditions. Optimum operating tilted angle are deduced. [Copyright &y& Elsevier]

Abstract: Freckle formation was directly simulated in three dimensions with a microstructure level model for unidirectional solidified Pb–Sn alloys. The microscale solidification model resolves the complex interaction of solute partition, interdendritic thermosolutal convection, dendrite formation and remelting. These simulations show that it is competition between all off these phenomena at the microstructural level that determines both the initiation and survival of solute channels and, hence, the propensity for freckle formation. The solute enriched interdendritic flow remelts both primary and secondary arms, and can deflect the primaries. This enables solutal channels to become self-sustaining, forming freckles. The microstructural model was then applied to predict the critical Rayleigh number, with excellent agreement with a large range of experimental data, demonstrating its potential for applications to new alloy systems and solidification processes. [Copyright &y& Elsevier]

Thermosolutal convection in a square cavity filled with a binary perfect gas mixture and submitted to an oriented magnetic field taking into account the effect of radiation heat transfer is numerically investigated. The cavity is heated and cooled along the active walls whereas the two other walls are adiabatic and insulated. Entropy generation due to heat and mass transfer, fluid friction and magnetic effect has been determined for laminar flow by solving numerically: The continuity, momentum energy and mass balance equations, using a Control Volume Finite-Element Method. The structure of the studied flows depends on five dimensionless parameters which are: The Grashof number, the buoyancy ratio, the Hartman number, the inclination angle of the magnetic field and the radiation parameter. [ABSTRACT FROM AUTHOR]

Abstract: Two- and three-dimensional numerical studies of double-diffusive natural convection in homogeneous and isotropic porous media, saturated with a binary fluid are analyzed. Top and bottom faces of the enclosures are subject respectively to different but uniform temperatures and concentrations, while its vertical boundaries are considered adiabatic and impermeable to mass transfer. The flow through the medium is governed by the Darcy model. The existence of multiple steady-state solutions is proved for both 2D and 3D models and all the solutions obtained are presented and described. Conjugate effects of the thermal Rayleigh number and the buoyancy ratio on heat transfer, mass transfer and existence range of each flow structure are discussed. Important differences in terms of heat and mass transfer are observed between different solutions. Depending on the governing parameters and the type of solution, the flow structure could be 2D or 3D. It is also found that all the 2D solutions are obtained with the 3D model but the latter has more degrees of freedom allowing a reorganization of the flow not possible in the case of 2D model. For Ra=200 and opposing thermal and species buoyancy forces, there exists a critical value of N (N =−0.6) below which the concentration stabilizes the flow by settling a bulk stratification. In such a case, the corresponding heat and mass transfer processes are ruled by pure diffusion solely (Nu=Sh=1). [Copyright &y& Elsevier]

Abstract: Heat and mass transfer by natural convection coupled to wall surface condensation or evaporation in a two-dimensional cavity subjected to uniform, but time-dependent wall temperatures is investigated numerically. At initial state, the cavity is filled with quiescent humid air at uniform temperature and density. By decreasing the wall temperature, condensation occurs at the four wall surfaces until an equilibrium thermodynamic state is reached. The walls are then heated and evaporation of the liquid water film is considered. Various time variations of the wall temperature were investigated. Since the mass of humid air and average pressure experience large changes during transient regimes, a weakly compressible formulation has been used. The model considers only condensation/evaporation under the thin film approximation. The computations were carried out for temperatures of humid air varying between 300 K and 350 K, and pressure variations around atmospheric pressure. The typical width of the cavities is L = 0.1 m. The results show that very different transient flow structures occur during condensation and evaporation processes. The thickness distributions of the water films condensed at the walls are discussed, and it is shown that the thicknesses reflect the flow structures. The effect of the cavity aspect ratio reveals more complicated results than for convection without phase change at the wall surfaces. [Copyright &y& Elsevier]

Abstract: The combined effects of the double-diffusion and of the viscous dissipation on the convective instability in a fluid-saturated porous medium with a basic horizontal throughflow are investigated. A horizontal porous layer with an impermeable adiabatic lower wall and an impermeable isothermal upper wall is considered. The parallel boundary walls are assumed to have uniform, but unequal, concentrations of the solute. A linear stability analysis is carried out both numerically and by a first-order perturbation method. General disturbances having the form of oblique rolls are considered, reducing either to longitudinal rolls or to transverse rolls in the special cases of roll axes parallel or orthogonal to the basic flow direction, respectively. It is shown that the combined effects of viscous dissipation and mass diffusion may lead to the instability of the basic horizontal flow. Either the longitudinal rolls or the transverse rolls may be the preferred modes of instability depending on the value of the viscous dissipation parameter Ξ. The longitudinal rolls are the most unstable when Ξ <61.86657. [ABSTRACT FROM AUTHOR]

Thermosolutal convection and macrosegregation formation during the solidification of steel ingots are numerically simulated in three dimensions. The simulation is based on a fully coupled model for mass, momentum, energy, and species conservation equations. The interdendritic flow in the mushy zone is governed by Darcy's law, and the permeability term is discretized using an interpolated liquid fraction method. The numerical results for a benchmark test of macrosegregation in a Pb-Sn alloy are compared with experimental data taken from the literature. The present model is applied to simulate the solidification of industrial steel ingots. Preliminary predictions are obtained, including the positive segregation in the hot top, and the conically shaped negative segregation zone at the bottom of the ingot. The predicted variation of the segregation ratio in carbon along the vertical centreline of an ingot is compared with measurements, and generally good agreement is observed. Future attention should be paid to the precision of prediction by considering complex solidification issues, such as the sedimentation of free equiaxed grains and the formation of shrinkage cavity. [ABSTRACT FROM AUTHOR]

Abstract: The influence of the thermosolutal convection of the liquid steel in the solidifying core of a 3.3-ton ingot on the formation of banded mesosegregates is investigated by a multiscale solidification model. We first show how the thermosolutal flow structure in the solidifying core depends on the relation between the interacting thermal and solutal buoyancy forces and the coupling by the phase-change kinetics. We further show that banded mesosegregates are triggered by instabilities of the solidification front, that their location is determined by flow instabilities, and that their “A” or “V” orientation depends on the global direction of the flow circulation. Moreover, the results show that local remelting is not necessary to develop a channel mesosegregate. Destabilization of the mushy zone with local variations of the solidification velocity is sufficient. [Copyright &y& Elsevier]

Abstract: 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 104. [Copyright &y& Elsevier]

Abstract: The present work deals with an analytical solution based on the parallel flow assumption obtained in the case of a vertical porous layer heated and salted from the long vertical sides with uniform fluxes of heat and mass, respectively. The study concerns a specific case for which the buoyancy ratio and the separation coefficient are identical. For this particular situation, the external mass flux is compensated by the Soret effect, which leads to zero concentration gradient on the vertical walls. The problem is first analyzed by solving numerically the full governing equations and the aspect ratio required to satisfy numerically the parallel flow conditions is determined. Analytical solutions for the pseudo-conductive and boundary layer regimes are proposed and discussed. The plane is divided into regions with specific behaviors and the results obtained are presented in terms of boundary layer thickness, heat transfer (Nusselt number), and mass transfer (Sherwood number) versus the main governing parameters. [Copyright &y& Elsevier]

This paper investigates the linear and weakly nonlinear stability of thermosolutal rotating porous convection subject to a lack of thermal equilibrium. Being an example of a triply diffusive fluid system in porous media with multiparameters, some novel results on the linear instability are delineated for a few isolated cases. The otherwise stabilizing factors such as stabilizing solute concentration and rotation are shown to destabilize the system under certain parameters space. Besides, the existence of a completely detached closed convex oscillatory neutral curve from that of the stationary neutral curve is uncovered indicating the requirement of three critical thermal Darcy-Rayleigh numbers to identify the linear instability criteria instead of the usual single critical value. The multivalued nature of the stability boundaries for the case of completely detached oscillatory neutral curves is displayed. The co-dimension-2 points are found through a stability map with the demarcation of stationary and oscillatory convective regions in a plane of Darcy-Taylor and scaled Vadasz numbers. Weakly nonlinear stability theory is carried out for a stationary case by using a modified multi-scale method and thereby the complex Ginzburg-Landau amplitude equation is derived identifying the pitchfork bifurcation. Moreover, heat and mass transport are quantified in terms of average thermal and solute Nusselt numbers, respectively. [ABSTRACT FROM AUTHOR]

Abstract: 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). [Copyright &y& Elsevier]

Abstract: Thermosolutal natural convection induced in a vertical porous layer heated and salted with uniform fluxes is studied analytically and numerically. The study is focused on a specific case where the separation coefficient is identical to the ratio of buoyancy forces. Analytical results, describing both pseudo-conductive and boundary layer regimes, are discussed. Specific behaviour, corresponding to this particular situation, is presented. To cite this article: M. Er-Raki et al., C. R. Mecanique 336 (2008). [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: In the present paper, a thorough analytical and numerical investigation is carried out to examine the double-diffusive convective instability within a horizontal porous layer heated and salted from below. A situation is considered where a lateral perturbing heat flux applied to the system is balanced by the horizontally induced Soret mass flux. The parameters governing this problem are the thermal Rayleigh number, R T; the Lewis Number, Le; the buoyancy ratio, N; the Soret parameter, M; the ratio of the horizontal to vertical heat flux, a; and the aspect ratio, A r; of the porous layer. The present investigation is focused on the situation where , which describes an equilibrium state between the induced Soret mass flux and the imposed heat flux. For this situation, a rest state solution is possible. The analytical solution, derived on the basis of the parallel flow approximation, is validated numerically using a finite difference method by solving the full governing equations. In the M ∗–Le plane , five distinct regions, describing different flow behaviors, are delineated and their location depends on the lateral heat flux parameter a. It is also demonstrated that supercritical and/or subcritical bifurcations are possible for specific ranges of M ∗ and Le. The effect of the lateral heating and the Soret parameter on the critical Rayleigh number, corresponding to the onset of parallel flow convection, is examined. The parameter a affects the flow and the heat transfer considerably, but its effect on the mass transfer is negligible. [Copyright &y& Elsevier]

Abstract: This paper deals with the theoretical investigation of the effect of rotation on a layer of a ferromagnetic fluid permeated with dust particles, heated and soluted from below subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained using a linearized stability theory and a normal mode analysis method. For the case of stationary convection, non-buoyancy magnetization and dust particles have always a destabilizing effect, whereas stable solute gradient and rotation have a stabilizing effect on the onset of instability. The critical wave number and the critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of buoyancy magnetization M 1. Graphs have been plotted by giving numerical values to the parameters, to depict the stability characteristics. It is observed that the critical magnetic thermal Rayleigh number is reduced solely because the heat capacity of clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid heated from below in the absence of dust particles, stable solute gradient and rotation. The oscillatory modes are introduced due to the presence of the dust particles, stable solute gradient and rotation, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained. [Copyright &y& Elsevier]

Abstract: This paper deals with the theoretical investigation of the effect of dust particles on the thermosolutal convection in ferrofluid subjected to a transverse uniform magnetic field. Using linearized stability theory and normal mode analysis, an exact solution is obtained for the case of two free boundaries. For the case of stationary convection, non-buoyancy magnetization and dust particles have a destabilizing effect, whereas stable solute gradient has a stabilizing effect on the onset of instability. The critical wave number and critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of buoyancy magnetic parameter and results are depicted graphically. It is observed that the critical magnetic thermal Rayleigh number is reduced because the heat capacity of clean fluid is supplemented by that of the dust particles. The principle of exchange of stabilities is found to hold good for the ferrofluid heated from below in the absence of dust particles and stable solute gradient. The oscillatory modes are introduced due to the presence of the dust particles and stable solute gradient, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained. The paper also reaffirms the qualitative findings of earlier investigations which are, in fact, limiting cases of the present study. [Copyright &y& Elsevier]

Abstract: A layer of a ferromagnetic fluid permeated with dust particles heated and soluted from below saturating a porous medium is considered in the presence of a transverse uniform magnetic field. Using linearized stability theory and normal mode analysis, an exact solution is obtained for the case of two free boundaries. For the case of stationary convection, medium permeability, dust particles and non-buoyancy magnetization have a destabilizing effect whereas stable solute gradient has a stabilizing effect on the onset of instability. The critical wave number and critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of buoyancy magnetic parameter M 1 and results are depicted graphically. It is observed that the critical magnetic thermal Rayleigh number is reduced because of the specific heat of the dust particles. The principle of exchange of stabilities is found to hold true for a ferromagnetic fluid saturating a porous medium heated from below in the absence of dust particles and stable solute gradient. The oscillatory modes are introduced due to the presence of the dust particles and stable solute gradient, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained. The paper also reaffirms the qualitative findings of earlier investigations which are, in fact, limiting cases of the present study. [Copyright &y& Elsevier]

A three phase volume averaging model has been developed to simulate globular equiaxed solidification, the three phases being liquid, solid and air. The basic conservation equations of mass, momentum and enthalpy have been solved for each phase, and the thermal and mechanical (drag force) interactions among the phases have been considered. Grain nucleation, growth rate (mass exchange), solute partitioning at the liquid/solid interface and solute transport have also been accounted for. Due to its low density, the air phase floats always at the top region, forming a definable air/liquid melt interface, i.e. free surface. By tracking this free surface, the shrinkage cavity in an open casting system can be modelled. As the temperature and concentration dependent density and solidification shrinkage are explicitly included, the thermosolutal convection, together with feeding flow and grain movement can be taken into account. This paper focuses on the model description; preliminary results on a benchmark ingot casting (Al–4Cu) are presented and discussed. [ABSTRACT FROM AUTHOR]

Abstract: The Soret effect on thermal natural convection within a horizontal porous enclosure heated uniformly from below by a constant heat flux is studied analytically and numerically using the Brinkman–extended Darcy model along with the Boussinesq approximation. The governing parameters of the problem are the Darcy–Rayleigh number, R T, the Lewis number, Le, the separation parameter, ϕ, the Darcy number, Da, and the aspect ratio of the enclosure, A r. In the limit of a shallow enclosure, an analytical closed form solution is obtained on the basis of a parallel flow approximation. A numerical study is also conducted to validate the results of the analytical predictions. The thresholds for the onset of stationary and finite amplitude convection are determined explicitly as function of Le, ϕ and Da. The obtained results show that in the ϕ–Le plane there exists three regions that correspond to different parallel flow regimes, namely the subcritical and supercritical flows. The separation parameter has a strong effect on the threshold of instabilities and on the characteristics of heat and mass transfer. The threshold for Hopf bifurcation is obtained on the basis of the linear stability analysis. The wavenumber and the oscillation frequency at the onset of instabilities are null for an infinite extent layer. [Copyright &y& Elsevier]

Abstract: 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 M1 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. [Copyright &y& Elsevier]

Abstract: This paper deals with natural convection driven by combined thermal and solutal buoyancy forces in a binary fluid. The configuration under study is a confined enclosure partially filled with a vertical porous layer. The mathematical description of the problem is based on a one-domain formulation of the conservation equations. The set of numerical results presented here quantitatively shows the influence of the porous layer on the flow structure and on heat and species transfer in the enclosure. The paper is focused on the analysis of the influence of the characteristic parameters governing double diffusive convection, namely the ratios of solutal and thermal parameters: the diffusivities and the buoyancy forces. Heat and mass transfer is analyzed as a function of the permeability of the porous layer. It is shown that the coupling of the flow penetration in the porous layer with the combined buoyancy forces induces a specific behavior of the flow structure and average heat transfer in the enclosure. [Copyright &y& Elsevier]

This paper reports numerical results of two-dimensional double-diffusive natural convection in a square porous cavity partially heated from below while its upper surface is cooled at a constant temperature. The vertical walls of the porous matrix are subjected to a horizontal concentration gradient. The parameters governing the problem are the thermal Rayleigh number (Ra=100 and 200), the Lewis number (Le=0.1, 1 and 10), the buoyancy ratio (-10N10) and the relative position of the heating element with respect to the vertical centerline of the cavity (δ=0 and 0.5). The effect of the governing parameters on fluid characteristics is analyzed. The multiplicity of solutions is explored and the existence of asymmetric bicellular flow is proved when the heated element is shifted towards a vertical boundary (δ=0.5). The solutal buoyancy forces induced by horizontal concentration gradient lead to the elimination of the multiplicity of solutions obtained in pure thermal convection when N reaches some threshold value which depends on Le and Ra. [Copyright &y& Elsevier]

• Freckle formation is found in two regions before and after peritectic reaction. • Effects of both TGZM and G-T effects on dendritic evolution is confirmed. • Less/More freckle in deceleration/acceleration growth is explained. • Rayleigh number R aP which predicts two maximum in peritectic alloy is proposed. [Display omitted] 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 R aP in consideration of both effects was proposed. The prediction of R aP 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 R aP. [ABSTRACT FROM AUTHOR]

• Heat and moisture transportations in a thermal storage wall system have been investigated. • Soret and Dufour effects have been considered in double diffusion convection and entropy generation. • Soret effect has mild influence on heat transfer whereas promoted moisture transportation. • Dufour effect has enhanced heat transfer while weakened moisture transfer. • Heatlines and masslines taking Soret and Dufour effects into consideration have been obtained. An analysis of thermosolutal convection and entropy generation considering Soret and Dufour effects inside an inclined rectangular enclosure attached with porous wall has been investigated. The physical parameters are in the ranges such as buoyancy ratio (-10 ≤ N ≤ 10), permeability (10−9 ≤ Da ≤ 10−1) and thickness (0 ≤ d ≤ 1.0) of porous wall, enclosure inclination angle (0° ≤ Φ ≤ 90°), Soret number (0 ≤ Sr ≤ 1.5) and Dufour number (0 ≤ Du ≤ 1.5). Heatlines and masslines taking Soret and Dufour effects into account are obtained as effective tools that visualizes the heat and species transported paths. Results demonstrate that Soret number has mild impact on heat transfer rate and promotes moisture transportation, while increasing Dufour number could boot heat transfer rate and reduce moisture transfer rate. Furthermore, increasing Soret numbers enhances entropy generation induced by heat transfer and fluid friction, while entropy generation induced by fluid friction, heat and moisture transfer is dropped with stronger Dufour effect. Present work can benefit the heat and moisture transfer performance and entropy generation minimization of thermal storage wall system. [Display omitted] [ABSTRACT FROM AUTHOR]

Numerous investigations on the macrosegregation and channel segregation formation have been reported during eutectic solidification. However, the current analyses are focused on the non-faceted dendritic mushy zone while the faceted mushy zone during solidification has not been reported before. In this work, the macrosegretion and channel segregation formation are studied in directional solidification of Sn-15at.%Ni hypereutectic alloy where the solidified Ni 3 Sn 4 phase exhibits faceted lath-type growth morphology. Similar to previous works on non-faceted mushy zone, more channels can be witnessed in the (Liquid + Ni 3 Sn 4) mushy zone at lower growth velocities. More obvious macrosegregation can be formed in samples at lower growth velocities where the thermosolutal convection is more severe. However, distinct from previous reports, the effective solute distribution coefficient is not constant in the faceted mushy zone of this alloy. Besides, the experimental results demonstrates that the convection can reduce the interspacing of Ni 3 Sn 4 phase during directional solidification. In addition, the thermosolutal convection has more significant influence on the interspacing of Ni 3 Sn 4 phase than the solutal buildup at the tip of Ni 3 Sn 4 phase. [Display omitted] • More freckles/channels is found at lower growth velocities in faceted mushy zone. • Freckle/channel is described by R a when secondary branch is not considered.. • The effective distribution coefficient is not constant in the faceted mushy zone. • Convection in mushy zone is more important than solutal buildup at tip. [ABSTRACT FROM AUTHOR]