Abstract

This study proposes a machine learning-enhanced framework to predict and analyze magnetohydrodynamic (MHD) natural convection within nanofluid-filled trapezoidal cavities featuring variable obstacle shapes, wall corrugations, and inclination angles. A comprehensive dataset was generated through Finite Element Method (FEM) simulations covering a wide parametric range, including Rayleigh numbers (10³ - 10⁶) and Hartmann numbers (0 - 50), for Cu-H₂O nanofluids. Key thermal performance metrics Nusselt number (Nu), entropy generation (S_T), and Ecological Coefficient of Performance (ECOP), were extracted and used to train supervised machine learning models: Support Vector Regression (SVR), Decision Tree (DT), and Random Forest (RF). Among them, the RF model achieved superior performance, yielding R² scores of 0.991 (Nu), 0.982 (St), and 0.989 (ECOP), with mean prediction errors under 1.5%. The results show excellent agreement between FEM and ML outputs across diverse configurations, including different obstacle geometries (star, square, triangular), wall undulations (sinusoidal, square, triangular), and inclination angles (15˚, 30˚, 45˚). The integration of ML significantly reduced computational cost while preserving high accuracy, thus demonstrating its viability for rapid prediction and optimization. This hybrid FEM-ML methodology offers a powerful tool for real-time thermal system analysis and design in advanced MHD nanofluid applications.

Keywords

Magnetohydrodynamic (MHD) Convection, Nanofluid, Machine Learning Prediction, Trapezoidal Cavity, Thermal Performance Analysis

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1. Introduction

Natural convection in enclosures has long been a cornerstone topic in thermal engineering, owing to its critical role in a wide range of applications, including solar thermal collectors, electronic cooling systems, and energy-efficient architectural designs [1]. These systems depend heavily on passive heat transfer mechanisms to regulate temperature and ensure operational efficiency. Over the past few decades, the performance of such systems has been significantly improved through the introduction of nanofluids engineered by dispersing nanoparticles in traditional base liquids. These nanoparticle suspensions enhance the thermophysical properties of the fluid, particularly thermal conductivity and heat capacity, thereby improving the rate and uniformity of convective heat transfer within enclosures [2] [3].

The behavior of nanofluid-based convection is not only influenced by the base fluid and nanoparticle type but also by a range of physical and geometric parameters. Studies have demonstrated that internal features such as heated bodies or obstructions, the inclination of the cavity, external magnetic fields, and specific boundary conditions all play crucial roles in shaping the flow and thermal fields [4]-[6]. For example, incorporating internal elliptical or circular obstacles within a square cavity with a wavy top wall has been shown to disrupt conventional flow patterns, resulting in enhanced mixing and heat transfer [7]. These findings highlight the importance of internal geometry and boundary configuration in optimizing convective behavior.

As engineering systems continue to increase in complexity, particularly in microscale and compact heat exchanger designs, the use of nanofluids in non-standard and irregular enclosures has attracted growing research attention. Among the computational tools employed, the Finite Element Method (FEM) stands out for its capacity to handle complex boundaries and heterogeneous materials with high numerical precision [8]. FEM-based simulations allow detailed resolution of the velocity, temperature, and pressure fields in domains where analytical or simplified numerical methods fall short. However, the primary limitation of FEM is its high computational cost, especially when applied to broad parametric studies or real-time optimization tasks.

To address this limitation, Machine Learning (ML) has emerged as a powerful and complementary approach. By training on datasets generated through high-fidelity simulations such as FEM, ML models can serve as surrogate predictors that provide fast and accurate estimations of thermal behavior without the need for solving complex differential equations repeatedly [9]-[11]. In particular, ML has proven effective in predicting nanofluid thermal properties and convective performance across various geometrical and operational settings [12]. Motivated by these developments, the current study proposes a hybrid framework that integrates FEM and ML to analyze the heat transfer characteristics and entropy generation in trapezoidal cavities filled with Cu-H₂O nanofluid and containing internally heated star-shaped, square, and triangular obstacles. The framework also incorporates magnetohydrodynamic (MHD) effects by applying external magnetic fields across the cavity.

The geometry of internal obstacles has a pronounced impact on fluid motion and heat transfer. Shapes such as circular, elliptical, square, and triangular affect vortex formation, thermal gradients, and Nusselt number distribution, all of which contribute to the overall thermal resistance of the system [13]-[17]. Furthermore, the number, placement, and orientation of these obstacles also influence entropy generation and convective flow structure [18] [19]. Another critical factor is the inclination of the cavity itself. Varying the inclination angle modifies the effective direction of buoyancy forces, thereby altering flow intensity, thermal stratification, and energy dissipation due to thermodynamic irreversibility [20]-[24]. Some studies have even identified specific inclination angles that optimize thermal efficiency by enhancing heat transfer while suppressing entropy production [25].

External magnetic fields introduce yet another layer of control, quantified by the Hartmann number. The Lorentz force generated by these fields tends to suppress fluid motion, shifting the heat transfer mechanism from convection-dominated to conduction-dominated regimes [26]-[31]. However, when optimized, MHD effects can lead to improved temperature uniformity and thermal regulation, especially in electrically conducting nanofluids like Cu-H₂O [32].

Beyond internal obstacles and magnetic fields, wall modifications such as introducing wavy or corrugated top boundaries have also been shown to disrupt the thermal boundary layer, generate secondary vortices, and amplify convective mixing [33]-[36]. These geometric perturbations further contribute to enhanced heat transfer and entropy management, underscoring the multifactorial nature of natural convection in nanofluid-filled cavities.

Recent studies have demonstrated the effectiveness of machine learning (ML) in modeling complex heat transfer phenomena. ML has been used to predict radiative nanofluid behavior [37], replace CFD for forced convection with high accuracy and efficiency [35], and estimate Nusselt numbers in natural convection within helical coils [38]. ML models have also shown strong predictive power in systems involving counter-rotating cylinders [39] and microchannel heat sinks with complex geometries [40]. These efforts support the use of ML in capturing nonlinear thermal behaviors, complementing conventional numerical methods.

The present study aims to bridge this research gap by developing a machine learning-driven predictive framework trained on FEM-generated simulation data to model the coupled effects of magnetic field intensity, wall corrugation, obstacle geometry, and inclination angle on heat transfer, entropy generation, and overall thermal performance in Cu-H₂O nanofluid-filled trapezoidal cavities. By combining the precision of FEM with the computational efficiency of ML, this work provides a hybrid platform for accurate, fast, and scalable analysis of MHD natural convection, thereby contributing to the advancement of intelligent thermal management and optimization systems.

2. Materials and Methods

This study explores natural convection and entropy generation in a two-dimensional trapezoidal enclosure filled with a Cu-H₂O nanofluid (φ = 0.02). The trapezoidal cavity features a fixed base angle of γ = 15˚ and is inclined at angles λ = 15˚, 30˚, and 45˚ to investigate the influence of gravitational orientation on thermal and flow characteristics. Two internal heat-generating solid obstacles, designed as star, square, or triangle shapes, are symmetrically embedded at the center of the cavity. The top boundary is geometrically modified using sinusoidal, square, or triangular corrugations to assess the impact of surface undulations on convective flow. Figures 1-3 illustrate these configurations.

Figure 1. Different geometry with sinusoidal corrugation.

Figure 2. Different geometry with square corrugation.

Figure 3. Different geometry with triangular corrugation.

A uniform horizontal magnetic field is applied across the cavity to induce magnetohydrodynamic (MHD) effects. The top and vertical side walls are maintained at a constant cold temperature, while the inclined bottom wall is adiabatic. All walls, including embedded solid surfaces, obey no-slip boundary conditions. The embedded obstacles serve as internal volumetric heat sources, uniformly generating thermal energy.

The governing equations are based on the conservation of mass, momentum, energy, and entropy generation, accounting for nanofluid properties, buoyancy-driven flow, and MHD interactions are described in below ([5] [17] [31] [32]):

For Sinusoidal wavy top wall:

$y=H+a\sin \left(\frac{2\pi fx}{L_t}\right)$ (1)

For Square wavy top wall:

$y=H+\frac{4a}{\pi }\left(\sin \left(\frac{2\pi fx}{L_t}\right)+\frac{1}{3}\sin \left(\frac{6\pi fx}{L_t}\right)+\frac{1}{5}\sin \left(\frac{10\pi fx}{L_t}\right)+\cdots \right)$ (2)

For Triangular wavy top wall:

$y=H+\frac{2a}{\pi }{\sin }^{-1}\left(\sin \left(\frac{2\pi fx}{L_t}\right)\right)$ (3)

here, $H$ : Height of the cavity, $a$ : Amplitude of the wave, $f$ : Frequency (number of waves), $L_t$ : Total length of the top wall, and $x$ is the position along the wall.

Gravitational acceleration g acts vertically downward, but is decomposed into components along the inclined axis:

$g_x=g\sin \left(\lambda \right),g_y=-g\cos \left(\lambda \right)$ (4)

Fluid domain:

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (5)

$\begin{array}{l}{\rho }_{nf}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)\\ =-\frac{\partial p}{\partial x}+{\mu }_{nf}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)+{\rho }_{nf}g{\beta }_{nf}\left(T_{nf}-T_c\right)\sin \left(\lambda \right)-{\sigma }_{nf}{B}_0^{2}u\end{array}$ (6)

${\rho }_{nf}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{\partial p}{\partial y}+{\mu }_{nf}\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)+{\rho }_{nf}g{\beta }_{nf}\left(T_{nf}-T_c\right)\cos \left(\lambda \right)$ (7)

${\rho }_{nf}{C}_{p,nf}\left(u\frac{\partial T_{nf}}{\partial x}+v\frac{\partial T_{nf}}{\partial y}\right)=k_{nf}\left(\frac{{\partial }^2{T}_{nf}}{\partial x^2}+\frac{{\partial }^2{T}_{nf}}{\partial y^2}\right)$ (8)

Star-shaped solid domains:

$k_s\left(\frac{{\partial }^2{T}_s}{\partial x^2}+\frac{{\partial }^2{T}_s}{\partial y^2}\right)+Q=0$ (9)

here, u and v denote velocity components in the x- and y-directions, respectively, and p and T represent pressure and temperature, respectively. The fluid properties are mass density (ρ), thermal conductivity (k), specific heat at constant pressure (C_p), volumetric thermal expansion coefficient (β), and electrical conductivity (σ).

Dimensional Boundary Conditions:

Top Wall:

$u=v=0$ , $T=T_c$ (10)

Inclined Side Wall:

$u=v=0$ , $T=T_c$ (11)

Bottom Wall:

$u=v=0$ , $\frac{\partial T}{\partial x}=0$ (12)

Heated Block:

An internal heat-generation rate of Q = 1 × 10⁶ W/m³ was used for obstacles, consistent with prior studies [37] on thermally active enclosures. Sensitivity analysis showed <3% variation in Nusselt number across typical Q ranges, confirming this value’s suitability. The continuity of temperature and heat flux at fluid-solid interfaces is enforced:

$T_{Fluid}=T_{Solid}$ , $k_{nf}\frac{\partial T}{\partial n_{Fluid}}=k_s\frac{\partial T}{\partial n_{Solid}}$ (13)

${\rho }_{nf}=\left(1-\varphi \right){\rho }_f+\varphi {\rho }_s$ (14)

${\left(\rho c_p\right)}_{nf}=\left(1-\varphi \right){\left(\rho c_p\right)}_f+\varphi {\left(\rho c_p\right)}_s$ (15)

${\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\varphi \right)}^{2.5}}$ (16)

$k_{nf}=k_f\left[\frac{k_s+2k_f-2\varphi \left(k_f-k_s\right)}{k_s+2k_f+\varphi \left(k_f-k_s\right)}\right]$ (17)

${\beta }_{nf}=\frac{\left(1-\varphi \right){\rho }_f{\beta }_f+\varphi {\rho }_s{\beta }_s}{{\rho }_{nf}}$ (18)

Entropy production reflects energy loss from irreversible effects like heat transfer, friction, and MHD forces. In buoyancy-driven MHD flow, entropy is generated through heat transfer, viscous dissipation, and magnetic fields. The local entropy generation due to heat transfer ($S_{ht}$ ) in solid and fluid domains is given by:

$S_{ht}=\frac{k_s}{T_s^2}\left[{\left(\frac{\partial T_s}{\partial x}\right)}^2+{\left(\frac{\partial T_s}{\partial y}\right)}^2\right]+\frac{k_{nf}}{T_{nf}^2}\left[{\left(\frac{\partial T_{nf}}{\partial x}\right)}^2+{\left(\frac{\partial T_{nf}}{\partial y}\right)}^2\right]+\frac{Q_{gen}}{T_s}$ (19)

The local volumetric entropy production due to viscous flow dissipation ($S_{ff}$ ) and external magnetic effects ($S_{mf}$ ) can be described using the following formulas [5] [17] [31] [32]:

$S_{ff}=\frac{{\mu }_{nf}}{T_{nf}}\left[2{\left(\frac{\partial u}{\partial x}\right)}^2+2{\left(\frac{\partial v}{\partial y}\right)}^2+{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2\right]$ (20)

$S_{mf}={\beta }_0^2\frac{{\sigma }_{nf}}{T_{nf}}{\nu }^2$ (21)

To get the non-dimensional governing equations, the following scales are used:

$X=\frac{x}{L},Y=\frac{y}{L},U=\frac{uL}{{\alpha }_f},V=\frac{vL}{{\alpha }_f},\theta =\frac{T-T_c}{T_h-T_c},P=\frac{p{L}^2}{{\mu }_{nf}},\text{Here},{\alpha }_f=\frac{k_f}{{\rho }_f{c}_{p,f}}$ (22)

$Ra=\frac{g{\beta }_{nf}\left(T_h-T_c\right)L^3}{{\nu }_{nf}{\alpha }_f},Pr=\frac{{\nu }_f}{{\alpha }_f},Ha={\beta }_{0}L\sqrt{\frac{{\sigma }_{nf}}{{\mu }_{nf}}},Q^{*}=\frac{Q{L}^2}{k_{nf}\left(T_h-T_c\right)}$ (23)

$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$ (24)

$U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{{\mu }_{nf}}{{\mu }_f}\left(\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}\right)+RaPr\frac{{\rho }_{nf}{\beta }_{nf}}{{\rho }_f{\beta }_f}\theta \sin \left(\lambda \right)-H{a}^{2}U$ (25)

$U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{{\mu }_{nf}}{{\mu }_f}\left(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}\right)+RaPr\frac{{\rho }_{nf}{\beta }_{nf}}{{\rho }_f{\beta }_f}\theta \cos \left(\lambda \right)$ (26)

$U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{k_{nf}}{k_f}\frac{1}{Pr}\left(\frac{{\partial }^2\theta }{\partial X^2}+\frac{{\partial }^2\theta }{\partial Y^2}\right)$ (27)

$\frac{k_s}{k_{nf}}\left(\frac{{\partial }^2{\theta }_s}{\partial X^2}+\frac{{\partial }^2{\theta }_s}{\partial Y^2}\right)+Q^{*}=0$ (28)

Non-Dimensional Boundary Conditions:

Top Wall:

$U=V=0$ , $T=T_c$ (29)

Inclined Side Wall:

$U=V=0$ , $T=T_c$ (30)

Bottom Wall:

$U=V=0$ , $\frac{\partial T}{\partial X}=0$ (31)

Heated Block:

Internal volumetric heat generation applied in the solid domain, Q > 0; continuity of temperature and heat flux at fluid-solid interfaces are enforced:

$T_{Fluid}=T_{Solid}$ , $k_{nf}\frac{\partial T}{\partial N_{Fluid}}=k_s\frac{\partial T}{\partial N_{Solid}}$ (32)

Non-Dimensional Nanofluid Properties:

$\frac{{\rho }_{nf}}{{\rho }_f}=\left(1-\varphi \right)+\varphi \frac{{\rho }_s}{{\rho }_f}$ (33)

$\frac{{\left(\rho c_p\right)}_{nf}}{{\left(\rho c_p\right)}_f}=\left(1-\varphi \right)+\varphi \frac{{\left(\rho c_p\right)}_s}{{\left(\rho c_p\right)}_f}$ (34)

$\frac{{\mu }_{nf}}{{\mu }_f}=\frac{1}{{\left(1-\varphi \right)}^{2.5}}$ (35)

$\frac{k_{nf}}{k_f}=\frac{k_s+2k_f-2\varphi \left(k_f-k_s\right)}{k_s+2k_f+\varphi \left(k_f-k_s\right)}$ (36)

${\beta }_{nf}{\rho }_{nf}=\left(1-\varphi \right){\rho }_f{\beta }_f+\varphi {\rho }_s{\beta }_s$ (37)

The thermal behavior of the chamber under different operating conditions is assessed by analyzing the Nusselt number (Nu) of the heated strips and the average fluid temperature (Θ_av) inside the domain. The definitions of these quantities are as follows:

$\text{Nu}=\frac{L}{L_s}\left[-{\displaystyle \underset{L_0/L}{\overset{2L_0/L}{\int }}{\frac{\partial \Theta }{\partial Y}|}_{Y=0}\text{d}X}-{\displaystyle \underset{3L_0/L}{\overset{4L_0/L}{\int }}{\frac{\partial \Theta }{\partial Y}|}_{Y=0}\text{d}X}\right],{\Theta }_{av}=\frac{1}{A}{\displaystyle \underset{A}{\int }\Theta \text{d}A},$ (38)

here, A represents the non-dimensional surface area of the fluid domain, X and Y are the dimensionless Cartesian coordinates, U and V indicate dimensionless velocity components, and P and Θ are the non-dimensional pressure and temperature of the nanofluid, respectively.

The total entropy generation, expressed as a dimensionless quantity, can be obtained using the following expression:

$S_T=\frac{T_c^2{L}^2}{k_f\Delta T^{2}A}{\displaystyle \underset{A}{\int }\left(S_{ht}+S_{ff}+S_{mf}\right)\text{d}A}$ (39)

where A represents the surface area of the computational domain.

The Ecological Coefficient of Performance (ECOP) can be defined as follows to provide a relative estimate of total entropy production associated with heat transfer:

$\text{ECOP}=\frac{S_T}{Nu}$ (40)

The working fluid is a Cu-H₂O nanofluid with a fixed nanoparticle volume fraction of φ = 0.02, selected for its superior thermal conductivity and MHD compatibility. Simulations are conducted across a Rayleigh number range of 10³ to 10⁶ and Hartmann number (Ha) range from 0 to 50. The working fluid’s properties at a reference temperature of 300  K are presented in Table 1(a). These properties are essential for modeling buoyancy-driven convection, MHD behavior, and entropy generation.

The Galerkin Finite Element Method (FEM) is employed to numerically solve the coupled governing equations under the defined boundary and initial conditions. To complement the Finite Element Method (FEM) simulations and accelerate the prediction of thermal performance metrics, a supervised Machine Learning (ML) framework was developed. The aim was to model the relationship between physical/geometric parameters and target thermal responses, namely, Nusselt number (Nu), Entropy Generation (S_T), and Ecological Coefficient of Performance (ECOP).

To ensure the numerical accuracy of the FEM simulations, a detailed mesh analysis was conducted for different obstacle shapes: square, star, and triangular blocks. As shown in Table 1(b), each mesh configuration includes a mix of triangular and quadrilateral elements, with total element counts ranging from 17,986 to 21,964. The average element quality remains above 0.77 for all cases, with the square block mesh achieving the highest quality at 0.8026.

Table 1. (a): Thermo-physical properties of Water and Cu at T_m = 300K ([5] [17] [31] [32]); (b): Mesh Statistics for Wavy-Top Trapezoidal Cavity with different Heated Blocks.

The minimum element quality and element area ratios confirm that the meshes are sufficiently refined for accurate thermal field resolution. Additionally, the total mesh area remains consistent across cases (3.178 × 10⁻⁷ m²), indicating geometric fidelity. Time-step convergence was validated separately using steps of 0.01, 0.005, and 0.002 s, showing negligible variation in Nusselt number predictions (<1.5%), confirming temporal resolution adequacy.

The dataset for training and testing was generated from 324 FEM simulations, covering diverse combinations of input parameters, including Rayleigh number (Ra), Hartmann number (Ha), nanoparticle volume fraction (φ), obstacle shape, wall corrugation type, and cavity inclination angle (λ). These features spanned broad ranges—Ra = 10³ - 10⁶, Ha = 0 - 50, and λ = 15˚ - 45˚ ensuring comprehensive ML coverage. FEM-derived Nusselt number (Nu), entropy generation (S_T), and energy conversion performance (ECOP) were used as target variables. Prior to training, continuous features were normalized, and categorical features were one-hot encoded. The dataset was randomly split into 80% training (259 cases) and 20% testing (65 cases) to evaluate generalization. Three regression models were employed in this study: Random Forest, Support Vector Regression, and Extreme Gradient Boosting.

1) Support Vector Regression (SVR) using a radial basis function (RBF) kernel;

2) Decision Tree Regression (DT) for rule-based partitioning of the feature space;

3) Random Forest Regression (RF) as an ensemble learning method for robust nonlinear prediction.

Model development was performed using Python’s scikit-learn library. Hyperparameters were optimized via grid search and five-fold cross-validation to prevent overfitting and ensure stability. The trained models were subsequently used to establish a predictive framework that offers a fast and accurate alternative to traditional FEM-based simulations for evaluating the impact of MHD, nanofluid dynamics, and cavity geometry on thermal performance.

3. Validation

To ensure the accuracy and reliability of the present numerical model, a validation study was conducted by comparing the simulation results with the benchmark work of Abdelmalek et al. [17]. The comparison focused on thermal profiles and average Nusselt number values under equivalent conditions. As illustrated in Figure 4, both studies exhibit closely matching isotherm distributions around star-shaped internal obstacles at a Rayleigh number of Ra = 10⁴. The present simulation demonstrates smoother and more symmetric thermal contours, which can be attributed to the use of a finer computational mesh and higher spatial resolution. This strong agreement confirms the validity of the adopted numerical approach in accurately capturing the key thermofluidic behaviors in MHD-assisted nanofluid convection within trapezoidal enclosures.

Table 2 reinforces the validation results, showing that the deviation in average Nusselt numbers remains below 1.5% across the Rayleigh number range of 10³ to 10⁶. This confirms both the accuracy and numerical stability of the present FEM-based model. The findings also emphasize the critical influence of geometric complexity and cavity inclination on heat transfer performance in magnetically actuated nanofluid systems. Notably, configurations combining wall corrugation with non-circular obstacles, such as star or triangular shapes, exhibit improved passive thermal regulation, demonstrating their effectiveness in enhancing natural convection under MHD conditions.

Abdelmalek et al. [17] Present Work

Figure 4. Isotherms for different values of Ra = 10⁴ when N = 8, A = 0.15, ϕ = 2%.

Table 2. Comparison of Nu between present work and Abdelmalek et al. [17].

4. Results and Discussion

This section presents the outcomes of the numerical and data-driven investigations into MHD natural convection in nanofluid-filled trapezoidal cavities. The results are divided into two parts: the first outlines the detailed Finite Element Method (FEM) simulations conducted across various geometric and physical configurations; the second discusses the implementation and evaluation of machine learning models trained on FEM data to predict key thermal performance indicators. Both approaches aim to assess the effects of obstacle shape, wall corrugation, inclination angle, and magnetic field strength on heat transfer (Nu), entropy generation (S_T), and ecological performance (ECOP).

4.1. FEM Simulation Results Analysis

The finite element simulations generated extensive data capturing the influence of geometric and physical parameters on heat transfer and entropy-related behavior in MHD-driven nanofluid convection. The results are structured into three key performance indicators: Nusselt number (Nu), Entropy generation (S_T), and Ecological Coefficient of Performance (ECOP), each analyzed based on variations in obstacle geometry, wall corrugation, inclination angle, Rayleigh number, and Hartmann number. A comprehensive summary of these results is provided in Tables 3-29.

Table 3. Nu for star-shaped obstacles with different Wavy Walls at λ = 15˚.

Table 4. Nu for Square-Shaped Obstacles with different Wavy Walls at λ = 15˚.

Table 5. Nu for Triangular-Shaped Obstacles with different Wavy Walls at λ = 15˚.

Table 6. Nu for Star-Shaped Obstacles with different Wavy Walls at λ = 30˚.

Table 7. Nu for Square-Shaped Obstacles with different Wavy Walls at λ = 30˚.

Table 8. Nu for Triangular-Shaped Obstacles with different Wavy Walls at λ = 30˚.

Table 9. Nu for Star-Shaped Obstacles with different Wavy Walls at λ = 45˚.

Table 10. Nu for Square-Shaped Obstacles with different Wavy Walls at λ = 45˚.

Table 11. Nu for Triangular-Shaped Obstacles with different Wavy Walls at λ = 45˚.

The average Nusselt number, indicative of heat transfer enhancement, exhibited strong sensitivity to both geometry and operating conditions. As seen in Tables 3-11, the highest Nu values were consistently observed for cavities with star-shaped obstacles and sinusoidal wall corrugation, especially at higher Rayleigh numbers (e.g., Ra = 10⁶). This is attributed to the sharper corners and expanded heat exchange surfaces of the star geometry, which promoted vigorous fluid circulation and boundary layer disruption. Increasing the inclination angle up to 30^∘ further intensified buoyancy-driven convection, leading to optimal Nu values. However, at higher Hartmann numbers (e.g., Ha = 50), the Lorentz force induced magnetic damping, thereby suppressing convective motion and decreasing Nu.

Tables 12-20 present the results for total entropy generation across all test configurations. S_T was notably reduced when smooth sinusoidal walls were employed in combination with moderate magnetic fields and lower inclination angles. The star obstacle configuration, while thermally effective, resulted in relatively higher entropy levels at high Ra, due to increased flow complexity and viscous dissipation. Conversely, the triangular obstacle, though less efficient in heat transfer, produced more thermodynamically ordered flow fields with lower entropy production. The data clearly demonstrate that minimizing S_T requires a delicate balance between geometric optimization and MHD control, especially under high thermal gradients.

Table 12. S_T for Star-Shaped Obstacles with different Wavy Walls at λ = 15˚.

Table 13. S_T for Square-Shaped Obstacles with different Wavy Walls at λ = 15˚.

Table 14. S_T for Triangular-Shaped Obstacles with different Wavy Walls at λ = 15˚.

Table 15. S_T for Star-Shaped Obstacles with different Wavy Walls at λ = 30˚.

Table 16. S_T for Square-Shaped Obstacles with different Wavy Walls at λ = 30˚.

Table 17. S_T for Triangular-Shaped Obstacles with different Wavy Walls at λ = 30˚.

Table 18. S_T for Star-Shaped Obstacles with different Wavy Walls at λ = 45˚.

Table 19. S_T for Square-Shaped Obstacles with different Wavy Walls at λ = 45˚.

Table 20. S_T for Triangular-Shaped Obstacles with different Wavy Walls at λ = 45˚.

The system’s thermodynamic efficiency was further evaluated using ECOP, as shown in Tables 21-29. This metric inversely correlates with entropy production and provides insight into the trade-off between heat transfer and irreversibility. Among all configurations, cavities with sinusoidal corrugation, star-shaped obstacles, and an inclination of 30˚ delivered the highest ECOP values, particularly at moderate Ha and Ra = 10⁵. The inclusion of magnetic effects (e.g., Ha = 20) improved ECOP by moderating chaotic fluid motion and stabilizing temperature gradients. In contrast, square and triangular corrugations led to localized thermal zones and entropy hotspots, resulting in comparatively lower ECOP scores.

Table 21. ECOP for Star-Shaped Obstacles with different Wavy Walls at λ = 15˚.

Table 22. ECOP for Square-Shaped Obstacles with different Wavy Walls at λ = 15⁰.

Table 23. ECOP for Triangular-Shaped Obstacles with different Wavy Walls at λ = 15˚.

Table 24. ECOP for Star-Shaped Obstacles with different Wavy Walls at λ = 30˚.

Table 25. ECOP for Square-Shaped Obstacles with different Wavy Walls at λ = 30˚.

Table 26. ECOP for Triangular-Shaped Obstacles with different Wavy Walls at λ = 30˚.

Table 27. ECOP for Star-Shaped Obstacles with different Wavy Walls at λ = 45˚.

Table 28. ECOP for Square-Shaped Obstacles with different Wavy Walls at λ = 45˚.