Numerical Investigation of Magnetohydrodynamic (MHD) Natural Convection in a Nanofluid Filled Trapezoidal Cavity Considering the Use of Obstacle Shape, Wall Corrugation, and Inclination Angle

Sree Pradip Kumer Sarker^*, Md. Mahmud Alam

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

DOI: 10.4236/am.2025.1611040                

Abstract

This study presents a numerical investigation of magnetohydrodynamic (MHD) natural convection in a nanofluid-filled trapezoidal cavity, focusing on the combined effects of internal obstacle shape, wall corrugation, and inclination angle. Using Cu-H₂O nanofluid and the finite element method, simulations were performed for Rayleigh numbers ranging from 10³ to 10⁶, Hartmann numbers from 0 to 50, and inclination angles of 15˚, 30˚, and 45˚. Results show that both obstacle geometry and wall corrugation profile strongly influence thermal performance. Square-shaped obstacles with triangular wavy walls achieved the highest Nusselt numbers, while sinusoidal walls provided superior thermodynamic efficiency, particularly at lower inclination angles. Increasing Hartmann numbers suppressed convective motion, as reflected by decreased Nu and ECOP values across all cases. The influence of geometric features, specifically obstacle shape and wall corrugation, was found to be most significant at low Rayleigh numbers, where buoyancy-driven flow is weaker and geometric modulation governs heat transport. The optimal configuration for enhanced heat transfer and energy efficiency was identified as square obstacles with sinusoidal or triangular walls at a 15˚ inclination and low Ha. These findings offer valuable guidance for designing thermally efficient enclosures in electronics cooling, solar collectors, and microfluidic systems, where controlling convection under magnetic effects is essential.

Keywords

MHD Natural Convection, Nanofluid Heat Transfer, Trapezoidal Enclosure, Wall Corrugation, Entropy Generation

1. Introduction

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.

2. Materials and Methods

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.

Figure 1. Different geometry with sinusoidal corrugation.

Figure 2. Different geometry with square corrugation.

Figure 3. Different geometry with triangular corrugation.

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).

Table 1. Thermo-physical properties of Water and Cu at T_m = 300 K [5] [17] [31] [32].

Name of Property	Symbol	Unit	Water	Cu
Mass Density	ρ	kgm−3	996.6	8933
Specific Heat at Constant Pressure	Cp	Jkg−1K−1	4179.2	385
Thermal Conductivity	k	Wm−1K−1	0.6102	401
Volumetric Thermal Expansion Coefficient	β	K−1	26.6 × 10−5	49.9 × 10−6
Electrical Conductivity	σ	Sm−1	0.05	59.6 × 10−6
Dynamic viscosity	μ	kgm−1s−1	8.538 × 10−4	-
Prandtl Number	Pr	-	5.856	-

3. Results

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.

3.1. Obstacle Shapes and Wall Corrugation Effects

The influence of obstacle geometry and wall corrugation on natural convection heat transfer in a nanofluid-filled trapezoidal cavity under magnetic fields is distinctly evident across Table 2-Table 10. For a fixed inclination angle of 15˚, 30˚, and 45˚, and over a range of Hartmann (Ha) and Rayleigh (Ra) numbers, the triangular corrugated wall consistently exhibits superior thermal performance. Particularly, in all inclination cases, the square-shaped obstacles in combination with triangular wavy walls yield the highest Nusselt numbers (Nu), indicating enhanced convective heat transfer. For instance, at λ = 45˚ and Ra = 10⁶, the triangular wall with square obstacles achieves Nu = 6.2401 at Ha = 0 (Table 10), outperforming sinusoidal and square corrugations for the same Ra and obstacle shape (Nu = 6.4137 and 6.2401, respectively).

Table 2. Nu for star-shaped obstacles with different wavy walls at λ = 15˚.

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

Table 3. Nu for square-shaped obstacles with different wavy walls at λ = 15˚.

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

Table 4. Nu for triangular-shaped obstacles with different wavy walls at λ = 15˚.

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

Table 5. Nu for star-shaped obstacles with different wavy walls at λ = 30˚.

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

Table 6. Nu for square-shaped obstacles with different wavy walls at λ = 30˚.

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

Table 7. Nu for triangular-shaped obstacles with different wavy walls at λ = 30˚.

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

Table 8. Nu for star-shaped obstacles with different wavy walls at λ = 45˚.

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

Table 9. Nu for square-shaped obstacles with different wavy walls at λ = 45˚.

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

Table 10. Nu for triangular-shaped obstacles with different wavy walls at λ = 45˚.

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

Figure 4. Nu vs Ra for star-shaped obstacles, with sinusoidal wavy wall at (a) λ = 15˚, (b) 30˚, (c) 45˚.

Figure 5. Nu vs Ra for star-shaped obstacles, with square wavy wall at (a) λ = 15˚, (b) 30˚, (c) 45˚.

Figure 6. Nu vs Ra for star-shaped obstacles, with triangular wavy wall at (a) λ = 15˚, (b) 30˚, (c) 45˚.

Figure 7. Nu vs Ra for square-shaped obstacles, with sinusoidal wavy wall at (a) λ = 15˚, (b) 30˚, (c) 45˚.

Figure 8. Nu vs Ra for square-shaped obstacles, with square wavy wall at (a) λ = 15˚, (b) 30˚, (c) 45˚.

Figure 9. Nu vs Ra for square-shaped obstacles, with triangular wavy wall at (a) λ = 15˚, (b) 30˚, (c) 45˚.

Figure 10. Nu vs Ra for triangular-shaped obstacles, with sinusoidal wavy Wall at (a) λ = 15˚, (b) 30˚, (c) 45˚.

Figure 11. Nu vs Ra for triangular-shaped obstacles, with square wavy wall at (a) λ = 15˚, (b) 30˚, (c) 45˚.

Figure 12. Nu vs Ra for triangular-shaped obstacles, with triangular wavy wall at (a) λ = 15˚, (b) 30˚, (c) 45˚.

Comparatively, across all wall shapes and inclinations, the trend shows that increasing Ha reduces Nu, confirming the suppressive effect of magnetic fields on natural convection. The square wavy walls slightly underperform compared to triangular ones, while sinusoidal walls offer moderate enhancement. As seen in Table 3 and Table 6, square obstacles generally produce higher Nu values than star or triangular shapes, suggesting the role of flow disturbance and vortex formation around angular features in augmenting heat transfer. Overall, the interplay between obstacle shape and wall geometry substantially dictates the convective behavior within MHD-driven nanofluid systems.

Figure 4-Figure 12 illustrate the variation of the Nusselt number (Nu) with Rayleigh number (Ra) for different obstacle shapes (star, square, triangular) and wall corrugation types (sinusoidal, square, triangular) at inclination angles of 15˚, 30°, and 45°, under varying Hartmann numbers (Ha = 0, 15, 30, 50). Across all configurations, Nu increases with Ra, indicating stronger natural convection as thermal buoyancy becomes dominant.

Among the obstacle shapes, square-shaped obstacles consistently yield the highest Nusselt number (Nu), demonstrating superior thermal performance due to strong corner-induced vortex intensification that enhances fluid mixing and heat transfer. Triangular obstacles follow closely, while star-shaped obstacles generally produce lower Nu values under similar conditions.

Regarding wall corrugation, triangular wavy walls promote stronger thermal mixing and circulation zones within the cavity, leading to more vigorous convective motion compared to square or sinusoidal patterns. This enhancement becomes even more pronounced when paired with square or triangular obstacles, as the sharp edges and varying surface gradients jointly intensify local heat transfer and streamline deformation.

Furthermore, the applied magnetic field (Ha) suppresses convective motion across all configurations, leading to a decrease in Nu with higher Ha, illustrating the magnetic damping effect. Nevertheless, cavities featuring square obstacles maintain relatively higher Nu even under stronger magnetic fields, underscoring the significant influence of obstacle geometry and wall corrugation in optimizing thermal performance.

Table 11. S_T for star-shaped obstacles with different wavy walls at λ = 15˚.

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

Table 12. S_T for square-shaped obstacles with different wavy walls at λ = 15˚.

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

Table 13. S_T for triangular-shaped obstacles with different wavy walls at λ = 15˚.

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

Table 14. S_T for star-shaped obstacles with different wavy walls at λ = 30˚.

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

Table 15. S_T for square-shaped obstacles with different wavy walls at λ = 30˚.

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

Table 16. S_T for triangular-shaped obstacles with different wavy walls at λ = 30˚.

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

Table 17. S_T for star-shaped obstacles with different wavy walls at λ = 45˚.

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

Table 18. S_T for square-shaped obstacles with different wavy walls at λ = 45˚.

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

Table 19. S_T for triangular-shaped obstacles with different wavy walls at λ = 45˚.

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

The effects of obstacle shapes and wall corrugation on the total entropy generation (S_T) were systematically analyzed at varying inclination angles (λ = 15˚, 30˚, and 45˚), Rayleigh numbers (Ra), and Hartmann numbers (Ha), as presented in Table 11-Table 19. The S_T values consistently decreased with increasing Ra across all configurations, indicating enhanced thermal diffusion dominance over natural convection, which is consistent with prior studies in MHD nanofluid flow regimes.

For all cases, the triangular-shaped wavy walls generally led to slightly higher S_T values at Ra = 10³ due to increased surface area and flow disturbances enhancing irreversibility. For example, at λ = 15˚, Ra = 10³, and Ha = 0, the S_T for triangular-shaped obstacles reached 17.206 (Table 13), compared to 17.074 for square and 16.898 for star-shaped ones (Table 11). A similar trend was observed at λ = 30˚ and λ = 45˚ (Table 14 and Table 17), supporting the assertion that triangular geometries induce more vigorous convective behavior.

Regarding wall corrugation, square-shaped wavy walls slightly enhanced entropy generation compared to sinusoidal and triangular types, particularly at lower Ha. For instance, at λ = 30˚, Ra = 10⁴, Ha = 0, and square obstacles, S_T = 0.66077 for square walls, while it was lower for sinusoidal (0.63838) and triangular (0.65439) (Table 14). This difference became negligible as Ra increased, with S_T values converging at 10⁶ due to MHD dominance.

At higher Ha values (Ha = 50), the influence of both obstacle shape and wall form diminished, with S_T values becoming nearly invariant. For example, across all obstacle shapes and wall types at Ra = 10³, S_T values remained unchanged as Ha increased from 0 to 50 (Table 11-Table 19), confirming magnetic damping of convection.

In conclusion, triangular-shaped obstacles with square wavy walls promote maximum entropy generation at low Ra and Ha, while MHD effects suppress these geometric influences at high Ra and Ha, making flow behavior more uniform.

The ECOP values presented in Table 20-Table 28 reveal that obstacle shape, wall corrugation, and inclination angle (λ) significantly influence thermal performance. At low Ra (10³), ECOP values remain nearly constant across all configurations, indicating conduction-dominated behavior. As Ra increases to 10⁶, ECOP rises sharply, especially for square obstacles with sinusoidal walls, which consistently yield the highest ECOP (e.g., 4177.5 in Table 21 at λ = 15˚, Ha = 0).

Table 20. ECOP for star-shaped obstacles with different wavy walls at λ = 15˚.

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

Table 21. ECOP for square-shaped obstacles with different wavy walls at λ = 15˚.

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

Table 22. ECOP for triangular-shaped obstacles with different wavy walls at λ = 15˚.

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

Table 23. ECOP for star-shaped obstacles with different wavy walls at λ = 30˚.

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

Table 24. ECOP for square-shaped obstacles with different wavy walls at λ = 30˚.

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

Table 25. ECOP for triangular-shaped obstacles with different wavy walls at λ = 30˚.

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

Table 26. ECOP for star-shaped obstacles with different wavy walls at λ = 45˚.

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

Table 27. ECOP for square-shaped obstacles with different wavy walls at λ = 45˚.

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

Table 28. ECOP for triangular-shaped obstacles with different wavy walls at λ = 45˚.

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

Sinusoidal walls outperform square and triangular ones due to their smoother profiles, enhancing natural convection. Among obstacles, square shapes offer slightly better ECOP than triangular and star-shaped ones. Increasing Ha reduces ECOP across all cases, reflecting magnetic suppression of convection. Additionally, a lower inclination angle (λ = 15˚) results in higher ECOP compared to 30˚ or 45˚, emphasizing the thermal advantage of minimal wall tilt. Overall, the optimal configuration for maximizing ECOP is square-shaped obstacles with sinusoidal wavy walls at low Ha and low λ.

3.2. Influence of Inclination Angle

The inclination angle (λ) of wavy walls significantly influences heat transfer behavior, entropy generation, and overall system performance characterized by the Entropy Generation-Based Cooling Performance (ECOP). The present study investigates the impact of three inclination angles 15˚, 30˚, and 45˚ on various obstacle shapes (star, square, and triangular) combined with different wall corrugation patterns (sinusoidal, square, and triangular), over a range of Rayleigh numbers (Ra) from 10³ to 10⁶ and Hartmann numbers (Ha) from 0 to 50.

A key observation is that the Nusselt number (Nu), which indicates convective heat transfer, consistently decreases with increasing inclination. This trend is due to the inclination weakening the buoyancy-driven convection mechanism by displacing the thermal boundary layers and misaligning the temperature gradients with gravity. For instance, square-shaped obstacles with sinusoidal walls at Ra = 10⁶ and Ha = 0 exhibited a high Nu of 83.64 at λ = 15˚ (Table 5), which decreased to 77.71 at λ = 45˚ (Table 16). A similar decline is evident for triangular-shaped obstacles with square wavy walls, where Nu dropped from 81.63 at λ = 15˚ (Table 8) to 76.59 at λ = 45˚ (Table 19). This behavior was consistent across all geometries and wall types, confirming that a lower inclination angle fosters stronger thermal convection.

The total entropy generation (S_T) also showed a downward trend with increasing inclination, which is attributed to the reduced convective activity and the subsequent decrease in temperature and velocity gradients. Star-shaped obstacles with triangular wavy walls, for example, showed a notable decrease in S_T from λ = 15˚ (Table 4) to λ = 45˚ (Table 17). Additionally, higher magnetic field strengths (Ha) accelerated this reduction, as the Lorentz force further suppressed convective currents, compounding the effect of inclination. The entropy generation due to heat transfer was more sensitive to these changes than that due to fluid friction, especially at higher Ra.

Among the three parameters analyzed, ECOP was most effective in revealing the thermodynamic performance under varying inclination angles. At λ = 15˚, ECOP values were highest across all obstacle shapes and wall types, signifying the most energy-efficient operation. For triangular obstacles with sinusoidal wavy walls at Ra = 10⁶ and Ha = 0, ECOP peaked at 4233.7 (Table 22). However, this value slightly declined to 4260.8 at λ = 30˚ (Table 25) and remained the same at λ = 45˚ (Table 28), highlighting a gradual deterioration in cooling performance. Although this reduction was modest in numerical terms, it indicates a systemic decline in efficiency caused by the misalignment of thermal gradients with the gravitational vector.

Square-shaped obstacles consistently delivered higher ECOP values than their star- and triangular-shaped counterparts. For instance, at Ra = 10⁶ and Ha = 0, ECOP for square obstacles with sinusoidal walls was 4177.5 at λ = 15˚ (Table 21), slightly decreasing to 4229 at λ = 30˚ (Table 24) and further to 4362.5 at λ = 45˚ (Table 27). Although ECOP values remained relatively high, the declining trend underscores the impact of inclination on overall system efficiency. For star-shaped obstacles, ECOP rose slightly from 3896.9 (Table 20) at λ = 15˚ to 4044.6 (Table 26) at λ = 45˚, indicating a minor improvement in thermodynamic performance with increased inclination.

Comparative analysis across the tables further revealed that sinusoidal walls generally yielded the highest ECOP values across all inclination angles, likely due to their smoother transitions that facilitate better fluid motion and heat transfer. Among the obstacle shapes, square-shaped geometries proved to be more resilient to inclination-induced performance drops. Star-shaped obstacles, with their pointed edges and asymmetric geometry, were more susceptible to changes in flow dynamics induced by inclination. For triangular shapes, the performance decline was moderate, positioned between the square and star configurations.

Notably, the combination of low inclination (λ = 15˚), high Ra (10⁶), and low Ha (0) consistently produced optimal thermal and energy performance across all geometries. For instance, ECOP values at this configuration were 4233.7 for triangular (Table 22), 4177.5 for square (Table 21), and 4044.6 for star obstacles (Table 20), indicating that λ = 15˚ is the most thermodynamically favorable orientation. As Ha increased, the damping effect of the magnetic field further compounded the performance loss due to inclination. At higher Ha values, the synergistic suppression of convective motion from both inclination and magnetic forces was especially evident.

In conclusion, increasing the inclination angle from 15˚ to 45˚ consistently reduced the system’s thermal and thermodynamic efficiency. This was evidenced by decreasing values of Nu, S_T, and ECOP across all obstacle and wall configurations. The results confirm that maintaining a lower inclination angle is essential for optimizing natural convection-driven systems, especially when coupled with minimal magnetic interference. Among the configurations studied, square-shaped obstacles with sinusoidal walls demonstrated the highest resilience and performance stability under inclined conditions, making them the most suitable choice for applications requiring high energy efficiency and reliable heat transfer in magnetohydrodynamic environments.

4. Discussion

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.

Figure 13. Isotherms for different values of Ra = 10⁴ when N = 8, A = 0.15, ϕ = 2%.

Table 29. Comparison of Nu between present work and Abdelmalek et al. [17].

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

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.

5. Conclusions

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.

Acknowledgements

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

Conflicts of Interest

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

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