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The purpose of this paper is to discuss the need to attend correctly to the accuracy and the manner in which the value of the streamfunction is determined when two or more impermeable boundaries are present. This is discussed within the context of the paper by Nandalur et al. (2019), which concerns the effect of a centrally located conducting square block on convection in a square sidewall-heated porous cavity. Detailed solutions are also presented which allow the streamfunction to take the natural value on the surface of the internal block.

Steady solutions are obtained using finite difference methods. Three different ways in which insulating boundary conditions are implemented are compared. Detailed attention is paid to the iterative convergence of the numerical scheme and to its overall accuracy. Error testing and Richardson’s extrapolation have been used to obtain very precise values of the Nusselt number.

The assumption that the streamfunction takes a zero value on the boundaries of both the cavity and the embedded block is shown to be incorrect. Application of the continuity-of-pressure requirement shows that the block and the outer boundary take different constant values.

The Darcy–Rayleigh number is restricted to values at or below 200; larger values require a finer grid.

This paper serves as a warning that one cannot assume that the streamfunction will always take a zero value on all impermeable surfaces when two or more are present. A systematic approach to accuracy is described and recommended.

The onset of convection in a ferrofluid-saturated porous layer has been investigated using a local thermal nonequilibrium (LTNE) model by allowing the solid phase to transfer heat via a Cattaneo heat flux theory while the fluid phase to transfer heat via usual Fourier heat-transfer law. The flow in the porous medium is governed by modified Brinkman-extended Darcy model. The instability of the system is discussed exactly for stress-free boundaries, while for rigid-ferromagnetic/paramagnetic boundaries the results are obtained numerically using the Galerkin method. The presence of Cattaneo effect introduces oscillatory convection as the preferred mode of instability contrary to the occurrence of instability via stationary convection found in its absence. Besides, oscillatory ferroconvection is perceived when the solid thermal relaxation time parameter exceeds a threshold value and increase in its value is to hasten the oscillatory onset. The effect of different boundary conditions on the instability of the system is noted to be qualitatively same. The paper aims to discuss these issues.

The investigators would follow the procedure of Straughan (2013) to obtain the expression for Rayleigh number. The Brinkman-extended Darcy model is used to describe the flow in a porous medium. The investigators have used a Galerkin method to obtain the numerical results for rigid-ferromagnetic/paramagnetic boundaries, while the instability of the system is discussed exactly for stress-free boundaries.

The Cattaneo–LTNE porous ferroconvection has been analyzed for different velocity and magnetic boundary conditions. The Brinkman-extended Darcy model is used to describe the flow in a porous medium. The effect of different types of velocity and magnetic boundary conditions on the instability of the system has been highlighted. The instability of the system is discussed exactly for stress-free boundaries, while for rigid-ferromagnetic/paramagnetic boundaries the results are obtained numerically using the Galerkin method.

The novelty of the present paper is to combine LTNE and second sound effects in solids on thermal instability of a ferrofluid-saturated porous layer by retaining the usual Fourier heat-transfer law in the ferrofluid. The Brinkman-extended Darcy model is used to describe the flow in a porous medium. The effect of different types of velocity and magnetic boundary conditions on the instability of the system is discussed.

The purpose of this paper is to determine the manner in which a yield stress fluid begins convecting when it saturates a porous medium. A sidewall-heated rectangular cavity is selected as the testbed for this pioneering work.

Steady solutions are obtained using a second order accurate finite difference method, line relaxation based on the Gauss-Seidel smoother, a Full Approximation Scheme multigrid algorithm with V-cycling and a regularization of the Darcy-Bingham model to smooth the piecewise linear relation between the Darcy flux and the applied body forces.

While Newtonian fluids always convect whenever the Darcy-Rayleigh number is nonzero, Bingham fluids are found to convect only when the Darcy-Rayleigh number exceeds a value which is linearly dependent on both the Rees-Bingham number and the overall perimeter of the rectangular cavity. Stagnation is always found in the centre of the cavity and in regions close to the four corners. Care must be taken over the selection of the regularization constant.

The Darcy-Rayleigh number is restricted to values which are at or below 200.

This is the first investigation of the effect of yield stress on nonlinear convection in porous media.

Several studies find that there is little sex gap in wages at labor market entry, and that the sex gap in wages emerges (and grows) with time in the labor market. This evidence is consistent with (i) there is little or no sex discrimination in wages at labor market entry, and (ii) the emergence of the sex gap in wages with time in the labor market reflects differences between women and men in human capital investment (and other decisions), with women investing less early in their careers. Indeed, some economists explicitly interpret the evidence this way. We show that this interpretation ignores two fundamental implications of the human capital model, and that differences in investment can complicate the interpretation of both the starting sex gap in wages (or absence of a gap), and the differences in “returns” to experience. We then estimate stylized structural models of human capital investment and wage growth to identify the effects of discrimination (or other sources of a starting pay gap) and differences in human capital investment.

This paper aims to perform a linear and nonlinear analysis of the stability of a chemically reacting Newtonian fluid in a Darcy porous medium. The purpose of selecting both analyses is to investigate the probability of subcritical instability resulting from combustion.

The chemical reaction problem in a Darcy porous medium with Arrhenius kinetics is considered. The effect of the Frank-Kamenetskii number on the linear and nonlinear stability is analysed. The critical eigenvalue is obtained numerically by the Chebyshev pseudospectral method for both analyses.

The inference from the two analyses is that in the presence of combustion, the situation in the Darcy−Bénard convection problem can lead to subcritical instability. It is found that the value of the critical Frank-Kamenetskii number keeps on changing as the lower boundary temperature changes, beyond the critical value of the Frank-Kamenetskii number where the system splits, going from a steady condition to an explosive state.

The Chebyshev pseudospectral approach has been applied to address the combustion problem in this research. The normal mode methodology and energy method are used for linear and nonlinear analyses, and the effects of nonlinear factors are examined by comparing the outcomes.

It is well known that the mixed convection process is the combined effect of the presence of both the forced and the free convection processes. In several applications such as environmental chambers, IC engines, etc. the forced convection is brought in by multiple suction/injection (S/I) effect. Study of mixed convection in a vertical square fluid saturated porous cavity with multiple S/I effect greatly contributes to such an understanding. So far, not much research work has reported in this direction. Hence, the purpose of this paper is to investigate such a mixed convection process in a fluid saturated vertical porous square cavity.

In this study, the authors numerically solved the couple partial differential equations governing the mixed convection process in a fluid saturated vertical square porous cavity by finite element method. The study is parametric in nature wherein the authors cover a large range of values for different parameters arising the mathematical model governing the problem under consideration.

The influence of multiple S/I effects on mixed convection is analyzed for a wide range of controlling parameters such as S/I flow velocities (a), S/I window size (D/H) and Rayleigh number (Ra). Both the flow and temperature fields are highly sensitive to magnitude of S/I velocity, S/I window slit size and “Ra”. While heat fluxes along the isothermal left vertical wall decrease with increasing S/I velocities they are formed to increase with increasing “(D/H)” and “Ra”. Nusselt numbers increase with increasing “Ra” and increasing size of S/I window slit size. Multi‐cellular circulation pattern and thermal boundary layers are seen to manifest in flow and temperature fields, respectively.

The study is based on 2D model, but the model is generic in nature; also it is fully numerical in nature. Due to lack of apt literature no experimental support is provided. The mathematical model used in the study is based on certain assumptions such as isotropic porous medium, fluid is viscous in nature and follows Newtonian laws and the porous structure is saturated with fluid, etc. Regarding future work, 3D modelling and simulation is in progress and attempts are also being made to collaborate with experimental groups on the problem under investigation.

The results from the work are relevant to the context of heat and fluid flow studies in IC engines, influence of mixed convection process on bacterial growth process in environmental chambers and cooling of electronic devices, etc.

The paper describes a mathematical model, especially the boundary treatment, for describing the influence of multiple S/I effects on mixed convection flow in a vertical square enclosure filled with a Darcian fluid saturated homogeneous porous medium. To understand the physics behind the mixed convection process in the proposed configuration, extensive numerical simulations have been carried out for the first time for different values of the important governing parameters arising from the model.

This study aims to elucidate the role of curved walls in the presence of identical mass of porous bed with identical heating at a wall for two heating objectives: enhancement of heat transfer to fluid saturated porous beds and reduction of entropy production for thermal and flow irreversibilities.

Two heating configurations have been proposed: Case 1: isothermal heating at bottom straight wall with cold side curved walls and Case 2: isothermal heating at left straight wall with cold horizontal curved walls. Galerkin finite element method is used to obtain the streamfunctions and heatfunctions associated with local entropy generation terms.

The flow and thermal maps show significant variation from Case 1 to Case 2 arrangements. Case 1 configuration may be the optimal strategy as it offers larger heat transfer rates at larger values of Darcy number, Da_m. However, Case 2 may be the optimal strategy as it provides moderate heat transfer rates involving savings on entropy production at larger values of Da_m. On the other hand, at lower values of Da_m (Da_m ≤ 10⁻³), Case 1 or 2 exhibits almost similar heat transfer rates, while Case 1 is preferred for savings of entropy production.

The concave wall is found to be effective to enhance heat transfer rates to promote convection, while convex wall exhibits reduction of entropy production rate. Comparison between Case 1 and Case 2 heating strategies enlightens efficient heating strategies involving concave or convex walls for various values of Da_m.

The material point method (MPM)is a particle-based numerical method suitable for solid–liquid simulation and large deformation problems. However, MPM is generally used in solid deformation at present, to develop a multi-physics coupling MPM; the purpose of this study is to extend the MPM to simulate the heat and fluid flow and address the thermal-hydrological (TH) coupling problems.

The porous medium was discretized into two sets of Lagrangian points, and the motion of fluid points follows the Darcy’s law. Two sets of heat transport equations were established for the heat conduction and heat exchange in the pore fluid and solid skeleton. Fractures were considered by adding the porosity gradient term in the governing equations; also a transition function was introduced to smoothen the fracture boundary.

Four cases of heat and fluid flow in porous medium and fractures were presented to verify the feasibility of the proposed method. And the effects of fractures on heat and fluid flow were investigated. Additionally, a case of geothermal extraction was solved and the importance of the interstitial convective heat transfer coefficient was analyzed.

The proposed method extends the conventional MPM, using two sets of material points and two sets of heat transport equations to simulate the heat and fluid flow and address the TH coupling problems, which can be applied in both porous medium and fractures.

This study aims to numerically analyze natural convection of alumina-water nanofluid in a differentially-heated square cavity partially filled with a heat-generating porous medium. A single-phase nanofluid model with experimental correlations for the nanofluid viscosity and thermal conductivity has been considered for the description of the nanoparticles transport effect in the present study. Local thermal non-equilibrium approach for the porous layer with the Brinkman-extended Darcy model has been used.

Dimensionless governing equations formulated using stream function, vorticity and temperature have been solved by the finite difference method. The effects of the Rayleigh number, Ostrogradsky number, Nield number and nanoparticles volume fraction on nanofluid flow, heat and mass transfer have been analyzed.

It has been revealed that the dimensionless heat transfer coefficient at the fluid/solid matrix interface can be a very good control parameter for the convective flow and heat transfer intensity. The present results are original and new for the study of non-equilibrium natural convection in a differentially-heated nanofluid cavity partially filled with a porous medium.

The results of this paper are new and original with many practical applications of nanofluids in the modern industry.

The simultaneous effects of local thermal non-equilibrium (LTNE) and vertical heterogeneity in permeability on the onset of ferromagnetic convection in a Brinkman porous medium are analyzed in the presence of a uniform vertical magnetic field. The eigenvalue problem is solved numerically using shooting method for isothermal rigid-ferromagnetic boundaries for various forms of vertically stratified permeability function Γ(z). The effect of vertically stratified permeability is found to either hasten or delay the onset of ferromagnetic convection. The deviation in the critical Rayleigh number between different forms of Γ(z) is found to be not so significant with an increase in the Darcy number. It is observed that the general quadratic variation of Γ(z) has more destabilizing effect on the system when compared to the constant permeability porous medium case. Besides, the influence of LTNE and magnetic parameters on the criterion for the onset of ferromagnetic convection has been assessed in detail. The paper aims to discuss these issues.

Ferroconvection in a porous medium has been analyzed considering heterogeneity in the permeability of the porous medium. The resulting eigenvalue problem has been solved numerically using shooting method as well as Galerkin method for realistic boundary conditions.

The novelty of the present study lies in understanding the effect of heterogeneity in the permeability of the porous medium on control of ferroconvection in a porous medium. In analyzing the problem, realistic boundary conditions are considered and the resulting eigenvalue problem is solved numerically using shooting method as well as Galerkin method.

Control of ferroconvection in a porous medium is an important feature in heat transfer-related problems and many mechanisms are being used to understand this aspect in the literature. The novelty of the present study lies in recognizing the effect of heterogeneity in the permeability of the porous medium on control of ferroconvection. This fact has been analyzed in detail for various forms of heterogeneity functions using numerical techniques by considering realistic boundary conditions.