High-fidelity FEM–PARDISO simulation of entropy generation in MHD nanofluid flow through fractal-structured porous enclosures: Toward advanced thermal management in energy and biomedical systems

1. Introduction

Interactive convective heat transfer refers to the bidirectional exchange of thermal energy between engaging fluid streams or phases, wherein each stream simultaneously influences and responds to the thermal and hydrodynamic behavior of the other. This interdependent mechanism is vital in engineering systems such as heat exchangers, chemical reactors, and energy devices, where efficient thermal interaction directly impacts overall system performance. Introducing fractal-designed barriers adds layers of complexity and precision to these systems. Their irregular, multiscale geometry disrupts laminar flows, enhances turbulence and mixing, and breaks down thermal boundary layers, thereby boosting heat transfer efficiency. These structures are particularly impactful in porous media, biomedical technologies, and advanced thermal management setups, where optimizing flow pathways and energy distribution through fractal patterns enables tailored, high-performance solutions. Numerical research by (Ahmad et al., 2023) investigates the generation of vortices in tri-hybrid nanofluid flows under the effect of localized magnetic fields. The findings show that localized application of magnetic fields contributes to an intensified vortex generation that can significantly enhance the heat transfer characteristics. A separate numerical investigation reveals that introducing multiple localized magnetic fields in hybrid nanofluid flow will intensively increase the occurrence of vortices, optimizing thermal performance. (Ali et al., 2022) studied molecular interaction and magnetic dipole effects in a under duct via the finite volume method, dep-team that magnetic dipole effects tend to enhance the mechanism between solid particles and fluid, and thus heat transfer in fully developed flows of nanofluids. Just as with that, the same authors reported on the effects of the localization of a magnetic field concerning vortex generation in a hybrid nanofluid flow, and they confirmed that a precise placement of such a magnetic field positively influenced vortex intensity and stability of flow. (Kai et al., 2023) studied the manipulation of the vortex formation by altering the strengths of magnetic fields in the hybrid nanofluid flow of Ag-TiO₂. They indicated that thermal energy and magnetic field strength are significant in the optimization of heat transfer, as highlighted in their results. In a pseudo-transient numerical study for dipole interaction in single-phase nanofluid flow in an enclosure, (Ayub et al., 2022) demonstrated an increase in heat transfer efficiency due to the magnetic dipoles. For thermal management of electronic systems, (Elshaer et al., 2024) investigated the insertion of open-cell aluminum foam (OCAF) in phase change materials (PCMs) as thermal regulators, indicating quite high improvements in the thermal performance of electronic systems due to increased heat dissipation. (Lv et al., 2024), for instance, reviewed the different thermal management technologies for spacecraft engine systems, driving home the importance of efficient heat storage systems in space environments. (Chakraborty et al., 2024) also provided a good review process by which different coolants for electronic thermal management were selected using multiple-criteria decision-making (MCDM) approaches. Finally, numerically simulated, phase change material (PCM)-partitioned cavities with fins for photovoltaic cell cooling were studied by (Hong et al., 2024) showing that the configuration indeed reduces temperature rise and enhances system efficiency. Considerable interest has addressed the recent development in studying vortex generation, mixed convection, and nanofluid flows in cavities and ducts. (Ahmad et al., 2022) predicted the generation of new vortices in single-phase nanofluid flows due to the dipole interaction of an external secondary field that can be used to enhance heat transfer. (Z. H. Khan et al., 2024) did a computational study of magnetohydrodynamic (MHD) convective flow in trapezoidal cavity with multiple obstacles using finite element analysis. It was shown that an affecting factor to the flow structure and heat transfer rates is that multiple obstacles are incorporated into systems, with the primary obstacle considered. (Mehrizi et al., 2012) performed an investigation on the mixed convection within a ventilated cavity with a heated obstacle installed, to find out the effects on the cooling performance of nanofluid and outlet port location. (Selimefendigil, 2019) considered mixed convection in a lid-driven cavity filled with single and multi-walled carbon nanotube nanofluids incorporating an inner elliptic obstacle. The results demonstrated that nanofluids containing multi-walled carbon nanotubes exhibited superior heat transfer performance compared to other nanofluids. (Alsabery et al., 2022a) investigated the influence of partial slip conditions on mixed convection within a lid-driven wavy cavity featuring an embedded solid body. Their findings highlighted the significance of partial slip conditions for optimizing heat transfer. Similar studies by (Ismael et al., 2014) also considered mixed convection under partial slip conditions in a lid-driven square cavity, whereby both discovered that effects of altered partial slip conditions differ from one state to another by modifying flow field and thermal distribution. (Rashad et al., 2017) inclined the phenomena of mixed convection within the realm of nanofluids under MHD conditions through the use of porous lid-driven cavities with a heat source-sink, by considering bottom heating being uniform and the non-uniform. The differences observed signified a non-uniform heating scheme could arguably alter the heat transfer rate significantly. However, (Rashad et al., 2019) analyzed the entropy generation and MHD mixed convection in a U-shaped cavity, which had inside heat generation and partial slip. The results indicated that partial slip significantly reduces entropy generation while being efficient in heat transfer. (Alsabery et al., 2022b) conducted a study on the convective flow and the generation of entropy in a wavy cubic container filled with nanofluids and a membranous embedded cylinder. These have proven that wavy walls increased thermal performance due to an increase in surface area for heat transfer. Finally, (Zarei et al., 2022) described the heat transfer in a square cavity filled with nanofluids concerning the influence of sinusoidal wavy walls with varied wavelengths and amplitudes. They synthesized that the wavelength and amplitude of the wavy walls have a substantial effect on the boreal thermal behavior and the flow structure.

Recent studies have broadly addressed the issue of convection and heat transfer in nanofluids and perforated geometries. (Shao et al., 2024) explores the simultaneous numerical analysis of thermal and entropy characteristics of Al₂O₃-H₂O nanofluid in a porous diamond-shaped container with an L-shaped obstacle and demonstrates improved thermal performance because of the obstacle.

(Nayak et al., 2023) investigated natural convection of aluminum oxide–water nanofluid within a hexagonal enclosure embedded with cold diamond-shaped obstacles, incorporating the influence of a spatially periodic magnetic field. The periodic magnetic field was found to enhance heat transfer as a function of the flow structure it caused. (Aly et al., 2023) analyzed thermosolutal convection in heterogeneous porous cavities comprising dual-rotating circular cylinders. According to their findings, the coupling between the rotation of the cylinders has a steeper influence on the mixing and flow patterns. Another study was carried about on natural convection in an H-shaped cavity filled with nanofluid and having three center gates via the ISPH technique by (Aly, 2020), and ultimately accentuated that fluid-particle mixing has improved and convective heat transfer is also enhanced. (Rashid et al., 2023) evaluated the effect of a fixed circular obstacle on heat transfer in square cavity nanofluid filled, revealing the way obstacle placement affects the flow dynamic and thermal performance. (Hassen et al., 2024) analytically investigated EHD (electrohydrodynamic) flow and heat transfer of hybrid nanofluid in a free surface cavity fitted with an internal hot obstacle and concluded that it improves due to superior thermophysical properties concerning hybrid nanofluids for heat transfer. (Akrour et al., 2023) conducted a study on electrothermo-convection in an inclined cavity, filled with a dielectrically fluid emphasizing the inclining of the cavity with regard to its heat transfer efficiency. The numerical treatment of electrothermo-convection in a square enclosure, energized with a hot solid body, was done by (Hassen et al., 2021) to show that electro-thermo effects increase the thermal distribution in the enclosure. In fact, many have also investigated the influence of the heated obstacle position on the magneto-hybrid nanofluid flow through a lid-driven porous cavity using Cattaneo-Christov heat flux. (Jakeer et al., 2021) highlighted the significance of obstacle placement and applied heat flux on the flow dynamics and thermal performance within porous enclosures. More recently, (Hussain and Geridonmez, 2022) investigated mixed bioconvective flow of an Ag-MgO/water nanofluid influenced by oxytactic bacteria under an inclined periodic magnetic field. Their findings demonstrated that the combined influence of bioconvection and magnetic field notably enhances heat transfer, accompanied by pronounced alterations in bacterial distribution patterns.

(M. S. Khan et al., 2024b) carried out a numerical investigation into a hybrid nanofluid natural convection in a permeable quadrantal enclosure surrounded by an internal source of heat. Based on their findings, they observed that convective heat transfer was enhanced with Rayleigh number and porosity when used experimentally whereas hybrid nanofluids also increased the thermal performance because of the higher thermal conductivity. (M. S. Khan et al., 2024a) conducted a similar study of natural convection of a heat-conducting, non-Newtonian fluid, subjected to a double-diffusive MHD environment between concentric cylinders. They discovered that convection is inhibited with increased thermal conductivity and mass transport reduced in magnetic fields, and that the more intense the thermophysical interaction between objects, the greater the rate of entropy generation. These studies show more significance attested to geometry, magnetic field, properties of fluids in maximizing thermal systems.

Despite the growing interest in nanofluid heat transfer within porous enclosures, the combined effects of magnetic fields, nonlinear porous resistance, and complex internal geometries, particularly multiscale fractal barriers remain insufficiently addressed. Traditional studies often employ simple obstacle shapes, overlooking how geometrical complexity influences convective flow and entropy generation. Motivated by this research gap, the present study focuses on evaluating the thermal and thermodynamic performance of a copper-based nanofluid within a porous enclosure embedded with fractal-shaped barriers under magnetohydrodynamic conditions. This work aims to provide deeper insights into optimizing convective heat transfer and minimizing entropy generation, with potential applications in energy systems, electronic cooling, and biomedical thermal management.

1.1 Mathematical formulation

The present work focuses on the simulation of MHD nanofluid flow under steady-state conditions within a porous medium that includes thermally conductive fractal obstructions of varying scales. The enclosure is subjected to a uniform magnetic field oriented transversely. To accurately capture the fluid behavior in the porous region, the Darcy–Brinkman–Forchheimer model is employed, incorporating both viscous drag and inertial resistance. The heat transfer process is further affected by the presence of internal heat generation and thermal radiation effects. Fig. 1. illustrates the physical layout of the computational domain. For computational efficiency, several well-justified assumptions are introduced to simplify the governing equations.

• The nanofluid flow is presumed to be steady-state, laminar, incompressible, and temporally invariant, thereby eliminating any transient effects from the governing equations.

• Local thermal equilibrium (LTE) is assumed between the suspended nanoparticles and the base fluid, while the no-slip condition is imposed along all solid–fluid interfaces.

• The porous medium is considered homogeneous and isotropic, with momentum transport governed by the generalized Darcy–Brinkman–Forchheimer model, which captures both viscous (Darcy–Brinkman) and inertial (Forchheimer) resistance mechanisms.

• A uniform static magnetic field is externally applied in the transverse direction, i.e., normal to the primary flow axis, introducing Lorentz forces in the momentum balance.

• The thermophysical properties of the nanofluid are treated as temperature-independent constants, except for the fluid density, whose temperature dependence drives buoyancy-induced convection under the framework of the Boussinesq approximation.

• Buoyancy and reference temperature are defined consistently with the non-dimensional temperature $\text{θ}$JK1015_01.eps] to ensure proper scaling of the governing equations.

• Fractal barriers are modeled as impermeable, thermally conductive solids with complex morphological boundaries, characterized by finite thermal conductivity and scale-invariant geometric features.

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Fig. 1.

Physical sketch of the flow.

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This formulation couples electromagnetic, hydrodynamic, and thermal phenomena, enabling a comprehensive exploration of entropy generation, convective heat transfer augmentation, and irreversibility dynamics in multiscale porous systems under multifield interactions.

Based on the generalized Darcy–Brinkman–Forchheimer model, the governing equations for flow in the porous medium are formulated as follows (Radouane et al., 2020) Eqs. (1-7):

(1)

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$

(2)

$\begin{array}{c}\frac{1}{{ϵ}^2}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=-\frac{1}{{\rho }_{nf}}\frac{\partial P}{\partial x}+\frac{{\nu }_{nf}}{ϵ}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)\\ -\frac{{\nu }_{nf}}{K}u-\frac{F_c}{\sqrt{K}}u\left(u\right),\end{array}$

(3)

$\begin{array}{c}\frac{1}{{ϵ}^2}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{1}{{\rho }_{nf}}\frac{\partial P}{\partial y}+\frac{{\nu }_{nf}}{ϵ}\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)-\frac{{\nu }_{nf}}{K}v\\ -\frac{F_c}{\sqrt{K}}v\left(u\right)+{\beta }_{nf}g\left(T-T_c\right)-\frac{{\sigma }_{nf}}{{\rho }_{nf}}{B}_0^{2}v\end{array}$

(4)

$\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)={\text{α}}_{\text{nf}}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)$

The permeability $\text{K}$ is related to the porosity ϵ through the Kozeny–Carman model:

(5)

$K=\frac{{ϵ}^3{d}_m^2}{150{(1-ϵ)}^2},$

This relation is substituted into the Darcy number:

(6)

$Da=\frac{K}{L^2}=\frac{{ϵ}^3}{150{(1-ϵ)}^2}{\left(\frac{d_m}{L}\right)}^2$

Similarly, the Forchheimer coefficient becomes:

(7)

$F_c=\frac{b}{\sqrt{a}{ϵ}^{\frac{3}{2}}},$

Here, a = 150 and b = 1.75 are empirical constants used to estimate the effective thermal conductivity of a porous medium saturated with nanofluid Eqs. (8-10).

$u=v=0,T_c=0,\left(ForUpperstraightwall\right)$

$u=v=0,\frac{\partial T}{\partial n}=0,(FoFractal-StructuredInternalBarriers)$

(8)

$u=v=0,T=T_h,\left(Forsidecurvewalls\right)$

(9)

$\begin{array}{c}X=\frac{x}{L},Y=\frac{y}{L},U=\frac{uL}{{\alpha }_{bf}},V=\frac{vL}{{\alpha }_{bf}},\\ \theta =\frac{T-T_c}{T_h-T_c},P=\frac{\left(p+{\rho }_{bf}gy\right)L^2}{{\text{ρ}}_{\text{bf}}{\alpha }_{bf}^2}.\end{array}$

(10)

$\begin{array}{c}Ra=\frac{{\beta }_{bf}g\left(T_h-T_f\right)L^3}{{\alpha }_{bf}{\nu }_{bf}},Ha=L{B}_0\sqrt{\frac{{\sigma }_{bf}}{{\mu }_{bf}}},\\ Da=\frac{K}{L^2},Pr=\frac{{\nu }_{bf}}{{\alpha }_{bf}}.\end{array}$

By substituting the aforementioned dimensionless variables into the governing Eqs. (1)–(4), the transformed equations are obtained as follows Eqs. (11-14):

(11)

$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0,$

(12)

$\begin{array}{c}\frac{1}{{ϵ}^2}\frac{{\rho }_{nf}}{{\rho }_{bf}}\left(U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}\right)=-\frac{\partial P}{\partial X}+\frac{1}{ϵ}\frac{{\nu }_{nf}}{{\nu }_{bf}}\text{Pr}\left(\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}\right)\\ -\frac{{\nu }_{nf}}{{\nu }_{bf}}\frac{\text{Pr}}{Da}-\frac{F_c}{\sqrt{Da}}\left(u\right)U,\end{array}$

(13)

$\begin{array}{c}\frac{1}{{ϵ}^2}\left(U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}\right)\\ =-\frac{\partial P}{\partial Y}+\frac{1}{ϵ}\frac{{\nu }_{nf}}{{\nu }_{bf}}\text{Pr}\left(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}\right)-\frac{{\nu }_{nf}}{{\nu }_{bf}}\frac{\text{Pr}}{Da}-\frac{F_c}{\sqrt{Da}}\left(u\right)V\\ +\frac{{\text{β}}_{\text{nf}}}{{\text{β}}_{\text{f}}}\text{Pr}\cdot Ra\theta +\frac{{\text{σ}}_{\text{nf}}}{{\text{σ}}_{\text{nf}}}\frac{{\text{ρ}}_{\text{f}}}{{\text{ρ}}_{\text{nf}}}\frac{\text{Pr }H{a}^2}{ϵ\sqrt{Ra}}V,\end{array}$

(14)

$U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{{\alpha }_{nf}}{{\alpha }_{bf}}\left(\frac{{\partial }^2\theta }{\partial X^2}+\frac{{\partial }^2\theta }{\partial Y^2}\right).$

2. Grid Convergence Analysis

The grid convergence analysis (Table 1) quantifies the evolution of the average Nusselt number (${\text{Nu}}_{\text{Avg}}$) across progressive mesh refinement levels (R.L = 1 to 9). As refinement increases, the element count and degrees of freedom (DOF) escalate, driving a monotonic increase in ${\text{Nu}}_{\text{Avg}}$ from 4.0909 (R.L = 1) to 7.9243 (R.L = 8). At R.L = 8 and R.L = 9, ${\text{Nu}}_{\text{Avg}}$ plateaus (7.9243) despite a 150% rise in mesh density (13,629 to 34,121 elements), confirming attainment of grid independence. This stagnation indicates negligible sensitivity to further refinement, establishing R.L = 8 as the computationally optimal level, balancing accuracy and resource efficiency. The progressively smaller increments in ${\text{Nu}}_{\text{Avg}}$ With successive refinements (R.L = 1 to 8) further validate numerical convergence. The analysis thus demonstrates solution reliability and mesh independence beyond R.L = 8, ensuring robust simulation outcomes. The Gridded Geometry for three cases is shown in Fig. 2.

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Fig. 2.

Gridded geometry for the three different cases.

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3. Method of Solution and Code Validation

The finite element method (FEM), empowered by advances in computational tools, has become a robust technique for solving complex problems involving nonlinearities and intricate geometries. In this study, COMSOL Multiphysics was utilized to solve the governing nonlinear equations. The domain was discretized into finite elements, with velocity, pressure, and temperature fields approximated via suitable interpolation functions. Model setup involved geometry construction, parameter specification, and iterative solution of the system until convergence, defined by |(λⁿ⁺¹ − λⁿ)/λⁿ| ≤ 10⁻⁶. Validation against (Mahapatra et al., 2013) showed strong agreement in Nusselt number profiles Fig. 3, and numerical data, Table 2, confirming the accuracy and reliability of the FEM-based approach. A schematic of the FEM procedure is depicted in Fig. 4.

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Fig. 3.

(a-b) Graphical comparison when uniform temperature, Pr = 0.7, Le = 2.0, A = 1, and Ra = 10³.

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Fig. 4.

Schematic diagram of the finite element method.

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Table 3 provides a comparative analysis between the current numerical results and those of (Mahapatra et al., 2013) under uniform temperature boundary conditions, with Pr = 0.7, Le = 2.0, aspect ratio A = 1, and Nusselt number N = 1.0, across various Rayleigh numbers (Ra). The comparison reveals excellent agreement, with the maximum deviation of approximately 0.03 occurring at Ra = 10⁵, which is within acceptable numerical tolerance. The results also confirm the expected increase in Nusselt number with rising Ra, due to enhanced buoyancy-driven convection. This validation reinforces the accuracy and reliability of the present numerical model.

4. Results and Discussion

This study investigates the behavior of magnetohydrodynamic natural convection within a fractal-structured porous enclosure filled with a Cu–water nanofluid. The flow is assumed to be steady, laminar, incompressible, and Newtonian, occurring through a uniformly permeable, saturated porous medium. High-fidelity finite element simulations using the PARDISO solver were employed to ensure computational accuracy and efficiency. The analysis focuses on key non-dimensional parameters such as the Rayleigh number (Ra), Prandtl number (Pr), the permeability parameter (γ = 1/Da), and the Hartmann number (Ha). Unless otherwise noted, standard values of Ra = 10⁶, Pr = 6.2, Da = 0.01 (γ = 100), Ha = 50, and nanoparticle volume fraction φ = 0.05 are used. The results demonstrate a strong enhancement in heat transfer performance due to the presence of Cu nanoparticles, with notable changes in isotherms and streamline distributions. A detailed parametric analysis is presented in the following figures.

Fig. 5, illustrates the influence of the Hartmann number (Ha) on streamlines for three distinct geometrical configurations. At Ha = 0, in the absence of a magnetic field, the flow is solely driven by buoyancy forces due to temperature gradients, resulting in well-defined, vigorous convective circulation cells. These cells are characterized by strong rotational motion, which enhances convective heat transfer. As Ha increases to 50, the imposed transverse magnetic field introduces Lorentz forces, which act as a resistive damping force opposing fluid motion. This magnetic damping suppresses the kinetic energy of the fluid, weakening the convective vortices and resulting in reduced circulation strength. At Ha = 100, the Lorentz force becomes dominant over buoyancy forces, severely inhibiting fluid motion. Consequently, the streamlines become more aligned and elongated, indicating suppressed recirculation and a transition toward conduction-dominated heat transfer. The flow becomes nearly stagnant across all geometries, demonstrating that higher magnetic field strengths effectively suppress natural convection due to increased electromagnetic resistance. This magnetic suppression effect is more pronounced in geometries with constricted flow paths, highlighting the interplay between geometric confinement and magnetic damping in controlling convective transport.

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Fig. 5.

Illustrate the streamlines at different $Ha$ values.

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Fig. 6, illustrates the variation in isotherm patterns under different Hartmann numbers (Ha), effectively capturing the interplay between magnetic field effects and natural convection within three distinct enclosure geometries. The analysis reveals that the Hartmann number significantly governs the dominance of convective versus conductive heat transfer, with clear implications on thermal distribution and stability. At Ha = 0, the absence of magnetic influence allows buoyancy-driven convection to dominate. This is evident from the highly distorted and crowded isotherms near the heated boundary, indicating intense thermal gradients and strong convective currents. The non-uniform temperature field reflects vigorous thermal mixing, particularly near vertical walls, where rising plumes form due to thermal buoyancy. These features are typical of classical natural convection behavior. As the Hartmann number increases to 50, the imposed magnetic field begins to exert a Lorentz force that resists fluid motion. This damping effect reduces the intensity of convective currents, leading to less distorted and more evenly spaced isotherms. The transition marks a partial suppression of thermal convection, where both conduction and weakened convection contribute to heat transfer. The isotherm pattern becomes smoother, indicating lower velocity magnitudes and reduced fluid agitation. At a high Hartmann number (Ha = 100), the magnetic field dominates the flow physics. The isotherms become almost parallel and symmetric, a clear signature of conduction-dominated heat transfer. The Lorentz force significantly suppresses fluid motion, thereby inhibiting the buoyancy-induced circulation. This suppression stabilizes the thermal field, and the temperature distribution becomes more uniform across the enclosure. Importantly, the extent of this suppression varies with geometry. In Case-1 and Case-2, featuring simpler and more symmetric obstacles, the transition from convection to conduction is smooth and spatially uniform. In contrast, Case-3, with its asymmetric and intricate internal boundary, shows localized suppression zones where convection persists longer due to geometric trapping of hot fluid or weaker Lorentz force alignment. This leads to non-uniform isotherm deformation, revealing that internal complexity affects how magnetic fields modulate heat transfer mechanisms. Overall, the observed trends confirm that increasing Ha stabilizes the thermal field by counteracting buoyancy forces through Lorentz damping, thereby reshaping the energy transport mechanism from convection-driven to conduction-dominated. The geometric dependence of this suppression underlines the critical role of domain configuration in magnetothermal convection problems.

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Fig. 6.

Illustrates the isotherm line at different $Ha$ values.

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The given Table 4, demonstrates the influence of escalating Hartmann numbers (Ha) on the average Nusselt number ($N{u}_{Avg}$), total entropy generation ($S_{Total}$), and average Bejan number (Be_Avg) across three distinct cases. As Ha increases from 0 to 100, all parameters exhibit a consistent decline, reflecting the MHD suppression of convective and irreversible processes. In Case-1, Table 2 highlights reductions of 0.166% in $N{u}_{Avg}$, 0.154% in $S_{Total}$, and 0.095% in Be_Avg, attributed to the Lorentz force dampening fluid motion, thereby weakening thermal convection and entropy generation. Case-2 follows a similar trend, with $N{u}_{Avg}$ and $S_{Total}$ decreasing by 0.146% and 0.136%, respectively, and Be_Avg dropping by 0.095%, signifying restrained buoyancy-driven flow and diminished thermal-frictional irreversibilities. Case-3 shows slightly attenuated effects, with $N{u}_{Avg}$, $S_{Total}$, and Be_Avg declining by 0.124%, 0.112%, and 0.076%, respectively, further confirming the magnetic field’s role in stabilizing hydrodynamic instabilities and reducing advective heat transfer. The progressive reduction in Be_Avg across all cases, as quantified in Table 2, suggests a relative dominance of thermal irreversibility over viscous dissipation, despite an overall decrease in total entropy generation. These trends underscore the magnetic field’s capacity to curtail fluid motion, suppress convective heat transfer, and mitigate system-wide thermodynamic losses, with the extent of damping contingent on case-specific geometric and thermal boundary conditions.

The Fig. 7, presents the influence of the Rayleigh number (Ra) on streamline patterns for three geometries. At low Ra (10³–10²), buoyancy effects are weak, and Case-1 exhibits symmetric, slow-moving counter-rotating vortices, indicating conduction-dominated flow. As Ra increases to 10⁵, buoyancy-driven convection strengthens, enhancing circulation and flow complexity. In Case-2, the star-shaped geometry promotes earlier and stronger convection due to increased surface area and sharp edges that intensify vortex formation. Case-3, with its intricate star-like structure, shows localized flow at low Ra but transitions to highly dynamic circulation at higher Ra, driven by strong buoyancy interaction with complex boundaries. Overall, increasing Ra shifts the system from diffusion-dominated to convection-dominated flow, with geometry playing a critical role. Case-3 achieves the most efficient mixing and heat transfer at high Ra due to its irregular shape and enhanced fluid interaction.

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Fig. 7.

Effect of Rayleigh number on streamlines.

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Fig. 8, illustrates the effect of Rayleigh number (Ra) on isotherm distributions across three geometries. At low Ra (10³), heat transfer is conduction-dominated, with nearly parallel and undistorted isotherms. As Ra increases to 10⁴, buoyancy-driven convection intensifies, causing noticeable isotherm bending near the heated bottom wall and around internal obstacles. At Ra = 10⁵, convection becomes dominant, producing significant isotherm distortions and enhancing thermal mixing. Case-3, with its complex star-shaped boundary, shows the most pronounced local temperature variations due to strong fluid recirculation. In contrast, Case-1 retains more uniform gradients due to its simpler triangular shape. Case-2 and Case-3 exhibit disrupted heat paths and localized hot/cold zones, driven by their geometrical complexity. Overall, increasing Ra transitions the system from conduction to convection-dominated heat transfer, with internal geometry playing a key role in modulating thermal distortion, circulation strength, and localized heat transfer enhancement.

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Fig. 8.

Effect of rayleigh number on isothermlines.

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The Fig. 9, shows the influence of the Darcy number (Da) on streamline patterns across three geometries. At high permeability (Da = 10⁻²), buoyancy forces dominate over porous resistance, resulting in strong convective flow and well-defined vortices around the inner boundaries. As Da decreases to 10⁻³, reduced permeability increases flow resistance, weakening convection and reducing vortex strength. At very low permeability (Da = 10⁻⁵), porous drag suppresses fluid motion almost entirely, shifting the system toward conduction-dominated heat transfer. Among the cases, Case-3 exhibits the greatest flow restriction due to its complex star-like geometry, which amplifies local resistance and reduces circulation. In contrast, Cases 1 and 2 show more gradual reductions in flow strength due to their simpler forms. Overall, decreasing Da significantly dampens convective motion, and the degree of suppression is further modulated by the inner geometry. The results confirm that flow behavior in porous media is strongly governed by the combined effects of permeability and internal obstacle complexity.

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Fig. 9.

Effect of Darcy number on streamline.

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Fig. 10, illustrates the effect of Darcy number (Da) on isotherm distributions for three geometries with different internal obstacles. At high permeability (Da = 10⁻²), convection dominates, resulting in bent and distorted isotherms near the heated boundaries. As Da decreases to 10⁻⁵, porous resistance suppresses fluid motion, and conduction becomes dominant, yielding smoother, nearly parallel isotherms. In Case-1 (triangular obstacle), the thermal field remains relatively uniform, with isotherms transitioning smoothly from distorted (convective) to aligned (conductive) as Da decreases. Case-2, with its star-shaped obstacle, introduces surface roughness that enhances local thermal disturbances. At high Da, convection leads to isotherm distortion near the tips, while lower Da restores symmetry through conduction. Case-3, featuring the most complex geometry, exhibits the greatest influence on the thermal field. At Da = 10⁻², the intricate shape enhances convective mixing and produces steep thermal gradients. As Da reduces, convection diminishes, and conduction prevails, smoothing the isotherm distribution around the irregular boundary. Overall, decreasing Da reduces convective strength, increases conduction dominance, and amplifies the role of internal geometry in shaping thermal behavior across all configurations.

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Fig. 10.

Effect of darcy number on isothermlines.

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The given results in Table 5, illustrate the impact of increasing Rayleigh number (Ra) on heat transfer, entropy generation, and the Bejan number in three different cases. As Ra increases from ${10}^1$ to ${10}^6$, the buoyancy-driven convection becomes more dominant, enhancing heat transfer and flow circulation. This is reflected in the rise of the average Nusselt number ($N{u}_{Avg}$across all cases, with the highest increase (16.40%) in Case-1, followed by 14.20% in Case-2, and 12.84% in Case-3. The total entropy generation ($S_{Total}$) also rises due to intensified thermal and fluid irreversibilities, showing a maximum increase of 21.82% in Case-1, with lower but still significant increases in Case-2 (19.23%) and Case-3 (17.47%). Meanwhile, the Bejan number (Be_Avg) decreases in all cases, with the highest reduction (16.62%) in Case-1, indicating that as Ra increases, heat transfer irreversibilities become more dominant over fluid friction irreversibilities. The trend across cases suggests that while higher Ra enhances convective heat transfer, it also increases entropy generation, leading to a trade-off between thermal performance and thermodynamic efficiency. The differences among the cases may arise from variations in boundary conditions, fluid properties, or geometrical factors, influencing the extent to which natural convection affects the system.

Fig. 11, presents the effect of porosity (ϵ = 0.2, 0.6, 1) on streamline patterns across the three geometries. At low porosity (ϵ = 0.2), limited permeability restricts fluid motion, resulting in weak, localized vortices near the heated wall. Case-1 forms a single dominant vortex, while Cases 2 and 3 show more complex but constrained circulations due to geometrical barriers. As porosity increases to ϵ = 0.6, permeability improves, enabling stronger convective flow. Vortex size and circulation intensity increase across all cases. Case-1 displays broader recirculation zones near the heated surface, whereas Cases 2 and 3 exhibit enhanced vortex interactions around their respective star-shaped structures, promoting better fluid mixing. At full porosity (ϵ = 1), the domain behaves like a pure fluid region. Case-1 shows an organized flow dominated by a strong single vortex. Case-2 generates multiple symmetric vortices around the obstacle, while Case-3 demonstrates the most complex flow structure with high-velocity circulation and intensified vortex formation due to its intricate geometry. Overall, increasing porosity enhances convective strength and flow complexity, with Case-3 achieving the most vigorous circulation due to its geometrical influence on flow pathways.

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Fig. 11.

Demonstrate the effect of porosity characteristics change between the three cases (Case-1, Case-2, and Case-3).

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Fig. 12, illustrates the impact of porosity (ϵ = 0.2, 0.6, 1) on isotherm distributions across three geometries. At low porosity (ϵ = 0.2), permeability is minimal, leading to conduction-dominated heat transfer. Isotherms are compressed near the heated wall, with minimal distortion. Case-1 shows slight bending around the triangular obstacle, while Cases 2 and 3 exhibit localized thermal gradients around their more complex boundaries. With moderate porosity (ϵ = 0.6), convection becomes more pronounced. Isotherms spread across the domain, and thermal gradients intensify near the heated wall and obstacles. The complex geometries of Cases 2 and 3 promote stronger mixing and thermal distortion, improving overall heat transfer. At full porosity (ϵ = 1), the flow behaves like a pure fluid, and convection fully dominates. Isotherms become smoother and more uniformly distributed. Case-2 shows enhanced thermal activity around the star-shaped boundary, while Case-3 exhibits the most significant isotherm distortion and gradient complexity due to its multi-armed geometry. In summary, increasing porosity shifts the heat transfer mechanism from conduction to convection, with higher porosity yielding better thermal mixing and more uniform temperature fields especially pronounced in geometrically complex configurations like Case-3.

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Fig. 12.

Demonstrate the effect of porosity characteristics change between the three cases (Case-1, Case-2, and Case-3).

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Table 6, demonstrates the influence of porosity parameter (ϵ) on thermal-fluid characteristics, quantified through the average Nusselt number (Nu_Avg), total entropy generation ($S_{Total}$), and average Bejan number (Be_Avg) for three distinct cases. As ϵ escalates from 0.2 to 1.0, Nu_Avg and $S_{Total}$ exhibit consistent upward trends, while Be_Avg declines uniformly across all cases. In Case-1, Nuₐᵥg and $S_{Total}$rise by 2.31% and 2.79%, respectively, signifying amplified convective heat transfer and entropy production due to enhanced fluid mobility in a more porous medium; Be_Avg decreases by 0.98%, reflecting heightened thermal irreversibility dominance. Case-2 shows marginal increments (Nu_Avg: 0.21%, $S_{Total}$: 0.30%) and a minor Be_Avg reduction (0.18%), indicating subdued porosity effects. Case-3 displays stronger responses, with Nu_Avg and $S_{Total}$ increasing by 1.86% and 2.24%, respectively, and Be_Avg declining by 0.71%, reaffirming porosity-driven thermal enhancement and entropy growth. Collectively, higher ϵ elevates convective transport and system-wide entropy but diminishes the relative contribution of fluid friction irreversibility in entropy generation.

The Fig. 13, illustrates the influence of porosity ε on the velocity components, shear rate, and rotation rate along the arc length of a porous medium. As $\text{ε}$ increases from 0.2 to 2.0, the u-velocity exhibits intensified peaks and more pronounced flow reversals, indicating enhanced permeability which allows greater axial fluid acceleration and deceleration. The v-velocity becomes increasingly negative with higher ε, suggesting stronger transverse flow due to reduced resistance in the porous matrix. The shear rate increases significantly with porosity, especially near the inlet and central regions, reflecting steeper velocity gradients and elevated viscous dissipation as fluid flows more freely through the medium. Similarly, the rotation rate rises with increasing ε, indicating greater local vorticity and fluid element rotation facilitated by higher permeability. These trends underscore that increasing porosity reduces flow resistance, thereby intensifying fluid motion, deformation, and mixing characteristics in porous structures relevant to applications in biofluid transport, filtration, and enhanced heat/mass transfer systems.

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Fig. 13.

(a-d) Illustrates the influence of porosity ε on the velocity components, shear rate, and rotation rate along the arc length of a porous medium.

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Fig. 14, The figure illustrates the impact of the Hartmann number (Ha) on the flow characteristics namely, u-velocity, v-velocity, shear rate, and rotation rate along the arc length of a MHD flow domain. As the Hartmann number increases from 0 to 100, a consistent damping effect is observed across all parameters due to the influence of the Lorentz force, which acts as a resistive body force opposing the motion of the conducting fluid. The u-velocity profile shows a significant reduction in peak magnitude with increasing Ha, indicating that the axial flow is strongly suppressed under the influence of a magnetic field. Similarly, the v-velocity, which represents the transverse motion, decreases in intensity, suggesting that the magnetic field inhibits secondary flow structures and cross-stream circulation. The shear rate diminishes as Ha increases, particularly in regions of high velocity gradients, reflecting reduced momentum transfer and viscous stresses due to magnetic damping. Additionally, the rotation rate, which quantifies local fluid element rotation or vorticity, also decreases with increasing Ha, indicating the magnetic field’s stabilizing role in suppressing vortices and chaotic flow behavior. Overall, the magnetic field exerts a stabilizing and suppressive influence on the flow, leading to reduced velocities, deformation rates, and rotational motion an effect particularly important in applications involving electrically conducting fluids such as blood-based nanofluids or liquid metals under magnetic fields.

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Fig. 14.

(a-d) The figure illustrates the impact of the Hartmann number (Ha) on the flow characteristics.

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The Fig. 15, demonstrates the influence of Darcy number (Da) on flow behavior in a porous medium by analyzing variations in u-velocity, v-velocity, shear rate, and rotation rate along the arc length. As Da increases from 10⁻⁶ to 10⁻², the permeability of the medium improves, enabling greater fluid mobility. At higher Da values, both u and v-velocities increase significantly, indicating enhanced convective flow. Correspondingly, the shear and rotation rates rise, reflecting intensified velocity gradients and stronger rotational behavior due to diminished resistance from the porous matrix. In contrast, at low Da, flow remains weak and highly suppressed, characteristic of conduction-dominated, nearly stagnant conditions. Overall, the results underscore that increasing Darcy number enhances momentum transport and flow activity, with the porous medium transitioning from a diffusion-limited regime to one dominated by convection as permeability improves.

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Fig. 15.

(a-d) Illustrates the impact of the darcy number (Da).

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5. Conclusions

This study conducts a comprehensive numerical analysis of coupled convective heat transfer in an MHD nanofluid-saturated porous enclosure incorporating fractal-shaped internal obstructions. The principal outcomes of the investigation are summarized as follows:

• Enhanced Heat Transfer: Fractal-shaped barriers disturb the flow symmetry and intensify convective circulation, resulting in up to a 16.40% increase in the average Nusselt number (Nu_Avg) for Case-1 compared to simpler geometries.

• Effect of Rayleigh Number: Increasing Ra strengthens buoyancy-driven convection, enhancing heat transfer but also increasing entropy generation. Nu_Avg increases by up to 16.40%, while S_Total rises by up to 21.82%, indicating a trade-off between thermal performance and thermodynamic efficiency.

• Magnetic Field Influence: As Ha increases, Lorentz force suppresses convective motion, shifting heat transport toward conduction. This results in reductions of 0.166% in Nu_Avg, 0.154% in S_Total, and 0.095% in Be_Avg in Case-1.

• Porosity and Darcy Number Effects: Higher porosity and Darcy numbers reduce flow resistance and enhance convective motion. Nu_Avg increases by 2.31%, S_Total by 2.79%, and Be_Avg decreases by 0.98%, suggesting reduced frictional irreversibility.

• Bejan Number Behavior: The Bejan number consistently decreases with increasing Ra and porosity, indicating a shift toward thermal irreversibility dominance.

• Geometrical Complexity Impact: Among the studied configurations, the fractal geometry (Case-1) achieves the highest heat transfer performance but also exhibits the most significant entropy generation, underscoring the balance between enhanced convection and thermodynamic losses.

This study provides a robust computational approach to assess MHD nanofluid-based thermal systems in complex porous enclosures. It offers valuable insights for optimizing heat transfer in advanced engineering applications involving magnetic fields and porous structures.