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Thermo-magnetic convective flow analysis under the impact of thermal radiation for heat and entropy generation phenomena is an active research field for understanding the efficiency of thermodynamic systems in various engineering sectors. This study aims to examine the characteristics of convective heat transport and entropy generation within an inverted T-shaped porous enclosure saturated with a hybrid nanofluid under the influence of thermal radiation and magnetic field.

The mathematical model incorporates the Darcy-Forchheimer-Brinkmann model and considers thermal radiation in the energy balance equation. The complete mathematical model has been numerically simulated through the penalty finite element approach at varying values of flow parameters, such as Rayleigh number (Ra), Hartmann number (Ha), Darcy number (Da), radiation parameter (Rd) and porosity value (e). Furthermore, the graphical results for energy variation have been monitored through the energy-flux vector, whereas the entropy generation along with its individual components, namely, entropy generation due to heat transfer, fluid friction and magnetic field, are also presented. Furthermore, the results of the Bejan number for each component are also discussed in detail. Additionally, the concept of ecological coefficient of performance (ECOP) has also been included to analyse the thermal efficiency of the model.

The graphical analysis of results indicates that higher values of Ra, Da, e and Rd enhance the convective heat transport and entropy generation phenomena more rapidly. However, increasing Ha values have a detrimental effect due to the increasing impact of magnetic forces. Furthermore, the ECOP result suggests that the rising value of Da, e and Rd at smaller Ra show a maximum thermal efficiency of the mathematical model, which further declines as the Ra increases. Conversely, the thermal efficiency of the model improves with increasing Ha value, showing an opposite trend in ECOP.

Such complex porous enclosures have practical applications in engineering and science, including areas like solar power collectors, heat exchangers and electronic equipment. Furthermore, the present study of entropy generation would play a vital role in optimizing system performance, improving energy efficiency and promoting sustainable engineering practices during the natural convection process.

To the best of the authors’ knowledge, this study is the first ever attempted detailed investigation of heat transfer and entropy generation phenomena flow parameter ranges in an inverted T-shaped porous enclosure under a uniform magnetic field and thermal radiation.

This study aims to analyze the impact of fractional derivatives on heat transfer and entropy generation during transient free convection inside various complex porous enclosures, such as triangle, L-shape and square-containing wavy surfaces. These porous enclosures are saturated with Cu-water nanofluid and subjected to the influence of a uniform magnetic field.

In the present study, Darcy’s model is used for the momentum transport equation in the porous matrix. Additionally, the Caputo time fractional derivative is introduced in the energy equation to assess the heat transfer phenomenon. Furthermore, the total entropy generation has been computed by combining the entropy generation due to fluid friction (S_ff), heat transfer (S_ht) and magnetic field (S_mf). The complete mathematical model is further simulated using the penalty finite element method, and the Caputo time derivative term is approximated using the L1 scheme. The study is conducted for various ranges of the Rayleigh number (102≤Ra≤104), Hartmann number (0≤Ha≤20) and fractional order parameter (0<α<1) with respect to time.

It has been observed that the fractional order parameter α governs the characteristics of entropy generation and heat transfer within the selected range of parameters. The Bejan number associated with heat transfer (Be_ht), fluid friction (Be_ff) and magnetic field (Be_mf) further demonstrate the dominance of flow irreversibilities. It becomes evident that the initial evolution state of streamlines, isotherms and local entropy varies according to the choice of α. Additionally, increasing Ra values from 10² to 10⁴ shows that the heat transfer rate increases by 123.8% for a square wavy enclosure, 7.4% for a triangle enclosure and 69.6% for an L-shape enclosure. Moreover, an increase in the value of Ha leads to a reduction in heat transfer rates and entropy generation. In this case, Bemf→1 shows the dominance of the magnetic field irreversibility in the total entropy generation.

Recently, fractional-order models have been widely used to express numerous physical phenomena, such as anomalous diffusion and dispersion in complex viscoelastic porous media. These models offer a more accurate representation of physical reality that classical models fail to capture; this is why they find a broad range of applications in science and engineering.

The fractional derivative model is used to illustrate the flow pattern, heat transfer and entropy-generating characteristics under the influence of a magnetic field. Furthermore, to the best of the author’s knowledge, a fractional-derivative-based mathematical model for the entropy generation phenomenon in complex porous enclosures has not been previously developed or studied.

The purpose of the study was to construct a Leadership Mindset Scale (LMS) and to assess its reliability and construct validity. Participants were 100 employees in a variety of leadership and non-leadership positions at various organizations in three states. An item and factor analysis on the 13 LMS items led to a scale with 11 items (Cronbach α = .80). A Principal Axis Factor analysis with Promax rotation suggested three factors: Leadership Mindset Teachability (LMS-T), a belief in leadership teachability; Leadership Mindset Improvability (LMS-I), a belief in leadership improvability over time; and Leadership Mindset Predictability (LMS-P), a belief that leadership cannot be predicted at an early age. Convergent validity of LMS-Total and Teachability was evidenced by significant correlations with the implicit theories of intelligence and anxiety scales, and developmental leadership and transactional leadership scales. Divergent validity was evidenced by a non-significant correlation with social desirability. The results suggest that the LMS measures a construct different from those of other leadership scales used in the study. The LMS can be helpful in leadership training programs to promote a growth mindset about the trainability of leadership skills.

The purpose of this study is to derive a physics based complete-flux approximation scheme by solving suitable nonlinear boundary value problems (BVP) for finite volume method for mixed convection problems, to study the mixed convection phenomenon inside partially and differentially heated cavity for various sets of flow parameters. And, to study the impact of source terms on the cell-face fluxes for various sets of flow parameters for mixed convection problems.

The governing equations have been discretized by finite volume method on a staggered grid, and the cell-face fluxes have been approximated by local nonlinear BVP. The cell-face flux is represented as a sum of homogeneous and an inhomogeneous flux term. The proposed flux approximation is fully physics based as it considers the pressure gradient term, thermal buoyancy term and the other source terms in the cell-face flux calculation. The scheme comes out to be second order accurate in space tested with known solution. Also, the scheme has been implemented to study the mixed convection problems in a partially and differentially heated cavity.

The numerical order of convergence study shows that the proposed scheme is of second order in space. The scheme is first validated with existing benchmark literature for the mixed convection problem. As the proposed cell-face flux approximation scheme is a homogeneous part and an inhomogeneous part, this study quantifies the influence of the several source terms on the cell-face flux with the help of the inhomogeneous flux term. Then, the mixed convection problems in a partially and differentially heated cavity has been studied. Also, the effect of heat transfer rate at the hot wall is studied for different height of the heat source with different directions of wall movement. The numerical findings show that the local Nusselt number at the left wall is higher when the top and bottom walls move in opposite directions compared to when they move in the same direction, regardless of the Richardson number. In addition, the heat transfer rate at the hot portion of the left wall increases uniformly as the Richardson number decreases when the walls move in opposite directions. However, when the top and bottom walls move in the same direction, the increase in heat transfer rate is not uniform due to the formation of secondary re-circulation of the fluid near the bottom wall.

In this work, the flux approximation is conducted through local nonlinear BVPs, an approach that, to the authors’ knowledge, has not been previously applied to mixed convection problems. One of the strong advantages of the proposed scheme is that it can quantify the influence of source terms, namely, pressure gradient, cross-flux and the thermal buoyancy force, on the cell face fluxes required in the finite volume methods. Furthermore, the study explores mixed convection in a partially and differentially heated cavity, which is also novel within the current literature. These factors contribute to the originality and scientific value of the research.

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 investigate numerically the impact of the three-dimensional convective nanoliquid flow on a rotating frame embedded in the non-Darcy porous medium in the presence of activation energy. The cross-diffusion effects, i.e. Soret and Dufour effects, and heat generation are included in the study. The convective heating condition is applied on the bounding surface.

The control model consisted of a system of partial differential equations (PDE) with boundary constraints. Using suitable similarity transformation, the PDE transformed into an ordinary differential equation and solved numerically by the Runge–Kutta–Fehlberg method. The obtained results of velocity, temperature and solute concentration characteristics plotted to show the impact of the pertinent parameters. The heat and mass transfer rate and skin friction are also calculated.

It is found that both Biot numbers enhance the heat and mass distribution inside the boundary layer region. The temperature increases by increasing the Dufour number, while concentration decreases by increasing the Dufour number. The heat transfer is increased up to 8.1% in the presence of activation energy parameter (E). But, mass transfer rate declines up to 16.6% in the presence of E.

The applications of combined Dufour and Soret effects are in separation of isotopes in mixture of gases, oil reservoirs and binary alloys solidification. The nanofluid with porous medium can be used in chemical engineering, heat exchangers and nuclear reactor.

This study is mainly useful for thermal sciences and chemical engineering.

The uniqueness in this research is the study of the impact of activation energy and cross-diffusion on rotating nanoliquid flow with heat generation and convective heating condition. The obtained results are unique and valuable, and it can be used in various fields of science and technology.

A porous cavity flow field generates entropy owing to energy and momentum exchange within the fluid and at solid barriers. The heat transport and viscosity effects on fluid and solid walls irreversibly generate entropy. This numerical study aims to investigate convective heat transfer together with entropy generation in a partially heated wavy porous cavity filled with a hybrid nanofluid.

The governing equations are nondimensionalized and the domain is transformed into a unit square. A second-order finite difference method is used to have numerical solutions to nondimensional unknowns such as stream function and temperature. This numerical computation is conducted to explore a wide range of regulating parameters, e.g. hybrid nano-particle volume fraction (σ = 0.1%, 0.33%, 0.75%, 1%, 2%), Rayleigh–Darcy number (Ra = 10, 10², 10³), dimensionless length of the heat source (ϵ = 0.25, 0.50,1.0) and amplitude of the wave (a = 0.05, 0.10, 0.15) for a number of undulations (N = 1, 3) per unit length.

A thorough analysis is conducted to analyze the effect of multiple factors such as thermal convective forces, heat source, surface corrugation factors, nanofluid volume fraction and other parameters on entropy generation. The flow and temperature fields are studied through streamlines and isotherms. The average Bejan number suggested that entropy generation is entirely dominated by irreversibility due to heat transport at Ra = 10, and the irreversibility due to the viscosity effect is severe at Ra = 10³, but the increment in s augments irreversibility due to the viscosity effect over the heat transport at Ra = 10².

To the best of the authors’ knowledge, this numerical study, for the first time, analyzes the influence of surface corrugation on the entropy generation related to the cooling of a partial heat source by the convection of a hybrid nanofluid.

This study aims to investigate entropy generation through natural convection and examine heat transfer properties within a partially heated and cooled enclosure influenced by an angled magnetic field. The enclosure, subjected to consistent heat production or absorption, contains a porous medium saturated with a hybrid nanofluid blend of Cu-Fe₃O₄ and MoS₂-Fe₃O₄.

The temperature and velocity equations are converted to a dimensionless form using suitable non-dimensional quantities, adhering to the imposed constraints. To solve these transformed dimensionless equations, the finite-difference method, based on the MAC (Marker and Cell) technique, is used. Comprehensive numerical simulations address various control parameters, including nanoparticle volume fraction, Rayleigh number, heat source or sink, Darcy number, Hartmann number and slit position. The results are illustrated through streamlines, isotherms, average Nusselt numbers and entropy generation plots, offering a clear visualization of the impact of these parameters across different scenarios.

Results obtained show that the Cu-Fe₃O₄hybrid nanofluid exhibits higher entropy generation than the MoS₂-Fe₃O₄ hybrid nanofluid when comparing them at a Rayleigh number of 10⁶ and a Darcy number of 10^–1. The MoS₂ hybrid nanofluid demonstrates a low permeability, as evidenced by an average Darcy number of 10^–3, in comparison to the Cu hybrid nanofluid. The isothermal contours for a Rayleigh number of 10⁴are positioned parallel to the vertical walls. Additionally, the quantity of each isotherm contour adjacent to the hot wall is being monitored. The Cu and MoS₂ nanoparticles exhibit the highest average entropy generation at a Rayleigh number of 10⁵ and a Darcy number of 10^–1, respectively. When a uniform heat sink is present, the temperature gradient in the central part of the cavity decreases. In contrast, the absence of a heat source or sink leads to a more intense temperature distribution within the cavity. This differs significantly from the scenario where a uniform heat sink regulates the temperature.

The originality of this study is to examine the generation of entropy in natural convection within a partially heated and cooled enclosure that contains hybrid nanofluids. Partially heated corners are essential for optimizing heat transfer in a wide range of industrial applications. This enhancement is achieved by increasing the surface area, which improves convective heat transfer. These diverse applications encompass fields such as chemical engineering, mechanical engineering, surface research, energy production and heat recovery processes. Researchers have been working on improving the precision of heated and cold corners using various methods, such as numerical, experimental and analytical approaches. These efforts aim to enhance the broad utility of these corners further.

Scientists have been conducting trials to find ways to reduce fuel consumption and enhance heat transfer rates to make heating systems more efficient and cheaper. Adding solid nanoparticles to conventional liquids may greatly improve their thermal conductivity, according to the available evidence. This study aims to examine the influence of external magnetic flux on the flow of a mixed convective Maxwell hybrid non-Newtonian nanofluid over a linearly extending porous flat plate. The investigation considers the effects of thermal radiation, Dufour and Soret.

The mathematical model is formulated based on the fundamental assumptions of mass, energy and momentum conservation. The implicit models are epitomized by a set of interconnected nonlinear partial differential equations, which include a suitable and comparable adjustment. The numerical solution to these equations is assessed for approximate convergence by the Runge−Kutta−Fehlberg method based on the shooting technique embedded with the MATLAB software.

The findings are presented through graphical representations, offering a visual exploration of the effects of various dynamic parameters on the flow field. These parameters encompass a wide range of factors, including radiation, thermal and Brownian diffusion parameters, Eckert, Lewis and Soret numbers, magnetic parameters, Maxwell fluid parameters, Darcy numbers, thermal and solutal buoyancy factors, Dufour and Prandtl numbers. Notably, the authors observed that nanoparticles with a spherical shape exerted a significant influence on the stream function, highlighting the importance of nanoparticle geometry in fluid dynamics. Furthermore, the analysis revealed that temperature profiles of nanomaterials were notably affected by their shape factor, while concentration profiles exhibited an opposite trend, providing valuable insights into the behavior of nanofluids in porous media.

A distinctive aspect of the research lies in its novel exploration of the impact of external magnetic flux on the flow of a mixed convective Maxwell hybrid non-Newtonian nanofluid over a linearly extending porous flat plate. By considering variables such as solar radiation, external magnetic flux, thermal and Brownian diffusion parameters and nanoparticle shape factor, the authors ventured into uncharted territory within the realm of fluid dynamics. These variables, despite their significant relevance, have not been extensively studied in previous research, thus underscoring the originality and value of the authors’ contribution to the field.

Analyzing and reducing entropy generation is useful for enhancing the thermodynamic performance of engineering systems. This study aims to explore how polymers and nanoparticles in the presence of Lorentz forces influence the fluid behavior and heat transfer characteristics to lessen energy loss and entropy generation.

The dispersion model is initially used to examine the behavior of polymer additives over a magnetized surface. The governing system of partial differential equations (PDEs) is subsequently reduced through the utilization of similarity transformation techniques. Entropy analysis is primarily performed through the implementation of numerical computations on a non-Newtonian polymeric FENE-P model.

The numerical simulations conducted in the presence of Lorentz forces provide significant insights into the consequences of adding polymers to the base fluid. The findings suggest that such an approach minimizes entropy in the flow region. Through the utilization of polymer-MHD (magnetohydrodynamic) interactions, it is feasible to reduce energy loss and improve the efficiency of the system.

This study’s primary motivation and novelty lie in examining the significance of polymer additives as agents that reduce entropy generation on a magnetic surface. The author looks at how nanofluids affect the development of entropy and the loss of irreversibility. To do this, the author uses the Lorentz force, the Soret effect and the Dufour effect to minimize entropy. The findings contribute to fluid mechanics and thermodynamics by providing valuable insights for engineering systems to increase energy efficiency and conserve resources.