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This study aims to carry out the analysis of Rayleigh-Bénard convection within enclosures with curved isothermal walls, with the special implication on the heat flow visualization via the heatline approach.

The Galerkin finite element method has been used to obtain the numerical solutions in terms of the streamlines (ψ ), heatlines (Π), isotherms (θ), local and average Nusselt number ( Nut¯) for various Rayleigh numbers (10³ ≤ Ra ≥ 10⁵), Prandtl numbers (Pr = 0.015 and 7.2) and wall curvatures (concavity/convexity).

The presence of the larger fluid velocity within the curved cavities resulted in the larger heat transfer rates and thermal mixing compared to the square cavity. Case 3 (high concavity) exhibits the largest Nut¯ at the low Ra for all Pr. At the high Ra, Nut¯ is the largest for Case 3 (high concavity) at Pr = 0.015, whereas at Pr = 7.2, Nut¯ is the largest for Case 1 (high concavity and convexity).

The results may be useful for the material processing applications.

The study of Rayleigh-Bénard convection in cavities with the curved isothermal walls is not carried out till date. The heatline approach is used for the heat flow visualization during Rayleigh-Benard convection within the curved walled enclosures for the first time. Also, the existence of the enhanced fluid and heat circulation cells within the curved walled cavities during Rayleigh-Benard heating is illustrated for the first time.

The effects of anisotropy and radiation cannot be considered negligible while investigating the stability of the fluid in convection. Hence, the purpose of this paper is to analyze how these effects could affect the system while considering a couple-stress dielectric fluid. Therefore, the study establishes the effect of thermal radiation in a couple-stress dielectric fluid with an anisotropic porous medium using Goody's approach (Goody, 1956).

To analyze the effect of radiation on the onset of convection, the Milne–Eddington approximation is employed to convert radiative heat flux to thermal heat flux. The equations are further developed to approximate for transparent and opaque medium. Stability of the quiescent state within the framework of linear theory is performed. The principle of exchange of stabilities is shown to be valid by means of single-term Galerkin method. Large values of conduction–radiation and absorptivity parameters are avoided as fluid is considered as liquid rather than gas.

The radiative heat transfer effect on a couple-stress dielectric fluid saturated anisotropic porous medium is examined in terms of Milne–Eddington approximation. The effect of couple-stress, dielectric, anisotropy and radiation parameters are analyzed graphically for both transparent and opaque medium. It is observed that the conduction–radiation parameter stabilizes the system; in addition, the critical Darcy–Rayleigh number also shows a stabilizing effect in the absence of couple-stress, dielectric and anisotropy parameters, for both transparent and opaque medium. Furthermore, the absorptivity parameter stabilizes the system in the transparent medium, whereas it exhibits a dual effect in the case of an opaque medium. It was also found that an increase in thermal and mechanical anisotropy parameters shows an increase in the cell size, whereas the increase in Darcy–Roberts number and conduction–radiation parameter decreases the cell size. The validity of principle of exchange of stability is performed and concluded that marginal stability is the preferred mode than oscillatory.

The effects of anisotropy and radiation on Rayleigh–Bénard convection by considering a couple-stress dielectric fluid has been analyzed for the first time.

This paper aims to investigate the role of shapes of containers (nine different containers) on entropy generation minimization involving identical cross-sectional area (1 sq. unit) in the presence of identical heating (isothermal). The nine containers are categorized into three classes based on their geometric similarities (Class 1: square, tilted square and parallelogram; Class 2: trapezoidal type 1, trapezoidal type 2 and triangular; Class 3: convex, concave and curved triangular).

Galerkin finite element method is used to solve the governing equations for a representative fluid (engine oil: Pr = 155) at Ra = 10³–10⁵. In addition, finite element method is used to solve the streamfunction equation and evaluate the entropy generation terms (S_ψ and S_θ). Average Nusselt number ( Nub¯) and average dimensionless spatial temperature ( θ^) are also evaluated via the finite element basis sets.

Based on larger Nub¯, larger θ^ and optimal S_total values, containers from each class are preferred as follows: Class 1: parallelogrammic and square, Class 2: trapezoidal type 1 and Class 3: convex (larger θ^, optimum S_total) and concave (larger Nub¯). Containers with curved walls lead to enhance the thermal performance or efficiency of convection processes.

Comparison of entropy generation, intensity of thermal mixing ( θ^) and average heat transfer rate give a clear picture for choosing the appropriate containers for processing of fluids at various ranges of Ra. The results based on this study may be useful to select a container (belonging to a specific class or containers with curved or plane walls), which can give optimal thermal performance from the given heat input, thereby leading to energy savings.

This study depicts that entropy generation associated with the convection process can be reduced via altering the shapes of containers to improve the thermal performance or efficiency for processing of identical mass with identical heat input. The comparative study of nine containers elucidates that the values of local maxima of S_ψ (S_ψ,max), S_θ (S_θ,max) and magnitude of S_total vary with change in shapes of the containers (Classes 1–3) at fixed Pr and Ra. Such a comparative study based on entropy generation minimization on optimal heating during convection of fluid is yet to appear in the literature. The outcome of this study depicts that containers with curved walls are instrumental to optimize entropy generation with reasonable thermal processing rates.

The purpose of this paper is to focus on the advanced solution of the parametric non-linear model related to the Rayleigh-Benard laminar flow involved in the modeling of natural thermal convection. This flow is fully determined by the dimensionless Prandtl and Rayleigh numbers. Thus, if one could precompute (off-line) the model solution for any possible choice of these two parameters the analysis of many possible scenarios could be performed on-line and in real time.

In this paper both parameters are introduced as model extra-coordinates, and then the resulting multidimensional problem solved thanks to the space-parameters separated representation involved in the proper generalized decomposition (PGD) that allows circumventing the curse of dimensionality. Thus the parametric solution will be available fast and easily.

Such parametric solution could be viewed as a sort of abacus, but despite its inherent interest such calculation is at present unaffordable for nowadays computing availabilities because one must solve too many problems and of course store all the solutions related to each choice of both parameters.

Parametric solution of coupled models by using the PGD. Model reduction of complex coupled flow models. Analysis of Rayleigh-Bernard flows involving nanofluids.

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.

This study aims to show the importance of natural convection of supercritical fluid in an inclined cavity. The heat transfer performance of natural convection can be improved.

A model of an inclined cavity was set up to simulate the natural convection of supercritical fluid. The influence of inclined angles (30 to approximately 90°) and pressures (8 to approximately 12 MPa) are analyzed. To ascertain flow and heat transfer of supercritical fluid natural convection, this paper conducts a numerical investigation using the lattice Boltzmann method (LBM), which is proven to be precise and convenient.

The results show that the higher heat transfer performance can be obtained with an inclined angle of 30°. It is also presented that the heat transfer performance under pressure of 10 MPa is the best. In addition, common criterion number correlations of average Nusselt number are also fitted.

These study results can provide a theoretical reference for the study of heat transfer of supercritical fluid natural convection in engineering.

Two‐dimensional finite difference calculations are carried out to study laminar flow in longitudinal and transverse convection rolls for three different geometries: a single rectangular cavity with high aspect ratio; a double cavity with a thin separation sheet; and a double cavity with a separation sheet and a honeycomb structure. The equations for the convection‐diffusion in the fluid and conduction in the solid region are solved simultaneously. Good agreement with experimental data is achieved for Rayleigh numbers not too high above the critical value for the onset of secondary convection rolls (Ra < 8500 for vertical and Ra < 2700 for horizontal cavities filled with air). Simulation fails for inclined cavities, where the flow structure is essentially three‐dimensional.

The purpose of this paper is to study thermal (natural) convection in nine different containers involving the same area (area= 1 sq. unit) and identical heat input at the bottom wall (isothermal/sinusoidal heating). Containers are categorized into three classes based on geometric configurations [Class 1 (square, tilted square and parallelogram), Class 2 (trapezoidal type 1, trapezoidal type 2 and triangle) and Class 3 (convex, concave and triangle with curved hypotenuse)].

The governing equations are solved by using the Galerkin finite element method for various processing fluids (Pr = 0.025 and 155) and Rayleigh numbers (10³ ≤ Ra ≤ 10⁵) involving nine different containers. Finite element-based heat flow visualization via heatlines has been adopted to study heat distribution at various sections. Average Nusselt number at the bottom wall ( Nub¯) and spatially average temperature (θ^) have also been calculated based on finite element basis functions.

Based on enhanced heating criteria (higher Nub¯ and higher θ^), the containers are preferred as follows, Class 1: square and parallelogram, Class 2: trapezoidal type 1 and trapezoidal type 2 and Class 3: convex (higher θ^) and concave (higher Nub¯).

The comparison of heat flow distributions and isotherms in nine containers gives a clear perspective for choosing appropriate containers at various process parameters (Pr and Ra). The results for current work may be useful to obtain enhancement of the thermal processing rate in various process industries.

Heatlines provide a complete understanding of heat flow path and heat distribution within nine containers. Various cold zones and thermal mixing zones have been highlighted and these zones are found to be altered with various shapes of containers. The importance of containers with curved walls for enhanced thermal processing rate is clearly established.

The purpose of this paper is to analyze the natural convection of a yield stress fluid in a square enclosure with differentially heated side walls. In particular, the Casson model is considered which is a commonly used model.

The coupled conservation equations of mass, momentum and energy related to the two-dimensional steady-state natural convection within square enclosures are solved numerically by using the Galerkin's weighted residual finite element method with quadrilateral, eight nodes elements.

Results highlight a small degree of the shear-thinning in the Casson fluids. It is shown that the yield stress has a stabilizing effect since the convection can stop for yield stress fluids while this is not the case for Newtonian fluids. The heat transfer rate, velocity and Yc obtained with the Casson model have the smallest values compared to other viscoplastic models. Results highlight a weak dependence of Yc with the Rayleigh number: Yc∼Ra0.07. A supercritical bifurcation at the transition between the convective and the conductive regimes is found.

The originality of the present study concerns the comprehensive and detailed solutions of the natural convection of Casson fluids in square enclosures with differentially heated side walls. It is shown that there exists a major difference between the cases of Casson and Bingham models, and hence using the Bingham model for analyzing the viscoplastic behavior of the fluids which follow the Casson model (such as blood) may not be accurate. Finally, a correlation is proposed for the mean Nusselt number Nu¯.

A few earlier studies presented infeasible heatline trajectories for natural convection within annular domains involving an inner circular cylinder and outer square/circular enclosure. The purpose of this paper is to revisit and illustrate the correct heatline trajectories for various test cases.

Galerkin finite element based methodology and space adaptive grid have been used to simulate natural convective flows within the annular domains. The prediction of heatlines involves derivatives at the nodes, which are evaluated based on finite element basis functions and contributions from neighboring elements.

The heatlines in the earlier work indicate infeasible heat flow paths such as heat flow from one portion to the other of isothermal hot walls and heat flow across the adiabatic walls. Current results illustrate physically consistent heat flow paths involving perpendicularly emerging heatlines from hot to cold walls for conductive transport, long heat flow paths around the closed-loop heatline cells for convective transport and parallel layout of heatlines to the adiabatic walls. Results also demonstrate complex heatlines involving multiple flow vortices and complex flow structures.

Current work translates heatfunctions from energy flux vectors, which are determined by using basis sets. This work demonstrates the expected heatline trajectories for various scenarios involving conductive and convective heat transport within enclosures with an inner hot object as a first attempt, and the results are precursors for the understanding of energy flow estimates.