Energy consumption has increased significantly during the past few decades. Tremendous efforts have been made to pacify the ever-increasing consumption rate or find an appropriate, cheap, and environment-friendly energy resource. Using nanotechnology is one of the main candidates for increasing the performance of systems. Nanofluids seem promising candidates for energy consumption reduction strategies due to their improved thermo-physical properties. In recent years, the study of nanofluids has garnered significant attention within the scientific community due to their superior thermal properties and potential applications in various engineering fields. Nanofluids, which are fluids embedded with nanoscale particles, have been extensively researched for their ability to enhance heat transfer in comparison to conventional fluids. This enhancement is primarily due to the increased thermal conductivity imparted by the nanoparticles. The nanofluids utilizations are successfully investigated in different industries and applications, including combustion [1,2], lubrication [3], medicine [4], heat exchangers [5], heating and tempering processes [6], microelectronic cooling [7] and refrigeration [8].

Early research efforts focused on the theoretical and experimental analysis of nanofluids, laying the groundwork for understanding their unique properties. However, as computational techniques have advanced, the numerical solution of nanofluid flow has become a critical area of study. Recent studies have explored a wide range of configurations, including flow over flat plates, through channels, and around complex geometries. These investigations have employed various numerical methods, such as finite difference, finite volume, and finite element methods, to solve the governing equations of nanofluid flow. One area of particular interest is the study of boundary layer flow, where the presence of nanoparticles can significantly alter the thermal boundary layer and, consequently, the overall heat transfer characteristics. Researchers have applied numerical techniques to solve the nonlinear partial differential equations governing boundary layer flow, often incorporating effects such as variable viscosity, thermal radiation, and magnetic fields to better understand the behavior of nanofluids under practical conditions. Nawaz et al. [9] modified the Adams–Bashforth method and applied this scheme to the mixed convection flow problem inside a square cavity with the simultaneous effects of thermal radiation and magnetic. Meanwhile, Cengizci et al. [10] incorporated the pressure-stabilizing/Petrov–Galerkin (PSPG) and the streamline-upwind/Petrov–Galerkin (SUPG) into the Galerkin finite element method (GFEM) formulation in order to solve the magnetohydrodynamic (MHD)-natural convection problem in the nanofluid-filled quadrantal cavity. Other references that highlighted the numerical schemes to solve nanofluid flow problems can be found here (see [11–15]).

The effects of hybrid nanofluids on heat transfer performances were recently investigated under different conditions. For example, Abdel Nour et al. [16] considered the natural convection of alumina-copper-water hybrid nanofluids under a magnetic field in a porous cavity. The porosity and nanoparticle concentration were in the range of 0.1–0.9, and 0%–0.08%, respectively. A Ra of 10³ to 10⁶, a Da range of 10⁻⁵ to 10⁻², and a Ha range of 0 to 100 were set in their simulations. It was concluded that the convective heat transfer mechanism is stronger at higher Ras. Mliki et al. [17] utilized the Lattice Boltzmann Method (LBM) to investigate the natural convection of hybrid nanofluids of Al₂O₃-CuO suspended in water in a heated incinerator-shaped cavity under magnetic force effects. Their findings show that Ra and Ha have a direct and reverse effect on the heat transfer rate, respectively. The same trend was also observed for the entropy generation. The use of nanoparticles was found to be beneficial in enhancing heat transfer and reducing the produced irreversibility. A numerical investigation of radiation effects on unsteady MHD hybrid nanofluids’ natural convection in an inclined wavy porous 2D enclosure was performed by Nabwey et al. [18]. A square obstacle was also included in the cavity. They found out that the heat transfer was enhanced by increasing the domain porosity, especially at higher rates of heat fluxes.

Ashorynejad et al. [19] simulated the free convection of hybrid nanofluids in wavy open cavities under Lorentz force effects by using the LBM. It was noticed that the heat transfer is reversely proportional to Ha. Meanwhile, Ra and nanoparticle volume fraction have direct effects on heat transfer. Sajjadi et al. [20] studied the effects of various parameters on the heat transfer during the natural convection of hybrid nanofluids subjected to a magnetic field using the double Multiple-Relaxation-Time (MRT) Lattice Boltzmann Method (LBM). The results indicate a clear rise in the Nu as both Ra and nanoparticle volume concentration increase. Similarly, the Darcy number and porosity directly affect the rate of heat transfer.

Ahmed et al. [21] delved into a numerical investigation of the intricate interplay between two distinct heat transfer mechanisms—surface thermal radiation and natural convection—in a micropolar fluid laden with Al₂O₃ and Cu nanoparticles, confined within a porous cavity under the influence of magnetic field inclination. Their study incorporated entropy analysis and employed both the first and second laws of thermodynamics to comprehensively scrutinize the physical system. The governing equations describing the 2-D steady-state flow were solved using the Alternating Direction Implicit (ADI) method within the framework of the finite volume method. The parameters considered in the investigation include the heat source length (0.2 ≤ B ≤ 0.8), heat generation/absorption (−2 ≤ Q ≤ 2), radiation factor (0 ≤ Rd ≤ 1.0), vortex viscosity parameter (0 ≤ k ≤ 2), Hartmann number (0 ≤ Ha ≤ 100), and inclination angle (0 ≤ θ ≤ π). Graphical representations of the computed results illustrated noteworthy trends, such as the impact of the vortex parameter (k) on the average Nusselt number and the role of the total solid volume fraction in influencing heat transfer. Moreover, their results highlighted the reinforcing effect of the radiation factor (Rd) on global entropy generation within the system. They found that the radiation parameter amplifies the convective processes, thereby elevating the nanofluid flow dynamics and enhancing the rate of heat transfer.

Roy [22] applied multiple heat sources to an enclosure filled with hybrid nanofluids under MHD natural convection conditions. He employed the finite difference method for the governing equations and validated the results against experimental and numerical data. The validation shows a good agreement. His results indicate significant alterations in flow patterns with variations in the magnetic field parameter, angle of the magnetic field, number and width of heat sources, and Rayleigh number. The stream function’s intensity is pronounced for higher Rayleigh numbers and smaller magnetic field parameters. Notably, when the volume fraction of Cu nanoparticles exceeds 6%, distinct patterns in streamlines and isotherms emerge compared to lower values. Additionally, as the number of heat sources, magnetic field angle, and Rayleigh number increase, the average Nusselt number is found to rise. Tayebi et al. [23] also conducted an interesting numerical analysis on the MHD natural convection of hybrid nanofluid of Al₂O₃-Cu/water in a square cavity, while a wavy circular conductive cylinder presents in the center of the domain.

They employed the Corcione correlations to calculate the effective thermal conductivity and viscosity of the hybrid nanofluid, incorporating the Brownian motion of nanoparticles. Then, a comprehensive numerical parametric study was conducted, exploring various values of nanoparticle volumetric concentration, Hartmann number, Rayleigh number, and the ratio of fluid to solid thermal conductivities. Their findings emphasize the significant influence of the corrugated conductive block in regulating both convective flow characteristics and the heat transfer rate within the system. Rashad et al. [24] also investigated the effect of multiple heat sources from the bottom and the center of a triangular cavity filled with hybrid nanofluids under MHD natural convection process. The inclined side of the cavity underwent isothermal cooling, while the remaining sides were thermally insulated. The numerical solution employed the finite difference method with the stream function-vorticity formulation for the governing equations. The problem involved several key parameters, including the size and position of the heater element (B = 0.2–0.8 and D = 0.3–0.7), the Rayleigh number (Ra = 10²–10⁶), the Hartmann number (Ha = 0–100), the volume fraction of suspended nanoparticles (ϕ = 0–0.2), and the heat generation parameter (Q = 0–6). Their results highlight a substantial impact of increasing the volume fraction of the hybrid nanofluid when natural convection is minimal.

Numerous studies have explored the convection of nanofluids in confined spaces. Most of them used Al₂O₃/Cu/water as hybrid nanofluids under different heating conditions and various cavity geometries [25–27]. Geometries including square [28], cylinder [29], triangular [30], trapezoidal [31], etc. However, there’s a noticeable gap in understanding the unsteady magnetohydrodynamic (MHD) natural convection of hybrid nanofluids. This research aims to bridge that gap by numerically investigating the impact of radiation on unsteady natural convection in MHD hybrid nanofluids. The study focuses on a unique scenario: a U-shaped square wavy porous cavity under a constant inclined magnetic field. It’s worth mentioning that the cavity is partially heated, and the flow is primarily driven by buoyancy forces.

Fig. 1 depicts the layout of the computational domain considered in the current investigation, featuring a U-shaped cavity with a side length denoted as H. The vertical walls of this cavity exhibit a wavy pattern with a specific wavelength, λ. The cavity is filled with a porous medium saturated with an Al₂O₃-Cu/water hybrid nanofluid. The sinusoidal right and left walls are cooled at a temperature denoted as T_c. Meanwhile, a specific section along the bottom wall, indicated by a length ‘b’, experiences heating at a temperature T_h, where T_h is greater than T_c. The rest of the enclosure’s walls are assumed to be adiabatic, meaning they do not exchange heat with their surroundings. The force of gravity acts downward in the cavity, and a constant magnetic field denoted as B₀, exerts its influence at an angle Φ along the horizontal axis.

The hybrid nanofluid (Al₂O₃-Cu/water) is Newtonian, and the flow is laminar, unsteady, and incompressible. Table 1 includes some of the properties of the host fluid and nanoparticles. The density variation was approximated using the Boussinesq approximation. Therefore, the governing equations are as follows (see Nabwey et al. [18]):

(1)∂u∂x+∂v∂y=0 (2)1ε2(∂u∂t+u∂u∂x+v∂u∂y)=−1ρhnf∂p∂x+νhnfε(∂2u∂x2+∂2u∂y2)+gβhnf(T−Tc)sin α−νhnfKu+σhnfB02ρhnf(v sin Φ cos Φ−u sin2Φ) (3)1ε2(∂v∂t+u∂v∂x+v∂v∂y)=−1ρhnf∂p∂y+νhnf(∂2v∂x2+∂2v∂y2)+gβhnf(T−Tc) cos α−νhnfKν+σhnfB02ρhnf(u sin Φ cos Φ−v cos2Φ) (4)1ε(∂T∂t+u∂T∂x+v∂T∂y)=[αeff.hnf+16σ∗Tc33k∗(ρcp)hnf].∇2T+Q0ε(ρcp)hnf

The considered boundary conditions are (see Nabwey et al. [18]): t<0:u=v=T=0, 0≤x≤H, 0≤y≤H t≥0:u=v=0, 0≤y≤H, 0≤x≤H at all walls (5)T=Th, D−0.5B≤yH≤D+0.5B, ∂T∂x=0,otherwise at wall x=0 T=Tc, x=H−AH[1− cos (2πλyH)], 0≤y≤H T=Tcx=AH[1− cos (2πλyH)], 0≤y≤H

An iterative Finite Difference Method (FDM) is used to solve the governing equations. The convective terms were approximated by using a second-order upwind finite difference scheme. The first point in the solution methodology is writing the previous system in the following general form: (26)U∂Ω∂X+V∂Ω∂Y=ΓΩ∇2Ω+SΩ

Here it should be mentioned that S refers to the source terms (including the pressure gradients and buoyancy terms in the momentum equations). The central differences scheme is used to estimate the first and second derivatives as: (27)∂Ω∂X=Ωi+1, j−Ωi−1, jXi+1−Xi−1, ∂Ω∂Y=Ωi, j+1−Ωi, j−1YJ+1−YJ−1∂2Ω∂X2+∂2Ω∂Y2=Ωi+1, j−2Ωi, j+Ωi−1, j(ΔX)2+Ωi, j−1−2Ωi, j+Ωi, j+1(ΔY)2

Eq. (27) refers to the dependent variables U, V, N and θ and ΔX=Xi+1−Xi, ΔY=Yj+1−Yj, Using the previous forms, the following algebraic system is obtained: (28)ApΩi, j=AEΩi+1, j+AWΩi−1, j+ANΩi, j+1+ASΩi, j−1+SP

Eq. (29) shows the FD approximation of the energy equation (see Nabwey et al. [18]): (29)θi, jn+1=θi, jn+Δτσ{(αnfαf)[θi+1,jn−2θi,jn+θi−1,jn(ΔX)2+θi,j+1n−2θi,jn+θi,j−1n(ΔY)2]+ Q(ρcp)f(ρcp)nfθi,jn−Ui,jnθi+1,jn−θi−1,jn2ΔX−Vi, jnθi, j+1n−θi, j−1n2ΔY}

A grid of 61 × 61 and a time step of 10⁻⁶ were considered to perform all the computations. To ensure the validity of the numerical model, a comparison with the findings of Ghasemi et al. [35] is performed (see Fig. 2). The comparison shows a very good concordance.

The simulation results of this work, including the flow structure, temperature field, and local and average Nu, are provided in this section. Fig. 3 shows the streamlines and isotherms of the hybrid nanofluids at different wave amplitudes of the walls. The results show that the higher amplitudes increase the significance of streamlines and reduce the significance of the isotherms. The variation of the local Nu vs. the x direction at different amplitudes is presented in Fig. 4. At the lowest amount of the amplitude (0.05), which is the closest value to a straight wall, the local Nu is minimum at the center, and it rises sharply by getting far from the center of the domain. The trend changes at higher amplitudes. The minimum local Nu moves from the center to the left corner. While, the maximum local Nu is still on both walls. The average Nu against nanoparticle volume concentration at different amplitudes is shown in Fig. 5. The higher the amplitudes, the higher average Nu. A possible reason could be the more significant streamlines at higher amplitudes.

Fig. 6 shows the streamlines and isotherms by changing the dimensionless heat source length at the bottom. It was observed that the streamlines are more significant by increasing the dimension less heat source length. A more uniform temperature distribution is expected with a larger heat source. This is shown in the results.

Fig. 7 demonstrates the local Nu variation vs. x direction for different heat source lengths. The minimum local Nu is achieved at the center of the domain (x = 0.5) for all the studied cases, and it is increased sharply by changing the location at both directions. The sharp rise is however encountered at the edges of the heat source. The same trend (U-shape) was observed at different heat source lengths. Nevertheless, the minimum local Nu decreased with any increase in the heat source length. On the other hand, the maximum local Nu increased by increasing the heat source dimensions.

The average Nu against the nanoparticle volume concentration for various dimensionless heat source lengths is shown in Fig. 8. The findings show that the average Nu is not dependent on the nanoparticle load. However, it is a strong function of the heat source length. The smaller the dimensionless heat source, the higher amount of average Nu. Fig. 9 shows the streamlines (left) an isotherms of the hybrid nanofluids (right) for various Ha. As it is shown, increasing Ha does not change the streamlines and isotherms significance considerably. Increasing Ha means increasing the effect of the magnetic field order of magnitude on the flow. Therefore, the magnitude of the magnetic field does not significantly affect the streamlines and isotherms.

The variation of the local Nusselt number (Nu) along the x-direction at different Hartmann numbers (Ha) is illustrated in Fig. 10. The local Nu reaches its minimum value at the center of the cavity and increases sharply as one moves away from the center, consistent with the behavior observed in previous cases. Interestingly, while Ha does not significantly influence the local Nu distribution, it has a substantial impact on the average Nu. As Ha increases, the average Nu decreases. This behavior suggests that a magnetic field (indicated by higher Ha values) suppresses convective heat transfer, allowing conduction to become the dominant heat transfer mechanism. Without a magnetic field (Ha = 0), the viscous forces are less effective in counteracting the convective currents, leading to a higher average Nu.

Another key observation is the impact of nanoparticle concentration on heat transfer. As shown in Fig. 11, increasing the nanoparticle volume fraction results in a decrease in the average Nu. This effect is more pronounced at lower Ha values, implying that the magnetic field moderates the influence of nanoparticles on the flow dynamics. Physically, this could be because, at lower Ha, the nanoparticles enhance the effective viscosity of the nanofluid, thereby reducing the convective heat transfer rate. However, as Ha increases, the magnetic forces dampen fluid motion, diminishing the nanoparticles’ ability to alter the flow field significantly. Furthermore, the simulation results reveal that the streamline patterns and isotherms of the nanofluids remain largely unaffected by thermal radiation. However, thermal radiation exerts a noticeable influence on both local and average Nu, as depicted in Figs. 12 and 13. Higher radiation levels lead to an increase in both local and average Nu, which can be attributed to thermal radiation enhancing the energy transport within the fluid, thereby boosting the overall heat transfer rate. This finding underscores the importance of considering radiative effects in applications where high precision in thermal management is required.

Fig. 14 shows the effect of heat absorption (negative values) or generation (positive values) on the streamlines and isotherms of the hybrid nanofluids. While the flow pattern and the significance of the streamlines are not dependent on heat absorption or generation, the isotherms and the temperature fields heavily depend on heat absorption or generation. Fig. 15 also shows that the minimum local Nu for heat generation is less than that of a heat absorption case. Fig. 16 proves that the nanoparticle volume concentration is not considerably effective the average Nu at a given amount of heat absorption or generation.

The effects of Ra on the flow and temperature fields are shown in Fig. 17. Varying Ra does not affect the significance of the streamlines. The flow pattern remains almost unchanged at different Ras. However, it affects the isotherms significance and pattern. Fig. 18 demonstrates that increasing Ra does not change the previously observed trend of the local Nu, except for Ra of 10⁵. In this case, the local Nu is considerably higher than in the other cases, especially in the left corner. Fig. 19 shows that the average Nu is almost fixed vs. the nanoparticles volume fraction at each given Ra. However, the average Nu decreases slightly with any increase in the nanoparticle load, only at Ra of 10⁵. A general observation is that the higher Ras yields higher average Nu.

Fig. 20 represents the temperature and flow fields of the hybrid nanofluid for various wavelengths of the walls. Although the streamlines’ significance is not affected heavily by changing the wavelength, the flow pattern is changed considerably. The flow pattern is elongated to the left at higher values of wavelength. The same trend was observed for the isotherms. The lower values of wavelengths, the more uniform temperature field. Fig. 21 shows a similar trend of the local Nu against the direction: the minimum value at the center and the maximum at both walls. Changing the wavelength does not change the local Nu considerably. The average Nu is almost constant with any change in the nanoparticles’ volume fraction at each given wavelength. However, a higher value of wavelength leads to a higher average Nu (see Fig. 22).

A series of numerical simulations were conducted in order to study the natural convection and radiation of alumina-copper/water hybrid nanofluids in a U-shaped wavy square porous cavity under a constant inclined magnetic field. The results are analyzed and provided in regard to flow structures, temperature fields, and heat transfer rates at various operational conditions by changing the wave amplitude of the wall, wall’s wavelength, Ha number, Ra number, heat source length, heat absorption/generation, and thermal radiation. The main conclusions are as follows: –

The higher amplitudes of the wall increase the significance of streamlines and reduce the significance of the isotherms.

The primary limitations of this study include the assumption of a constant and inclined magnetic field and a homogeneous, isotropic porous medium, which may not fully capture real-world conditions. Our future research will address these aspects to broaden the study’s scope and relevance.