Natural convection in enclosures is a fundamental mechanism widely utilized in thermal management systems such as electronic cooling devices, solar energy collectors, and energy-efficient building designs [1] . The performance of these systems has seen considerable improvement through the introduction of nanofluids, fluids containing suspended nanoparticles that enhance thermal conductivity and heat transport characteristics [2] . Due to their superior thermophysical properties, nanofluids have been effective in optimizing convective flow behavior across various boundary and geometric configurations [3] .

The integration of internal heat-generating obstacles within cavities has a pronounced impact on natural convection patterns. These effects are closely tied to the obstacle’s shape, size, and thermal loading [4] . Common geometries such as circular, square, elliptical, and triangular obstacles have demonstrated their ability to influence local flow recirculation, vortex formation, and overall heat transfer performance [5] . Additionally, the inclination of the cavity introduces a gravitational asymmetry that modifies buoyancy-driven forces, further affecting flow circulation and entropy generation [6] .

When magnetic fields are applied, magnetohydrodynamic (MHD) effects arise, characterized by the Hartmann number. These effects tend to suppress convective motion due to the Lorentz force, thereby altering the dominant heat transfer mode from convection to conduction [7] . Nonetheless, careful tuning of magnetic field intensity can lead to improved temperature control and thermal uniformity in nanofluid-filled enclosures [8] . The shape and placement of internal obstacles, such as square or triangular blocks, have been shown to enhance boundary layer disturbance, leading to better convective performance [9] [10] . Inclined cavities filled with nanofluids are of particular interest as changes in inclination affect the alignment between thermal gradients and gravity, thus influencing heat transfer effectiveness and entropy generation [11] [12] . These effects become more complex in the presence of magnetic fields, where interactions between geometric design, thermal forces, and electromagnetic suppression must be considered simultaneously [13] .

In recent studies, cavity wall modifications, especially wavy or corrugated surfaces, have demonstrated the ability to create secondary vortices, thereby improving heat transfer rates and flow mixing [14] [15] . At the same time, the use of hybrid nanofluids, composed of multiple types of nanoparticles, offers enhanced control over thermophysical properties like conductivity and viscosity [16] . Composite cavities with porous structures or layered domains further enhance energy transport efficiency by introducing porosity and permeability effects [17] [18] . Performance optimization has also been linked to nanoparticle shape, volume fraction, and conductivity ratio [19] , while entropy generation remains a key indicator of system irreversibility under conduction, viscous, and magnetic effects [20] . Surface undulations, sinusoidal heating, and wall topologies further alter flow and thermal fields, improving both heat transfer and entropy-based metrics [21] - [23] .

Studies of hybrid and multi-phase cavities [24] [25] , particularly under MHD influence in trapezoidal geometries, are still limited [26] - [28] . Meanwhile, advanced modeling, including Brownian motion and thermophoresis [29] , and finite element techniques [30] , has improved accuracy. Recent validation confirms that obstacle shape and wall undulation significantly affect convective behavior in nanofluid systems [31] [32] .

Despite extensive research, limited studies have concurrently addressed the combined influence of internal obstacle shape, wall corrugation geometry, and cavity inclination under magnetohydrodynamic (MHD) conditions in trapezoidal nanofluid-filled enclosures. These factors are crucial in engineering applications where precise control of thermal behavior is required under complex boundary conditions. Notably, geometric modifications such as internal obstructions and wall undulations alter flow recirculation, thermal stratification, and entropy generation, particularly when coupled with the damping effects of an external magnetic field.

To fill this research gap, the present study numerically investigates MHD natural convection in a trapezoidal cavity filled with Cu-H₂O nanofluid, incorporating variations in obstacle shapes (square, triangular, and star), wall corrugation profiles (triangular, square, sinusoidal), and inclination angles (15˚, 30˚, 45˚). The Galerkin finite element method is employed to solve the coupled fluid-thermal equations. The study systematically evaluates heat transfer via the Nusselt number (Nu), entropy generation (S_T), and entropy generation-based cooling performance (ECOP). Through comprehensive parametric analysis, the findings aim to identify thermally optimized configurations for advanced passive cooling systems influenced by magnetic and geometric effects.

This study investigates natural convection and entropy generation within a two-dimensional trapezoidal enclosure filled with Cu-H₂O nanofluid. The cavity features a base angle γ = 15˚ and is inclined at λ =15˚, 30˚, and 45˚, to evaluate gravitational reorientation effects. Two uniformly heated solid blocks configured as a star, a square, or a triangle are centrally embedded in the cavity. The top boundary is modeled with sinusoidal, square, or triangular corrugations to examine surface geometry effects. Figure 1- Figure 3 illustrate these geometric configurations.

A uniform horizontal magnetic field is applied to account for magnetohydrodynamic (MHD) effects. The top and side walls are maintained at a constant cold temperature, while the inclined bottom wall is adiabatic. All cavity walls and embedded solid surfaces satisfy the no-slip condition. The internal obstacles act as uniform volumetric heat sources. The governing mathematical equations and their non-dimensional forms, which account for natural convection, MHD effects, and nanofluid behavior, boundary conditions are detailed 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:

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

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

Internal volumetric heat generation applied in the solid domain, Q > 0; 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)

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

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

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

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

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

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

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

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

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

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

where A represents the surface area of the computational domain.

The Ecological Coefficient of Performance (ECOP) represents the balance between heat transfer enhancement and thermodynamic irreversibility. A higher ECOP indicates more efficient thermal performance, where greater heat transfer occurs with lower entropy generation. In essence, it quantifies the trade-off between energy utilization and system losses, reflecting the overall thermodynamic sustainability of the convective process. It can be defined as follows to provide a relative estimate of total entropy production associated with heat transfer:

$ECOP=\frac{S_T}{Nu}$ (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. The simulations cover a Rayleigh number (Ra) range of 10³ - 10⁶ and a Hartmann number (Ha) range of 0 - 50. Thermophysical properties of the nanofluid components, including density, thermal conductivity, specific heat, viscosity, and electrical conductivity, are listed in Table 1 at a reference temperature of 300 K. These properties are essential for modeling buoyancy-driven convection, MHD behavior, and entropy generation.

The Galerkin Finite Element Method (FEM) is used to solve the governing conservation equations for mass, momentum, and energy. Performance is evaluated using the Nusselt number (Nu), Entropy generation (S_T), and Ecological Coefficient of Performance (ECOP).

This section presents the numerical results for natural convection heat transfer in a nanofluid-filled trapezoidal cavity under magnetohydrodynamic (MHD) effects. It is divided into two key analyses: Firstly, it examines the influence of different obstacle shapes (square, star, triangular) and wall corrugation types (sinusoidal, square, triangular) on thermal performance; and secondly, it evaluates the impact of varying wall inclination angles (15˚, 30˚, 45˚). The outcomes are discussed in terms of Nusselt number (Nu), entropy generation (S_T), and energy performance (ECOP) across a range of Rayleigh and Hartmann numbers to identify optimal configurations for enhanced heat transfer.

This numerical study examined MHD natural convection in a nanofluid-filled trapezoidal cavity, focusing on the roles of obstacle shape, wall corrugation, and inclination angle. The simulation results were validated against Abdelmalek et al. [17] , showing strong agreement in thermal profiles and Nusselt number values. As shown in Figure 13 , both studies display similar isotherm patterns around star-shaped obstacles at Ra = 10⁴, with the present work achieving smoother and more symmetrical contours, likely due to finer grid resolution.

Table 29 further supports this validation, with Nusselt number deviations below 1.5% across Ra = 10³ - 10⁶, confirming the accuracy and stability of the current model. These findings underline the impact of geometric complexity and inclination on heat transfer in magnetically influenced nanofluid systems. In particular, the combination of corrugated walls and non-circular obstacles enhances passive thermal control.

The validated model provides a reliable framework for interpreting the thermofluidic behavior of nanofluids under magnetic influence, allowing meaningful insights into practical applications. The results indicate that geometric configuration plays a decisive role in determining whether a system favors maximum heat transfer or optimal thermodynamic efficiency. In engineering practice, systems such as electronic cooling units, high-power microchips, or microchannel heat exchangers prioritize rapid heat dissipation to prevent thermal buildup, thus requiring configurations that yield the highest Nusselt number (Nu). For these purposes, square-shaped obstacles combined with triangular corrugated walls proved most effective, generating strong vortices and high convective flow intensity that enhance local and average heat transfer rates.

In contrast, applications such as solar thermal collectors, heat storage systems, and sustainable energy devices often prioritize maximum Ecological Coefficient of Performance (ECOP), which reflects the trade-off between heat transfer and irreversibility (entropy generation). In such cases, square obstacles with sinusoidal walls offer the best thermodynamic performance, maintaining efficient energy utilization with minimal entropy production. Hence, while configurations optimizing Nu are ideal for intensive cooling, those maximizing ECOP are more suitable for long-term, energy-efficient thermal management. This distinction highlights how the present model can guide geometry-based optimization of MHD nanofluid systems across a wide range of real-world thermal applications.

This numerical investigation explored the effects of obstacle shape, wall corrugation, and inclination angle on magnetohydrodynamic (MHD) natural convection in a nanofluid-filled trapezoidal cavity. The study analyzed heat transfer (Nu), entropy generation (S_T), and thermodynamic efficiency (ECOP) under varying Rayleigh (Ra) and Hartmann (Ha) numbers to identify optimal geometric and operating conditions.

The key findings can be summarized as follows:

1) Heat Transfer: Square-shaped obstacles combined with triangular wavy walls achieved the highest Nusselt numbers, indicating superior convective heat transfer performance.

2) Magnetic Field Effect: Increasing Hartmann number (Ha) reduced Nu and ECOP in all cases, confirming the damping influence of magnetic fields on convective motion.

3) Entropy Generation: Triangular corrugated walls produced slightly higher S_T at low Ra, while magnetic effects minimized geometric influence at high Ha.

4) Energy Efficiency (ECOP): Sinusoidal walls with square obstacles provided the highest ECOP at λ = 15˚ and Ha = 0, showing the most efficient energy conversion.

5) Inclination Angle: Lower inclination angles (λ = 15˚) consistently enhanced heat transfer and efficiency, while higher angles weakened buoyancy effects.

In summary, the optimal configuration for MHD natural convection in nanofluid-filled cavities is achieved with square obstacles, triangular or sinusoidal wavy walls, low inclination (15˚), and minimal magnetic influence. These findings provide practical insights for designing efficient thermal systems in applications such as electronics cooling, solar collectors, and microfluidic devices.

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.