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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.

The purpose of the paper is to study natural convection within porous square and triangular geometries (design 1: regular isosceles triangle, design 2: inverted isosceles triangle) subjected to discrete heating with various locations of double heaters along the vertical (square) or inclined (triangular) arms.

Galerkin finite element method is used to solve the governing equations for a wide range of modified Darcy number, Da_m = 10⁻⁵–10⁻² with various fluid saturated porous media, Pr_m = 0.015 and 7.2 at a modified Rayleigh number, Ra_m = 10⁶ involving the strategic placement of double heaters along the vertical or inclined arms (types 1-3). Adaptive mesh refinement is implemented based on the lengths of discrete heaters. Finite element based heat flow visualization via heatlines has been adopted to study heat distribution at various portions.

The strategic positioning of the double heaters (types 1-3) and the convective heatline vortices depict significant overall temperature elevation at both Da_m = 10⁻⁴ and 10⁻² compared to type 0 (single heater at each vertical or inclined arm). Types 2 and 3 are found to promote higher temperature uniformity and greater overall temperature elevation at Da_m = 10⁻². Overall, the triangular design 2 geometry is also found to be optimal in achieving greater temperature elevation for the porous media saturated with various fluids (Pr_m).

Multiple heaters (at each side [left or right] wall) result in enhanced temperature elevation compared to the single heater (at each side [left or right] wall). The results of the current work may be useful for the material processing, thermal storage and solar heating applications.

The heatline approach is used to visualize the heat flow involving double heaters along the side (left or right) arms (square and triangular geometries) during natural convection involving porous media. The heatlines depict the trajectories of heat flow that are essential for thermal management involving larger thermal elevation. The mixing cup or bulk average temperature values are obtained for all types of heating (types 0-3) involving all geometries, and overall temperature elevation is examined based on higher mixing cup temperature values.

This paper is aimed to study natural convection in enclosures with curved (concave and convex) side walls for porous media via the heatline-based heat flow visualization approach.

The numerical scheme involving the Galerkin finite element method is used to solve the governing equations for several Prandtl numbers (Pr_m) and Darcy numbers (Da_m) at Rayleigh number, Ra_m = 10⁶, involving various wall curvatures. Finite element method is advantageous for curved domain, as the biquadratic basis functions can be used for adaptive automated mesh generation.

Smooth end-to-end heatlines are seen at the low Da_m involving all the cases. At the high Da_m, the intense heatline cells are seen for the Cases 1-2 (concave) and Cases 1-3 (convex). Overall, the Case 1 (concave) offers the largest average Nusselt number ( Nur¯) at the low Da_m for all Pr_m. At the high Da_m, Nur¯ for the Case 1 (concave) is the largest involving the low Pr_m, whereas Nur¯ is the largest for Case 1 (convex) involving the high Pr_m.

Thermal management for flow systems involving curved surfaces which are encountered in various practical applications may be complicated. The results of the current work may be useful for the material processing, thermal storage and solar heating applications

The heatline approach accompanied by energy flux vectors is used for the first time for the efficient heat flow visualization during natural convection involving porous media in the curved walled enclosures involving various wall curvatures.

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.

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.

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 aim of this paper is to introduce a new technique for convection visualization. This is similar to Bejan's heatlines and is even an exact match to Landau and Lifshitz's energy streamlines for two‐dimensional geometries.

The work benefits from a combination of numerical and analytical tools to show that, in two‐dimensional space, heatlines and energy streamlines are effectively the same. More importantly, the energy flux vectors are tracing both of them accurately; as verified for some cases of free and forced convection problems in this paper.

The new technique is easier to implement compared to the existing counterparts which are available in the literature. More specifically, the advantage of this new technique is that, contrary to heatlines and energy streamlines, it does not require further numerical analysis in addition to solving momentum and energy equations.

Energy flux vectors offer higher resolution compared to existing visualization tools.

This paper aims to perform the lattice Boltzmann simulation of conjugate natural convection heat transfer, heat flow visualization via heatlines approach and entropy generation in a partitioned medium filled with Ag-MgO (15-85%)/water.

The lattice Boltzmann method (LBM) is used to predict the dynamic and thermal behaviors. Experimental correlations for dynamic viscosity and thermal conductivity versus solid volume fraction are used. The study is conducted for the ranges of Rayleigh number 10³ ≤ Ra ≤ 10⁶, the partitioner thickness 0.01 ≤ δ ≤ 0.9, its position 0.15 ≤ X_s ≤ 0.85 and the hybrid nano-suspensions volume fraction 0% ≤ ϕ ≤ 2%.

The effects of varying of controlling parameters on the convective flow patterns, temperature contours, heat transfers, the heatlines and the entropy generation are presented. It has been found that the maximum rate of heat transfer enhancement occurs for low Ra numbers (10³) and is close to 13.52%. The solid thickness d and its horizontal position X_s have a substantial influence on the heat transfer rate, flow structure, heatline, total entropy generation and Bejan number. Besides, the maximum heat transfer is detected for high Ra and δ ≈ 1 and the percentage of augmentation is equal to 65.55% for ϕ = 2%. According to the horizontal position, the heat transfer remains invariant for Ra = 10³ and takes a maximum value near the active walls for Ra ≥ 10⁴. The total entropy generation increases with Ra and decreases with ϕ for Ra = 10⁶. The increase of ϕ from 0 to 2% leads to a reduction in close to 40.76%. For this value of Ra, the entropy is the maximum for δ = 0.4 and X_s = 0.35 and X_s = 0.65%. Moreover, as the Ra increases the Bejan number undergoes a decrease. The Bejan number is the maximum for Ra = 10³ independently to δ and X_s. The superior thermal performance manifests at low Ra and high value of δ independently to the positions of the conducting body.

The originality of this paper is to analyze the hybrid nano-additive effects on the two-dimensional conjugate natural convection in a partitioned medium using the LBM. The experimental correlations used for the effective thermal conductivity and dynamic viscosity give credibility to our study. Different approaches such as heatlines and entropy generation are used.

This paper aims to examine the thermal transmission and entropy generation of hybrid nanofluid filled containers with solid body inside. The solid body is seen as being both isothermal and capable of producing heat. A time-dependent non-linear partial differential equation is used to represent the transfer of heat through a solid body. The current study’s objective is to investigate the key properties of nanoparticles, external forces and particular attention paid to the impact of hybrid nanoparticles on entropy formation. This investigation is useful for researchers studying in the area of cavity flows to know features of the flow structures and nature of hybrid nanofluid characteristics. In addition, a detailed entropy generation analysis has been performed to highlight possible regimes with minimal entropy generation rates. Hybrid nanofluid has been proven to have useful qualities, making it an attractive coolant for an electrical device. The findings would help scientists and engineers better understand how to analyse convective heat transmission and how to forecast better heat transfer rates in cutting-edge technological systems used in industries such as heat transportation, power generation, chemical production and passive cooling systems for electronic devices.

Thermal transmission and entropy generation of hybrid nanofluid are analysed within the enclosure. The domain of interest is a square chamber of size L, including a square solid block. The solid body is considered to be isothermal and generating heat. The flow driven by temperature gradient in the cavity is two-dimensional. The governing equations, formulated in dimensionless primitive variables with corresponding initial and boundary conditions, are worked out by using the finite volume technique with the SIMPLE algorithm on a uniformly staggered mesh. QUICK and central difference schemes were used to handle convective and diffusive elements. In-house code is developed using FORTRAN programming to visualize the isotherms, streamlines, heatlines and entropy contours, which are handled by Tecplot software. The influence of nanoparticles volume fraction, heat generation factor, external magnetic forces and an irreversibility ratio on energy transport and flow patterns is examined.

The results show that the hybrid nanoparticles concentration augments the thermal transmission and the entropy production increases also while the augmentation of temperature difference results in a diminution of entropy production. Finally, magnetic force has the significant impact on heat transfer, isotherms, streamlines and entropy. It has been observed that the external magnetic force plays a good role in thermal regulations.

Hybrid nanofluid is a desirable coolant for an electrical device. Various nanoparticles and their combinations can be analysed. Ferro-copper hybrid nanofluid considered with the help of prevailing literature review. The research would benefit scientists and engineers by improving their comprehension of how to analyses convective heat transmission and forecast more accurate heat transfer rates in various fields.

Due to its helpful characteristics, ferrous-copper hybrid nanofluid is a desirable coolant for an electrical device. The research would benefit scientists and engineers by improving their comprehension of how to analyse convective heat transmission and forecast more accurate heat transfer rates in cutting-edge technological systems used in sectors like thermal transportation, cooling systems for electronic devices, etc.

Entropy generation is used for an evaluation of the system’s performance, which is an indicator of optimal design. Hence, in recent times, it does a good engineering sense to draw attention to irreversibility under magnetic force, and it has an indispensable impact on investigation of electronic devices.

An efficient numerical technique has been developed to solve this problem. The originality of this work is to analyse convective energy transport and entropy generation in a chamber with internal block, which is capable of maintaining heat and producing heat. Effects of irreversibility ratio are scrutinized for the first time. Analysis of convective heat transfer and entropy production in an enclosure with internal isothermal/heat generating blocks gives the way to predict enhanced heat transfer rate and avoid the failure of advanced technical systems in industrial sectors.

This study aims to undertake a comprehensive examination of heat transfer by convection in porous systems with top and bottom walls insulated and differently heated vertical walls under a magnetic field. For a specific nanofluid, the study aims to bring out the effects of different segmental heating arrangements.

An existing in-house code based on the finite volume method has provided the numerical solution of the coupled nondimensional transport equations. Following a validation study, different explorations include the variations of Darcy–Rayleigh number (Ra_m = 10–10⁴), Darcy number (Da = 10^–5–10^–1) segmented arrangements of heaters of identical total length, porosity index (ε = 0.1–1) and aspect ratio of the cavity (AR = 0.25–2) under Hartmann number (Ha = 10–70) and volume fraction of φ = 0.1% for the nanoparticles. In the analysis, there are major roles of the streamlines, isotherms and heatlines on the vertical mid-plane of the cavity and the profiles of the flow velocity and temperature on the central line of the section.

The finding of a monotonic rise in the heat transfer rate with an increase in Ra_m from 10 to 10⁴ has prompted a further comparison of the rate at Ra_m equal to 10⁴ with the total length of the heaters kept constant in all the cases. With respect to uniform heating of one entire wall, the study reveals a significant advantage of 246% rate enhancement from two equal heater segments placed centrally on opposite walls. This rate has emerged higher by 82% and 249%, respectively, with both the segments placed at the top and one at the bottom and one at the top. An increase in the number of centrally arranged heaters on each wall from one to five has yielded 286% rate enhancement. Changes in the ratio of the cavity height-to-length from 1.0 to 0.2 and 2 cause the rate to decrease by 50% and increase by 21%, respectively.

Further research with additional parameters, geometries and configurations will consolidate the understanding. Experimental validation can complement the numerical simulations presented in this study.

This research contributes to the field by integrating segmented heating, magnetic fields and hybrid nanofluid in a porous flow domain, addressing existing research gaps. The findings provide valuable insights for enhancing thermal performance, and controlling heat transfer locally, and have implications for medical treatments, thermal management systems and related fields. The research opens up new possibilities for precise thermal management and offers directions for future investigations.