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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]

The current work establishes research into the non-Newtonian Casson fluid's linear and weakly non-linear stability analysis under the influences of internal heating and gravitational modulation. The study is confined to stationary convection, where marginal stability is determined by the critical Rayleigh number, which is derived from linear stability analysis. The marginal stability curves are plotted to observe the onset of convection due to different parameters that exist in the problem. Heat and mass transfer are measured in terms of the Nusselt number (Nu) and Sherwood number (Sh), respectively, in the non-linear stability analysis. These measurements are based on the Ginzburg-Landau (GL) equation. One of the main outcomes of this stability analysis is that the internal Rayleigh number and Casson parameter behave similarly for mass transport in a fluid whereas it operates oppositely for heat transfer in a fluid. [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, , the power-law index, , the cavity aspect ratio, , the Lewis number, , and the buoyancy ratio, . An analytical solution is obtained for shallow enclosures ( ) 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.

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]

A linear stability analysis is carried out to investigate the onset of thermosolutal convection in a fluid layer overlying a fluid-saturated porous layer under the high-frequency small-amplitude translational vibration with different direction in the gravitational field. Distinct temperatures and concentrations are applied at the external boundaries of the two-layered fluid-porous domain in such a way that the buoyancy ratio (the ratio of density drop due to concentration difference to that due to temperature difference) has a positive value. The numerical calculations show that transverse (vertical) vibration suppresses convection by delaying its onset when the domain is heated from below. There is a jump-like transition from local to large-scale convective regimes with intensifying vibration. In the case of longitudinal (horizontal) vibration the convection onset value varies non-monotonically: it increases initially, reaches a maximum, and then decreases. The noticeable enhancement of convection is observed at the vibrational Rayleigh-Darcy number close to its value in weightlessness. The longitudinal vibration in contrast to the transverse one is additionally capable of creating convection when the two-layered domain is heated from above. In such a situation the flow has exclusively thermosolutal vibrational nature. With strengthening longitudinal vibration, the critical flow patterns replace each other in the following order: local flows – large-scale flows – long-wave flows with the system of vertically ordered vortexes – “super-shortwave” flows. If one applies at least a small concentration difference across the layers, all of the mentioned vibration effects manifest themselves at less temperature difference and vibration acceleration than they are in the case of pure thermal convection. The thermal and concentration density gradients reinforce each other at positive buoyancy ratios, so heat and mass transfer become most effective. [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]

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 present paper reports on an analytical and numerical study of combined Soret and Dufour effects on thermosolutal convection in a horizontal porous cavity saturated with an electrically conducting binary fluid under a magnetic field. The horizontal walls of the system are subject to vertical uniform fluxes of heat and mass, whereas the vertical walls are assumed to be adiabatic and impermeable. The main governing parameters of the problem are the Rayleigh, the Hartmann, the Soret, the Dufour and the Lewis numbers, the buoyancy ratio, the enclosure aspect ratio, and the normalized porosity of the porous medium. An asymptotic parallel flow approximation is applied to determine the onset of subcritical nonlinear convection. In addition, a linear stability analysis is performed to predict explicitly the thresholds for the onset of stationary, overstable and oscillatory convection, and the Hopf bifurcation as functions of the governing parameters. The combined effect of a magnetic field, Soret and Dufour parameters have a noticeable influence on the intensity of the convective flow, the heat and mass transfer rates, and the thresholds of linear convection. It is found that the imposition of a magnetic field delays the onset of convection and its intensification can lead to the total suppression of the convective currents. The heat transfer rate increases with the Dufour number and decreases with the Soret number and vice versa for the mass transfer rate. [ABSTRACT FROM AUTHOR]

The thermosolutal stability of a layer of the Rivlin-Ericksen fluid in a porous medium is considered under varying gravity conditions. It is found that for stationary convection, medium permeability and suspended particles have a destabilizing/stabilizing effect when gravity increases/decreases. The stable solute gradient has a stabilizing effect on the system.

The present paper reports on an analytical and numerical study of combined Soret and Dufour effects on thermosolutal convection in a horizontal porous cavity saturated with an electrically conducting binary fluid under a magnetic field. The horizontal walls of the system are subject to vertical uniform fluxes of heat and mass, whereas the vertical walls are assumed to be adiabatic and impermeable. The main governing parameters of the problem are the Rayleigh, the Hartmann, the Soret, the Dufour and the Lewis numbers, the buoyancy ratio, the enclosure aspect ratio, and the normalized porosity of the porous medium. An asymptotic parallel flow approximation is applied to determine the onset of subcritical nonlinear convection. In addition, a linear stability analysis is performed to predict explicitly the thresholds for the onset of stationary, overstable and oscillatory convection, and the Hopf bifurcation as functions of the governing parameters. The combined effect of a magnetic field, Soret and Dufour parameters have a noticeable influence on the intensity of the convective flow, the heat and mass transfer rates, and the thresholds of linear convection. It is found that the imposition of a magnetic field delays the onset of convection and its intensification can lead to the total suppression of the convective currents. The heat transfer rate increases with the Dufour number and decreases with the Soret number and vice versa for the mass transfer rate.

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]

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 BVP4 c . 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. [ABSTRACT FROM AUTHOR]

Formation of spurious grains during directional solidification (DS) of Al-7 wt.% Si and Al-19 wt.% Cu alloys through an abrupt increase in cross-sectional area has been examined by experiments and by numerical simulations. Stray grains were observed in the Al-19 wt.% Cu samples and almost none in the Al-7 wt.% Si. The locations of the stray grains correlate well where numerical solutions indicate the solute-rich melt to be flowing up the thermal gradient faster than the isotherm velocity. It is proposed that the spurious grain formation occurred by fragmentation of slender tertiary dendrite arms was enhanced by thermosolutal convection. In Al-7 wt.% Si, the dendrite fragments sink in the surrounding melt and get trapped in the dendritic array growing around them, and therefore they do not grow further. In the Al-19 wt.% Cu alloy, on the other hand, the dendrite fragments float in the surrounding melt and some find conducive thermal conditions for further growth and become stray grains. [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]

The thermosolutal convection in Walters’ (Model B') elastico-viscous rotating fluid permeated with suspended particles (fine dust) and variable gravity field in porous medium in hydromagnetics is considered. By applying normal mode analysis method, the dispersion relation has been derived and solved numerically. It is observed that the rotation, magnetic field, gravity field, suspended particles and viscoelasticity introduce oscillatory modes. For stationary convection, Walters’ (Model B') elastico-viscous fluid behave like an ordinary Newtonian fluid and it is observed that rotation and stable solute gradient has stabilizing effects and suspended particles are found to have destabilizing effect on the system, whereas the medium permeability has stabilizing or destabilizing effect on the system under certain conditions. The magnetic field has destabilizing effect in the absence of rotation, whereas in the presence of rotation, magnetic field has stabilizing or destabilizing effect under certain conditions. The effect of rotation, suspended particles, magnetic field, stable solute gradient and medium permeability has also been shown graphically. AMS subject classifications are 76A10, 76E07, 76E25 and 76S05.

The thermosolutal convection in a couple-stress fluid layer heated and soluted from below in porous medium is considered to include the effects of uniform vertical magnetic field and uniform vertical rotation. Following the linearized stability theory and normal mode analysis, the dispersion relation is obtained. For the case of stationary convection, the stable solute gradient and rotation have stabilizing effects on the system. In the presence of rotation, the medium permeability has a destabilizing (or stabilizing) effect whereas magnetic field and couple-stress parameter have stabilizing (or destabilizing) effect on the system. On the other hand, in the absence of rotation, medium permeability has a destabilizing effect whereas magnetic field and couple-stress parameter have a stabilizing effect. The dispersion relation is also analyzed numerically. The stable solute gradient, rotation and magnetic field introduce oscillatory modes in the system, which were nonexistent in their absence. A condition for the system to be stable is obtained by using Rayleigh-Ritz inequality

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.

The instability of thermosolutal convection coupled with crossdiffusion of an electrically conducting fluid completely confined in an arbitrary region bounded by rigid wall in the presences of a uniform magnetic field applied in an arbitrary direction is investigated. Some general qualitatively results concerning the character of marginal state, stability of oscillatory motions and limitations on the oscillatory motions of growing amplitude are derived. The results for the thermosolutal convection problems with or without the individual consideration of Dufour and Soret effects follow as a consequence.

We use the energy method to obtain the non-linear stability threshold for thermosolutal convection porous media of Brinkman type with reaction. The obtained non-linear boundaries for different values of the reaction terms are compared with the relevant linear instability boundaries obtained by Wang and Tan (Phys Lett A 373:776-780, 2009). Using the energy theory we obtain the non-linear stability threshold below which the solution is globally stable. The compound matrix numerical technique is implemented to solve the associated system of equations with the corresponding boundary conditions. Two systems are investigated, the heated below salted above case and the heated below salted below case. The effect of the reaction terms and Brinkman term on the Rayleigh number is discussed and presented graphically. [ABSTRACT FROM AUTHOR]

In this study, we investigated numerically two- and three-dimensional convective heat and mass transfer in a horizontal rectangular enclosure filled with heterogeneous porous media. The main goal is to underline the limitation of the widely used classical parallel flow assumption. The considered configuration is Cartesian. The horizontal and vertical walls are submitted to different mass and heat transfer. The Darcy model and the Boussinesq approximation are considered. The governing parameters which control the problem are the Darcy-Rayleigh number, Rt, the buoyancy ratio, N, the enclosure aspect ratio, A, the local permeability ratio, Kr and the Lewis number (fixed to ten in the present study). The obtained results with two-dimensional (2D) and three-dimensional (3D) approaches are compared to underline similarities and differences. We demonstrate the limit validity of 2D solution and the transition to 3D solutions when the convective forces or the domain permeable heterogeneity increases. The flow intensity, heat and mass transfer increases with the domain permeable heterogeneity. [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]

Based on the developed coupled model of electromagnetism, heat and solute transportation, the macrosegregation formation and effect of secondary cooling water ratio on macrosegregation degree in strand during round bloom continuous casting process have been investigated. The solute segregation degree fluctuates from a positive to a negative value with distance from strand surface in the initial solidified shell region within thickness of 20 mm. A negative segregation region in concave shape and an irregular positive segregation zone are presented in the fixed and loosened side of strand respectively due to the gravity and thermosolutal convection. As the secondary cooling water ratio decreases from 0.25 to 0.15 L kg− 1, the solidification ratio at final electromagnetic stirring (F-EMS) centre increases from 73.14 to 77.83%. For the steel grade of 50Mn casted by round bloom casting within diameter of 0.35 m, the optimal solidification ratio at F-EMS centre is 75.05%, where the radial centre crack and shrinkage cavity at strand cross-section are removed. [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]

The thermosolutal convection in Walters' (Model B') elastico-viscous rotating fluid permeated with suspended particles (fine dust) and variable gravity field in porous medium in hydromagnetics is considered. By applying normal mode analysis method, the dispersion relation has been derived and solved numerically. It is observed that the rotation, magnetic field, gravity field, suspended particles and viscoelasticity introduce oscillatory modes. For stationary convection, Walters' (Model B') elastico-viscous fluid behave like an ordinary Newtonian fluid and it is observed that rotation and stable solute gradient has stabilizing effects and suspended particles are found to have destabilizing effect on the system, whereas the medium permeability has stabilizing or destabilizing effect on the system under certain conditions. The magnetic field has destabilizing effect in the absence of rotation, whereas in the presence of rotation, magnetic field has stabilizing or destabilizing effect under certain conditions. The effect of rotation, suspended particles, magnetic field, stable solute gradient and medium permeability has also been shown graphically. AMS subject classifications are 76A10, 76E07, 76E25 and 76S05. [ABSTRACT FROM AUTHOR]

The thermosolutal convection in a couple-stress fluid layer heated and soluted from below in porous medium is considered to include the effects of uniform vertical magnetic field and uniform vertical rotation. Following the linearized stability theory and normal mode analysis, the dispersion relation is obtained. For the case of stationary convection, the stable solute gradient and rotation have stabilizing effects on the system. In the presence of rotation, the medium permeability has a destabilizing (or stabilizing) effect whereas magnetic field and couple-stress parameter have stabilizing (or destabilizing) effect on the system. On the other hand, in the absence of rotation, medium permeability has a destabilizing effect whereas magnetic field and couple-stress parameter have a stabilizing effect. The dispersion relation is also analyzed numerically. The stable solute gradient, rotation and magnetic field introduce oscillatory modes in the system, which were nonexistent in their absence. A condition for the system to be stable is obtained by using Rayleigh-Ritz inequality. [ABSTRACT FROM AUTHOR]

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]