Search

In the current study, cross-temperature and concentration gradients are used to model the in a binary fluid contained in an angled square cavity. Using a method, the , , and conservation equations were numerically solved. The inclined cavity under equal solutal buoyancy and thermal forces was the subject of the study . Since the horizontal components of the thermal and singular volume forces were equal but opposed to one another, an equilibrium solution for this situation that corresponds to the rest state of the immobile fluid is feasible. However, this equilibrium solution becomes unstable above a specific critical value of the , leading to vertical density stratification inside the enclosure. The results are shown using the and as well as the and for the flow intensity. The existence of the commencement of convection is demonstrated in this work, and both natural and anti-natural flow solutions are obtained. Subcritical convection has also been seen for the natural solution when the is more or less than unity. For the start of supercritical and subcritical convection, the number's critical values are identified. As the climbed, so did the flow's intensity and the rates at which heat and mass were transferred. Reducing flow intensity and accelerating mass transfer are the results of raising the . Different flow patterns are shown for an aspect ratio of 4, and the existence interval of the oscillatory solutions is calculated.

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 fractional of a time derivative for the incompressible smoothed particle hydrodynamics (ISPH) method is adopted to handle thermal radiation on thermosolutal convection in a porous cavity barred by nanoencapsulated phase change materials (NEPCMs). A novel work on the rotational velocities of two rods during thermosolutal convection of NEPCMs within a novel cavity of two connected circular cylinders is introduced. The management of heat-mass transfer and velocity magnitude inside distinct designed materials below various boundary conditions carries improvements for energy efficiency. The physical factors are thermal radiation [Formula: see text], Fusion temperature [Formula: see text], Rayleigh number [Formula: see text], hot source length [Formula: see text], Darcy number [Formula: see text], and fractional order parameter [Formula: see text]. This study reported the role of thermal/solutal conditions in varying the dual convection flow and heat capacity contour. The lower Da provides an elevated porous resistance which decelerates the nanofluid velocity. The fusion temperature alters a heat capacity contour.

In the presence of a magnetic field, a thermosolutal convection is postulated to occur in a compressible Rivlin-Ericksen viscoelastic fluid in a porous media. The dispersion relation is found by using the linear stability theory and the normal mode analysis approach, respectively. In the scenario of stationary convection, it was discovered that compressibility, magnetic fields, and steady solute gradients all serve to delay the beginning of the convection process, but medium permeability serves to speed up the beginning of the convection process. In addition to this, it has been discovered that the system is reliable for [equation not detected by OJS] and under the condition [equation not detected by OJS]. The system goes into an unstable state. Overstability has also been looked at from the perspective of a scenario in which sufficient circumstances are met to rule out the possibility of the phenomenon occurring. It has been discovered that the steady gradient of the solute and the magnetic field both induce oscillatory modes into the system.

The onset of thermosolutal convection in a uniformly rotating horizontal nanofluid layer is investigated. By employing an order-1 Galerkin residual method, an approximation of the instability threshold has been determined. A sufficient condition for the onset of steady convection has been found.

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

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

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 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.

Species separation in heterogeneous porous media is a field of interest of many industrial activities. In our investigation, the effect of a single discrete fracture on the thermosolutal convection coupled with the Soret effect have been analyzed. The main results show that the fracture can greatly affect the behavior of the thermogravitational flow and might play a positive role to the separation caused by the Soret effect. Furthermore, the fracture tilted to the cold wall causes a large separation compared to the one tilted to the hot wall with the same angle. Therefore, the separation process could be greatly improved.

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.

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.

This work discusses the dual diffusion of a sloshing wavy rod in a nanofluid-filled cavity. The top zone of a cavity is suspended by a porous medium. The boundary treatment in the ISPH method employs a kernel renormalization function. The wavy rod sloshes by an excitation frequency [Formula: see text] and it carries [Formula: see text] and [Formula: see text]. The rigid cavity walls are fixed, adiabatic (horizontal walls), and [Formula: see text] and [Formula: see text] (vertical walls). The measurements of pertinent parameters are the Soret number ([Formula: see text] Sr [Formula: see text]), nanoparticles parameter ([Formula: see text]), Rayleigh number ([Formula: see text] Ra [Formula: see text]), sinusoidal function ([Formula: see text]), Dufour number ([Formula: see text] Du [Formula: see text]), a slope angle of a magnetic field ([Formula: see text]), Hartmann number ([Formula: see text]), and Darcy parameter ([Formula: see text] Da [Formula: see text]). The performed simulations revealed that the nanofluid flow is accelerated by the Sort number, the sloshing of the rod, and the sinusoidal wavy of the rod. Besides, the dual convection is enhanced by the sinusoidal wavy function and the sloshing of the rod. There is almost no nanofluid flow in a top porous zone of a cavity when Da [Formula: see text].

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.

In this paper, we study the problem of thermosolutal convection in a Navier–Stokes–Voigt fluid when the layer is heated from below and simultaneously salted from above or below. This problem is studied under the effects of Soret and slip boundary conditions. Both linear and nonlinear stability analyses are employed. When the layer is heated from below and salted from above, the boundaries exhibit great concordance, resulting in a very narrow region of probable subcritical instabilities. This proves that linear analysis is reliable enough to forecast the beginning of convective motion. The Chebyshev collocation technique and QZ algorithm have been used to solve systems of linear and nonlinear theories. For thermal convection in a dissolved salt field with a complex viscoelastic fluid of the Navier–Stokes–Voigt type, instability boundaries are computed. When the convection is of the oscillatory type, the Kelvin–Voigt parameter is observed to play a crucial role in functioning as a stabilizing agent. This effect's quantitative size is shown.

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

In this paper, 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 results of a study of convection of moist air in conditions close to atmospheric are presented. Numerical estimates of dimensionless gas-dynamic parameters characteristic of the problem of convection of dry and moist air are presented and analyzed. Distinctive features of convection caused by a change in the concentration of water vapor as a result of its evaporation (condensation) are described. The effect of water vapor concentration on the convective stability of moist air is considered from the point of view of concentration convection, characterized by the concentration Rayleigh number. The technical characteristics of an experimental setup using the principle of holographic interferometry to study the thermoconcentration convection of moist air in the temperature range from 5 to 45 C are shown. The ability to distinguish convective heat and mass transfer in humid air under laboratory conditions is demonstrated.

In this paper, we studied the rotation effect on the thermosolutal convection in visco-elastic nanofluid in the presence of porous medium using Walters` (model B`). To solve the conservation equation, we used the normal mode technique and Galerkin weighted residual method. For stationary convection, the onset criterion derived analytically and experiential that visco-elastic nanofluid behaves as a regular Newtonian nanofluid. The effect of rotation, thermo-nanofluid Lewis number, thermosolutal Lewis number and solutal Rayleigh number analyze analytically and graphically.

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

An investigation of the stability of an micropolar nanofluid overlying a sparsely packed porous medium and implanted in a parallel conduit is reviewed. Linear and also nonlinear terms are incorporated for the study. A Darcy-Brinkman-Forchheimer drag force model is deployed. To evaluate nanoscale effects the Buongiorno model is employed. The equations for mass, momentum, angular momentum, energy and nanoparticle species conservation with correlated wall conditions are non-dimensionalized. Modified diffusivity ratio and Lewis number stable the system, the micropolar parameters concentration Rayleigh number destable system for stationary convection. Concentration Rayleigh number, micropolar parameters stabilize and Lewis number destabilizes the system for oscillatory convection. Applications of the study include micro/nano-fluidic devices, nano-doped energy systems and packed beds in chemical engineering.

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

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

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.

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

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.

The objective of the current paper is to investigate the thermosolutal flow along with heat and mass dissipation rates in an upright porous annular space subjected to discrete heating and salting along the vertical boundaries with insulated and impermeable horizontal boundaries. To solve the model equations, an implicit finite difference scheme with over relaxation technique has been implemented. The numerical predictions focus specifically on the effect of buoyancy ratio, Lewis number, Darcy number and source-sink arrangements on fluid flow behavior, thermal and solutal characteristics, average Nusselt and Sherwood numbers. The simulations were performed for an extensive range of dimensionless parameters (−10 ≤ N ≤ 10, 0.1 ≤ Le ≤ 1, 10−5 ≤ Da ≤ 10−1) by considering two different types of source-sink arrangements and found that the flow circulation strength is maximum with in-line arrangement of source and sink compared to alternative arrangement. Also, we found that in case-I, regardless of Lewis number, the maximum heat and mass dissipation rates takes place through top source during opposing flow, and through bottom source for aided flow. However, in case-II arrangement, irrespective to type of flow, the greater amount of heat and mass transport is found through the bottom source at the inner cylinder compared to top source placed at outer cylinder.

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

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

The main purpose of our investigation is to show the impact of pertinent parameters; such Lewis and porous thermal Rayleigh numbers as well as the buoyancy and the aspect ratios; on the double-diffusive convection phenomena which occur within a porous annulus; found between a cold (and less concentric) outer circular cylinder and a hot (and concentric) inner one, to come out with global correlations which predict the mean transfer rates in 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. Summarizing the numerical predictions, global correlations of overall transfer within the porous annulus as a function of the governing studied parameters are set forth which predict within ±2% the numerical results. These correlations may count as a complement to previous researches done in the case a Newtonian-fluid annulus. It is to note that the validity of the computing code used was ascertained by comparing our results with the experimental data and numerical ones already available in the literature.

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