The optimization of heat transfer in enclosures with complex geometries using nanofluids and magnetohydrodynamic (MHD) effects has attracted extensive research interest due to its broad applications in thermal energy systems, electronic cooling, and advanced heat exchangers. Trapezoidal cavities, owing to their structural flexibility, have become a popular geometry in recent thermal studies, particularly when subjected to natural or mixed convection using various working fluids [1] - [4] . Nanofluids, such as Cu-water and hybrid formulations, offer enhanced thermal conductivity and flow characteristics that support more efficient convective mechanisms [5] - [8] . Investigations combining MHD effects and nanofluid convection in trapezoidal or wavy enclosures reveal that magnetic fields significantly influence boundary layer thickness, flow circulation, and heat transfer uniformity [9] - [13] . The incorporation of non-standard geometries—like corrugated, sinusoidal, and wavy boundaries—further modifies thermal gradients and promotes stronger vortex formations, especially in cases involving inclined or undulated walls [14] - [19] . Recent studies have also examined internal features such as heated cylinders, triangular blocks, and solid obstacles to understand conjugate heat transfer behavior and localized fluid motion [20] - [24] . Hybrid nanofluids with phase-change materials or metallic nanoparticles in porous and double-diffusive environments have shown marked improvements in thermal efficiency under both steady and oscillating magnetic fields [25] - [28] . Simulations using finite element and finite volume methods have demonstrated strong agreement with experimental results in capturing complex thermal-fluid interactions inside cavities with internal heat generation and varying thermal boundary conditions [29] - [33] . Star-shaped internal heaters are an emerging focus due to their geometric complexity, increased surface area, and ability to intensify convective currents and thermal mixing [5] [17] [34] - [36] . Multiphysics models that incorporate thermal radiation, entropy generation, and MHD damping illustrate how advanced control parameters can improve heat transfer performance in energy systems [37] - [39] . Meanwhile, parametric studies using response surface methodologies and sensitivity analysis offer quantitative strategies for optimization across varying geometric, fluidic, and magnetic configurations [40] - [42] . Building upon these insights, recent simulations have focused on zigzag wall structures, ferrofluids, and rotating elements to further explore the interplay between geometry, nanofluid properties, and external forces [43] [44] . The numerically investigation of conjugate mixed convection heat transfer in a lid-driven cavity featuring a spinning heat-generating solid cylinder to analyze thermal performance enhancements [45] .

In this context, the current study contributes to the literature by numerically investigating MHD-driven conjugate free convection of Cu-H₂O nanofluid within an inclined trapezoidal enclosure featuring a corrugated top wall and internally heat-generating star-shaped copper obstacles. The research aims to uncover how variations in Rayleigh number, Hartmann number, and nanoparticle volume fraction affect flow dynamics and heat transfer, thereby offering design guidelines for efficient thermal management in advanced engineering applications.

The physical model considered in this study consists of a two-dimensional inclined trapezoidal enclosure filled with a Cu–H₂O nanofluid and subjected to free convective heat transfer under the influence of a uniform transverse magnetic field. As shown in Figure 1 , the top boundary of the enclosure is sinusoidally corrugated, while the side walls are inclined at a base angle γ, forming a trapezoidal geometry. Two identical internal star-shaped copper obstacles are embedded symmetrically within the nanofluid domain, each acting as a localized heat source.

The coupled heat transfer process includes conduction within the solid copper obstacles and convection–conduction within the nanofluid, forming a conjugate heat transfer problem. The nanofluid is assumed to be incompressible and Newtonian, and the flow is driven by buoyancy forces due to internal heating, with the Boussinesq approximation applied to account for density variations.

The governing equations are formulated for mass, momentum, and energy conservation and are solved numerically using the Galerkin finite element method (FEM). The system is subject to the following thermal boundary conditions, summarized in Table 1 and elaborated below:

Top Wavy Wall: Maintained at a constant cold temperature T = T_m, this boundary facilitates upward thermal diffusion and mimics a cooled surface. Its sinusoidal profile increases surface area, enhancing local heat exchange and promoting complex flow structures such as vortices.

Bottom Wall: Also kept isothermal at the cold reference temperature T = T_m, the bottom wall acts as a second cooling surface. The thermal symmetry created by the top and bottom boundaries encourages the rise of heated fluid and the fall of cooled fluid, forming closed-loop natural convection currents.

Left and Right Inclined Walls: These boundaries are thermally insulated, implying zero heat flux across them, mathematically expressed as n⋅∇T = 0. This adiabatic condition confines the thermal activity within the interior domain, ensuring that heat transfer is driven purely by internal generation and vertical heat dissipation.

Star-Shaped Obstacles: Each copper obstacle is modeled as an internally heat-generating solid, with uniform volumetric heating characterized by a source term Q. These localized heaters represent embedded thermal devices and induce strong buoyancy-driven flows within the surrounding nanofluid, making them critical to the dynamics of the convective patterns observed.

This boundary condition setup ensures a vertically symmetric thermal gradient with dominant internal heating. Combined with the wavy top surface, it produces a highly interactive flow environment ideal for investigating the effect of magnetic field strength, Rayleigh number, and nanoparticle volume fraction on the system’s thermal behavior.

This study investigates the conjugate magnetohydrodynamic (MHD) free convection within a nanofluid-saturated trapezoidal enclosure containing internal heat-generating star-shaped obstacles. The base fluid is water, while copper (Cu) nanoparticles are dispersed to form the Cu-H₂O nanofluid. The governing physical model incorporates the conservation equations of mass, momentum, and energy under the Boussinesq approximation. The enclosure is subjected to a uniform magnetic field perpendicular to the slanted side, introducing Lorentz force effects on the convective flow.

Governing Equations:

Assuming laminar, incompressible, and steady-state flow conditions, the governing dimensional and non-dimensional equations (1) to (22) are as follows:

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (1)

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

${\rho }_{nf}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{\partial p}{\partial y}+{\mu }_{nf}\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)+{\rho }_{nf}g{\beta }_{nf}\left(T-T_c\right)\cos \left(\gamma \right)-{\sigma }_{nf}{\beta }_0^{2}v$(3)

${\rho }_{nf}{c}_{p,nf}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=k_{nf}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)+Q$ (4)

Nanofluid Property:

${\rho }_{nf}=\left(1-\varphi \right){\rho }_f+\varphi {\rho }_s$ (5)

${\left(\rho c_p\right)}_{nf}=\left(1-\varphi \right){\left(\rho c_p\right)}_f+\varphi {\left(\rho c_p\right)}_s$ (6)

${\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\varphi \right)}^{2.5}}$ (7)

$k_{nf}=k_f\left[\frac{k_s+2k_f-2\varphi \left(k_f-k_s\right)}{k_s+2k_f+\varphi \left(k_f-k_s\right)}\right]$ (8)

${\beta }_{nf}=\frac{\left(1-\varphi \right){\rho }_f{\beta }_f+\varphi {\rho }_s{\beta }_s}{{\rho }_{nf}}$ (9)

Non-Dimensional Equations:

$X=\frac{x}{L},Y=\frac{y}{L},U=\frac{uL}{{\alpha }_f},V=\frac{vL}{{\alpha }_f},\theta =\frac{T-T_c}{T_h-T_c},P=\frac{p{L}^2}{{\mu }_{nf}},\text{Here},{\alpha }_f=\frac{k_f}{{\rho }_f{c}_{p,f}}$ (10)

$Ra=\frac{g{\beta }_{nf}\left(T_h-T_c\right)L^3}{{\nu }_{nf}{\alpha }_f},Pr=\frac{{\nu }_f}{{\alpha }_f},Ha={\beta }_{0}L\sqrt{\frac{{\sigma }_{nf}}{{\mu }_{nf}}},Q^{*}=\frac{Q{L}^2}{k_{nf}\left(T_h-T_c\right)}$ (11)

$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$ (12)

$U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{{\mu }_{nf}}{{\mu }_f}\left(\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}\right)+RaPr\frac{{\rho }_{nf}{\beta }_{nf}}{{\rho }_f{\beta }_f}\theta \sin \left(\gamma \right)-H{a}^{2}U$(13)

$U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{{\mu }_{nf}}{{\mu }_f}\left(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}\right)+RaPr\frac{{\rho }_{nf}{\beta }_{nf}}{{\rho }_f{\beta }_f}\theta \cos \left(\gamma \right)-H{a}^{2}U$(14)

$U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{k_{nf}}{k_f}\frac{1}{Pr}\left(\frac{{\partial }^2\theta }{\partial X^2}+\frac{{\partial }^2\theta }{\partial Y^2}\right)+Q^{*}$ (15)

Here, $Q^{*}$ represents the internal heat generation within the star-shaped obstacles

and is scaled using the characteristic temperature difference $\Delta T=T_h-T_c$

Nanofluid Property:

$\frac{{\rho }_{nf}}{{\rho }_f}=\left(1-\varphi \right)+\varphi \frac{{\rho }_s}{{\rho }_f}$ (16)

$\frac{{\left(\rho c_p\right)}_{nf}}{{\left(\rho c_p\right)}_f}=\left(1-\varphi \right)+\varphi \frac{{\left(\rho c_p\right)}_s}{{\left(\rho c_p\right)}_f}$ (17)

$\frac{{\mu }_{nf}}{{\mu }_f}=\frac{1}{{\left(1-\varphi \right)}^{2.5}}$ (18)

$\frac{k_{nf}}{k_f}=\frac{k_s+2k_f-2\varphi \left(k_f-k_s\right)}{k_s+2k_f+\varphi \left(k_f-k_s\right)}$ (19)

${\beta }_{nf}{\rho }_{nf}=\left(1-\varphi \right){\rho }_f{\beta }_f+\varphi {\rho }_s{\beta }_s$ (20)

Local Nusselt number (Nu) and Average Nusselt number (Nu_avg) inside the enclosure are assessed and expressed as performance parameters of the current system in the following ways:

$N{u}_{Local}=-{\frac{\partial \theta }{\partial n}|}_{\text{Wavy-Wall}}$ (21)

$\bar{N}u=\frac{1}{L_w}{\displaystyle {\int }_0^{L_w}N{u}_{Local}\text{d}X}=-\frac{1}{L_w}{\displaystyle {\int }_0^{L_w}\frac{\partial \theta }{\partial n}\text{d}X}$ (22)

Here, $L_w$ is the arc length of the top wavy wall.

Thermophysical Parameters

The thermophysical properties of copper nanoparticles and pure water are summarized in Table 2 , while the nanofluid properties for varying volume fractions are detailed in Table 3 .

Table 2 shows the thermophysical properties of copper nanoparticles and pure water. Copper has high density, thermal, and electrical conductivity, making it ideal for enhancing heat transfer. In contrast, water has a much higher specific heat and expansion coefficient, supporting strong buoyancy effects.

Table 3 presents nanofluid properties at different nanoparticle volume fractions. As copper content increases, thermal conductivity, viscosity, and density rise, while specific heat and thermal expansion decrease. This highlights the trade-off between improved heat transfer and fluid resistance.

This study systematically analyzed the convective heat transfer behavior of Cu-water nanofluid in an inclined trapezoidal enclosure featuring a wavy top wall and internally placed star-shaped heat sources under the influence of a magnetic field. Through parametric variation of Rayleigh numbers (10³ - 10⁶), Hartmann numbers (15, 50), and nanoparticle volume fractions (0.01 - 0.03), and employing the finite element method, the investigation highlighted how geometric complexity and magnetic control collectively influence thermal transport and fluid flow characteristics.

Key Findings:

1) Rayleigh number enhancement consistently increased the Nusselt number and promoted stronger convective currents across all nanoparticle concentrations and magnetic field strengths.

2) Higher nanoparticle volume fractions improved the thermal conductivity of the fluid and elevated convective heat transfer, with ϕ = 0.03 yielding the highest performance.

3) Magnetic field application (Ha) suppressed convection due to Lorentz force damping, resulting in reduced Nusselt numbers, especially at higher Ha = 50 compared to Ha = 15.

4) Velocity and temperature contours revealed that higher Ra and lower Ha produce more vigorous circulation and better thermal mixing, while increasing ϕ reduced the average fluid temperature due to enhanced thermal transport.

5) A maximum enhancement of heat transfer was observed at Ra = 10⁶, ϕ = 0.03, and Ha = 15, demonstrating the synergistic effect of thermal conductivity and buoyancy dominance under weaker magnetic damping.

Future Work:

To extend the current research, the following directions are suggested:

1) Investigate the transient behavior of nanofluid convection in time-dependent heating applications.

2) Incorporate non-Newtonian nanofluids and hybrid nanoparticles to assess advanced thermal performance.

3) Evaluate the effects of variable wall temperature, different obstacle shapes, or moving heat sources for more complex real-world modeling.

4) Explore machine learning-based surrogate models to predict optimal configurations with reduced computational cost.

5) Consider the impact of three-dimensional geometries and inclination angles to improve generalization for practical designs.

The findings demonstrate that integrating Cu-water nanofluids with star-shaped internal heaters in a wavy top trapezoidal cavity under magnetic influence provides a highly effective approach for thermal enhancement. By fine-tuning key parameters Rayleigh number, nanoparticle volume fraction, and Hartmann number, the system achieves improved convective mixing and enhanced heat transfer while maintaining controlled flow behavior. The complex geometry amplifies thermal performance through increased surface area and strategic flow disruption. This makes it particularly well-suited for applications like data center cooling and compact heat exchangers, where managing localized heat buildup is essential for maintaining reliability and operational efficiency.

We sincerely thank the Department of Mathematics, DUET, Gazipur, for their continuous support, guidance, and facilities throughout this research work.