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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.

In this work, the incompressible smoothed particle hydrodynamics (ISPH) method is utilized to simulate thermosolutal convection in a novel annulus barred by NEPCMs. The novel annulus is formed between a horizontal curved rectangle connected to a vertical rectangle containing a vertical ellipse. It is the first attempt to investigate the heat and mass transmission of NEPCM in such a unique annulus. NEPCM’s sophisticated designs of closed domains during heat/mass transfer can be applied in energy savings, electrical device cooling, and solar cell cooling. The ISPH method solved the fractional time derivative of governing partial differential equations. The artificial neural network (ANN) is integrated with the ISPH results to predict the average Nusselt Nu¯and Sherwood numbers Sh¯. The scales of physical parameters are Hartmann number (Ha = 0–80), buoyancy ratio parameter (N = − 10–20), Dufour/Soret numbers (Du = 0–0.4 & Sr = 0–0.8), Rayleigh number (Ra=103–105), fractional time derivative (α= 0.85–1), nanoparticle parameter (φ = 0–0.15), and fusion temperature (θf= 0.05–0.95). The main findings showed the importance of buoyancy ratio and Rayleigh number in enhancing the buoyancy-driven convection which accelerates the velocity field and strengths the isotherms and isoconcentration. The velocity field decreases according to an enhancement in Hartmann number and nanoparticle parameter. The exact agreement of the ANN model prediction values with the goal values demonstrates that the created ANN model can predict the Nu¯and Sh¯values properly. The complicity of a closed domain by carving the horizontal rectangle and inserting the ellipse inside a vertical rectangle can be utilized into cooling equipment, solar cells, and heat exchangers.

Mesmerizing utilizations of thermosolutal convection (TSC) have been found in metallurgical and petrochemical procedures, crystal growth, thermal exchangers, nuclear waste disposal, and building environment control. In the current study, we considered three heat and solute sources on the left boundary of the cavity, which raises the significance of studying multiple chemical and civil engineering phenomena. Therefore, the prime motivation to commence this work is to attain the optimization of thermosolutal attributes characterized by non-Newtonian (power–law) liquid in the presence of installed sources. For this purpose, the natural convective flow governed by the nonpersistent impact of buoyancy forces is encompassed in the square enclosure. In addition to the source location, the vertical boundaries are considered to be cold, whereas the horizontal boundaries are kept insulated. Owing to the free convective flow, no compositional and thermal gradients are induced by lid movement, and all walls are prescribed with no-slip constraints. To examine the transport mechanism, structuring of problem is attained in the form of dimensional governing expressions by employing conservation laws. Subsequently, constitutive equations in a dimensionless format are accessed through similar variables. The finite element approach is employed to find the solution using COMSOL Multiphysics software. The deviations in the thermal and solutal fields are examined against the involved physical parameters by illustrating graphs and tables. Quantities of interest, such as the mean Nusselt and Sherwood numbers, are also computed against sundry parameters. The assurance of the computed results is validated by constructing a comparison with published studies. A grid sensitivity test is also executed to ensure the credibility of the built-in software (COMSOL). It is observed that the averaged heat and mass flux coefficient diminishes with an elevation in the magnitude of the power-law index (n), irrespective of the change in other involved parameters. The average Sherwood number tends to increase for a positive magnitude of (N) in contrast to negative values of (N). The average heat flux and average mass flux coefficients increase until the Prandtl number (Pr) reaches a value of 5 in shear thinning case at Ra≥105,whereas, for n > 1 there is no substantial effect on either the average heat flux coefficient or average mass flux coefficient.

The characteristics of gas−liquid flows with evaporation at the thermocapillary interface in an infinite rectangular duct, with a linearly distributed thermal load being applied on the upper and lower walls, are studied. The theoretical research of the three-dimensional convective flows is carried out within the framework of a two-sided model of evaporative convection based on the Navier−Stokes equations in the Oberbeck−Boussinesq approximation. A solution of a special type of governing stationary equations is used for describing the heat and mass transfer in a system of two immiscible fluids. We investigate the influence of the working (equilibrium) temperature of the system and intensity of the external thermal load on the structure of the velocity and temperature fields, as well as on changes in the evaporation mass flow rate and vapor content in the gas phase. The simulations are performed for the ethanol−air system. Based on the comparison of the calculated and experimental data, an effective way of nondimensionalization is proposed that allows one to consistently take into account the impact of the gas pumping velocity being a controlled parameter in experiments. It provides correct matching of the mathematical model to the experiment conditions, as well as a better qualitative and quantitative agreement between theoretical and measured values of evaporative mass flow rate. The results of the present study can aid in developing a theoretical basis for experimental research methods of evaporative convection and also in designing equipment for thermal coating or drying.

The present paper deals with the onset of convection in a horizontal layer of a porous material saturated with a viscoelastic nanofluid under the influence of rotation and magnetic field. The momentum equation in porous media was simulated using the Darcy model. The rheological behavior of viscoelastic nanofluids was described using a Jeffrey model−type constitutive equation. The viscoelastic nanofluid model contains Brownian motion and thermophoresis effects. The normal mode approach is used in a linear stability study, and the onset criteria for stationary and oscillatory convection are derived analytically. The effects of the Jeffrey parameter, solutal Rayleigh number, medium porosity, nanoparticle Rayleigh number, thermosolutal Lewis number, Chandrasekhar number, Taylor number, Dufour parameter and Soret parameter on the stability of the system were investigated. Stationary and oscillatory convection regimes were calculated and analyzed in detail for various parameters.

Linear stability analysis is used for the commencement of thermosolutal convection in Jeffrey nanofluid layer with magnetic field, while the Darcy model is used for the porous medium. Brownian motion and thermophoresis are both used in the Jeffrey nanofluid layer. Furthermore, the thermal energy equations contain regular diffusion and cross-diffusion terms. The normal mode technique is used to study stationary convection. The effects of the Jeffrey parameter, solutal Rayleigh number, medium porosity, nanoparticle Rayleigh number, thermo-nanofluid Lewis number, thermosolutal Lewis number, modified diffusivity ratio, Dufour parameter, Soret parameter, and magnetic field on the system's stability have been examined. The magnetic field is discovered to stabilize the Jeffrey nanofluid layer for stationary convection, and Jeffrey parameter is found to destabilize the Jeffrey nanofluid layer for stationary convection. The effect of various parameters on thermal Darcy–Rayleigh number has been presented graphically.

Nanoparticle-Enhanced Phase Change Materials (NePCM) have garnered significant attention in engineering literature due to their enhanced thermo-physical properties. However, their behavior during phase change process, such as melting or solidification, remains inadequately understood. This study focuses on investigating the melting process of NePCM in a square cavity, exploring distinct cases of melting from both the top and bottom sides. Notably, this work delves into the effect of thermosolutal convection on NePCM melting for the first time in the literature. The NePCM comprises copper nanoparticles (2 nm in size) suspended in water. We examine various combinations of constant temperature boundary conditions and particle volume fractions. Employing a numerical model based on the one-fluid mixture approach combined with the single-domain enthalpy-porosity model, we capture the phase change process and particles’ interaction with the solid–liquid interface. In our investigation, the hot side temperature ranges between 290 K and 300 K, while the cold side temperature remains fixed at 270 K. The mass fraction of particles (ϕw) varies from 0.5% to 10%. When melting NePCM from the top side, convection effects are suppressed, resulting in a melting process primarily governed by conduction. Both NePCM and pure water melt at the same rate under these conditions. However, melting NePCM from the bottom side induces convection-dominated melting. In the case of pure water, thermal convection leads to the formation of convection cells during melting. In contrast, melting NePCM triggers thermosolutal convection due to temperature and particle concentration gradients. The flow cells formed from thermosolutal convection in NePCM differ from those in pure water driven by pure thermal convection. Our simulations reveal that thermosolutal convection contributes to decelerating the solid–liquid interface, thereby prolonging NePCM melting compared to pure water. For example, for a mass fraction of particles (ϕw= 10%), NePCM melts 6% slower compared to pure water. Surprisingly, the increase in viscosity of NePCM plays a minimal role in the deceleration process, contrary to prior literature attributing slowdowns of the NePCM melting process primarily to increased viscosity.

The model of double diffusive convection in a bidisperse porous medium was examined. Brinkman effects and slip were allowed to be used in macropores. In the micropores, only the Darcy effects are held. In equilibrium, the concentration of solutes is thought to be a linear function of temperature. We performed both a linear and a nonlinear stability analysis, with a particular emphasis on the impact of slip boundary conditions on the system's instability and stability. After deciding when the instability started, we determined the critical Rayleigh number as a function of the slip coefficient. Numerical findings for stability/instability thresholds were also presented.

The subject under study is the matter of double diffusive convection in a fluid-saturated bidispersive porous medium when the medium is rotating about an axis orthogonal to the layer in the direction of gravity. Nield and Kuznetsov's very interesting recent results are elaborated upon and scrutinized in detail, notably concerning a wide range of temperature and salt boundary conditions which facilitate the combination of prescribed heat flux and temperature. The linear instability threshold will be shown to be the same as that of the nonlinear stability one if the layer is salted above. This indicates that the linear theory completely encapsulates the physics of the onset of thermal convection. The numerical solution has been carried out via the Chebyshev collocation method and the accuracy of this method has been checked. A detailed examination of the behavior of the transition from stationary to oscillatory convection is also undertaken, as the boundary conditions vary from the prescribed temperature and salt concentration to those of prescribed heat flux and salt flux.

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

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

The numerical analysis based on incompressible smoothed particle hydrodynamics (ISPH) is introduced to examine the impacts of Cattaneo-Christov (Ca-Ch) heat flux and exothermic chemical reaction on thermosolutal convection of nano-enhanced phase change materials (NEPCM) in Bézier-annulus. The used annulus is formed between inner connected Bézier curves and outer connected spline-Bézier curves. The inner shape of connected Bézier curves is maintained at Th&Ch, left/right walls of spline-Bézier curves are kept at Tc&Ccand other walls are adiabatic. The governing equations, after being converted into non-dimensional form, have been solved by ISPH which is an accurate meshless algorithm the treatment of internal flows inside complex geometries. The simulations are executed for Frank-Kamenetskii number FK, Ca-Ch heat flux δC, buoyancy ratio parameter N, Soret-Dufour numbers SrDu, Rayleigh number Ra, and nanoparticle parameter φon thermosolutal convection of a suspension fluid. From the numerical simulation values for the average Nusselt and Sherwood numbers (Nu¯, Sh¯) were obtained for some fluid flow scenarios. Then, based on those values an artificial neural network (ANN) model was developed to predict the values of Nu¯, and Sh¯without the need to perform the ordinary simulations again for the new cases which is a high cost compared to ANN. From the obtained simulations, it was concluded that the ANN model is an accurate tool to be used to predict the needed values. Also, the Frank-Kamenetskii number significantly influences the enhancement process of the temperature distributions and velocity field as well as phase change material in Bézier-annulus.

An incompressible Oldroydian viscoelastic fluid layer heated and soluted from below in the presence of suspended (dust) particles and uniform vertical magnetic field to include the effect of Hall currents in porous medium is considered. Following the linearized stability theory and normal mode analysis, the dispersion relation is obtained. For the case of stationary convection, Oldroydian viscoelastic fluid behaves like an ordinary Newtonian fluid. Dust particles and Hall currents are found to have a destabilizing effect on the thermosolutal convection, whereas magnetic field is found to have a stabilizing effect on the thermosolutal convection. Medium permeability has both stabilizing and destabilizing effect on the thermosolutal convection under certain conditions. Graphs have been plotted by giving numerical values to the parameters to depict the stability characteristics. The case of overstability is also considered.

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.

Purpose: The purpose of this paper is to work out the magnetic forces on heat/mass transmission in a cavity filled with a nanofluid and wavy porous medium by applying the incompressible smoothed particle hydrodynamics (ISPH) method. Design/methodology/approach: The cavity is filled by a nanofluid and an undulating layer of a porous medium. The inserted two circular cylinders are rotated around the cavity's center by a uniform circular velocity. The outer circular cylinder has four gates, and it carries two different boundary conditions. The inner circular cylinder is carrying Th and Ch. The Lagrangian description of the dimensionless regulating equations is solved numerically by the ISPH method. Findings: The major outcomes of the completed numerical simulations illustrated the significance of the wavy porous layer in declining the nanofluid movements, temperature and concentration in a cavity. The nanofluid movements are declining by an increase in nanoparticle parameter and Hartmann number. The variations on the boundary conditions of an outer circular cylinder are changing the lineaments of heat/mass transfer in a cavity. Originality/value: The originality of this study is investigating the dual rotations of the cylinders on magnetohydrodynamics thermosolutal convection of a nanofluid in a cavity saturated by two wavy horizontal porous layers. [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.

The thermosolutal free convection flow in a saturated porous medium past a vertical porous plate exposed to three thermal states of the surface and a constant concentration is numerically analyzed in the presence of chemical reaction, uniform suction, or injection and Soret-Dufour effects. The mathematical model of the problem is reduced into a set of coupled nonlinear ordinary differential equations, using unique similarity transformations. The resulting equations are solved numerically by a computational code based on the fifth-order Runge-Kutta scheme along with the shooting iteration technique. A comparison to previously published work is performed, and the results were in excellent agreement. The obtained results are displayed graphically for various controlling parameters. It was found that the chemical reaction has a relevant effect on the concentration field. Also, the thermal state of the plate surface has an important effect on the temperature and concentration profiles as on the wall heat and mass transfer rates.

The thermosolutal convection in a couple-stress fluid in a porous medium in the presence of uniform rotation and uniform vertical magnetic field (separately) is studied. Using linearized stability theory and normal mode analysis, the dispersion relation is obtained in each case. In the case of magneto thermosolutal convection, the stable solute gradient, magnetic field and couple-stress are found to have stabilizing effects, whereas medium permeability has a destabilizing effect on the system for stationary convection. For rotatory thermosolutal convection, the stable solute gradient and rotation are found to have stabilizing effects, whereas medium permeability and couple-stress have both stabilizing and destabilizing effects for stationary convection. Further, the solute gradient, magnetic field, and rotation are found to introduce oscillatory modes, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained.

In this paper, we present a numerical study along with an exhaustive adsorption investigation in a binary dilute mixture model nearby the solvent’s critical point in a configuration relevant for soil remediation. By means of this model, mass and heat transfer efficiency were qualitatively and quantitatively discussed through this work. The convergence of the solution was evaluated on the values of the Nusselt and Sherwood numbers. The results reveal intense convection expanding into the cavity close to the critical point, thus enabling homogeneous adsorption of the solute. Moreover, the mass fraction perturbation isolines exhibit the existence, along the adsorbent plate, of a thin boundary layer which becomes thinner when approaching the critical point.

The impact of thermosolutal convection during directional solidification is explored via results of numerical investigations. Results from fully transient numerical simulations of directional solidification in a differentially heated cavity under terrestrial conditions and Bridgman cytstal growth in space are discussed. The pivowl role ofboththermalandsolutal convection in the solidifaction process is illustrated by examining these two cases. In particular, radical and longitudinal macrosegregation resulting from this thermosolutal convection is discussed.

The effect of a uniform vertical magnetic field on thermosolutal convection in a layer of an electrically conducting couple-stress fluid heated and soluted from below is considered. For the case of stationary convection, the stable solute gradient, magnetic field, and couple-stress parameter have stabilizing effect on the system. It is also observed that a stable solute gradient and a magnetic field introduce oscillatory modes in the system, but in the absence of a stable solute gradient and a magnetic field, oscillatory modes are not allowed and the principle of exchange of stabilities is valid.

As a response to “call for contribution to a numerical problem for 2D columnar solidification of binary alloys” [Bellet et al., Int. J. Therm. Sci., Vol. 48(11)(2009), p. 2013], the macrosegregation in a Pb-18wt.%Sn benchmark casting is numerically studied with a two-phase columnar solidification model developed by the current authors. The studies were done with 2D calculations in response to the call, and a 3D calculation was performed to confirm the consistency with the 2D case. A grid-sensitivity study was done to ensure the reliability and accuracy of the present results. The segregation mechanism due to thermosolutal convection was analyzed and the uncertainties resulting from the inaccurate thermophysical properties, modelling and process parameters are discussed. The numerical model was evaluated by comparison with experiments.

The present study analyzes numerically the mixed thermosolutal convection in a vertical partly porous annular cavity. The fluid motion in the porous matrix is described by the generalized Darcy-Brinkman-Forchheimer model. The results show that the composite porous layer tends to improve the heat transfer and moderately reduce the transfer of aqueous solution compared to the case of a homogeneous porous partition. This conclusion is valid for all characteristic values (Da, ) or the flow modes set up in the porous matrix.

ABSTRACTThis study attempts to examine convection caused by the combined effects of temperature and concentration gradients in an annular enclosure with steps. An experimental study of thermosolutal convection of a CuSO4+ H2SO4+ H2O solution in a horizontal annular enclosure with steps is conducted using an electrochemical system that enables either opposing or cooperating in temperature and concentration gradients to be imposed. The shadowgraph recording technique is utilized to visualize and analyze the flow field phenomenon. The large difference between the thermal and solutal diffusion rates causes the flow to exhibit double-diffusive characteristics. In both cooperating and opposing cases, some finger-type flow can be visualized in the layered flow structure. Furthermore, Sh (mass transfer rate) increased with increasing Grtfor fixed Grmand S*. Sh is larger in the cooperating case than in the opposing case for given Grtand Grmvalues.

A comparative study on fluid flow and heat and mass transfer, induced by Soret-driven thermosolutal convection in a horizontal porous layer subject to constant fluxes of heat, is conducted for the cases where both horizontal boundaries are stress-free or rigid. The problem formulation is based on the Brinkman-extended Darcy model with the Boussinesq approximation, and the governing equations are solved, analytically and numerically, using a finite-difference method. The critical Rayleigh number for the onset of supercritical and subcritical convection is predicted explicitly as a function of the governing parameters. The threshold for oscillatory flows is also predicted, and the stability diagram, defining various convective regimes, is illustrated. As expected, both free and rigid boundaries yield results that are identical to those predicted by the Darcy model when the Darcy number is small enough. For sufficiently large values of the separation parameter φ, the Nusselt and Sherwood numbers corresponding to free and rigid boundaries exhibit asymptotic behaviors. It is also found that, as expected, the flow intensity and heat and mass transfer rates are more affected by thermal diffusion effects in the case of free boundaries in comparison to rigid ones.

Abstract The effect of magnetic field dependent viscosity on thermosolutal convection in a ferromagnetic fluid saturating a porous medium is considered for a fluid layer heated and soluted from below in the presence of uniform magnetic field. Using linearized stability theory and normal mode analysis, an exact solution is obtained for the case of two free boundaries. For case of stationary convection, medium permeability has a destabilizing effect, whereas a stable solute gradient and magnetic field dependent viscosity have a stabilizing effect on the system. In the absence of magnetic field dependent viscosity, the destabilizing effect of non-buoyancy magnetization is depicted but in the presence of magnetic field dependent viscosity non-buoyancy magnetization may have a destabilizing or 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 parameter M1 and the results are depicted graphically. The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid saturating a porous medium 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. The paper also reaffirms the qualitative findings of earlier investigations which are, in fact, limiting cases of the present study.

The thermosolutal convection in ferrofluid saturating a porous medium is considered for a ferromagnetic fluid layer heated and soluted from below in the presence of a uniform magnetic field. Using the linearized stability theory and normal mode analysis, the exact solutions are obtained for the case of two free boundaries. For the case of stationary convection, the medium permeability and non-buoyancy magnetization both have destabilizing effects on the system, whereas a stable solute gradient has a stabilizing effect on the system. The critical magnetic thermal Rayleigh number for the onset of instability is also determined. Graphs have been plotted by giving numerical values to the parameters, to depict the stability characteristics. The principle of exchange of stabilities is found to hold true for the ferromagnetic fluid saturating a porous medium heated from below in the absence of a stable solute gradient. The oscillatory modes are introduced due to the presence of the stable solute gradient, which were nonexistent in its absence. A sufficient condition for the nonexistence of overstability is also obtained.

Abstract: A numerical analysis has been carried out to investigate the influences of thermosolutal convection on the heat and mass transfer and solute segregation in crystals grown by the vertical Bridgman technique. The governing equations are solved by a finite-volume method using the power law scheme and the SIMPLE algorithm in which body-fitted coordinate system has been used. A primary convective cell driven by thermal gradients forms in the bulk of the domain, while a secondary convective cell driven by solutal gradients forms near interface. As the solutal Rayleigh number increases, secondary cell becomes to be stronger and has a great influence on the radial concentration along the interface.

When convection occurs in a finite box, successive modes become unstable as the Rayleigh number is increased, giving rise to branches of nonlinear solutions that are linked by mixed–mode branches emerging from Hopf, transcritical or pitchfork bifurcations. The fundamental mode, in particular, gives rise to resonant interactions. Several specific examples of these generic features have already been described. Here we study resonant interactions in two–dimensional thermosolutal convection (with point symmetry imposed), where behaviour is enriched by the presence of oscillations and of subcritical steady convection. Steady m–roll solutions are obtained numerically and their stability properties are investigated by computing the leading real and complex eigenvalues. We find that the branch of fundamental m 1 solutions for a box with aspect ratio λ 1.5 develops loops that touch the m 3 branch in transcritical bifurcations. As λ is reduced these loops detach themselves from the m 1 branch and eventually join the stacked m 2 branch. Increasing λ, on the other hand, leads to complicated triple resonances involving m 1, m 3 and m 5 solutions. These resonant interactions are associated with pattern selection and the transfer of stability from single–roll to multiple–roll solutions in wider boxes.

The thermosolutal convection in a layer of Maxwellian viscoelastic fluid heated and soluted from below in porous medium is considered. The effects of uniform magnetic field and uniform rotation on the thermosolutal convection are also considered. For stationary convection, the Maxwellian viscoelastic fluid behaves like a Newtonian fluid. The sufficient conditions for the nonexistence of overstability are obtained. The critical Rayleigh number is found to increase with the increase in magnetic field, rotation and stable solute gradient.

Purpose: The purpose of this study is to simulate the thermo-solutal convection resulting from a circular cylinder hanging in a rod inside a ∧-shaped cavity. Design/methodology/approach: The two dimensional ∧-shaped cavity is filled by Al2O3-water nanofluid and saturated by three different levels of heterogeneous porous media. An incompressible smoothed particle hydrodynamics (ISPH) method is adopted to solve the governing equations of the present problem. The present simulations have been performed for the alteration of buoyancy ratio (−2≤N≤2) , radius of a circular cylinder (0.05≤Rc≤0.3) , a height of a rod (0.1≤Lh≤0.4) , Darcy parameter (10−3≤Da≤10−5) , Lewis number (1≤Le≤40) , solid volume fraction (0≤ϕ≤0.06) , porous levels (0≤η1=η2≤1.5) and various boundary-wall conditions. Findings: The performed numerical simulations indicated the importance of embedded shapes on the distributions of temperature, concentration and velocity fields inside ∧-shaped cavity. Increasing buoyancy ratio parameter enhances thermo-solutal convection and nanofluid velocity. Adiabatic conditions of the vertical-walls of ∧-shaped cavity augment the distributions of the temperature and concentration. Regardless the Darcy parameter, a homogeneous porous medium gives the lowest values of a nanofluid velocity. Originality/value: ISPH method is used to simulate thermo-solutal convection of a nanofluid inside a novel ∧-shaped cavity containing a novel embedded shape and heterogeneous porous media. [ABSTRACT FROM AUTHOR]

The present work aims to study convection and heat transfer and mass in a porous cubic cavity. The configuration considered is a cavity cube with vertical walls left and right are subjected to temperatures required while others are impermeable and adiabatic. We realized that the results depend on several characteristic parameters, and general correlations are established for the calculation of heat and mass transfer, according to various studied parameters. The study focuses on the influence of the control parameters on the structure of the flow, heat and mass transfer.

Abstract: During solidification of a binary alloy at constant velocity vertically upward, thermosolutal convection can occur if the solute rejected at the crystal-melt interface decreases the density of the melt. We assume that the crystal-melt interface remains planar and that the flow field is periodic in the horizontal direction. The time-dependent nonlinear differential equations for fluid flow, concentration, and temperature are solved numerically in two spatial dimensions for small Prandtl numbers and moderately large Schmidt numbers. For slow solidification velocities, the thermal field has an important stabilizing influence: near the onset of instability the flow is confined to the vicinity of the crystal-melt interface. Further, for slow velocities, as the concentration increases, the horizontal wavelength of the flow decreases rapidly — a phenomenon also indicated by linear stability analysis. The lateral in-homogeneity in solute concentration due to convection is obtained from the calculations. For a narrow range of solutal Rayleigh numbers and wavelengths, the flow is periodic in time.

The effects of natural convection on temperature and solute distributions, growth rate and growth uniformity along the interface are studied for CdTe single crystal growth by the traveling heater method (THM). The study is carried out by developing a thermosolutal convection model to simulate the coupled heat, solute and fluid flow fields in the growth ampoule. The model takes both thermal and solutal convections into account for this material growth and is implemented by the finite element method using the penalty function formulation. Simulation results show that the solute distribution in the liquid zone is significantly influenced by the convective mass transfer and the growth rate increases substantially. This is consistent with experimental observations. The simulations point out the importance of controlling convection for uniform growth. The uniformity of growth along the growing interface is closely related to the effect of the natural convection there. Weaker convection in the interface region is desired for the uniform growth. The heater thermal gradient and the asymmetric distance (offset) between the axial positions of heater peak temperature and solution zone center affect the natural convective state and are investigated through modeling. It is found that a smaller heater thermal gradient and a larger offset moderate the effect of natural convection in the substrate interface region, allowing uniform growth. The model predicts the heater thermal gradient and related growth rate for the crystal growth industry.

Abstract: Numerical simulations of a binary metal alloy (Pb-Sn) undergoing solidification phase change are performed using a continuum model for conservation of total mass, momentum, energy, and species. The system is contained in an axisymmetric, annular mold which is cooled along its outer vertical wall. Results show that thermosolutal convection in the melt and mushy zones is strongly coupled and that macrosegregation is reduced with increased cooling rate. For low cooling rates, solutally induced convection in the mushy zone favors the development of channels, which subsequently spawn macrosegregation in the form of A-segregates. With increasing solidification rate, however, thermosolutal interactions in the melt contribute to reducing the formation of channels and A-segregates.

The onset of thermosolutal convection driven by a temperature and concentration dependent surface tension is investigated for a thin layer of fluid having a deformable free surface. It is shown that there exist two Crispation numbers (Cr) for oscillatory modes of instability. It is further shown that Cr and the frequency of the oscillatory mode are strongly coupled for large values of Cr. It is found that Cr destabilizes the system for both cases, salted from below and salted from above

The effect of surface deformation on the onset of thermosolutal convection in a horizontal thin layer heated from below and salted from above is examined, using linear stability theory. It is found that two critical Crispation numbers Cr1and Cr2exist. For Cr ≦ Cr1the instability mechanism remains unefTected by the deformation of the free surface, whereas for Cr1< Cr < Cr2the critical Marangoni number (Mc) decreases, showing instability due to deformation. If Cr = Cr2, Mc is obtained for two values of the wave number. When Cr > Cr2, Mc decreases as the wave number tends to zero. Further, the effect of the Marangoni number, Biot number, Bond number etc. on the stability characteristics of the problem is discussed.

Abstract: The essential principles of thermosolutal convection are outlined, and how convection provides a transport mechanism between the mushy region of a casting and the open bulk liquid is illustrated. The convective flow patterns which develop assist in heat exchange and macroscopic solute segregation during solidification; they also provide a mechanism for the transport of dendritic fragments from the mushy region into the bulk liquid. Surviving fragments become nuclei for equiaxed grains and so lead to blocking of the parental columnar, dendritic growth front from which they originated. The physical steps in such a sequence are considered and some experimental data are provided to support the argument.

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

A mathematical model of thermosolutal convection in directionally solidified dendritic alloys has been developed that includes a mushy zone underlying an all-liquid region. The model assumes a nonconvective initial state with planar and horizontal isotherms and isoconcentrates that move upward at a constant solidification velocity. The initial state is perturbed, nonlinear calculations are performed to model convection of the liquid when the system is unstable, and the results are compared with the predictions of a linear stability analysis. The mushy zone is modeled as a porous medium of variable porosity consistent with the volume fraction of, interdendritic liquid that satisfies the conservation equations for energy and solute concentrations. Results are presented for systems involving lead-tin alloys (Pb-10 wt pct Sn and Pb-20 wt pct Sn) and show significant differences with results of plane-front solidification. The calculations show that convection in the mushy zone is mainly driven by convection in the all-liquid region, and convection of the interdendritic liquid is only significant in the upper 20 pct of the mushy zone if it is significant at all. The calculated results also show that the systems are stable at reduced gravity levels of the order of 10−4g0(g0=980 cm·s−1) or when the lateral dimensions of the container are small enough, for stable temperature gradients between 2.5≤Gl≤100 K·cm−1at solidification velocities of 2 to 8 cm·h−1.

Abstract: Macrosegregation in direct chill (DC) continuous casting of an Al-4.5 wt pct Cu round ingot is numerically simulated. The model incorporates descriptions of heat transfer, solute redistribution, and melt convection on the system scale with microscopic relations for grain growth, solutal undercooling, and microsegregation. Simulations are conducted to study the effects of mushy zone permeability, thermosolutal convection, and solidification contraction on the macrosegregation pattern in a DC casting. The results indicate that centerline segregation can be either positive or negative, depending upon the grain density in and permeability of the mush. In addition, it is shown that the flow induced by solidification contraction not only causes inverse segregation at the ingot surface, but also has a significant influence on the macrosegregation across the central portion of the ingot. A comparison with temperature and macrosegregation patterns measured in a previous experiment shows reasonable agreement. Several areas for future model improvements are identified.

Numerical experiments on two-dimensional thermosolutal convection reveal a transition from periodic oscillations to chaos through a sequence of period-doubling bifurcations. Within the chaotic region there are narrow periodic windows. This is the first example of period-doubling in solutions of partial differential equations. A truncated model indicates that this behaviour is associated with heteroclinic explosions.

Thermosolutal convection flow and its effect on the heat and the mass transfer in a square enclosure is studied experimentally. Both thermal and solute diffusion are induced from the sides, and natural convection is initiated by the combined thermal and solutal buoyancies, which either augment or oppose to each other. The solute diffusion is initiated in an electrochemical system that uses copper sulfate-sulfuric acid solution as an electrolyte. Depending on the magnitude of buoyancy ratio, three different kinds of flow regimes and structures can occur, which lead to different distributions of concentration in the enclosure. The formation and growth of layered flow structure is attributed to the solutal boundary-layer flow that can intrude and accumulate along the horizontal wall. The nearly stagnant layer that occurs can reduce the heat transfer rate. The Nusselt numbers at different flow regimes are measured and correlated in terms of relevant nondimensional parameters. This suggests the correlation of Sherwood number in different ranges of buoyancy ratio. The visualization of flow structures and measurements of both heat and mass transfer allow better understanding of the complicated system.

An attempt is made to investigate the influence of the solutal effect on convection in BiSbTe3melts as a representative for semiconductor melts of low PRANDTLnumber. Calculations have been performed for a 3D BRIDGMANconfiguration applying an experimentally measured temperature profile at the outer surface of the ampoule. The FIDAPTMFEM code was used to solve the transient hydrodynamic moving‐boundary problem. Results are presented for the excessive tellurium as a melt component as well as for an additive lead doping. It is shown that, according to the growth conditions, the melt components differently contribute to the thermosolutal convection. For Bridgman growth experiments in space, arbitrary orientations of the ampoule axis with respect to the residual gravity vector can occur. Calculations have been performed for certain orientations (angles) at a constant microgravity level. They show an influence of the solutal effect only for large deviations of the ampoule axis from the direction of the residual gravity vector.

Abstract: The formation of macrosegregation by multicomponent thermosolutal convection during the solidification of steel is simulated by simultaneously solving macroscopic mass, momentum, energy, and species conservation equations with full coupling of the temperature and concentrations through thermodynamic equilibrium at the solid/liquid interface. The flow field, solid fraction evolution, and macrosegregation patterns for four cases are presented. The results show both the formation of channel segregates and the formation of islands of mush surrounded by bulk melt. In examining the solidification of a ten-element steel, the global extent of macrosegregation of an element is found to be linearly dependent on its partition coefficient (more severe segregation for small partition coefficient), although such scaling is not possible locally. Results for the solidification of a binary Fe-C alloy (with the same carbon content as the ten-element alloy) are similar to those for the ten-element alloy due solely to the large contribution of carbon to buoyancy driven flow in the ten-element steel chosen for study. While including only those elements that make significant contributions to buoyancy driven flow reproduces the global extent of macrosegregation seen in the ten-element alloy, local differences in the predictions are visible. Finally, comparison of results for the solidification of the same ten-element steel using two different sets of data to describe the partition coefficients and change in liquidus temperature with concentration of the elements shows completely opposite behavior,i.e., upward flow through the mushy zone for one case and downward flow for the other. Thus, the need to have accurate phase-equilibrium data when modeling multicomponent macrosegregation is illustrated. Together, the results give an indication of what areas require more careful examination if accurate modeling of multicomponent solidification is to be accomplished.

Thermosolutal convection caused by the solute build up ahead of growing arrays of cells and dendrites results in macrosegregation along the length of Pb-Sn alloy (10 to 58 wt pct Sn) specimens when they are directionally solidified in a positive thermal gradient(melt on top, solid below, and gravity pointing down). At a constant thermal gradient, the extent of macro-segregation increases with decreasing growth speed as the microstructure changes from dendritic to cellular and to planar. An empirical parameter, effective partition coefficient (keff), obtained from the dependence of the longitudinal macrosegregation on fraction distance solidified can be used to represent the extent of macrosegregation. The extent of macrosegregation appears to be related to a parameter, [g(Ct- Co) Dl/RCo]1/2, where Cois the tin content of the alloy, Ctis the tin content of melt at the array tips, R is the growth speed, Dlis the diffusivity of tin in the melt, and g is the acceleration due to gravity.

Thermosolutal convection is considered in a fluid layer between poorly conducting horizontal boudaries. The horizontal scale of the motions is much greater than the depth of the fluid layer, provided the motion is not too vigorous, and this disparity between horizontal and vertical scales provides the basis foe an asymptotic expansion of the solution. Under the assumption of near-constant solute-flux at the horizontal boundaries, a pair of evolution equations is derived for the depth-averaged temperature and solute concentration fields.