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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]

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]

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]

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, 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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

The present study is aimed at understanding the interface kinetics during solute driven remelting in metallic alloys. Solid Al is placed in contact with a liquid Al-Mg alloy. As solid and liquid compositions at the interface are out of equilibrium, remelting takes place. The remelting rate is estimated as a function of time using a simple heat balance. The estimated velocity from the heat-balance calculations shows excellent agreement with the geometric velocity, directly measured from the remelted samples in each experiment. This confirms the accuracy and reliability of the heat-balance calculations and establishes this technique as a potential method for tracing the interface velocity during remelting. The results indicate that, at a constant temperature, an increase in liquid supersaturation leads to a linear increase in remelting velocity, as a result of an increasing driving force for remelting. At a constant liquid supersaturation, an increase in temperature results in an exponential increase in the remelting velocity, due to the enhanced mass transport at the higher temperatures. Semi-empirical relations are derived from these experimental observations and a combined analysis of the effects of driving force and kinetics yields a relation for remelting velocity as a function of temperature for a variety of boundary conditions. Remelting velocities predicted by this relation are in good agreement with the experimental observations. [ABSTRACT FROM AUTHOR]

The thermosolutal convection in a fluid layer heated and soluted from below in a porous medium is considered to include, separately, the effects of a uniform horizontal magnetic field and a uniform rotation. The solute gradient, magnetic field and rotation are found to introduce oscillatory modes which were non-existent in their absence. The stable solute gradient, magnetic field and rotation are found to have stabilizing effects whereas medium permeability has both destabilizing and stabilizing effects. The sufficient conditions for non-existence of overstability are obtained. [ABSTRACT FROM AUTHOR]