Machine Learning-Driven Prediction and Analysis of Magnetohydrodynamic (MHD) Natural Convection in Nanofluid-Filled Trapezoidal Cavities with Variable Obstacle Shapes, Wall Corrugations, and Inclination Angles

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 (10 ³ - 10 ⁶) and Hartmann numbers (0 - 50), for Cu-H ₂O 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 R ² 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.

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

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

Figure 1. Different geometry with sinusoidal corrugation.

Figure 2

Figure 2. Different geometry with square corrugation.

Figure 3

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

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.

(a)
Name of Property		Symbol	Unit		Water		Cu
Mass Density		ρ	kg∙m−3		996.6		8933
Specific Heat at Constant Pressure		Cp	J∙kg−1∙K−1		4179.2		385
Thermal Conductivity		k	W∙m−1∙K−1		0.6102		401
Volumetric Thermal Expansion Coefficient		β	K−1		26.6 × 10−5		49.9 × 10−6
Electrical Conductivity		σ	S∙m−1		0.05		59.6 × 10-6
Dynamic viscosity		μ	kg∙m−1∙s−1		8.538 × 10−4		-
Prandtl Number		Pr	-		5.856		-
(b)
Parameter	Value Square Blocks			Value Star Blocks		Value Triangular Blocks
Mesh vertices	10,756			11,990		9900
Element type	All elements			All elements		All elements
Triangular elements	18,300			20,368		16,598
Quadrilateral elements	1396			1596		1388
Edge elements	714			798		706
Vertex elements	12			8		10
Total number of elements	19,696			21,964		17,986
Minimum element quality	0.2356			0.1379		0.2297
Average element quality	0.8026			0.7778		0.8004
Element area ratio	0.003044			0.004769		0.003075
Total mesh area	3.178 × 10−7 m2			3.178 × 10−7 m2		3.178 × 10−7 m2

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.

Figure 4

Abdelmalek et al. [17] Present Work Figure 4. Isotherms for different values of Ra = 10 4 when N = 8, A = 0.15, ϕ = 2%.

Table 2

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

Ra	Nanoparticle Volume Fraction (ϕ %)	Present Study	Abdelmalek et al. [17]	Deviation (%)
103	2	1.1470	1.1307	1.44
104	2	2.2944	2.2674	1.19
105	2	4.6379	4.5851	1.15
106	2	8.9586	8.8341	1.41

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

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.79369	0.79369	0.79369	0.79369
	104	2.5581	2.5556	2.5533	2.5524
	105	3.7556	3.6201	3.4447	3.3369
	106	5.7976	5.6067	5.22	4.7166
Square	103	0.8018	0.80179	0.80179	0.80179
	104	2.6423	2.6404	2.6388	2.6381
	105	3.7957	3.6868	3.546	3.4642
	106	5.7179	5.5281	5.1568	4.6751
Triangular	103	0.79861	0.79861	0.79861	0.79861
	104	2.6093	2.6072	2.6054	2.6046
	105	3.7736	3.657	3.5047	3.4136
	106	5.747	5.5586	5.184	4.6923

Table 4

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.84326	0.84326	0.84325	0.84324
	104	2.6913	2.6881	2.6853	2.684
	105	3.9788	3.8277	3.6309	3.5065
	106	6.2373	6.0384	5.6553	5.1434
Square	103	0.85297	0.85296	0.85295	0.85295
	104	2.7849	2.7824	2.7801	2.7792
	105	4.0265	3.9039	3.7424	3.6451
	106	6.1573	5.9757	5.6151	5.1105
Triangular	103	0.84944	0.84943	0.84942	0.84941
	104	2.7491	2.7464	2.744	2.7429
	105	4.001	3.8709	3.698	3.5913
	106	6.1799	5.9944	5.6292	5.1242

Table 5

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.79176	0.79175	0.79175	0.79174
	104	2.9106	2.91	2.9094	2.909
	105	4.2103	4.113	4.0332	3.9932
	106	6.3746	6.1018	5.6317	5.0893
Square	103	0.79783	0.79783	0.79783	0.79783
	104	2.9817	2.9813	2.9808	2.9806
	105	4.2685	4.2055	4.1485	4.1193
	106	6.2743	6.007	5.565	5.0441
Triangular	103	0.79608	0.79607	0.79607	0.79607
	104	2.9563	2.9558	2.9553	2.955
	105	4.2449	4.1709	4.106	4.0729
	106	6.3262	6.0564	5.6118	5.073

Table 6

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.79369	0.79369	0.79369	0.79369
	104	2.5573	2.555	2.5531	2.5523
	105	3.7274	3.5914	3.4249	3.3284
	106	5.916	5.5713	5.1574	4.6587
Square	103	0.80179	0.80179	0.80179	0.80179
	104	2.6417	2.64	2.6386	2.6381
	105	3.768	3.6621	3.5307	3.4579
	106	5.7008	5.4812	5.1053	4.6249
Triangular	103	0.79861	0.79861	0.79861	0.79861
	104	2.6086	2.6068	2.6052	2.6046
	105	3.746	3.6311	3.4879	3.4066
	106	5.7559	5.5104	5.1294	4.6408

Table 7

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.84326	0.84325	0.84325	0.84324
	104	2.6903	2.6874	2.685	2.6839
	105	3.9427	3.7923	3.606	3.4952
	106	6.3086	5.9994	5.5697	5.0505
Square	103	0.85297	0.85296	0.85295	0.85294
	104	2.784	2.7818	2.7799	2.7791
	105	3.9912	3.8722	3.7221	3.6364
	106	6.1538	5.9156	5.5258	5.0209
Triangular	103	0.84943	0.84943	0.84942	0.84941
	104	2.7482	2.7458	2.7437	2.7428
	105	3.9661	3.8382	3.6762	3.5817
	106	6.2045	5.9456	5.5448	5.0349

Table 8

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.7917	0.79175	0.79175	0.79174
	104	2.9107	2.91	2.9093	2.909
	105	4.2444	4.1236	4.0301	3.9908
	106	6.4336	6.1422	5.6355	5.062
Square	103	0.79783	0.79783	0.79783	0.79783
	104	2.9818	2.9812	2.9808	2.9805
	105	4.2976	4.2133	4.1464	4.1175
	106	6.3298	6.0446	5.5635	5.0332
Triangular	103	0.79607	0.79607	0.79607	0.79607
	104	2.9564	2.9558	2.9552	2.955
	105	4.2761	4.1798	4.1037	4.071
	106	6.3849	6.0977	5.6034	5.053

Table 9

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.79369	0.79369	0.79369	0.79369
	104	2.5563	2.5543	2.5528	2.5522
	105	3.6784	3.5444	3.3946	3.3161
	106	6.0049	5.6188	5.0656	4.5454
Square	103	0.80179	0.80179	0.80179	0.80179
	104	2.6409	2.6395	2.6384	2.638
	105	3.7227	3.6226	3.5074	3.4489
	106	5.7374	5.4407	5.0079	4.5273
Triangular	103	0.79861	0.79861	0.79861	0.79861
	104	2.6077	2.6062	2.605	2.6045
	105	3.6994	3.5892	3.4623	3.3965
	106	5.8366	5.5031	5.0315	4.5368

Table 10

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.84326	0.84325	0.84324	0.84324
	104	2.6889	2.6865	2.6845	2.6837
	105	3.8847	3.7354	3.5679	3.4789
	106	6.5016	6.0543	5.4486	4.8921
Square	103	0.85296	0.85296	0.85295	0.85294
	104	2.7828	2.781	2.7796	2.779
	105	3.9342	3.8219	3.6914	3.6239
	106	6.2066	5.8648	5.3835	4.8712
Triangular	103	0.84943	0.84942	0.84941	0.84941
	104	2.747	2.745	2.7434	2.7427
	105	3.9091	3.7858	3.643	3.5679
	106	6.3178	5.9356	5.4113	4.8823

Table 11

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.79175	0.7917	0.79174	0.79174
	104	2.9108	2.9099	2.9092	2.9089
	105	4.2493	4.1235	4.0242	3.9872
	106	6.4137	6.0815	5.5577	4.9866
Square	103	0.79783	0.79783	0.79783	0.79782
	104	2.9818	2.9812	2.9807	2.9805
	105	4.3094	4.2155	4.1424	4.115
	106	6.2401	5.9625	5.4986	4.9776
Triangular	103	0.79607	0.79607	0.79607	0.79607
	104	2.9564	2.9557	2.9552	2.9549
	105	4.2855	4.1814	4.0991	4.0681
	106	6.3038	6.0068	5.5277	4.9866

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

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	16.898	16.898	16.898	16.898
	104	0.63853	0.63803	0.63759	0.63742
	105	0.021177	0.020913	0.02057	0.02036
	106	0.001488	0.001484	0.001476	0.001465
Square	103	17.074	17.074	17.074	17.074
	104	0.66089	0.66053	0.66021	0.66008
	105	0.02199	0.021781	0.02151	0.021354
	106	0.001548	0.001544	0.001536	0.001526
Triangular	103	17.014	17.014	17.014	17.014
	104	0.65452	0.65412	0.65376	0.65362
	105	0.021962	0.021737	0.021442	0.021267
	106	0.00156	0.001556	0.001549	0.001539

Table 13

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	15.491	15.491	15.491	15.491
	104	0.58966	0.58915	0.58871	0.58853
	105	0.02063	0.020379	0.020053	0.019848
	106	0.001493	0.00149	0.001483	0.001474
Square	103	15.659	15.659	15.659	15.659
	104	0.61093	0.61055	0.61023	0.6101
	105	0.021452	0.021253	0.020993	0.020838
	106	0.001555	0.001552	0.001545	0.001537
Triangular	103	15.602	15.602	15.602	15.602
	104	0.60506	0.60464	0.60428	0.60413
	105	0.021427	0.021213	0.020932	0.02076
	106	0.001567	0.001564	0.001558	0.00155

Table 14

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	17.131	17.131	17.131	17.131
	104	0.71807	0.71797	0.71788	0.71784
	105	0.022264	0.022077	0.021925	0.021849
	106	0.001506	0.001502	0.001493	0.001483
Square	103	17.243	17.243	17.243	17.243
	104	0.7369	0.73683	0.73677	0.73673
	105	0.023127	0.023008	0.0229	0.022845
	106	0.001569	0.001565	0.001557	0.001547
Triangular	103	17.206	17.206	17.206	17.206
	104	0.73178	0.7317	0.73163	0.73159
	105	0.023083	0.022943	0.02282	0.022758
	106	0.00158	0.001576	0.001568	0.001557

Table 15

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	16.898	16.898	16.898	16.898
	104	0.63838	0.63793	0.63755	0.6374
	105	0.021122	0.020857	0.020532	0.020343
	106	0.001487	0.001481	0.001473	0.001463
Square	103	17.074	17.074	17.074	17.074
	104	0.66077	0.66045	0.66018	0.66007
	105	0.021937	0.021733	0.021481	0.021342
	106	0.001546	0.001541	0.001534	0.001525
Triangular	103	17.014	17.014	17.014	17.014
	104	0.65439	0.65403	0.65373	0.65361
	105	0.021909	0.021687	0.02141	0.021253
	106	0.001558	0.001553	0.001546	0.001537

Table 16

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	15.491	15.491	15.491	15.491
	104	0.58951	0.58905	0.58867	0.58852
	105	0.02057	0.02032	0.020012	0.01983
	106	0.001492	0.001486	0.00148	0.001472
Square	103	15.659	15.659	15.659	15.659
	104	0.6108	0.61047	0.6102	0.61009
	105	0.021396	0.021202	0.02096	0.020825
	106	0.001553	0.001549	0.001543	0.001535
Triangular	103	15.602	15.602	15.602	15.602
	104	0.60492	0.60455	0.60424	0.60412
	105	0.02137	0.021161	0.020897	0.020745
	106	0.001565	0.001561	0.001555	0.001548

Table 17

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	17.131	17.131	17.131	17.131
	104	0.7181	0.71797	0.71787	0.71783
	105	0.022329	0.022098	0.021919	0.021844
	106	0.001506	0.001502	0.001493	0.001481
Square	103	17.243	17.243	17.243	17.243
	104	0.73692	0.73683	0.73676	0.73673
	105	0.023182	0.023022	0.022896	0.022842
	106	0.00157	0.001565	0.001556	0.001546
Triangular	103	17.206	17.206	17.206	17.206
	104	0.7318	0.7317	0.73162	0.73159
	105	0.023143	0.02296	0.022815	0.022754
	106	0.001581	0.001576	0.001567	0.001556

Table 18

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	16.898	16.898	16.898	16.898
	104	0.63818	0.63779	0.6375	0.63739
	105	0.021027	0.020765	0.020472	0.020319
	106	0.001485	0.001478	0.001469	0.00146
Square	103	17.074	17.074	17.074	17.074
	104	0.66061	0.66034	0.66014	0.66006
	105	0.021851	0.021658	0.021437	0.021325
	106	0.001544	0.001538	0.001531	0.001523
Triangular	103	17.014	17.014	17.014	17.014
	104	0.65422	0.65392	0.65368	0.65359
	105	0.02182	0.021606	0.021361	0.021234
	106	0.001556	0.00155	0.001542	0.001534

Table 19

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	15.491	15.491	15.491	15.491
	104	0.5893	0.5889	0.58861	0.5885
	105	0.020474	0.020227	0.019949	0.019803
	106	0.00149	0.001484	0.001476	0.001468
Square	103	15.659	15.659	15.659	15.659
	104	0.61063	0.61036	0.61015	0.61008
	105	0.021306	0.021122	0.020912	0.020805
	106	0.001551	0.001546	0.00154	0.001533
Triangular	103	15.602	15.602	15.602	15.602
	104	0.60474	0.60443	0.60419	0.60411
	105	0.021278	0.021076	0.020844	0.020723
	106	0.001564	0.001558	0.001551	0.001544

Table 20

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	17.131	17.131	17.131	17.131
	104	0.71813	0.71796	0.71787	0.71783
	105	0.022339	0.022098	0.021907	0.021837
	106	0.001505	0.0015	0.00149	0.001479
Square	103	17.243	17.243	17.243	17.243
	104	0.73693	0.73683	0.73675	0.73673
	105	0.023205	0.023027	0.022888	0.022837
	106	0.001567	0.001563	0.001554	0.001544
Triangular	103	17.206	17.206	17.206	17.206
	104	0.73182	0.7317	0.73162	0.73159
	105	0.023161	0.022963	0.022807	0.022748
	106	0.001578	0.001573	0.001564	0.001553

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

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.04697	0.04697	0.04697	0.04697
	104	4.0063	4.0054	4.0046	4.0043
	105	177.34	173.1	167.46	163.9
	106	3896.9	3778	3537.2	3218.6
Square	103	0.046959	0.046959	0.046959	0.046959
	104	3.9981	3.9974	3.9969	3.9966
	105	172.61	169.27	164.85	162.22
	106	3694.1	3581.2	3358.1	3063.4
Triangular	103	0.046938	0.046938	0.046938	0.046938
	104	3.9866	3.9859	3.9852	3.9849
	105	171.82	168.24	163.45	160.51
	106	3684.9	3572.4	3347.2	3048.4

Table 22

Table 22. ECOP for Square-Shaped Obstacles with different Wavy Walls at λ = 15 0 .

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.054436	0.054436	0.054435	0.054435
	104	4.5642	4.5627	4.5612	4.5605
	105	192.86	187.82	181.07	176.66
	106	4177.5	4053.6	3814.3	3488.9
Square	103	0.054471	0.054471	0.05447	0.05447
	104	4.5584	4.5572	4.5559	4.5553
	105	187.7	183.69	178.27	174.92
	106	3959.4	3851.2	3634	3325.3
Triangular	103	0.054443	0.054442	0.054442	0.054442
	104	4.5436	4.5423	4.5409	4.5402
	105	186.73	182.47	176.67	172.99
	106	3943.8	3832.8	3612.8	3305.8

Table 23

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.046218	0.046218	0.046217	0.046217
	104	4.0534	4.0531	4.0527	4.0525
	105	189.11	186.3	183.95	182.77
	106	4233.7	4063.9	3771	3433
Square	103	0.04627	0.04627	0.04627	0.04627
	104	4.0463	4.0461	4.0458	4.0456
	105	184.57	182.79	181.16	180.31
	106	3998.5	3839.2	3575.3	3261
Triangular	103	0.046268	0.046268	0.046267	0.046267
	104	4.0399	4.0396	4.0393	4.0391
	105	183.9	181.79	179.93	178.97
	106	4002.9	3843.6	3580.2	3258.8

Table 24

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.04697	0.04697	0.04697	0.04697
	104	4.006	4.0052	4.0045	4.0043
	105	176.47	172.19	166.81	163.61
	106	3979.7	3762.6	3501.7	3183.8
Square	103	0.046959	0.046959	0.046959	0.046959
	104	3.9979	3.9973	3.9968	3.9966
	105	171.76	168.5	164.36	162.02
	106	3688.2	3556.1	3328.2	3033.1
Triangular	103	0.046938	0.046938	0.046938	0.046938
	104	3.9864	3.9857	3.9851	3.9849
	105	170.98	167.44	162.91	160.29
	106	3695.3	3547.9	3317.1	3018.6

Table 25

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.054436	0.054436	0.054435	0.054435
	104	4.5637	4.5623	4.5611	4.5604
	105	191.67	186.62	180.19	176.26
	106	4229	4036.6	3764.6	3431.4
Square	103	0.054471	0.05447	0.05447	0.05447
	104	4.5579	4.5568	4.5558	4.5552
	105	186.54	182.63	177.58	174.62
	106	3962	3818.4	3581.1	3270.5
Triangular	103	0.054443	0.054442	0.054442	0.054441
	104	4.5431	4.5419	4.5408	4.5401
	105	185.59	181.38	175.92	172.65
	106	3964.3	3808.9	3565.2	3252.8

Table 26

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.046218	0.046218	0.046217	0.046217
	104	4.0534	4.053	4.0527	4.0524
	105	190.08	186.61	183.86	182.69
	106	4270.8	4090.6	3775.3	3417.3
Square	103	0.04627	0.04627	0.04627	0.04627
	104	4.0463	4.046	4.0458	4.0456
	105	185.38	183.01	181.1	180.26
	106	4032.6	3863	3575.5	3255.4
Triangular	103	0.046268	0.046268	0.046267	0.046267
	104	4.0399	4.0396	4.0393	4.0391
	105	184.77	182.05	179.86	178.91
	106	4039.3	3869.9	3576.7	3248.1

Table 27

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.04697	0.04697	0.04697	0.04697
	104	4.0056	4.0049	4.0044	4.0042
	105	174.94	170.69	165.81	163.2
	106	4044.6	3801.3	3448.4	3113.6
Square	103	0.046959	0.046959	0.046959	0.046959
	104	3.9976	3.9971	3.9967	3.9966
	105	170.36	167.26	163.61	161.73
	106	3716.8	3536.9	3271.5	2973.7
Triangular	103	0.046938	0.046938	0.046938	0.046938
	104	3.986	3.9855	3.9851	3.9849
	105	169.54	166.12	162.09	159.96
	106	3750.9	3549.9	3262.3	2957.2

Table 28

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

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.054436	0.054435	0.054435	0.054435
	104	4.563	4.5618	4.5608	4.5603
	105	189.74	184.68	178.85	175.67
	106	4362.5	4079.7	3692.5	3332.1
Square	103	0.054471	0.05447	0.05447	0.05447
	104	4.5573	4.5564	4.5555	4.5551
	105	184.66	180.94	176.52	174.18
	106	4001.3	3793.3	3497	3178.7
Triangular	103	0.054442	0.054442	0.054442	0.054441
	104	4.5424	4.5414	4.5405	4.54
	105	183.71	179.63	174.78	172.17
	106	4040.8	3809.8	3489.1	3161.7

Table 29

Table 29. ECOP for Triangular-Shaped Obstacles with different Wavy Walls at λ = 45˚.

Wavy Wall	Ra	Ha = 0	Ha = 15	Ha = 30	Ha = 50
Sinusoidal	103	0.046218	0.046218	0.046217	0.046217
	104	4.0533	4.0529	4.0526	4.0524
	105	190.22	186.6	183.69	182.59
	106	4260.8	4055.7	3729.7	3372.5
Square	103	0.04627	0.04627	0.04627	0.04627
	104	4.0462	4.046	4.0458	4.0456
	105	185.71	183.07	180.98	180.19
	106	3981.2	3815.9	3538	3223.1
Triangular	103	0.046268	0.046268	0.046267	0.046267
	104	4.0398	4.0395	4.0392	4.0391
	105	185.03	182.09	179.73	178.83
	106	3995.4	3819.4	3534.2	3210.6

In summary, the FEM simulation results highlight the intricate interdependence of obstacle geometry, wall corrugation, inclination angle, and magnetic field strength in shaping the thermofluidic behavior of nanofluid-filled trapezoidal cavities under MHD influence. Star-shaped obstacles consistently outperformed other geometries in enhancing convective heat transfer, while sinusoidal wall corrugation emerged as the most effective in promoting secondary flows and minimizing entropy generation. An inclination angle of 30˚ provided an optimal alignment for buoyancy forces, resulting in favorable thermal performance across most cases. While higher Hartmann numbers suppressed convection, they also contributed to reduced entropy and enhanced ECOP in certain configurations. These results, detailed in Tables 3-29 , offer a comprehensive reference for optimizing passive thermal systems and lay the groundwork for the development of machine learning-based predictive models addressed in the next section.

4.2. Machine Learning Model Predictions and Analysis

To overcome the computational cost associated with high-fidelity FEM simulations across a wide parametric space, this study integrates supervised machine learning (ML) techniques to develop surrogate models capable of accurately predicting key thermal performance indicators. By learning from the numerical dataset generated under various geometric and physical conditions, the ML models aim to forecast the average Nusselt number (Nu), entropy generation (S_T), and ecological coefficient of performance (ECOP) with minimal error and significantly reduced computation time. This section presents both the qualitative and quantitative evaluation of three distinct ML algorithms: Support Vector Regression (SVR), Decision Tree Regression (DT), and Random Forest Regression (RF). The objective is to assess their predictive accuracy, generalization capacity, and potential to serve as efficient alternatives to traditional numerical solvers in the context of MHD-influenced nanofluid convection in complex cavity geometries.

To complement the detailed FEM analysis and accelerate parametric exploration, machine learning models were trained on simulation data to predict key thermophysical performance metrics: the average Nusselt number (Nu), entropy generation (S_T), and ecological coefficient of performance (ECOP). Qualitatively, the predictions produced by Support Vector Regression (SVR), Decision Tree (DT), and Random Forest (RF) models demonstrated strong alignment with the FEM-derived trends across varying Rayleigh and Hartmann numbers, obstacle geometries, and wall corrugation profiles.

The Random Forest model, in particular, exhibited a robust ability to replicate the nonlinear relationships governing the thermal response. For instance, it accurately captured the enhanced heat transfer associated with star-shaped obstacles and sinusoidal wall corrugations, as well as the entropy-reducing effect of moderate magnetic field strength. The machine learning models also consistently reproduced the peak Nu and ECOP values around the 30˚ inclination angle observed in FEM simulations, confirming their ability to internalize spatial and physical patterns. These qualitative consistencies reinforce the validity of ML as a fast-response surrogate for predicting thermal behavior in complex geometrical enclosures.

Quantitative assessment of model performance was conducted using standard regression metrics: Mean Absolute Error (MAE), Mean Squared Error (MSE), and the coefficient of determination (R²). Table 30 summarizes the comparative performance of the three machine learning models across the three target outputs.

Among the models evaluated, Random Forest Regression achieved the highest predictive accuracy, with R² scores above 0.98 for all output variables. The model also maintained the lowest MAE and MSE, indicating not only precision but also reliability across a diverse dataset spanning geometric and thermal parameters. Decision Tree Regression followed closely, showing competitive results with less complexity. SVR, while still effective, underperformed slightly in capturing high-gradient behaviors, especially in configurations with sharp obstacle edges or strong magnetic damping.

Table 30

Table 30. Performance metrics of machine learning models.

Model	Metric	Nu	ST	ECOP
SVR	MAE	0.074	0.083	0.068
	MSE	0.011	0.014	0.010
	R2	0.972	0.964	0.971
Decision Tree	MAE	0.062	0.071	0.061
	MSE	0.009	0.012	0.008
	R2	0.978	0.971	0.975
Random Forest	MAE	0.041	0.050	0.045
	MSE	0.004	0.006	0.005
	R2	0.991	0.982	0.989

The random forest model developed in this study was fine-tuned using a grid search approach, with the optimal hyperparameters determined as follows: n_estimators = 200, max_depth = 12, and criterion = squared_error.

Table 31

Table 31. The feature importance for predicting the Nusselt number is summarized below.

Feature	Importance (%)
Rayleigh Number (Ra)	45
Hartmann Number (Ha)	20
Inclination Angle (λ)	17
Obstacle Shape	10
Wall Corrugation	8

As shown in Table 31 , feature importance analysis identified the Rayleigh number (Ra) as the most influential input, contributing 45% to the model’s prediction of Nusselt number, followed by the Hartmann number (Ha) at 20%, inclination angle (λ) at 17%, obstacle shape at 10%, and wall corrugation at 8%. These results clearly demonstrate that thermal and magnetic field effects are the dominant factors influencing convective heat transfer in the studied MHD cavity system.

Figures 5-13 illustrate the scatter plots comparing FEM-simulated and ML-predicted Nusselt Numbers (Nu) for various obstacle shapes and wavy wall configurations at inclination angles of λ = 15˚, 30˚, and 45˚. The predictions were generated using a trained machine learning model on a comprehensive dataset of FEM results. Across all figures, the data points align closely along the ideal prediction line (y = x), indicating a high level of agreement between FEM and ML results. The consistency in prediction accuracy is evident for all three obstacle shapes, star, square, and triangular, and for all wall geometries, sinusoidal, square, and triangular.

Figures 5-7 show results for star-shaped obstacles. These figures reveal minimal scatter and a near-perfect linear relationship, highlighting the model’s robustness under geometrically complex and thermally dynamic conditions.

Figures 8-10 correspond to square-shaped obstacles. The slight deviations from the ideal line remain within acceptable error margins, reinforcing the ML model’s capacity to generalize over regular-shaped geometries.

Figure 5

Figure 5. FEM vs ML Prediction of Nu for Star-Shaped Obstacles with different Wavy Walls at λ = 15˚.

Figure 6

Figure 6. FEM vs ML Prediction of Nu for Star-Shaped Obstacles with different Wavy Walls at λ = 30˚.

Figure 7

Figure 7. FEM vs ML Prediction of Nu for Star-Shaped Obstacles with different Wavy Walls at λ = 45˚.

Figure 8

Figure 8. FEM vs ML Prediction of Nu for Square-Shaped Obstacles with different Wavy Walls at λ = 15˚.

Figure 9

Figure 9. FEM vs ML Prediction of Nu for Square-Shaped Obstacles with different Wavy Walls at λ = 30˚.

Figure 10

Figure 10. FEM vs ML Prediction of Nu for Square-Shaped Obstacles with different Wavy Walls at λ = 45˚.

Figure 11

Figure 11. FEM vs ML Prediction of Nu for Triangular-Shaped Obstacles with different Wavy Walls at λ = 15˚.

Figure 12

Figure 12. FEM vs ML Prediction of Nu for Triangular-Shaped Obstacles with different Wavy Walls at λ = 30˚.

Figure 13

Figure 13. FEM vs ML Prediction of Nu for Triangular-Shaped Obstacles with different Wavy Walls at λ = 45˚.

Figures 11-13 depict results for triangular-shaped obstacles, which pose higher flow complexity. Even under these conditions, the ML model successfully captures the trends, maintaining low predictive error. Overall, the analysis of Figures 5-13 confirms the high predictive fidelity of the ML model across varying geometrical and thermal boundary conditions. This underscores its practical value in accelerating heat transfer predictions in magnetohydrodynamic nanofluid systems without compromising accuracy.

The comparative analysis between FEM and machine learning predictions for the Nusselt number across diverse geometric and thermal configurations demonstrates the strong predictive capability of the ML model. The scatter plots ( Figures 5-13 ) validate the model’s effectiveness in accurately estimating thermal performance for nanofluid-filled trapezoidal cavities influenced by obstacle shape, wall corrugation, and inclination angle under MHD conditions.

The ML model reliably reproduced FEM results across all configurations, with minimal deviations observed regardless of the complexity of obstacle geometries or wall undulations. This consistency underscores the robustness and generalization ability of the trained ML algorithm. Additionally, the model’s rapid inference capabilities make it a valuable surrogate for computationally expensive FEM simulations, enabling efficient design and optimization of thermal systems.

In conclusion, the machine learning approach provides a powerful, efficient, and accurate alternative to traditional numerical simulations for predicting natural convection behavior in complex nanofluidic enclosures, significantly accelerating the analysis while maintaining high fidelity.

5. Conclusions

This study proposed an integrated framework that combines high-fidelity finite element method (FEM) simulations with machine learning (ML) predictions to analyze magnetohydrodynamic (MHD) natural convection in nanofluid-filled trapezoidal cavities. Key variables included obstacle shape (star, square, triangular), top wall corrugation (sinusoidal, square, triangular), and inclination angle (λ = 15˚, 30˚, 45˚), using Cu-H₂O nanofluids to improve thermal performance.

FEM results demonstrated that obstacle geometry and wall undulations significantly influence heat transfer and entropy generation. Star and Square obstacles with sinusoidal or triangular wavy walls yielded the highest Nusselt numbers and lowest entropy generation, particularly at λ = 30˚ and 45˚, indicating enhanced thermodynamic efficiency and ecological performance.

The ML models, trained on the FEM dataset, showed excellent predictive accuracy for Nu, S_t, and ECOP, with deviations typically under 2%. The agreement across various configurations ( Figures 5-13 ) confirms ML’s effectiveness as a reliable and computationally efficient surrogate for FEM.

The trained ML models are reliable within the parameter ranges covered by the FEM dataset; however, they may fail to generalize accurately when applied to scenarios outside these bounds, such as Ra > 10⁶ or Ha > 50. This limitation arises because the models lack embedded physical laws and rely solely on learned patterns from the training data.

6. Key Findings

1) Optimal heat transfer occurred with sinusoidal walls, Star/Square obstacles, and λ = 30˚ - 45˚.

2) MHD effects, though dampening convection, can enhance uniformity and efficiency when geometry is optimized.

3) FEM offered deep physical insights, while ML delivered rapid, accurate predictions ideal for iterative design.

4) The hybrid FEM-ML approach supports scalable, real-time thermal analysis and control.

Future studies should explore the use of hybrid nanofluids, porous domains, and transient heating conditions to simulate more realistic systems. Moreover, incorporating deep learning architectures such as physics-informed neural networks (PINNs) could further improve accuracy and adaptability. These advancements would facilitate digital twin development and intelligent control strategies for energy-efficient systems.

Acknowledgements

The authors gratefully acknowledge the Department of Mathematics, Dhaka University of Engineering and Technology (DUET), Gazipur-1707, Bangladesh, for providing the necessary support and resources to carry out this research work.

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