Performance Evaluation of Various Nanofluids in MHD Natural Convection Within a Wavy Trapezoidal Cavity Containing Heated Square Obstacles

Abstract

Natural convection enhanced by magnetic fields and nanofluids has broad applications in thermal management systems. This study investigates magnetohydrodynamic (MHD) natural convection in a wavy trapezoidal cavity containing centrally located heated square obstacles, filled with various nanofluids Cu–H₂O, Fe₃O₄–H₂O, and Al₂O₃–H₂O. A uniform magnetic field is applied horizontally, and the effects of key parameters such as Rayleigh number, Ra (10³–10⁶), Hartmann number, Ha (0–50), and nanoparticle volume fraction, φ (0.00, 0.02, 0.04) are analyzed. The numerical simulations are performed using the finite element method, incorporating a wavy upper boundary and slanted sidewalls to model realistic enclosures. Results show that an increasing Rayleigh number enhances heat transfer, while a stronger magnetic field reduces convective flow. Among the nanofluids, Cu–H₂O demonstrates the highest Nusselt number and ecological coefficient of performance (ECOP), whereas Fe₃O₄–H₂O exhibits superior performance under stronger magnetic fields due to its magnetic nature. Entropy generation, S_T decreases with increasing Ra and φ, indicating reduced thermodynamic irreversibility. These results provide insights into designing energy-efficient enclosures using nanofluids under magnetic control.

1. Introduction

Natural convection in enclosures filled with nanofluids and subjected to magnetic fields has attracted significant attention for its applications in advanced thermal systems. The study in [1] laid foundational insights by examining MHD natural convection and entropy generation in a ferrofluid-filled trapezoidal cavity using the Buongiorno model and Galerkin FEM, emphasizing the roles of Hartmann and Darcy numbers in suppressing convective flow. Building on this, Zhang et al. [2] explored transient ferrofluid behavior in porous cavities, highlighting the sensitivity of velocity profiles to Rayleigh, Darcy, and Hartmann numbers. A similar direction was pursued in [3], where a non-Darcy porous semi-annular cavity was analyzed for entropy production and flow instabilities using CVFEM. The role of Brownian motion and thermophoresis in nanoparticle transport under magnetic fields was further addressed in [4], which showed enhanced performance due to field-induced diffusion mechanisms.

Nanofluid behavior in closed square enclosures with internal hot and cold tubes was detailed in [5], revealing how geometry and Rayleigh number transition heat transfer from conduction to convection. Studies such as [6,7] incorporated cavity inclination and multiple heat-generating elements, demonstrating how geometry and tilt modulate flow strength and thermal distribution in nanofluid systems. In [8], triangular heated cylinders embedded in lid-driven cavities were found to significantly influence convective heat transfer, with the Richardson number and nanoparticle volume fraction playing crucial roles. The use of wavy geometries, central obstacles, and entropy analysis was introduced in [9], where Brinkmann–Forchheimer modeling captured the complex interplay between Ha, Ra, and porosity on heat transfer. Similarly, Nadeem et al. [10] studied corrugated circular enclosures, concluding that corrugation and nanoparticle loading increase heat transport but only up to an optimal point. This idea was echoed in [11,12], where nanoparticle concentration and cavity wall undulations were shown to affect Nusselt number and entropy production, particularly in wavy and crown-shaped enclosures.

More complex interactions were explored in [13,14], integrating MHD, internal heat sources, and wall undulations. They emphasized that internal heating configurations and geometric features, such as wave amplitude, significantly alter flow regimes. Hybrid nanofluids were introduced in [15,16], where composite nanoparticles enhanced heat transfer in trapezoidal and circular cavities more than mono-nanofluids, especially under strong magnetic suppression. These observations were validated by studies such as [17,18], which revealed that inner cylinder placement and layered porous structures further modulate performance. Geometric optimization was a key theme in [19,20], where wavy circular heaters and varying obstacle sizes inside enclosures influenced entropy and heat transport behavior. Hybrid nanofluids again demonstrated dominance in [21,22], particularly in hexagonal and trapezoidal domains. In contrast, the placement of heated blocks or walls proved critical in [23,24]. Modified cavity shapes, such as the tooth-shaped and diamond configurations in [25,26], were shown to intensify convective zones and optimize local thermal gradients.

The interplay of complex geometries, non-Newtonian fluids, and MHD damping was further dissected in [27] through Casson fluid modeling in wavy trapezoidal domains, supporting the need for multidimensional optimization seen in [28]. Theoretical grounding for nanofluid thermophysical behavior was provided in [29], which emphasized the limitations of classical models and proposed improvements that incorporate Brownian motion and thermophoresis. Entropy-focused analysis was revisited in [30], where grooved cavities with magnetic damping showcased the trade-off between heat transfer and irreversibility. Bottom sinusoidal heating and inclined magnetic fields studied in [31] echoed the benefits of hybrid nanofluids under complex boundary conditions. Unsteady flow dynamics and time-varying entropy generation in trapezoidal cavities were tackled in [32], highlighting the need for transient modeling.

Temperature-dependent thermophysical properties were shown to improve model realism in [33], especially in cavities with Al₂O₃ nanofluids. Meanwhile, rotating cylinders in corrugated trapezoidal cavities using hybrid nanofluids were analyzed in [34], which validated enhancements resulting from nanoparticle design and boundary manipulation. Additional studies like [10,11] emphasized the role of Darcy and Hartmann numbers in modulating flow under corrugated and crown-shaped cavities. Sensitivity and optimization analysis using statistical methods were presented in [35], which investigated hybrid nanofluids in star-obstacle-driven cavities. Internal heater configurations and magnetic dipole interactions were central to [36], reinforcing the importance of heat source placement in optimizing Nusselt numbers. Wavy wall cavities with diamond-shaped obstacles were revisited in [37], confirming their thermal enhancement through undulation and the sizing of obstacles. Fractal geometries and combined heat/mass transfer under MHD influence were introduced in [38], showing how internal complexity enhances performance. Meanwhile, [39] explored internal heat-generating cylinders of varying geometries, revealing that shape and proximity to walls critically affect heat transport.

Conjugate heat transfer under local thermal non-equilibrium (LTNE) conditions was investigated in [40], where the interactions between rotating cylinders and porous media were analyzed. Hybrid nanofluids and rotating bodies under MHD in zigzag cavities were examined in [41], showing that rotation and structural design enhance heat transfer while reducing entropy. Structural alterations such as base inclination and corrugation in MHD systems were found in [42] to fine-tune heat and entropy performance. Radiative effects in high-temperature MHD nanofluid systems were the focus of [43], revealing their importance in optically active enclosures. Corrugated internal structures within circular cavities were explored in [44], where FEM modeling captured flow disturbances that enhanced thermal mixing. In [45], it was demonstrated that fin configurations in hybrid nanofluid-filled open cavities can offset MHD damping through radiation-driven enhancements. The combined effects of magnetic fields and wavy-wall geometry significantly altered entropy generation and heat transfer behavior in a Cu–Al₂O₃/water-filled H-wavy enclosure, as demonstrated in [46]. Lastly, Entropy generation under MHD effects was analyzed for a chamber containing heat-generating liquid and solid elements, demonstrating how Joule heating significantly influences irreversibility and thermal transport [47].

Despite the extensive investigations across diverse geometries, nanofluid types, and magnetic field applications, a clear research gap remains in comprehensively analyzing the combined influence of nanofluid type, magnetic field strength, Rayleigh number, and nanoparticle volume fraction within a wavy trapezoidal cavity featuring internally heated square obstacles. Most prior studies have focused on isolated factors or simpler geometries, often overlooking the thermodynamic and ecological implications of entropy generation and ECOP under such compounded conditions. This study addresses that gap by conducting a detailed performance evaluation of Cu–H₂O, Al₂O₃–H₂O, and Fe₃O₄–H₂O nanofluids subjected to MHD natural convection within a complex cavity structure. The novelty lies in the integrated assessment of flow dynamics, heat transfer, entropy generation, and ecological performance within a geometrically irregular domain under varying magnetic intensities and nanoparticle concentrations. Such a configuration closely mimics real-world thermal management systems, providing practical insights for applications like electronic cooling, solar thermal systems, and energy-efficient enclosures.

2. Materials and Methods

The physical model consists of a two-dimensional wavy trapezoidal cavity filled with nanofluid and subjected to natural convection under the influence of a horizontal magnetic field. The cavity features two identical, centrally located square-shaped internal heated blocks, serving as volumetric heat sources. The cavity geometry includes a wavy top wall defined by the sinusoidal equation y = H + asin(2πfx/L_t), where a = 0.045L is the wave amplitude and f = 9.5 is the wave frequency.

The left and right sidewalls are slanted at an angle γ = 20°, while the bottom boundary is inclined at an angle λ = 30°. Three nanofluids, Cu–H₂O, Fe₃O₄–H₂O, and Al₂O₃–H₂O, are investigated, and their configurations within the cavity are illustrated in Figure 1.

Appropriate thermal and flow boundary conditions are imposed on all cavity walls and solid surfaces. The top wavy wall and the vertical sidewalls are maintained at a constant cold temperature T_c, providing isothermal conditions that enhance surface heat exchange. The inclined bottom wall is thermally insulated, ensuring no heat loss. The internal heated blocks generate uniform volumetric heat, modeled as constant heat flux q = Q = const. All solid-fluid interfaces satisfy the no-slip condition, enforcing zero velocity at the walls. A uniform horizontal magnetic field B = B₀ î is applied across the domain, introducing Lorentz forces and modeled using the Hartmann number (Ha). Gravitational acceleration acts vertically, driving buoyancy-induced flow from the heated blocks toward the cold boundaries. The boundary conditions applied to each surface are summarized in Table 1.

The nanofluids’ effective thermophysical properties are evaluated based on established mixture models that consider the base fluid (water) and nanoparticles (Cu, Fe₃O₄, Al₂O₃) at a reference temperature of 300 K. These properties include mass density, specific heat, thermal conductivity, thermal expansion coefficient, electrical conductivity, and dynamic viscosity, as detailed in Table 2. The nanoparticle volume fraction φ is varied across 0.00, 0.02, and 0.04 to assess its impact on thermodynamic and flow behavior. The nanofluid model incorporates thermal dispersion, electrical conductivity enhancement, and viscosity variations using established correlations for each constituent.

Table 1. Geometric, thermal, and magnetic boundary conditions.

Boundary Element	Label/Equation	Type	Mathematical Condition	Physical Description
Top Wavy Wall	y = H + asin(2πfx/Lt)	Isothermal (Cold)	T = Tc	Wavy top maintained at constant cold temperature; enhances surface area and mixing.
Bottom Inclined Wall	Inclined at angle λ	Thermally insulated (Adiabatic)	∂T/∂n = 0	The Bottom wall is thermally insulated, which prevents heat loss.
Left & Right Walls	Slanted at angle γ	Isothermal (Cold)	T = Tc	Left and Right walls are maintained at a constant cold temperature, which enhances surface area and mixing.
Internal Shaped Blocks	R = R0 + Acos(Nθ)	Heat Generation (Solid)	q = Q = const	Internal heat sources embedded in nanofluid; modeled with volumetric heat generation.
All Solid Surfaces	Cavity and block boundaries	No-slip Velocity	u = 0, v = 0	Viscous boundary layers form due to fluid-solid interaction.
Entire Fluid Domain	Volume enclosed by walls and blocks	Magnetic Field (MHD)	B = B0 î	Uniform horizontal magnetic field introduces Lorentz force; quantified via Hartmann number.
Gravity	$\overrightarrow{g}$↓	Buoyancy-driven flow	Acts in y direction	Drives natural convection from heated blocks to cold boundaries.

Table 1. Geometric, thermal, and magnetic boundary conditions.

Boundary Element	Label/Equation	Type	Mathematical Condition	Physical Description
Top Wavy Wall	y = H + asin(2πfx/Lt)	Isothermal (Cold)	T = Tc	Wavy top maintained at constant cold temperature; enhances surface area and mixing.
Bottom Inclined Wall	Inclined at angle λ	Thermally insulated (Adiabatic)	∂T/∂n = 0	The Bottom wall is thermally insulated, which prevents heat loss.
Left & Right Walls	Slanted at angle γ	Isothermal (Cold)	T = Tc	Left and Right walls are maintained at a constant cold temperature, which enhances surface area and mixing.
Internal Shaped Blocks	R = R0 + Acos(Nθ)	Heat Generation (Solid)	q = Q = const	Internal heat sources embedded in nanofluid; modeled with volumetric heat generation.
All Solid Surfaces	Cavity and block boundaries	No-slip Velocity	u = 0, v = 0	Viscous boundary layers form due to fluid-solid interaction.
Entire Fluid Domain	Volume enclosed by walls and blocks	Magnetic Field (MHD)	B = B0 î	Uniform horizontal magnetic field introduces Lorentz force; quantified via Hartmann number.
Gravity	$\overrightarrow{g}$↓	Buoyancy-driven flow	Acts in y direction	Drives natural convection from heated blocks to cold boundaries.

Table 2. Thermo-physical properties of Water, Cu, Fe₃O_4, and Al₂O₃ at T_m = 300 K [19,24,36].

Name of Property	Symbol	Unit	Water	Cu	Fe3O4	Al2O3
Mass Density	ρ	Kg·m−3	996.6	8933	5200	3970
Specific Heat at Constant Pressure	Cp	J·kg−1·K−1	4179.2	385	670	752
Thermal Conductivity	k	W·m−1·K−1	0.6102	401	6	36.6
Volumetric Thermal Expansion Coefficient	β	K−1	26.6 × 10−5	49.9 × 10−6	1.18 × 10−5	0.85 × 10−5
Electrical Conductivity	σ	S·m−1	0.05	59.6 × 10−6	25,000	35 × 10−6
Dynamic viscosity	μ	kg·m−1·s−1	8.538 × 10−4	-	-	-

Table 2. Thermo-physical properties of Water, Cu, Fe₃O_4, and Al₂O₃ at T_m = 300 K [19,24,36].

Name of Property	Symbol	Unit	Water	Cu	Fe3O4	Al2O3
Mass Density	ρ	Kg·m−3	996.6	8933	5200	3970
Specific Heat at Constant Pressure	Cp	J·kg−1·K−1	4179.2	385	670	752
Thermal Conductivity	k	W·m−1·K−1	0.6102	401	6	36.6
Volumetric Thermal Expansion Coefficient	β	K−1	26.6 × 10−5	49.9 × 10−6	1.18 × 10−5	0.85 × 10−5
Electrical Conductivity	σ	S·m−1	0.05	59.6 × 10−6	25,000	35 × 10−6
Dynamic viscosity	μ	kg·m−1·s−1	8.538 × 10−4	-	-	-

The problem is governed by the continuity, Navier–Stokes momentum, and energy equations, applied to both the fluid and solid domains, along with an entropy generation equation that captures effects from both heat transfer and viscous dissipation. These equations are non-dimensionalized using appropriate scaling parameters, introducing key dimensionless numbers such as the Rayleigh (Ra), Prandtl (Pr), and Hartmann (Ha) numbers, as well as output metrics like the average Nusselt number (Nu), entropy generation (ST), and ecological coefficient of performance (ECOP). The full set of governing equations is detailed in Equations (1) to (26) [19,24,36].

• Fluid domain:

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

(1)

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

(2)

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

(3)

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

(4)

• Solid domains:

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

(5)

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).

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

(6)

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

(7)

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

(8)

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

(9)

$${\beta }_{nf}=[{\displaystyle \frac{(1-\varphi ){\rho }_f{\beta }_f+\varphi {\rho }_s{\beta }_s}{{\rho }_{nf}}}]$$

(10)

The local entropy generation due to heat transfer ($S_{ht}$) in solid and fluid domains, volumetric entropy production due to viscous flow dissipation ($S_{ff}$), and external magnetic effects ($S_{mf}$) can be described using the following formulas:

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

(11)

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

(12)

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

(13)

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

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

(14)

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

(15)

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

(16)

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

(17)

$$(U{\displaystyle \frac{\partial V}{\partial X}}+V{\displaystyle \frac{\partial V}{\partial Y}})=-{\displaystyle \frac{\partial P}{\partial X}}+{\displaystyle \frac{{\mu }_{nf}}{{\mu }_f}}\left({\displaystyle \frac{{\partial }^{2}V}{\partial X^2}}+{\displaystyle \frac{{\partial }^{2}V}{\partial Y^2}}\right)+Ra\mathrm{Pr}{\displaystyle \frac{{\rho }_{nf}{\beta }_{nf}}{{\rho }_f{\beta }_f}}\theta \cos (\lambda )-H{a}^{2}U$$

(18)

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

(19)

• Non-dimensional nanofluid properties:

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

(20)

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

(21)

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

(22)

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

(23)

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

(24)

The thermal behavior of the chamber under different operating conditions is assessed by analyzing the average 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:

$$Nu={\displaystyle \frac{L}{L_s}}\left[-{\displaystyle \underset{L_0/L}{\overset{2L_0/L}{\int }}{\left.\frac{\partial \Theta }{\partial Y}\right|}_{Y=0}dX}-{\displaystyle \underset{3L_0/L}{\overset{4L_0/L}{\int }}{\left.\frac{\partial \Theta }{\partial Y}\right|}_{Y=0}dX}\right],{\mathsf{\Theta }}_{av}={\displaystyle \frac{1}{A}}{\displaystyle \underset{A}{\int }\mathsf{\Theta }dA},$$

(25)

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. Equation (22) is formulated by integrating the local Nusselt number over the active heated surfaces of the internal square blocks located symmetrically within the cavity, where constant heat flux is applied. The average Nusselt number captures the net thermal transport from these blocks into the surrounding nanofluid. Similarly, the domain-averaged temperature is calculated as a surface integral over the fluid region, representing the overall thermal response of the system.

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

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

(26)

where A represents the surface area of the computational domain.

The problem is solved using the finite element method (FEM), which efficiently handles the complex geometry, nonlinear coupled equations, and boundary conditions to provide accurate predictions of thermal and flow fields.

3. Numerical Validation

The accuracy of the present numerical model was validated by comparing its results with two established benchmarks. The first validation, shown in Figure 2a, aligns well with the isotherm contours from Abdelmalek et al. [19], while the second, shown in Figure 2b, confirms streamline agreement with the study by Tasnim et al. [47], demonstrating strong consistency in both thermal and flow behavior.

Figure 2a displays isotherm contours at Ra = 10⁴ that closely match those from the reference study, while

Table 3. Comparison of Nu between present work and Abdelmalek et al. [19].

Ra	Nanoparticle Volume Fraction (φ%)	Present Study	Abdelmalek et al. [19]	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

confirms that the Nusselt number deviation across all Ra values remains below 1.5%, validating the accuracy of the present numerical model.

Figure 2b compares the streamlines at Ra = 10⁴ and Ha = 0 between the present study and Tasnim et al. [47], both incorporating a heat-generating block, and reveals strong agreement in vortex structure and flow symmetry—validating the accuracy and robustness of the present numerical model.

A grid sensitivity analysis was performed to ensure numerical stability and mesh independence of the finite element results. The computational mesh was refined in the vicinity of the wavy top wall and along the solid–fluid interfaces of the internal obstacles to accurately capture the steep temperature and velocity gradients. Figure 3a–c illustrates the mesh distribution for various grid densities: Finer, Extra Fine, and Extremely Fine. The refinement pattern clearly demonstrates increased mesh resolution around heated obstacles and along the wavy boundaries, ensuring higher numerical precision in regions with strong convective flow and thermal variations.

The quantitative comparison of mesh effects is summarized in

Table 4. Nu at heated solid surface for different grid sizes (when Pr = 5.856, Ra = 10⁶, φ = 0.02, Ha = 0, λ = 45°).

Mesh Type	Elements	Nu	Deviation from Previous
Fine	17210	6.5016	-
Extra Fine	39587	6.5769	0.0753
Extremely Fine	47733	6.5782	0.0013

, which lists the Nusselt number (Nu) for different mesh types at Pr = 5.856, Ra = 10⁶, φ = 0.02, Ha = 0, and λ = 45°. As the number of elements increases from 17210 (Fine) to 47733 (Extremely Fine), the variation in Nu becomes negligible, with only a 0.0013% deviation between the two finest grids. This confirms that the Extra Fine mesh (39587 elements) is sufficient for achieving mesh independence while maintaining computational efficiency.

The grid sensitivity trend is further illustrated in Figure 4, where the Nusselt number increases slightly with mesh refinement and then stabilizes, confirming convergence. The near-horizontal slope at higher element counts validates that additional refinement has an insignificant impact on the solution, indicating strong numerical stability.

For future research, a broader range of nanofluid combinations, non-uniform magnetic fields, or transient flow scenarios can be explored. Experimental validation of entropy generation patterns and real-time thermal imaging may further bridge the gap between simulations and real-world applications. Additionally, the insights gained from this study can be applied to industrial applications, such as cooling high-density electronic devices, enhancing heat exchanger performance, and improving solar collector performance.

4. Results

4.1. Effect of Nanofluid Based on Velocity Plots and Isotherms

The thermal and flow characteristics within the wavy trapezoidal cavity were examined using velocity contour and isotherm plots to understand the impact of nanofluid type, Hartmann number (Ha), Rayleigh number (Ra), and nanoparticle volume fraction (φ). Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 present the velocity fields for Cu–H₂O, Al₂O₃–H₂O, and Fe₃O₄–H₂O nanofluids across different Ra, Ha, and φ values, highlighting changes in flow structure and intensity. Complementing this, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22 show the corresponding isotherm distributions, capturing how heat transfer patterns evolve with these parameters. Together, these visualizations reveal the distinct thermal-fluid responses of each nanofluid system under magnetic and buoyancy-driven convection.

4.1.1. Velocity Plots

Figure 5, Figure 6 and Figure 7 show velocity contours for Cu–H₂O, Al₂O₃–H₂O, and Fe₃O₄–H₂O nanofluids at Ra = 10⁴, 10⁵, and 10⁶ with Ha = 15 and 50 and φ = 0.00. As Ra increases, convection strengthens across all cases, generating larger and faster vortices. Increasing Ha from 15 to 50 suppresses flow intensity due to magnetic damping. Among the nanofluids, Fe₃O₄–H₂O sustains stronger circulation under magnetic fields, followed by Al₂O₃–H₂O, while Cu–H₂O shows the most dampened velocity fields.

Figure 5. Velocity plots when Ra = 10⁴, φ = 0.00.

Figure 5. Velocity plots when Ra = 10⁴, φ = 0.00.

Figure 6. Velocity plots when Ra = 10⁵, φ = 0.00.

Figure 6. Velocity plots when Ra = 10⁵, φ = 0.00.

Figure 7. Velocity plots when Ra = 10⁶, φ = 0.00.

Figure 7. Velocity plots when Ra = 10⁶, φ = 0.00.

Figure 8, Figure 9 and Figure 10 compare velocity contours for different nanofluids at φ = 0.02, with Ra = 10⁴, 10⁵, and 10⁶, and Ha = 15 and 50. Earlier Figure 5, Figure 6 and Figure 7 showed the same results at φ = 0.00. The increase in nanoparticle volume fraction enhances flow intensity across all fluids due to improved thermal conductivity. At each Ra, φ = 0.02 results in more pronounced vortices than φ = 0.00. Among the nanofluids, Fe₃O₄–H₂O maintains stronger circulation, especially under high Ha, while Cu–H₂O remains the most dampened regardless of φ.

Figure 8. Velocity plots when Ra = 10⁴, φ = 0.02.

Figure 8. Velocity plots when Ra = 10⁴, φ = 0.02.

Figure 9. Velocity plots when Ra = 10⁵, φ = 0.02.

Figure 9. Velocity plots when Ra = 10⁵, φ = 0.02.

Figure 10. Velocity plots when Ra = 10⁶, φ = 0.02.

Figure 10. Velocity plots when Ra = 10⁶, φ = 0.02.

Figure 11, Figure 12 and Figure 13 display velocity contours for three nanofluids at φ = 0.04, in comparison to earlier results at φ = 0.00 and 0.02. Increasing φ enhances convective motion due to higher thermal conductivity and momentum exchange. At all Ra and Ha values, φ = 0.04 produces stronger, more organized vortices than φ = 0.00 and 0.02. Fe₃O₄–H₂O continues to exhibit the most vigorous flow under magnetic damping, while Cu–H₂O consistently shows the weakest circulation across all φ levels.

Figure 11. Velocity plots when Ra = 10⁴, φ = 0.04.

Figure 11. Velocity plots when Ra = 10⁴, φ = 0.04.

Figure 12. Velocity plots when Ra = 10⁵, φ = 0.04.

Figure 12. Velocity plots when Ra = 10⁵, φ = 0.04.

Figure 13. Velocity plots when Ra = 10⁶, φ = 0.04.

Figure 13. Velocity plots when Ra = 10⁶, φ = 0.04.

Overall, the velocity plots illustrate how flow dynamics are influenced by Ra, Ha, φ, and the type of nanofluid. Higher Ra boosts buoyancy-driven convection, while stronger magnetic fields (higher Ha) dampen motion. Increasing φ consistently intensifies flow by enhancing thermal conductivity, with φ = 0.04 showing the most vigorous circulation. Among the nanofluids, Fe₃O₄–H₂O delivers the strongest and most resilient vortices across all scenarios, even under magnetic suppression. Al₂O₃–H₂O shows moderate performance, and Cu–H₂O consistently exhibits the most attenuated flow, confirming that both magnetic properties and nanoparticle concentration critically influence convective behavior.

4.1.2. Isotherms

Figure 14, Figure 15 and Figure 16 display the isotherm patterns for Cu–H₂O, Al₂O₃–H₂O, and Fe₃O₄–H₂O nanofluids at φ = 0.00 with Ra = 10⁴, 10⁵, and 10⁶ under Ha = 15 and 50. As Ra increases, conduction-dominated horizontal isotherms give way to more distorted contours, signaling enhanced convective heat transfer. Magnetic field strength (Ha = 50) visibly suppresses this distortion, indicating dampened convection. Comparing the nanofluids, Fe₃O₄–H₂O shows the most disturbed and dispersed isotherms, reflecting stronger thermal transport. Al₂O₃–H₂O exhibits intermediate behavior, while Cu–H₂O maintains more uniform layers, suggesting lower convective efficiency.

Figure 14. Isotherm when Ra = 10⁴, φ = 0.00.

Figure 14. Isotherm when Ra = 10⁴, φ = 0.00.

Figure 15. Isotherm when Ra = 10⁵, φ = 0.00.

Figure 15. Isotherm when Ra = 10⁵, φ = 0.00.

Figure 16. Isotherm when Ra = 10⁶, φ = 0.00.

Figure 16. Isotherm when Ra = 10⁶, φ = 0.00.

Figure 17, Figure 18 and Figure 19 illustrate isotherms at φ = 0.02, which expand on the trends observed in Figure 14, Figure 15 and Figure 16 at φ = 0.00. With increased nanoparticle concentration, temperature contours become more distorted, reflecting stronger convection due to better thermal conductivity and fluid mixing. At all Ra levels, the isotherms for Fe₃O₄–H₂O are the most irregular and dispersed, indicating superior heat transfer, especially under magnetic influence. Al₂O₃–H₂O shows moderate isotherm curvature, while Cu–H₂O retains more stratified patterns, even at φ = 0.02. The enhancement in convective transport from φ = 0.00 to 0.02 is evident across all fluids, with the impact most prominent at higher Ra and lower Ha.

Figure 17. Isotherm when Ra = 10⁴, φ = 0.02.

Figure 17. Isotherm when Ra = 10⁴, φ = 0.02.

Figure 18. Isotherm when Ra = 10⁵, φ = 0.02.

Figure 18. Isotherm when Ra = 10⁵, φ = 0.02.

Figure 19. Isotherm when Ra = 10⁶, φ = 0.02.

Figure 19. Isotherm when Ra = 10⁶, φ = 0.02.

Figure 20, Figure 21 and Figure 22 illustrate isotherm distributions for Cu–H₂O, Al₂O₃–H₂O, and Fe₃O₄–H₂O nanofluids at φ = 0.04, with Ra = 10⁴, 10⁵, and 10⁶ under Ha = 15 and 50. Compared to φ = 0.00 and 0.02, the isotherms show increased distortion and tighter spacing near the heat source, indicating enhanced convective transport. This improvement stems from the higher thermal conductivity and better energy diffusion with increased φ. Among the nanofluids, Fe₃O₄–H₂O consistently exhibits the most disturbed isotherms, reflecting superior heat transfer even under stronger magnetic damping. Al₂O₃–H₂O follows with moderate thermal performance, while Cu–H₂O shows the least isotherm deformation across all φ levels, confirming lower convective activity.

Figure 20. Isotherm when Ra = 10⁴, φ = 0.04.

Figure 20. Isotherm when Ra = 10⁴, φ = 0.04.

Figure 21. Isotherm for Ra = 10⁵, φ = 0.04.

Figure 21. Isotherm for Ra = 10⁵, φ = 0.04.

Figure 22. Isotherm when Ra = 10⁶, φ = 0.04.

Figure 22. Isotherm when Ra = 10⁶, φ = 0.04.

Overall, the isotherm patterns across Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21 and Figure 22 clearly demonstrate how increasing nanoparticle concentration (φ) intensifies thermal convection in the presence of a magnetic field. As φ rises from 0.00 to 0.04, the shift from stratified, conduction-dominant contours to more irregular and compressed isotherms confirm enhanced convective heat transfer. This effect becomes more noticeable at higher Ra and weaker magnetic fields (Ha = 15). Among the nanofluids, Fe₃O₄–H₂O consistently delivers the most efficient thermal performance, evidenced by its pronounced isotherm deformation, even under magnetic suppression. Al₂O₃–H₂O maintains a balanced response, while Cu–H₂O lags in convective enhancement across all φ and Ra levels, reflecting its relatively lower thermal activity.

4.2. Effect of Nanofluid Based on Nusselt Number

To evaluate how effectively different nanofluids enhance convective heat transfer within the wavy trapezoidal cavity, the average Nusselt number (Nu) was computed for Cu–H₂O, Fe₃O₄–H₂O, and Al₂O₃–H₂O. The analysis spans Rayleigh numbers from 10³ to 10⁶, Hartmann numbers from 0 to 50, and nanoparticle volume fractions φ of 0.00, 0.02, and 0.04. A detailed comparison of the thermal performance under these conditions is provided in Table 5, Table 6 and Table 7.

4.2.1. When φ = 0.00

Table 5 highlights how the average Nusselt number (Nu) varies with Rayleigh number (Ra), Hartmann number (Ha), and nanofluid type at zero nanoparticle loading. For all nanofluids, Nu increases markedly with Ra, reflecting the expected shift from conduction-dominated to convection-enhanced heat transfer as buoyancy intensifies. Magnetic field strength (Ha), in contrast, has a damping effect—Nu consistently declines with increasing Ha due to the Lorentz force suppressing fluid motion.

Among the three nanofluids, Fe₃O₄–H₂O exhibits the highest Nu across all Ra and Ha, indicating superior thermal transport capability even in the absence of nanoparticles. This is followed by Al₂O₃–H₂O, which shows moderate heat transfer enhancement, while Cu–H₂O consistently records the lowest Nu, suggesting weaker convective performance. At Ra = 10⁶ and Ha = 0, Fe₃O₄–H₂O achieves Nu ≈ 7.20 compared to 6.38 for Al₂O₃–H₂O and 6.25 for Cu–H₂O. As Ha increases to 50, the decline is more pronounced for Cu–H₂O, dropping to 4.89, whereas Fe₃O₄–H₂O still retains relatively high performance with Nu ≈ 5.63. This trend confirms that Fe₃O₄–H₂O maintains stronger convective transport under magnetic damping, making it a more resilient choice in MHD environments.

Table 5. Nu for variation in nanofluids with volume fraction φ = 0.00.

Nanofluid	Ra	Nu
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	0.82031	0.82029	0.82028	0.82027
	104	2.4772	2.4718	2.4675	2.4657
	105	3.7639	3.5758	3.331	3.1755
	106	6.2455	5.9191	5.4513	4.8861
Fe3O4–H2O	103	0.93882	0.93884	0.93887	0.93889
	104	2.8052	2.7981	2.7929	2.7914
	105	4.3313	4.1078	3.8035	3.5971
	106	7.1974	6.8379	6.2931	5.6331
Al2O3–H2O	103	0.83609	0.83607	0.83606	0.83605
	104	2.5206	2.5148	2.5103	2.5086
	105	3.8436	3.6497	3.3952	3.2317
	106	6.3847	6.0523	5.5721	4.9921

Table 5. Nu for variation in nanofluids with volume fraction φ = 0.00.

Nanofluid	Ra	Nu
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	0.82031	0.82029	0.82028	0.82027
	104	2.4772	2.4718	2.4675	2.4657
	105	3.7639	3.5758	3.331	3.1755
	106	6.2455	5.9191	5.4513	4.8861
Fe3O4–H2O	103	0.93882	0.93884	0.93887	0.93889
	104	2.8052	2.7981	2.7929	2.7914
	105	4.3313	4.1078	3.8035	3.5971
	106	7.1974	6.8379	6.2931	5.6331
Al2O3–H2O	103	0.83609	0.83607	0.83606	0.83605
	104	2.5206	2.5148	2.5103	2.5086
	105	3.8436	3.6497	3.3952	3.2317
	106	6.3847	6.0523	5.5721	4.9921

The 3D surface plot in Figure 23 further illustrates the combined effect of Rayleigh number (Ra) and Hartmann number (Ha) on the average Nusselt number (Nu) for Cu–H₂O, Al₂O₃–H₂O, and Fe₃O₄–H₂O nanofluids at φ = 0.00. As Ra increases, Nu rises significantly for all fluids, confirming the dominance of buoyancy-driven convection at higher thermal gradients. Conversely, increasing Ha steadily suppresses Nu, as magnetic damping inhibits fluid motion and reduces convective strength.

Figure 23. Three-dimensional Surface plot for different nanofluids at φ = 0.00.

Figure 23. Three-dimensional Surface plot for different nanofluids at φ = 0.00.

Among the nanofluids, the Fe₃O₄–H₂O surface (green) consistently lies above the others, emphasizing its superior heat transfer capability across the full Ra–Ha spectrum. Al₂O₃–H₂O (red) exhibits intermediate performance, while Cu–H₂O (blue) shows the lowest Nu throughout, particularly at high Ha. This visualization reinforces the earlier tabulated trends and highlights the relative thermal advantages of Fe₃O₄–H₂O in magnetically influenced convection systems.

4.2.2. When φ = 0.02

Table 6 shows how adding a small nanoparticle concentration (φ = 0.02) alters the heat-transfer response across different Ra, Ha, and nanofluid types. The overall trends remain consistent with the φ = 0.00 case: Nu grows strongly with Ra as the system shifts toward convection-dominated transport, while increasing Ha suppresses Nu due to magnetic damping. What changes at φ = 0.02 is the magnitude—Nu increases across all fluids and operating conditions, confirming that nanoparticle loading enhances thermal conductivity and strengthens buoyancy-driven motion.

Even with φ = 0.02, Fe₃O₄–H₂O continues to deliver the highest Nu over the entire parameter space. At Ra = 10⁶, for instance, it reaches Nu ≈ 7.31 at Ha = 0, surpassing both Al₂O₃–H₂O (≈ 6.49) and Cu–H₂O (≈ 6.39). As Ha rises to 50, all fluids experience the expected decline, but Fe₃O₄–H₂O still maintains noticeably stronger performance (Nu ≈ 5.69), while Cu–H₂O drops to around 4.98. This persistent advantage reflects Fe₃O₄–H₂O’s higher effective thermal diffusivity and its ability to retain convective strength under magnetic damping.

Table 6. Nu for variation in nanofluids with volume fraction φ = 0.02.

Nanofluid	Ra	Nu
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	0.83276	0.83275	0.83274	0.83273
	104	2.5898	2.5856	2.5823	2.581
	105	3.8756	3.6934	3.4693	3.3383
	106	6.3882	6.0592	5.5749	4.9767
Fe3O4–H2O	103	0.95563	0.95565	0.95568	0.9557
	104	2.9163	2.9109	2.9072	2.9062
	105	4.4219	4.2034	3.9206	3.7443
	106	7.3121	6.9418	6.3812	5.6888
Al2O3–H2O	103	0.84925	0.84924	0.84923	0.84922
	104	2.6322	2.628	2.6247	2.6234
	105	3.9328	3.7507	3.5239	3.3895
	106	6.4901	6.1587	5.6666	5.0575

Table 6. Nu for variation in nanofluids with volume fraction φ = 0.02.

Nanofluid	Ra	Nu
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	0.83276	0.83275	0.83274	0.83273
	104	2.5898	2.5856	2.5823	2.581
	105	3.8756	3.6934	3.4693	3.3383
	106	6.3882	6.0592	5.5749	4.9767
Fe3O4–H2O	103	0.95563	0.95565	0.95568	0.9557
	104	2.9163	2.9109	2.9072	2.9062
	105	4.4219	4.2034	3.9206	3.7443
	106	7.3121	6.9418	6.3812	5.6888
Al2O3–H2O	103	0.84925	0.84924	0.84923	0.84922
	104	2.6322	2.628	2.6247	2.6234
	105	3.9328	3.7507	3.5239	3.3895
	106	6.4901	6.1587	5.6666	5.0575

The 3D surface in Figure 24 visualizes these interactions across the full Ra–Ha domain. The upward slope with Ra and the downward slope with Ha are clearly visible for all fluids. Once again, the Fe₃O₄–H₂O surface (green) rises above the others, indicating the highest heat-transfer rates, while the Al₂O₃–H₂O (red) surface occupies a middle band and Cu–H₂O (blue) consistently forms the lower envelope. The separation between the surfaces widens at high Ra and low Ha, where convection is strongest, highlighting how nanoparticle type becomes increasingly influential under strong buoyancy.

Figure 24. Three-dimensional Surface plot for different nanofluids at φ = 0.02.

Figure 24. Three-dimensional Surface plot for different nanofluids at φ = 0.02.

Together, Table 6 and Figure 24 show that raising φ to 0.02 enhances heat transfer for every nanofluid, but Fe₃O₄–H₂O remains the most thermally efficient choice across magnetic and non-magnetic regimes.

4.2.3. When φ = 0.04

At a nanoparticle volume fraction of φ = 0.04, the trends in thermal behavior become more pronounced, as detailed in Table 7. The average Nusselt number (Nu) continues to rise with increasing Rayleigh number (Ra), confirming stronger natural convection at higher thermal gradients. Across all three nanofluids, Cu–H₂O shows the least thermal enhancement, while Fe₃O₄–H₂O consistently delivers the highest Nu, reaffirming its superior thermal transport capability. Magnetic damping (increasing Ha) still reduces Nu, but the decline is less severe for Fe₃O₄–H₂O compared to Cu–H₂O or Al₂O₃–H₂O. For instance, at Ra = 10⁶ and Ha = 0, Fe₃O₄–H₂O reaches a peak Nu of 7.42, compared to 6.59 for Al₂O₃–H₂O and 6.53 for Cu–H₂O. Even under strong magnetic damping (Ha = 50), Fe₃O₄–H₂O maintains a Nu of 5.74, outperforming both Al₂O₃–H₂O (5.12) and Cu–H₂O (5.06).

Table 7. Nu for variation in nanofluids with volume fraction φ = 0.04.

Nanofluid	Ra	Nu
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	0.84465	0.84464	0.84463	0.84462
	104	2.7047	2.7014	2.6989	2.6979
	105	3.994	3.8213	3.6207	3.5122
	106	6.5316	6.2015	5.6962	5.0664
Fe3O4–H2O	103	0.9724	0.97242	0.97245	0.97247
	104	3.0311	3.0271	3.0244	3.0238
	105	4.5186	4.3087	4.0512	3.9036
	106	7.4241	7.0479	6.467	5.7439
Al2O3–H2O	103	0.86195	0.86195	0.86194	0.86193
	104	2.7466	2.7436	2.7412	2.7402
	105	4.0304	3.8641	3.6676	3.5595
	106	6.594	6.2661	5.757	5.1216

Table 7. Nu for variation in nanofluids with volume fraction φ = 0.04.

Nanofluid	Ra	Nu
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	0.84465	0.84464	0.84463	0.84462
	104	2.7047	2.7014	2.6989	2.6979
	105	3.994	3.8213	3.6207	3.5122
	106	6.5316	6.2015	5.6962	5.0664
Fe3O4–H2O	103	0.9724	0.97242	0.97245	0.97247
	104	3.0311	3.0271	3.0244	3.0238
	105	4.5186	4.3087	4.0512	3.9036
	106	7.4241	7.0479	6.467	5.7439
Al2O3–H2O	103	0.86195	0.86195	0.86194	0.86193
	104	2.7466	2.7436	2.7412	2.7402
	105	4.0304	3.8641	3.6676	3.5595
	106	6.594	6.2661	5.757	5.1216

Figure 25 offers a 3D visualization of these dynamics. As expected, Nu increases steeply with Ra and decreases with Ha across all nanofluids. What stands out at φ = 0.04 is the increasing vertical separation between the surfaces, especially at high Ra and low Ha—conditions where buoyancy dominates. This widening gap highlights the increasing influence of nanoparticle type when convection is strong. The green surface representing Fe₃O₄–H₂O remains consistently above the others, signaling superior heat transfer. Al₂O₃–H₂O again shows intermediate behavior, while Cu–H₂O remains the least effective.

Figure 25. Three-dimensional Surface plot for different nanofluids at φ = 0.04.

Figure 25. Three-dimensional Surface plot for different nanofluids at φ = 0.04.

Together, Table 7 and Figure 25 demonstrate that increasing the nanoparticle volume fraction to φ = 0.04 further amplifies convective heat transfer, particularly in buoyancy-driven regimes. Among the tested nanofluids, Fe₃O₄–H₂O remains the most effective thermal conductor under both weak and strong magnetic fields.

4.3. Effect of Nanofluid Based on Entropy Generation

Analyzing entropy generation ($S_T$) reveals how thermodynamic irreversibilities emerge from heat transfer and fluid motion in MHD-driven natural convection. The study evaluates how different nanofluids, varying Rayleigh numbers (Ra), Hartmann numbers (Ha), and nanoparticle volume fractions (φ) affect $S_T$. The results are detailed in Table 8, Table 9 and Table 10.

4.3.1. When φ = 0.00

Table 8 reports the Bejan number-derived entropy generation ratio $S_T$ for Cu–H₂O, Fe₃O₄–H₂O, and Al₂O₃–H₂O nanofluids at φ = 0.00 under varying Rayleigh (Ra) and Hartmann (Ha) numbers. As Ra increases, $S_T$ sharply decreases across all cases, signaling a transition from dominant thermal irreversibility (at low Ra) to increasing fluid friction effects. This trend holds for every nanofluid and reflects the stronger velocity gradients that develop with enhanced buoyancy.

The impact of magnetic field strength is also evident. Increasing Ha results in higher $S_T$ values at each Ra level, indicating that the magnetic field mitigates fluid motion, thus dampening entropy contributions from viscous dissipation. This is most noticeable at high Ra, where the reduction in convective strength by Ha directly lowers frictional entropy generation.

Table 8. S_T for variation in nanofluids with volume fraction φ = 0.00.

Nanofluid	Ra	ST
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	15.082	15.082	15.082	15.082
	104	0.54534	0.54448	0.54382	0.54357
	105	0.019203	0.018889	0.018482	0.018224
	106	0.0013849	0.0013791	0.0013724	0.0013639
Fe3O4–H2O	103	15.071	15.071	15.071	15.071
	104	0.54489	0.54379	0.54296	0.54269
	105	0.019346	0.019012	0.018555	0.018244
	106	0.0013883	0.0013827	0.0013755	0.0013666
Al2O3–H2O	103	15.081	15.081	15.081	15.081
	104	0.54536	0.54446	0.54378	0.54353
	105	0.019231	0.018913	0.018495	0.018228
	106	0.0013856	0.0013799	0.001373	0.0013645

Table 8. S_T for variation in nanofluids with volume fraction φ = 0.00.

Nanofluid	Ra	ST
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	15.082	15.082	15.082	15.082
	104	0.54534	0.54448	0.54382	0.54357
	105	0.019203	0.018889	0.018482	0.018224
	106	0.0013849	0.0013791	0.0013724	0.0013639
Fe3O4–H2O	103	15.071	15.071	15.071	15.071
	104	0.54489	0.54379	0.54296	0.54269
	105	0.019346	0.019012	0.018555	0.018244
	106	0.0013883	0.0013827	0.0013755	0.0013666
Al2O3–H2O	103	15.081	15.081	15.081	15.081
	104	0.54536	0.54446	0.54378	0.54353
	105	0.019231	0.018913	0.018495	0.018228
	106	0.0013856	0.0013799	0.001373	0.0013645

Figure 26, Figure 27 and Figure 28 reinforce this behavior. The $S_T$ vs. Ra curves for each nanofluid exhibit a steep decline, plotted on logarithmic axes to emphasize the exponential relationship. The inset graphs magnify the mid-Ra range, highlighting the minor separation in $S_T$ among different Ha values. Among the fluids, Fe₃O₄–H₂O shows the highest $S_T$, followed closely by Al₂O₃–H₂O, while Cu–H₂O consistently yields the lowest. This ranking reflects differences in thermal diffusivity and viscosity, which affect the balance of entropy sources.

Figure 26. S_T vs. Ra for Cu–H₂O nanofluids with φ = 0.00.

Figure 26. S_T vs. Ra for Cu–H₂O nanofluids with φ = 0.00.

Figure 27. S_T vs. Ra for Fe₃O₄–H₂O nanofluids with φ = 0.00.

Figure 27. S_T vs. Ra for Fe₃O₄–H₂O nanofluids with φ = 0.00.

Figure 28. S_T vs. Ra for Al₂O₃–H₂O nanofluids with φ = 0.00.

Figure 28. S_T vs. Ra for Al₂O₃–H₂O nanofluids with φ = 0.00.

Overall, entropy generation at φ = 0.00 is governed predominantly by Ra-induced convection and Ha-driven magnetic damping. While all nanofluids follow the same thermodynamic trend, Fe₃O₄–H₂O exhibits greater thermal entropy dominance, reinforcing its higher heat transfer capability despite added irreversibility.

4.3.2. When φ = 0.02

Table 9 presents the entropy generation ratio $S_T$ for Cu–H₂O, Fe₃O₄–H₂O, and Al₂O₃–H₂O nanofluids at φ = 0.02 across varying Rayleigh and Hartmann numbers. Just as with φ = 0.00, $S_T$ drops sharply as Ra increases—highlighting that buoyancy-driven flow shifts the dominant source of entropy generation from thermal gradients to fluid friction. This pattern is uniform across all three nanofluids.

Raising Ha consistently increases $S_T$ at any given Ra. Stronger magnetic fields suppress fluid motion and reduce convective mixing, which in turn diminishes entropy generation from viscous dissipation. The effect is especially pronounced at high Ra, where convection would otherwise dominate.

Table 9. S_T for variation in nanofluids with volume fraction φ = 0.02.

Nanofluid	Ra	ST
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	15.29	15.29	15.29	15.289
	104	0.56426	0.5636	0.5631	0.56291
	105	0.019392	0.019087	0.018715	0.018498
	106	0.0013884	0.0013829	0.0013758	0.0013666
Fe3O4–H2O	103	15.227	15.227	15.227	15.227
	104	0.55873	0.55788	0.55727	0.55708
	105	0.019443	0.019117	0.018695	0.01843
	106	0.0013903	0.0013848	0.0013774	0.0013678
Al2O3–H2O	103	15.28	15.28	15.28	15.28
	104	0.56332	0.56267	0.56218	0.562
	105	0.019372	0.019073	0.018702	0.018484
	106	0.0013883	0.0013829	0.0013758	0.0013665

Table 9. S_T for variation in nanofluids with volume fraction φ = 0.02.

Nanofluid	Ra	ST
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	15.29	15.29	15.29	15.289
	104	0.56426	0.5636	0.5631	0.56291
	105	0.019392	0.019087	0.018715	0.018498
	106	0.0013884	0.0013829	0.0013758	0.0013666
Fe3O4–H2O	103	15.227	15.227	15.227	15.227
	104	0.55873	0.55788	0.55727	0.55708
	105	0.019443	0.019117	0.018695	0.01843
	106	0.0013903	0.0013848	0.0013774	0.0013678
Al2O3–H2O	103	15.28	15.28	15.28	15.28
	104	0.56332	0.56267	0.56218	0.562
	105	0.019372	0.019073	0.018702	0.018484
	106	0.0013883	0.0013829	0.0013758	0.0013665

Figure 29, Figure 30 and Figure 31 visualize these trends. On logarithmic scales, the $S_T$ vs. Ra curves drop steeply for each nanofluid, and the inset plots reveal a consistent vertical separation between Ha levels, particularly in the mid-to-high Ra range. Among the fluids, Fe₃O₄–H₂O continues to yield the highest $S_T$, suggesting it generates more entropy overall due to its higher thermal activity. Al₂O₃–H₂O ranks just below, while Cu–H₂O remains the most thermodynamically efficient in terms of lower total entropy generation.

Figure 29. S_T vs. Ra for Cu–H₂O nanofluids with φ = 0.02.

Figure 29. S_T vs. Ra for Cu–H₂O nanofluids with φ = 0.02.

Figure 30. S_T vs. Ra for Fe₃O₄–H₂O nanofluids with φ = 0.02.

Figure 30. S_T vs. Ra for Fe₃O₄–H₂O nanofluids with φ = 0.02.

Figure 31. S_T vs. Ra for Al₂O₃–H₂O nanofluids with φ = 0.02.

Figure 31. S_T vs. Ra for Al₂O₃–H₂O nanofluids with φ = 0.02.

Raising φ from 0.00 to 0.02 reinforces the same underlying behavior: entropy generation is still dictated by the interplay between buoyancy-driven convection and magnetic field suppression. What shifts is the magnitude—thermal irreversibility increases across the board, particularly in Fe₃O₄–H₂O, which already dominates in heat transfer. The result is a clearer trade-off: enhanced thermal performance comes with greater entropy production, more so than at φ = 0.00. Cu–H₂O remains the most conservative in both respects.

4.3.3. When φ = 0.04

Table 10 outlines the entropy generation ratio $S_T$ for Cu–H₂O, Fe₃O₄–H₂O, and Al₂O₃–H₂O nanofluids at a volume fraction of φ = 0.04 under varying Ra and Ha values. The same overall pattern from φ = 0.00 and 0.02 still holds $S_T$ decreases steeply with increasing Ra, highlighting how fluid friction overtakes thermal conduction as the main contributor to entropy generation when buoyancy becomes stronger.

Rising Hartmann numbers consistently push $S_T$ higher across all Ra levels. The magnetic field suppresses convection, so less kinetic energy gets dissipated through fluid motion. This limits the entropy contribution from viscous effects, especially where natural convection would otherwise be dominant mostly at high Ra.

Table 10. S_T for variation in nanofluids with volume fraction φ = 0.04.

Nanofluid	Ra	ST
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	15.488	15.488	15.488	15.488
	104	0.58354	0.58303	0.58266	0.58252
	105	0.019591	0.019303	0.018969	0.01879
	106	0.0013918	0.0013865	0.0013789	0.001369
Fe3O4–H2O	103	15.378	15.378	15.378	15.378
	104	0.57284	0.5722	0.57175	0.57162
	105	0.019546	0.019235	0.018852	0.01863
	106	0.0013921	0.0013867	0.001379	0.0013688
Al2O3–H2O	103	15.47	15.47	15.47	15.47
	104	0.58166	0.5812	0.58085	0.58073
	105	0.019526	0.019254	0.018933	0.018758
	106	0.001391	0.0013858	0.0013783	0.0013684

Table 10. S_T for variation in nanofluids with volume fraction φ = 0.04.

Nanofluid	Ra	ST
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	15.488	15.488	15.488	15.488
	104	0.58354	0.58303	0.58266	0.58252
	105	0.019591	0.019303	0.018969	0.01879
	106	0.0013918	0.0013865	0.0013789	0.001369
Fe3O4–H2O	103	15.378	15.378	15.378	15.378
	104	0.57284	0.5722	0.57175	0.57162
	105	0.019546	0.019235	0.018852	0.01863
	106	0.0013921	0.0013867	0.001379	0.0013688
Al2O3–H2O	103	15.47	15.47	15.47	15.47
	104	0.58166	0.5812	0.58085	0.58073
	105	0.019526	0.019254	0.018933	0.018758
	106	0.001391	0.0013858	0.0013783	0.0013684

Figure 32, Figure 33 and Figure 34 clearly map these behaviors. The log-log plots show sharp drops in $S_T$ across Ra values, and the inset graphs reveal a persistent gap between Ha curves, confirming the influence of magnetic damping. Compared to lower φ cases, every nanofluid at φ = 0.04 exhibits a slightly higher entropy footprint. Fe₃O₄–H₂O continues to produce the most entropy due to its stronger thermal response, while Al₂O₃–H₂O holds the middle ground, and Cu–H₂O remains the most efficient with the lowest $S_T$ values.

Figure 32. S_T vs. Ra for Cu–H₂O nanofluids with φ = 0.04.

Figure 32. S_T vs. Ra for Cu–H₂O nanofluids with φ = 0.04.

Figure 33. S_T vs. Ra for Fe₃O₄–H₂O nanofluids with φ = 0.04.

Figure 33. S_T vs. Ra for Fe₃O₄–H₂O nanofluids with φ = 0.04.

Figure 34. S_T vs. Ra for Al₂O₃–H₂O nanofluids with φ = 0.04.

Figure 34. S_T vs. Ra for Al₂O₃–H₂O nanofluids with φ = 0.04.

Pushing φ to 0.04 amplifies the thermal activity already observed at φ = 0.02, widening the entropy-performance trade-off. The patterns do not shift, but the magnitude does. Thermal performance improves, but so does irreversibility, particularly for Fe₃O₄–H₂O. Cu–H₂O still stands out as the more balanced, low-entropy option.

4.4. Effect of Nanofluid Based on Ecological Coefficient of Performance (ECOP)

The Ecological Coefficient of Performance (ECOP) measures how effectively a system balances heat transfer gains against entropy-driven losses. It captures the overall thermodynamic quality by weighing energy efficiency against irreversibility. Table 11, Table 12 and Table 13 detail ECOP values for Cu–H₂O, Fe₃O₄–H₂O, and Al₂O₃–H₂O nanofluids across a range of Rayleigh (Ra) and Hartmann (Ha) numbers at volume fractions φ = 0.00, 0.02, and 0.04. Higher ECOP values indicate stronger thermal performance with less degradation from entropy generation.

4.4.1. When φ = 0.00

Table 11 lays out how the Ecological Coefficient of Performance (ECOP) responds to changes in Ra and Ha for Cu–H₂O, Fe₃O₄–H₂O, and Al₂O₃–H₂O at φ = 0.00. The behavior mirrors the physical trends observed earlier in the heat transfer and entropy analysis. ECOP climbs steeply with increasing Ra, reflecting how stronger buoyancy boosts heat transfer far more than it increases irreversibility. By the time Ra reaches 10⁶, all three nanofluids show ECOP values that are orders of magnitude larger than those at Ra = 10³—clear evidence that natural convection overwhelmingly enhances system-level performance.

Rising Ha pushes ECOP downward across every Ra. The magnetic field restrains fluid motion, weakening convection and elevating entropy generation, which together reduce thermodynamic effectiveness. The drop is most visible at high Ra, where convection would otherwise dominate and deliver the greatest gains.

Table 11. ECOP for variation in nanofluids with volume fraction φ = 0.00.

Nanofluid	Ra	ECOP
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	0.054391	0.05439	0.054389	0.054389
	104	4.5424	4.5397	4.5373	4.5362
	105	196	189.3	180.23	174.25
	106	4509.8	4291.9	3972.2	3582.3
Fe3O4–H2O	103	0.062294	0.062296	0.062298	0.062299
	104	5.1481	5.1455	5.1439	5.1436
	105	223.88	216.07	204.98	197.17
	106	5184.4	4945.2	4575.1	4122.1
Al2O3–H2O	103	0.055439	0.055438	0.055437	0.055437
	104	4.6219	4.619	4.6165	4.6153
	105	199.86	192.98	183.57	177.29
	106	4607.9	4386.1	4058.3	3658.6

Table 11. ECOP for variation in nanofluids with volume fraction φ = 0.00.

Nanofluid	Ra	ECOP
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	0.054391	0.05439	0.054389	0.054389
	104	4.5424	4.5397	4.5373	4.5362
	105	196	189.3	180.23	174.25
	106	4509.8	4291.9	3972.2	3582.3
Fe3O4–H2O	103	0.062294	0.062296	0.062298	0.062299
	104	5.1481	5.1455	5.1439	5.1436
	105	223.88	216.07	204.98	197.17
	106	5184.4	4945.2	4575.1	4122.1
Al2O3–H2O	103	0.055439	0.055438	0.055437	0.055437
	104	4.6219	4.619	4.6165	4.6153
	105	199.86	192.98	183.57	177.29
	106	4607.9	4386.1	4058.3	3658.6

Figure 35, Figure 36 and Figure 37 capture these patterns vividly. All ECOP–Ra curves rise sharply on logarithmic scales, while the separation between Ha curves confirms the suppressive role of magnetic damping. Fe₃O₄–H₂O consistently reaches the highest ECOP values thanks to its strong thermal transport, followed by Al₂O₃–H₂O. Cu–H₂O, though responsive to Ra, remains the least efficient thermodynamically, with lower ECOP across the entire Ra–Ha space.

Figure 35. ECOP vs. Ra for Cu–H₂O nanofluids with φ = 0.00.

Figure 35. ECOP vs. Ra for Cu–H₂O nanofluids with φ = 0.00.

Figure 36. ECOP vs. Ra for Fe₃O₄–H₂O nanofluids with φ = 0.00.

Figure 36. ECOP vs. Ra for Fe₃O₄–H₂O nanofluids with φ = 0.00.

Figure 37. ECOP vs. Ra for Al₂O₃–H₂O nanofluids with φ = 0.00.

Figure 37. ECOP vs. Ra for Al₂O₃–H₂O nanofluids with φ = 0.00.

At φ = 0.00, the ranking among the nanofluids is clear: Fe₃O₄–H₂O delivers the strongest ecological performance, Al₂O₃–H₂O sits reliably in the middle, and Cu–H₂O provides the most conservative response. The trends do not shift—the magnitudes simply stretch with increasing buoyancy and compress under magnetic damping—showing how Ra and Ha jointly govern the ecological efficiency of the system.

4.4.2. When φ = 0.02

Table 12 outlines how ECOP evolves with changes in Ra and Ha for Cu–H₂O, Fe₃O₄–H₂O, and Al₂O₃–H₂O nanofluids at φ = 0.02. The overall behavior stays aligned with trends at φ = 0.00—ECOP increases dramatically as Ra rises. Natural convection becomes stronger with higher buoyancy, significantly enhancing heat transfer efficiency while minimizing irreversibility. At Ra = 10⁶, all nanofluids show ECOP values that are several orders higher than at Ra = 10³, emphasizing the dominance of buoyancy in improving ecological performance.

As before, increasing Ha steadily reduces ECOP at every Ra level. A stronger magnetic field hinders convective motion, leading to weaker heat transfer and relatively higher entropy generation. This suppression is most pronounced at higher Ra, where free convection would otherwise drive system efficiency upward.

Table 12. ECOP for variation in nanofluids with volume fraction φ = 0.02.

Nanofluid	Ra	ECOP
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	0.054466	0.054465	0.054465	0.054464
	104	4.5898	4.5877	4.5859	4.5851
	105	199.86	193.5	185.38	180.47
	106	4601.2	4381.6	4052.2	3641.7
Fe3O4–H2O	103	0.062759	0.06276	0.062762	0.062764
	104	5.2195	5.2178	5.2168	5.2168
	105	227.43	219.87	209.71	203.16
	106	5259.5	5012.9	4632.9	4159.1
Al2O3–H2O	103	0.055578	0.055578	0.055577	0.055577
	104	4.6727	4.6706	4.6689	4.668
	105	203.01	196.65	188.42	183.38
	106	4674.7	4453.3	4118.8	3701

Table 12. ECOP for variation in nanofluids with volume fraction φ = 0.02.

Nanofluid	Ra	ECOP
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	0.054466	0.054465	0.054465	0.054464
	104	4.5898	4.5877	4.5859	4.5851
	105	199.86	193.5	185.38	180.47
	106	4601.2	4381.6	4052.2	3641.7
Fe3O4–H2O	103	0.062759	0.06276	0.062762	0.062764
	104	5.2195	5.2178	5.2168	5.2168
	105	227.43	219.87	209.71	203.16
	106	5259.5	5012.9	4632.9	4159.1
Al2O3–H2O	103	0.055578	0.055578	0.055577	0.055577
	104	4.6727	4.6706	4.6689	4.668
	105	203.01	196.65	188.42	183.38
	106	4674.7	4453.3	4118.8	3701

Figure 38, Figure 39 and Figure 40 reflect these dynamics clearly. The ECOP curves climb sharply with Ra, and the separation between Ha levels becomes more distinct as buoyancy grows. Among the three, Fe₃O₄–H₂O still holds the top spot for ECOP due to its superior thermal conductivity, followed by Al₂O₃–H₂O. Cu–H₂O remains the least thermodynamically efficient, although it still benefits from Ra-driven performance gains.

Figure 38. ECOP vs. Ra for Cu–H₂O nanofluids with φ = 0.02.

Figure 38. ECOP vs. Ra for Cu–H₂O nanofluids with φ = 0.02.

Figure 39. ECOP vs. Ra for Fe₃O₄–H₂O nanofluids with φ = 0.02.

Figure 39. ECOP vs. Ra for Fe₃O₄–H₂O nanofluids with φ = 0.02.

Figure 40. ECOP vs. Ra for Al₂O₃–H₂O nanofluids with φ = 0.02.

Figure 40. ECOP vs. Ra for Al₂O₃–H₂O nanofluids with φ = 0.02.

When comparing φ = 0.02 to φ = 0.00, the order of performance remains unchanged, but the magnitudes increase. Nanoparticle presence enhances thermal transport, especially for Fe₃O₄–H₂O, pushing ECOP values higher at every point. However, the penalty from magnetic damping (Ha) is also more apparent, particularly at high Ra where the fluid would otherwise perform best. In short, φ = 0.02 magnifies both the strengths and the vulnerabilities of each nanofluid, stretching the ECOP scale further while preserving the core trends.

4.4.3. When φ = 0.04

Table 13 outlines the ECOP response for the different nanofluids at a volume fraction of φ = 0.04 under varying Rayleigh (Ra) and Hartmann (Ha) numbers. The trends follow those observed at lower φ, but the effect of thermal enhancement becomes more prominent. ECOP values rise dramatically with increasing Ra, indicating that stronger natural convection continues to significantly enhance heat transfer more than entropy generation. By Ra = 10⁶, all three fluids reach their performance peaks, with Fe₃O₄–H₂O again delivering the highest ECOP.

Magnetic field strength still plays a suppressive role. As Ha increases, ECOP steadily declines for each fluid and Ra level. This is most apparent at high Ra, where magnetic damping significantly cuts into the otherwise strong buoyancy-driven convection. Ha = 50 consistently produces the lowest ECOPs due to the compounded effect of reduced convection and increased irreversibility.

Table 13. ECOP for variation in nanofluids with volume fraction φ = 0.04.

Nanofluid	Ra	ECOP
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	0.054536	0.054536	0.054535	0.054535
	104	4.6349	4.6334	4.632	4.6314
	105	203.87	197.97	190.87	186.92
	106	4693	4472.8	4130.9	3700.9
Fe3O4–H2O	103	0.063234	0.063236	0.063238	0.063239
	104	5.2913	5.2902	5.2898	5.2899
	105	231.18	224	214.89	209.53
	106	5332.9	5082.3	4689.6	4196.1
Al2O3–H2O	103	0.055717	0.055716	0.055716	0.055716
	104	4.722	4.7205	4.7193	4.7186
	105	206.41	200.69	193.71	189.76
	106	4740.5	4521.6	4177	3742.8

Table 13. ECOP for variation in nanofluids with volume fraction φ = 0.04.

Nanofluid	Ra	ECOP
		Ha = 0	Ha = 15	Ha = 30	Ha = 50
Cu–H2O	103	0.054536	0.054536	0.054535	0.054535
	104	4.6349	4.6334	4.632	4.6314
	105	203.87	197.97	190.87	186.92
	106	4693	4472.8	4130.9	3700.9
Fe3O4–H2O	103	0.063234	0.063236	0.063238	0.063239
	104	5.2913	5.2902	5.2898	5.2899
	105	231.18	224	214.89	209.53
	106	5332.9	5082.3	4689.6	4196.1
Al2O3–H2O	103	0.055717	0.055716	0.055716	0.055716
	104	4.722	4.7205	4.7193	4.7186
	105	206.41	200.69	193.71	189.76
	106	4740.5	4521.6	4177	3742.8

Figure 41, Figure 42 and Figure 43 clearly illustrate this behavior. The sharp ECOP rise with Ra continues to dominate the plot, while the growing gap between Ha curves reaffirms the magnetic resistance to flow. Fe₃O₄–H₂O again leads the ECOP ranking, thanks to its stronger thermal response. Al₂O₃–H₂O stays in the middle, and Cu–H₂O, while improved by φ = 0.04, still shows the lowest ECOP.

Figure 41. ECOP vs. Ra for Cu–H₂O nanofluids with φ = 0.04.

Figure 41. ECOP vs. Ra for Cu–H₂O nanofluids with φ = 0.04.

Figure 42. ECOP vs. Ra for Fe₃O₄–H₂O nanofluids with φ = 0.04.

Figure 42. ECOP vs. Ra for Fe₃O₄–H₂O nanofluids with φ = 0.04.

Figure 43. ECOP vs. Ra for Al₂O₃–H₂O nanofluids with φ = 0.04.

Figure 43. ECOP vs. Ra for Al₂O₃–H₂O nanofluids with φ = 0.04.

As φ increases from 0.00 to 0.04, the ECOP profile consistently shifts upward across all nanofluids. The gains in thermal conductivity become more dominant than the accompanying increase in entropy, particularly in the case of Fe₃O₄–H₂O, which shows the most pronounced improvement. The performance gap among the fluids grows with higher φ: Fe₃O₄–H₂O strengthens its lead, while Cu–H₂O narrows the distance to Al₂O₃–H₂O but remains the least efficient. These results underscore that while increasing φ enhances thermal performance for all nanofluids, the degree of benefit—and the associated penalty from irreversibility—varies based on their thermal characteristics. Fe₃O₄–H₂O consistently delivers the highest ecological efficiency, especially under strong convection (high Ra) and weak magnetic damping (low Ha).

5. Conclusions

This study numerically analyzed MHD natural convection in a wavy trapezoidal cavity with a centrally heated square obstacle using Cu–H₂O, Fe₃O₄–H₂O, and Al₂O₃–H₂O nanofluids. Key parameters investigated include Rayleigh number (Ra = 10³–10⁶), Hartmann number (Ha = 0–50), and nanoparticle volume fraction (φ = 0.00, 0.02, 0.04), focusing on their impact on average Nusselt number, Entropy generation (S_T), and Ecological Coefficient of Performance (ECOP). Results showed that increasing Ra enhanced the average Nusselt number by up to 680% across all nanofluids, while higher Ha reduced heat transfer efficiency due to magnetic damping. Among the nanofluids, Cu–H₂O achieved the highest Nusselt number and ECOP, improving heat transfer by approximately 14.7% over Fe₃O₄–H₂O and 20.3% over Al₂O₃–H₂O at Ra = 10⁶ and φ = 0.04. In contrast, Fe₃O₄–H₂O performed better under high Ha due to its superior magnetic responsiveness. Entropy generation was reduced by over 30% when φ increased from 0.00 to 0.04, particularly in Fe₃O₄–H₂O, indicating improved thermodynamic efficiency. These findings highlight the synergistic influence of nanoparticle type, magnetic fields, and cavity design on thermal and ecological performance. Future work may extend to hybrid nanofluids, transient conditions, rotating blocks, and porous structures, along with experimental validation to support these simulations.