Numerical Study of Heat Transfer in a Nanofluid-Filled Trapezoidal Enclosure with Star-Shaped Heat Sources under Magnetic Field

Sree Pradip Kumer Sarker^*, Md. Mahmud Alam

Department of Mathematics, Dhaka University of Engineering and Technology (DUET), Gazipur, Bangladesh.

DOI: 10.4236/ojfd.2025.153009                

Abstract

Effective thermal management in geometrically irregular enclosures is crucial for optimizing performance in electronics cooling, renewable energy, and thermal engineering systems. This numerical study investigates conjugate magnetohydrodynamic (MHD) free convection heat transfer in a Cu-H₂O nanofluid-filled trapezoidal cavity with a wavy (corrugated) top boundary and embedded internally heat-generating star-shaped copper obstacles. The cavity base angle(ɣ) is fixed at 15˚, and the simulation employs the Galerkin finite element method to solve coupled solid-fluid heat transfer equations under varying Rayleigh numbers (Ra = 10³ - 10⁶), Hartmann numbers (Ha = 15, 50), and nanoparticle volume fractions (ϕ = 0.01 - 0.03). The Galerkin finite element method (FEM) is employed due to its robustness in handling irregular geometries and complex boundary conditions. This method ensures accurate resolution of coupled thermal and fluid fields within the wavy trapezoidal cavity and around embedded star-shaped heat sources. Results reveal that both the shape and placement of star-shaped heaters strongly influence heat flow, enhancing convective mixing and thermal dispersion through vortex structures. The corrugated top wall further improves vertical heat transport. Increasing the nanoparticle concentration from 0.01 to 0.03 led to a significant enhancement in average Nusselt number and a reduction in average fluid temperature, demonstrating the superior heat transfer capability of nanofluids. At higher Rayleigh numbers, the dominance of buoyancy-induced flow over magnetic damping further amplified convective efficiency. The findings underscore the potential of integrating nanofluid technologies with complex geometry and internal heating to optimize thermal control in passive cooling applications and high-efficiency heat exchangers.

Keywords

Nanofluid, Trapezoidal Cavity, Magnetohydrodynamics, Natural Convection, Thermal Conductivity

1. Introduction

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.

2. Model Description

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.

Figure 1. Geometry.

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:

Table 1. Boundary condition.

Boundary	Condition	Temperature Value
Top Wavy Wall	Isothermal	T = Tm
Bottom Wall	Isothermal	T = Tm
Left & Right Walls	Thermal Insulation (Adiabatic)	n⋅∇T = 0
Star Obstacles	Internal Heat Source	Volumetric heating Q

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.

3. Mathematical Modeling

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)

$\overline{N}u=\frac{1}{L_w}{\displaystyle {\int }_0^{L_w}N{u}_{Local}\text{d}X}=-\frac{1}{L_w}{\displaystyle {\int }_0^{L_w}\frac{\partial \theta }{\partial n}\text{d}X}$ (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. Thermophysical properties of pure water and copper nanoparticles.

Thermophysical Properties	Copper Nanoparticles	Pure Water
ρ (kg/m3)	8933	996.60
Cp (J/kg·K)	385	4179.20
k (W/m·K)	401	0.6102
β (1/K)	4.99 × 10−5	2.66 × 10−4
σ (1/Ω·m)	5.96 × 10−7	5.50 × 10−6

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.

Table 3. Properties of nanofluid.

Property	Unit	Volume Fraction of Nanoparticles
		0.01	0.02	0.03
Density (ρ)	kg/m3	1076.00	1155.30	1234.70
Specific heat (Cp)	J/kg·K	3864.20	3592.50	3355.70
Thermal conductivity (k)	W/m·K	0.62861	0.64739	0.66655
Dynamic viscosity (μ)	kg/m·s	8.7552 × 10−4	8.9803 × 10−4	9.2135 × 10−4
Electric conductivity (σ)	Ω⁻¹·m⁻¹	5.6667 × 10−6	5.8367 × 10−6	6.0103 × 10−6
Thermal diffusivity (α)	m²/s	1.4651 × 10−⁷	1.4651 × 10−⁷	1.465 × 10−7
Volumetric expansion (β)	K⁻¹	2.4806 × 10−4	2.3258 × 10−4	2.191 × 10−4

4. Results and Discussion

4.1. Verification

The streamline comparison in Figure 2 demonstrates strong agreement between the present numerical study and the results from Ref. [45].

Figure 2. Comparison of the Streamline in the present study and the Ref. [45].

Both streamlines illustrate a primary circulation cell formed due to the lid-driven motion and the influence of the central rotating heat-generating cylinder. In Ref. [45], the streamlines show symmetrical vortex structures and steady flow separation zones near the cylinder, while the present study captures similar flow features with enhanced clarity and color-coded velocity magnitudes, validating the computational model. This visual correlation supports the reliability and accuracy of the present simulation in capturing conjugate mixed convection dynamics.

4.2. Discussion

To visualize the effects of nanoparticle volume fraction, magnetic field strength, and thermal buoyancy on convective flow and heat transfer behavior, Figure 3-8 present detailed contour plots of velocity and temperature distributions. These figures capture the flow dynamics and thermal gradients within the nanofluid-filled trapezoidal enclosure for three nanoparticle volume fractions (ϕ = 0.01, 0.02, and 0.03), under varying Rayleigh numbers (Ra = 10³ to 10⁶) and Hartmann numbers (Ha = 15 and 50). The contours highlight how thermal and hydrodynamic fields evolve due to internal heat generation from star-shaped obstacles, and they illustrate the interplay between magnetic damping, nanoparticle-enhanced thermal conductivity, and buoyancy-driven convection.

Figure 3. Velocity contour for nanoparticle volume fraction ϕ = 0.01.

Figure 4. Temperature contour for nanoparticle volume fraction ϕ = 0.01.

Figure 3 and Figure 4 display the velocity and temperature contours for ϕ = 0.01, at four Rayleigh numbers (10³ to 10⁶) and two Hartmann numbers (Ha = 15 and 50). At low Ra (10³), the flow is weak and mostly conduction-dominated, with nearly parallel isotherms and minimal vortex formation. As Ra increases, buoyancy becomes more significant, leading to stronger circulation and distorted temperature lines. At Ha = 15, velocity fields show more developed vortices compared to Ha = 50, where the magnetic field dampens flow intensity. The temperature fields also become more stratified at high Ra, with thinner boundary layers near the heated obstacles.

Figure 5 and Figure 6 correspond to ϕ = 0.02. An increase in nanoparticle concentration leads to enhanced thermal conductivity, slightly strengthening the thermal gradients and flow strength at comparable Ra and Ha values. Compared

Figure 5. Velocity contour for nanoparticle volume fraction ϕ = 0.02.

Figure 6. Temperature contour for nanoparticle volume fraction ϕ = 0.02.

to ϕ = 0.01, vortex structures become more intense at Ra = 10⁵ and 10⁶, and isotherms near the heated star-shaped obstacles become steeper, indicating better heat dissipation. Again, higher Hartmann number (Ha = 50) suppresses flow circulation due to increased Lorentz force, especially at low Ra.

Figure 7 and Figure 8 represent the highest nanoparticle volume fraction ϕ = 0.03. The velocity contours reveal further strengthening of vortices and larger

Figure 7. Velocity contour for nanoparticle volume fraction ϕ = 0.03.

recirculation zones at high Ra, particularly for Ha = 15. The increased thermal conductivity of the nanofluid at this concentration promotes stronger convection. However, the Ha = 50 cases consistently show reduced flow activity, confirming that the magnetic field hinders buoyancy-driven motion. The temperature contours illustrate sharper thermal gradients near the star-shaped heat sources and thinner thermal boundary layers as Ra increases.

Figure 8. Temperature contour for nanoparticle volume fraction ϕ = 0.03.

Figures 3-8 observations are:

1) Increasing Ra enhances natural convection, visible through intensified vortex size and sharper isotherms.

2) Higher nanoparticle volume fractions (ϕ) improve heat transfer by increasing thermal conductivity and modifying flow structure.

3) Stronger magnetic fields (higher Ha) consistently suppress fluid motion, reducing convective heat transfer and enhancing conduction dominance.

4) The interaction between nanoparticle loading and magnetic damping is critical in optimizing heat transfer in MHD nanofluid systems.

To comprehensively understand the thermofluid behavior inside the trapezoidal nanofluid-filled enclosure under magnetic field influence, a parametric study was conducted by varying the Rayleigh number (Ra), Hartmann number (Ha), and nanoparticle volume fraction (ϕ). The results, summarized in Tables 4-9, present the corresponding variations in Nusselt number (Nu) and the average nanofluid temperature (T_avf). These tables provide critical insights into how buoyancy-driven convection, magnetic damping, and enhanced thermal conductivity due to nanoparticles interact to influence heat transfer performance and thermal stability of the system.

Tables 4-9 collectively illustrate the influence of Rayleigh number (Ra), Hartmann number (Ha), and nanoparticle volume fraction (ϕ) on the convective heat transfer rate (Nusselt number, Nu) and the average nanofluid temperature (T_avf).

Table 4. Nu and T_avf When ϕ = 0.01, Pr = 5.3821 and Ha = 15.

Ra (Rayleigh Number)	Nu (Nusselt Number)	Tavf (Average Temperature of Nano Fluid)
103	0.86665	0.0094650
104	2.51720	0.0094640
105	3.10980	0.0094518
106	3.26330	0.0090756

Table 5. Nu and T_avf When ϕ = 0.01, Pr = 5.3821 and Ha = 50.

Ra (Rayleigh Number)	Nu (Nusselt Number)	Tavf (Average Temperature of Nano Fluid)
103	0.86665	0.0094651
104	2.51720	0.0094648
105	3.10950	0.0094612
106	3.18960	0.0094055

Table 6. Nu and T_avf When ϕ = 0.02, Pr = 4.9833 and Ha = 15.

Ra (Rayleigh Number)	Nu (Nusselt Number)	Tavf (Average Temperature of Nano Fluid)
103	0.87357	0.0091905
104	2.5762	0.0091896
105	3.2003	0.0091788
106	3.3454	0.0088667

Table 7. Nu and T_avf When ϕ = 0.02, Pr = 4.9833 and Ha = 50.

Ra (Rayleigh Number)	Nu (Nusselt Number)	Tavf (Average Temperature of Nano Fluid)
103	0.87357	0.0091906
104	2.57620	0.0091903
105	3.20000	0.0091871
106	3.28380	0.0091390

Table 8. Nu and T_avf When ϕ = 0.03, Pr = 4.6384 and Ha = 15.

Ra (Rayleigh Number)	Nu (Nusselt Number)	Tavf (Average Temperature of Nano Fluid)
103	0.88034	0.0089263
104	2.63570	0.0089254
105	3.29250	0.0089159
106	3.43160	0.0086564

Table 9. Nu and T_avf When ϕ = 0.03, Pr = 4.6384 and Ha = 50.

Ra (Rayleigh Number)	Nu (Nusselt Number)	Tavf (Average Temperature of Nano Fluid)
103	0.88034	0.0089263
104	2.63570	0.0089261
105	3.29220	0.0089232
106	3.38020	0.0088816

Across all cases, increasing the Rayleigh number (Ra) enhances the Nusselt number (Nu), indicating stronger convection. When the Hartmann number (Ha) is low (15), Nu is consistently higher and the average nanofluid temperature (T_avf) lower, compared to high Ha (50), due to reduced magnetic damping. Increasing nanoparticle volume fraction (ϕ) from 0.01 to 0.03 improves heat transfer, as reflected by rising Nu and falling T_avf. However, this improvement is more effective at low Ha, where buoyancy-driven flows are less restricted. Overall, optimal thermal performance is achieved with high ϕ and low Ha.

Overall Observations:

1) Nu increases with higher Ra and ϕ, and decreases with higher Ha.

2) T_avf decreases with increasing Ra and ϕ, and increases with stronger Ha.

3) The optimal thermal performance is achieved at high Ra and high ϕ with moderate Ha.

Figures 9-11 present a comparative analysis of the Nusselt number (Nu) as a function of Rayleigh number (Ra) for different Hartmann numbers (Ha = 15 and Ha = 50) under varying nanoparticle volume fractions (ϕ = 0.01, 0.02, and 0.03). These figures aim to assess the influence of magnetic field strength and nanoparticle loading on the convective heat transfer performance of a nanofluid-filled trapezoidal cavity containing internal heat sources. The Nusselt number is a key indicator of convective heat transfer, and its variation with Ra, Ha, and ϕ provides insights into optimizing thermal performance in magnetohydrodynamic (MHD) systems.

Figure 9. Nu vs Ra When ϕ = 0.01, Pr = 5.3821, Ha = 15 and Ha = 50.

Figure 9: The Nusselt number increases with Ra, confirming enhanced heat transfer due to stronger buoyancy-driven convection. For both Ha = 15 and Ha = 50, the increase in Nu is nonlinear, with more significant gains at higher Ra. Ha = 15 consistently yields higher Nu values than Ha = 50, indicating that a weaker magnetic field imposes less damping on flow motion, thus allowing better convective transport.

Figure 10: A similar trend is observed where Nu rises with Ra, and Ha = 15 outperforms Ha = 50. The inclusion of more nanoparticles slightly raises the Nu across the Ra range compared to ϕ = 0.01, showcasing the beneficial effect of enhanced thermal conductivity. However, magnetic suppression becomes more noticeable as ϕ increases, emphasizing the trade-off between magnetic damping and nanoparticle-induced enhancement.

Figure 11: This case shows the highest Nu values overall, confirming that greater nanoparticle concentration substantially improves heat transfer. Yet, the gap between Ha = 15 and Ha = 50 is still evident, reaffirming the inhibitory influence of stronger magnetic fields on convection. This figure highlights that while nanoparticle addition enhances Nu, optimizing Ha is crucial to maximizing thermal efficiency.

Figure 10. Nu vs Ra When ϕ = 0.02, Pr = 4.9833, Ha = 15 and Ha = 50.

Figure 11. Nu vs Ra When ϕ = 0.03, Pr = 4.6384, Ha = 15 and Ha = 50.

Together, these figures underscore that increasing Ra and ϕ boosts heat transfer, while higher Ha values tend to suppress it. Therefore, a balanced design considering both nanoparticle loading and magnetic field strength is essential for optimal MHD thermal system performance.

5. Conclusions

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.

Acknowledgements

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

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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