Mixed Convection Heat Transfer and Fluid Flow of Nanofluid/Porous Medium Under Magnetic Field Influence

Abstract

This study aims to investigate the effect of a constant magnetic field on heat transfer, flow of fluid, and entropy generation of mixed convection in a lid-driven porous medium enclosure filled with nanofluids (TiO₂-water). Uniform constant heat fluxes are partially applied to the bottom wall of the enclosure, while the remaining parts of the bottom wall are considered to be adiabatic. The vertical walls are maintained at a constant cold temperature and move with a fixed velocity. A sinusoidal wall is assumed to be fixed and kept adiabatic at the top enclosure. Three scenarios are considered corresponding to different directions of the moving isothermal vertical wall (±1). The influence of pertinent parameters on the heat transfer, flow of fluid, and entropy generation in an enclosure are deliberated. The parameters are the Richardson number (${\tilde{R}}_i$ = 1, 10, and 100), the Hartmann number (0 ≤ $\tilde{H}$a ≤ 75 with a 25 step), and the solid volume fraction of nanoparticles (0 ≤ $\tilde{\Phi }$ ≤ 0.15 with a 0.05 step). The Grashof and Darcy numbers are assumed to be constant at 10⁴ and 10⁻³, respectively. The finite element method, utilizing the variational formulation/weak form, is applied to discretize the main governor equations. Triangular elements have been employed within the studied envelope, with the elements adapting as needed. The results showed that the streamfunction and fluid temperature decreased as the solid volume fraction increased. The local $\tilde{\mathrm{N}}\mathrm{u}$ number increased by more than 50% at low values of $\tilde{\mathsf{\Phi }}$ (up to 0.1). This percentage decreases between 25% and 40% when $\tilde{\mathsf{\Phi }}$ is in the range of 0.1 to 0.15. As $\tilde{\mathrm{H}}$a increases from 0 to 75, these percentages increase at low values of the value of $\tilde{\mathrm{R}}\mathrm{i}=1$ and 10. These variations are primarily dependent on the value of the Richardson number.

1. Introduction

Mixed convection occurs when the buoyancy force and shear force affect the fluid flow and heat transfer. This phenomenon is significant for various convection applications and has been extensively studied in lid-driven cavities in several engineering and technology fields. These include the cooling of electronic devices, food processing, chemical processing equipment, polymer processing, energy extraction, manufacturing processes, and material processing [1]. The mixed convection of Newtonian fluids under various scenarios within cavities has been widely studied in research [2,3,4].

Researchers have found that adding nanoparticles to a base fluid can enhance heat transfer [5,6]. Buongiorno’s two-phase model, introduced in 2006, has been widely used to study nanofluid heat transfer. Tian et al. [7] employed Buongiorno’s nonhomogeneous model to analyze mixed convection in a two-dimensional inclined cavity filled with nanofluid, considering different aspect ratios. Their results showed that the cavity angle and aspect ratio significantly influenced the flow, heat and mass transfer, and nanoparticle concentration. A recent review by Alsabery et al. [8] compared one-phase and two-phase models for nanofluid research. While only 19% of the studies adopted the two-phase model, it provides valuable insights into nanoparticle accumulation on surfaces.

Many studies have considered using porous media inside enclosures to improve the mixed convection coefficient of heat transfer [9,10,11,12,13]. Bourantas et al. [14] numerically studied the free convection of a nanofluid in a square cavity saturated with a porous medium and addressed the impact of the presence of the porous medium on the nanofluid system cooling efficiency. Matin and Ghanbari [15] investigated the nanofluid convection heat transfer over a vertical channel filled with a porous medium and studied the effect of the main parameters, such as $\tilde{\mathrm{R}}\mathrm{i}$, Brinkman, $\tilde{D}$a, and Lewis numbers, on velocity, flow, and temperature.

The generation of entropy and the Bejan Number in incompressible laminar mixed convective fluid within a cavity has been addressed by Shuvo et al. [16]. Their study was performed on two configurations of tilted lid-driven trapezoidal cavities filled with water as the base fluid along with Al₂O₃ nanoparticles.

Recent studies have demonstrated that the use of a magnetic field can enhance heat transfer and reduce irreversibility (entropy) within an enclosure [17,18,19,20,21].

Islam et al. [22] performed a thermal performance analysis study on a hexagonal enclosure that was filled with TiO₂-H₂O nanofluid. Their results confirmed that the effect of the nanofluid on the thermal performance is enhanced by increasing the values of $\tilde{R}$e and $\tilde{\mathsf{\Phi }},$ and increasing the value of Ha. Based on their results, the optimal average Nusselt number was achieved at $\tilde{R}$e = 200 and $\tilde{\mathsf{\Phi }}$ = 0.1 at $\tilde{\mathrm{H}}$a = 0. Ahmed et al. [23] studied mixed convection in an odd-shaped cavity. The upper and right walls of the enclosure were heated partially, and various heating scenarios were considered based on the positions of the active parts. The main conclusions of their study indicated that for optimal mixed convection, it is recommended to place the sources of heat in the middle of the exterior orthogonal walls. Alomari et al. [24] investigated the mixed convection numerically in an enclosure filled with nanofluid by using a finite element method. Their results show that heat transfer is enhanced as the Reynolds number, nanoparticle volume fraction, and Richardson number increase. From another point, the heat transfer decreased with increasing Hartmann number, and the average Nusselt number and streamfunction strength increased with an increase in the Darcy number. Rajarathinam et al. [25] investigate the mixed convection heat transfer in an inclined porous cavity filled with nanofluid with the presence of a magnetic field. Experimental studies were used to verify their computational results. Their study established that the rate of heat transfer decreased with an increase in $\tilde{\mathrm{H}}$a. Also, their results pointed out that the optimum rate of heat transfer was in forced convection.

This study aims to analyze the influence of magnetic fields and nanoparticle volume fraction on mixed convection heat transfer within a porous medium saturated with TiO₂-water nanofluid. A numerical method utilizing the Finite Element Method and the Variational formulation/weak form. In addition, LU decomposition is considered to solve the matrices. A homemade program in C++ has been developed to solve the main governor equations. The current study addressed various values of the Richardson number ($\tilde{R}$i = 1, 10, 100), Hartmann number ($\tilde{\mathrm{H}}$a = 0 to 75 with 25-step increments), and nanoparticle solid volume fraction ($\tilde{\varphi }$ = 0 to 0.15 with 0.05 step increments). The Grashof number ($\tilde{G}$r) is assumed to be constant at 10⁴, and the Darcy number ($\tilde{D}$a) is 10⁻³. The effects of heatlines, Nusselt number, Bejan number, and entropy generation have been thoroughly addressed, topics that have not been widely covered in a single study in recent literature.

2. Problem Description

The graph in Figure 1 displays the physical envelope model with some major geometric parameters and dimensions. The model is a two-dimensional rectangular shape enclosure with a width (W) and height (H). On the base wall, a pair of uniform and constant heat sources with length (ε) are embedded, while the rest of the base wall is thermally insulated. The vertical wall has a fixed motion and cold temperature. Meanwhile, the top wall with sinusoidal corrugation is assumed to be fixed and adiabatic. The cavity is filled with a porous medium and TiO₂-water nanofluid. An external magnetic field (${\tilde{B}}_O$) is applied to the X-axis, as shown in Figure 1.

3. Mathematical Formulation

3.1. Assumptions

The current analysis is based on the following assumptions [26,27,28]:

• The porous medium is homogeneous and isotropic.
• The nanofluid with a nanoparticle volume fraction ≤ 0.15 is Newtonian and incompressible.
• Laminar flow.
• Regular shape of the nanoparticles.
• Thermal equilibrium is assumed between the base fluid and the nanoparticles, along with no-slip conditions.
• The thermophysical properties of the nanofluid are fixed, except for variations in density that cause the body force term in the vertical component of the momentum equation (Equation (3)).
• A fixed and uniform magnetic field.

3.2. Distributions of the Velocity and Temperature

Based on the assumptions in Section 3.1, the final dimensional form of steady-state two-dimensional governing equations for mixed convection porous medium saturated by nanofluids can be expressed as follows [25,27]:

$${\displaystyle \frac{\partial \tilde{\mathrm{u}}}{\partial \tilde{\mathrm{x}}}}+{\displaystyle \frac{\partial \tilde{\mathrm{v}}}{\partial \tilde{\mathrm{y}}}}=0$$

(1)

$${\tilde{\mathsf{\rho }}}_{\mathrm{n}\mathrm{f}}\left(\tilde{\mathrm{u}}{\displaystyle \frac{\partial \tilde{\mathrm{u}}}{\partial \tilde{\mathrm{x}}}}+\tilde{\mathrm{v}}{\displaystyle \frac{\partial \tilde{\mathrm{u}}}{\partial \tilde{\mathrm{y}}}}\right)=-{\displaystyle \frac{\partial \tilde{\mathrm{p}}}{\partial \tilde{\mathrm{x}}}}+{\tilde{\mathsf{\mu }}}_{\mathrm{n}\mathrm{f}}\left({\displaystyle \frac{{\partial }^2\tilde{\mathrm{u}}}{{\partial \tilde{\mathrm{x}}}^2}}+{\displaystyle \frac{{\partial }^2\tilde{\mathrm{u}}}{{\partial \tilde{\mathrm{y}}}^2}}\right)-{\tilde{\mathsf{\mu }}}_{\mathrm{n}\mathrm{f}}{\displaystyle \frac{\tilde{\mathrm{u}}}{\tilde{\mathrm{K}}}}$$

(2)

$${\tilde{\mathsf{\rho }}}_{\mathrm{n}\mathrm{f}}\left(\tilde{\mathrm{u}}{\displaystyle \frac{\partial \tilde{\mathrm{v}}}{\partial \tilde{\mathrm{x}}}}+\tilde{\mathrm{v}}{\displaystyle \frac{\partial \tilde{\mathrm{v}}}{\partial \tilde{\mathrm{y}}}}\right)=-{\displaystyle \frac{\partial \tilde{\mathrm{p}}}{\partial \tilde{\mathrm{y}}}}+{\tilde{\mathsf{\mu }}}_{\mathrm{n}\mathrm{f}}\left({\displaystyle \frac{{\partial }^2\tilde{\mathrm{v}}}{{\partial \tilde{\mathrm{x}}}^2}}+{\displaystyle \frac{{\partial }^2\tilde{\mathrm{v}}}{{\partial \tilde{\mathrm{y}}}^2}}\right)-{\tilde{\mathsf{\mu }}}_{\mathrm{n}\mathrm{f}}{\displaystyle \frac{\tilde{\mathrm{v}}}{\tilde{\mathrm{K}}}}-{\tilde{\mathsf{\sigma }}}_{\mathrm{n}\mathrm{f}}{\tilde{\mathrm{B}}\mathrm{o}}^2\tilde{\mathrm{v}}+{\left(\tilde{\mathsf{\rho }}\tilde{\mathsf{\beta }}\right)}_{nf}\tilde{\mathrm{g}}(\tilde{\mathrm{T}}-{\tilde{\mathrm{T}}}_c)$$

(3)

$$\left(\tilde{\mathrm{u}}{\displaystyle \frac{\partial \tilde{\mathrm{T}}}{\partial \tilde{\mathrm{x}}}}+\tilde{\mathrm{v}}{\displaystyle \frac{\partial \tilde{\mathrm{T}}}{\partial \tilde{\mathrm{y}}}}\right)={\tilde{\mathsf{\alpha }}}_{\mathrm{n}\mathrm{f}}\left({\displaystyle \frac{{\partial }^2\tilde{\mathrm{T}}}{{\partial \tilde{\mathrm{x}}}^2}}+{\displaystyle \frac{{\partial }^2\tilde{\mathrm{T}}}{{\partial \tilde{\mathrm{y}}}^2}}\right)$$

(4)

By introducing the following dimensionless variables and numbers

$$\tilde{\mathrm{X}}={\displaystyle \frac{\tilde{\mathrm{x}}}{{\tilde{\mathrm{L}}}_{\mathrm{c}}}},\tilde{\mathrm{Y}}={\displaystyle \frac{\tilde{\mathrm{y}}}{{\tilde{\mathrm{L}}}_{\mathrm{c}}}},\tilde{\mathrm{U}}={\displaystyle \frac{\tilde{\mathrm{u}}}{{\tilde{\mathrm{U}}}_{\mathrm{o}}}},\tilde{\mathrm{V}}={\displaystyle \frac{\tilde{\mathrm{v}}}{{\tilde{\mathrm{U}}}_{\mathrm{o}}}},\tilde{\mathrm{P}}={\displaystyle \frac{\tilde{\mathrm{p}}}{{\tilde{\mathsf{\rho }}}_{\mathrm{n}\mathrm{f}}{\tilde{\mathrm{U}}}_{\mathrm{o}}^2}},\tilde{\mathsf{\theta }}={\displaystyle \frac{\tilde{\mathrm{T}}-{\tilde{\mathrm{T}}}_{\mathrm{c}}}{\Delta \tilde{\mathrm{T}}}},\Delta \tilde{\mathrm{T}}={\displaystyle \frac{{\tilde{\mathrm{q}}}^{″}{\tilde{\mathrm{L}}}_{\mathrm{c}}}{{\tilde{\mathrm{k}}}_{\mathrm{f}}}},\tilde{\mathrm{G}}\mathrm{r}={\displaystyle \frac{\tilde{\mathrm{g}}{\tilde{\mathsf{\beta }}}_{\mathrm{f}}{\tilde{\mathrm{L}}}_{\mathrm{c}}^2\Delta \tilde{\mathrm{T}}}{{\tilde{\mathsf{\nu }}}_{\mathrm{f}}^2}},\tilde{\mathrm{P}}\mathrm{r}={\displaystyle \frac{{\tilde{\mathsf{\nu }}}_{\mathrm{f}}}{{\tilde{\mathsf{\alpha }}}_{\mathrm{f}}}},\tilde{\mathrm{R}}\mathrm{e}={\displaystyle \frac{{\tilde{\mathrm{U}}}_{\mathrm{o}}{\tilde{\mathrm{L}}}_{\mathrm{c}}}{{\tilde{\mathsf{\nu }}}_{\mathrm{f}}}},\tilde{\mathrm{R}}\mathrm{i}={\displaystyle \frac{\tilde{\mathrm{G}}\mathrm{r}}{{\tilde{\mathrm{R}}\mathrm{e}}^2}},\tilde{\mathrm{H}}\mathrm{a}={\tilde{\mathrm{B}}}_{\mathrm{o}}{\tilde{\mathrm{L}}}_{\mathrm{c}}\sqrt{{\displaystyle \frac{{\tilde{\mathsf{\sigma }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\nu }}}_{\mathrm{f}}{\tilde{\mathsf{\rho }}}_{\mathrm{n}\mathrm{f}}}}}$$

The final dimensionless governing equations form in terms of dimensionless variables can be written as:

$${\displaystyle \frac{\partial \tilde{\mathrm{U}}}{\partial \tilde{\mathrm{X}}}}+{\displaystyle \frac{\partial \tilde{\mathrm{V}}}{\partial \tilde{\mathrm{Y}}}}=0$$

(5)

$$\tilde{\mathrm{U}}{\displaystyle \frac{\partial \tilde{\mathrm{U}}}{\partial \tilde{\mathrm{X}}}}+\tilde{\mathrm{V}}{\displaystyle \frac{\partial \tilde{\mathrm{U}}}{\partial \tilde{\mathrm{Y}}}}=-{\displaystyle \frac{\partial \tilde{\mathrm{P}}}{\partial \tilde{\mathrm{X}}}}+{\displaystyle \frac{{\tilde{\mathsf{\nu }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\nu }}}_{\mathrm{f}}}}{\displaystyle \frac{1}{\tilde{\mathrm{R}}\mathrm{e}}}\left({\displaystyle \frac{{\partial }^2\tilde{\mathrm{U}}}{{\partial \tilde{\mathrm{X}}}^2}}+{\displaystyle \frac{{\partial }^2\tilde{\mathrm{U}}}{{\partial \tilde{\mathrm{Y}}}^2}}\right){\displaystyle \frac{{\tilde{\mathsf{\nu }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\nu }}}_{\mathrm{f}}}}{\displaystyle \frac{1}{\tilde{\mathrm{R}}\mathrm{e}}}{\displaystyle \frac{\tilde{\mathrm{U}}}{\tilde{\mathrm{D}}\mathrm{a}}}$$

(6)

$$\tilde{\mathrm{U}}{\displaystyle \frac{\partial \tilde{\mathrm{V}}}{\partial \tilde{\mathrm{X}}}}+\tilde{\mathrm{V}}{\displaystyle \frac{\partial \tilde{\mathrm{V}}}{\partial \tilde{\mathrm{Y}}}}=-{\displaystyle \frac{\partial \tilde{\mathrm{P}}}{\partial \tilde{\mathrm{Y}}}}+{\displaystyle \frac{{\tilde{\mathsf{\nu }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\nu }}}_{\mathrm{f}}}}{\displaystyle \frac{1}{\tilde{\mathrm{R}}\mathrm{e}}}\left({\displaystyle \frac{{\partial }^2\tilde{\mathrm{V}}}{{\partial \tilde{\mathrm{X}}}^2}}+{\displaystyle \frac{{\partial }^2\tilde{\mathrm{V}}}{{\partial \tilde{\mathrm{Y}}}^2}}\right)+{\displaystyle \frac{{\left(\tilde{\mathsf{\rho }}\tilde{\mathsf{\beta }}\right)}_{\mathrm{n}\mathrm{f}}}{{{\tilde{\mathsf{\rho }}}_{\mathrm{n}\mathrm{f}}\tilde{\mathsf{\beta }}}_{\mathrm{f}}}}\tilde{\mathrm{R}}\mathrm{i}\tilde{\mathsf{\theta }}-{\displaystyle \frac{{\tilde{\mathrm{H}}\mathrm{a}}^2}{\tilde{\mathrm{R}}\mathrm{e}}}\tilde{\mathrm{V}}-{\displaystyle \frac{{\tilde{\mathsf{\nu }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\nu }}}_{\mathrm{f}}}}{\displaystyle \frac{1}{\tilde{\mathrm{R}}\mathrm{e}}}{\displaystyle \frac{\tilde{\mathrm{V}}}{\tilde{\mathrm{D}}\mathrm{a}}}$$

(7)

$$\tilde{\mathrm{U}}{\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{X}}}}+\tilde{\mathrm{V}}{\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{Y}}}}={\displaystyle \frac{{\tilde{\mathsf{\alpha }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\alpha }}}_{\mathrm{f}}}}{\displaystyle \frac{1}{\tilde{\mathrm{R}}\mathrm{e}\tilde{\mathrm{P}}\mathrm{r}}}\left({\displaystyle \frac{{\partial }^2\tilde{\mathsf{\theta }}}{{\partial \tilde{\mathrm{X}}}^2}}+{\displaystyle \frac{{\partial }^2\tilde{\mathsf{\theta }}}{{\partial \tilde{\mathrm{Y}}}^2}}\right)$$

(8)

where ${\tilde{\mathsf{\rho }}}_{\mathrm{n}\mathrm{f}}=\left(1-\tilde{\mathsf{\Phi }}\right){\tilde{\mathsf{\rho }}}_{\mathrm{f}}+\tilde{\mathsf{\Phi }}{\mathsf{\rho }}_{\mathrm{p}},{\tilde{\mathsf{\mu }}}_{\mathrm{n}\mathrm{f}}={\displaystyle \frac{{\tilde{\mathsf{\mu }}}_{\mathrm{f}}}{{\left(1-\tilde{\mathsf{\Phi }}\right)}^{2.5}}},{\displaystyle \frac{{\tilde{\mathsf{\sigma }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\sigma }}}_{\mathrm{f}}}}=1+{\displaystyle \frac{3\left(\tilde{\mathsf{\gamma }}-1\right)\tilde{\mathsf{\Phi }}}{\left(\tilde{\mathsf{\gamma }}+2\right)-\left(\tilde{\mathsf{\gamma }}-1\right)\tilde{\mathsf{\Phi }}}},\tilde{\mathsf{\gamma }}={\displaystyle \frac{{\tilde{\mathsf{\sigma }}}_{\mathrm{p}}}{{\tilde{\mathsf{\sigma }}}_{\mathrm{f}}}},{\tilde{\mathsf{\alpha }}}_{\mathrm{n}\mathrm{f}}={\displaystyle \frac{{\tilde{\mathrm{k}}}_{\mathrm{n}\mathrm{f}}}{({\tilde{\mathsf{\rho }}{\tilde{\mathrm{C}}}_{\mathrm{p}})}_{\mathrm{n}\mathrm{f}}}},({\tilde{\mathsf{\rho }}\tilde{\mathsf{\beta }})}_{\mathrm{n}\mathrm{f}}=\left(1-\tilde{\mathsf{\Phi }}\right)({\tilde{\mathsf{\rho }}\tilde{\mathsf{\beta }})}_{\mathrm{f}}+\tilde{\mathsf{\Phi }}({\tilde{\mathsf{\rho }}\tilde{\mathsf{\beta }})}_{\mathrm{p}},{\tilde{\mathrm{k}}}_{\mathrm{n}\mathrm{f}}={\tilde{\mathrm{k}}}_{\mathrm{f}}{\displaystyle \frac{\left({\tilde{\mathrm{k}}}_{\mathrm{p}}+{2\tilde{\mathrm{k}}}_{\mathrm{f}}\right)-2\tilde{\mathsf{\Phi }}\left({\tilde{\mathrm{k}}}_{\mathrm{f}}-{\tilde{\mathrm{k}}}_{\mathrm{p}}\right)}{\left({\tilde{\mathrm{k}}}_{\mathrm{p}}+{2\tilde{\mathrm{k}}}_{\mathrm{f}}\right)+\tilde{\mathsf{\Phi }}\left({\tilde{\mathrm{k}}}_{\mathrm{f}}-{\tilde{\mathrm{k}}}_{\mathrm{p}}\right)}},{\tilde{\mathsf{\alpha }}}_{\mathrm{n}\mathrm{f}}={\tilde{\mathsf{\alpha }}}_{\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{s}},\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{y}=1,{\left(\tilde{\mathsf{\rho }}{\tilde{\mathrm{C}}}_{\mathrm{p}}\right)}_{\mathrm{n}\mathrm{f}}=\left(1-\tilde{\mathsf{\Phi }}\right)({\tilde{\mathsf{\rho }}{\tilde{\mathrm{C}}}_{\mathrm{p}})}_{\mathrm{f}}+\tilde{\mathsf{\Phi }}({\tilde{\mathsf{\rho }}{\tilde{\mathrm{C}}}_{\mathrm{p}})}_{\mathrm{p}}$.

The thermophysical properties of the base fluid and nanoparticles in the current study are given in

Table 1. Thermophysical properties, [21].

Properties	Pure Water	Titanium Oxide (TiO2)
${\tilde{\mathrm{C}}}_{\mathrm{p}}$ (J/kg·K)	4179	686.2
$\tilde{\mathrm{K}}$ (W/m.K)	0.613	8.953
$\tilde{\mathsf{\rho }}$ (kg/m3)	997.1	4250
$\tilde{\mathsf{\beta }}$ (1/K) × 105	21	0.9
Σ (Ω·m)−1	0.05	3.7 × 103
$\tilde{\mu }$ (pa·s)	0.00091
$\tilde{P}$r	6.2

[21].

The local Nusselt number for the heat sources is: ${\tilde{\mathrm{N}}\mathrm{u}}_{\mathrm{h}\mathrm{f}}\left(\tilde{\mathrm{x}}\right)={\displaystyle \frac{\tilde{\mathrm{h}}{\tilde{\mathrm{L}}}_{\mathrm{c}}}{{\tilde{\mathrm{k}}}_{\mathrm{f}}}}$ where $\tilde{\mathrm{h}}={\displaystyle \frac{{\tilde{\mathrm{q}}}^{″}}{{\tilde{\mathrm{T}}}_{\mathrm{s}}-{\tilde{\mathrm{T}}}_{\mathrm{c}}}}$. The local $\tilde{\mathrm{N}}\mathrm{u}$ number at the heat sources can be expressed by dimensionless temperature [1] as

$${\tilde{\mathrm{N}}\mathrm{u}}_{\mathrm{h}\mathrm{f}}\left(\tilde{\mathrm{X}}\right)={\displaystyle \frac{1}{{\tilde{\mathsf{\theta }}}_{\mathrm{h}\mathrm{f}}\left(\tilde{\mathrm{X}}\right)}}$$

(9)

where, ${\tilde{\mathsf{\theta }}}_{\mathrm{h}\mathrm{f}}$ is the dimensionless heat source temperature. The average $\tilde{\mathrm{N}}\mathrm{u}$ number over the heat source is determined by integrating the local $\tilde{\mathrm{N}}\mathrm{u}$ number as follows:

$${\overline{\tilde{\mathrm{N}}\mathrm{u}}}_{\mathrm{h}\mathrm{f}}={\displaystyle \frac{1}{\tilde{\epsilon }}}{\int }_0^{\tilde{\epsilon }}{\tilde{\mathrm{N}}\mathrm{u}}_{\mathrm{h}\mathrm{f}}\left(\tilde{\mathrm{X}}\right)\mathrm{d}\tilde{\mathrm{X}}={\displaystyle \frac{1}{\tilde{\epsilon }}}{\int }_0^{\tilde{\epsilon }}{\displaystyle \frac{\mathrm{d}\tilde{\mathrm{X}}}{{\tilde{\mathsf{\theta }}}_{\mathrm{h}\mathrm{f}}\left(\tilde{\mathrm{X}}\right)}}$$

(10)

3.3. Streamfunction and Heatfunction

The streamfunction ($\tilde{\mathsf{\psi }}$) is used to describe the fluid motion. The streamfunction equations in two-dimensional flows are derived from velocity components $\tilde{\mathrm{U}}$ and $\tilde{\mathrm{V}}$ [1].

$$\tilde{\mathrm{U}}={\displaystyle \frac{\partial \tilde{\mathsf{\psi }}}{\partial \tilde{\mathrm{Y}}}},\tilde{\mathrm{V}}=-{\displaystyle \frac{\partial \tilde{\mathsf{\psi }}}{\partial \tilde{\mathrm{X}}}},\hspace{1em}\hspace{1em}{\displaystyle \frac{{\partial }^2\tilde{\mathsf{\psi }}}{{\partial \tilde{\mathrm{X}}}^2}}+{\displaystyle \frac{{\partial }^2\tilde{\mathsf{\psi }}}{{\partial \tilde{\mathrm{Y}}}^2}}={\displaystyle \frac{\partial \tilde{\mathrm{U}}}{\partial \tilde{\mathrm{Y}}}}-{\displaystyle \frac{\partial \tilde{\mathrm{V}}}{\partial \tilde{\mathrm{X}}}}$$

(11)

Heat transfer can be traced and analyzed by introducing heatlines. These heatlines are represented mathematically by heatfunctions ($\tilde{\mathsf{\Pi }}$). The heatfunction is derived from conductive heat fluxes (diffusion heatfunction) (−∂$\tilde{\mathsf{\theta }}$/∂$\tilde{\mathrm{X}}$, −∂$\tilde{\mathsf{\theta }}$/∂$\tilde{\mathrm{Y}}$) as well as convective heat fluxes (convection heatfunction) ($\tilde{\mathrm{U}}\tilde{\mathsf{\theta }}$, $\tilde{\mathrm{V}}\tilde{\mathsf{\theta }}$) [29].

$${\displaystyle \frac{\partial \tilde{\mathsf{\Pi }}}{\partial \tilde{\mathrm{Y}}}}=\tilde{\mathrm{U}}\tilde{\mathsf{\theta }}-{\displaystyle \frac{{\tilde{\mathsf{\alpha }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\alpha }}}_{\mathrm{f}}}}{\displaystyle \frac{1}{\tilde{\mathrm{R}}\mathrm{e}\tilde{\mathrm{P}}\mathrm{r}}}{\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{X}}}},\hspace{1em}\hspace{1em}{\displaystyle \frac{\partial \tilde{\mathsf{\Pi }}}{\partial \tilde{\mathrm{X}}}}=\tilde{\mathrm{V}}\tilde{\mathsf{\theta }}-{\displaystyle \frac{{\tilde{\mathsf{\alpha }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\alpha }}}_{\mathrm{f}}}}{\displaystyle \frac{1}{\tilde{\mathrm{R}}\mathrm{e}\tilde{\mathrm{P}}\mathrm{r}}}{\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{Y}}}}$$

(12)

The heatfunction equations (Equation (12)) have to satisfy the energy equation (Equation (8)), so the final form of the heatfunction can be written as:

$${\displaystyle \frac{{\partial }^2\tilde{\mathsf{\Pi }}}{{\partial \tilde{\mathrm{X}}}^2}}+{\displaystyle \frac{{\partial }^2\tilde{\mathsf{\Pi }}}{{\partial \tilde{\mathrm{Y}}}^2}}={\displaystyle \frac{\partial }{\partial \tilde{\mathrm{Y}}}}\left(\tilde{\mathrm{U}}\tilde{\mathsf{\theta }}\right)-{\displaystyle \frac{\partial }{\partial \tilde{\mathrm{X}}}}\left(\tilde{\mathrm{V}}\tilde{\mathsf{\theta }}\right)$$

(13)

3.4. Entropy Generation

The local entropy rate in the cavity is generated from three sources: the heat flow, the viscous dissipation, and the magnetic field dissipation (magnetic field). The entropy generation can be expressed mathematically by the following equations [1,27]:

$${\tilde{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n}}={\tilde{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{T}}+{\tilde{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathsf{\mu }}+{\tilde{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n},\tilde{\mathrm{H}}\mathrm{a}}$$

(14)

$${\tilde{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{T}}={\displaystyle \frac{{\tilde{\mathrm{k}}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathrm{k}}}_{\mathrm{f}}}}\left[{\left({\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{X}}}}\right)}^2+{\left({\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{Y}}}}\right)}^2\right]$$

(15)

$${\tilde{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathsf{\mu }}=\tilde{\chi }\left\{\left({\tilde{\mathrm{U}}}^2+{\tilde{\mathrm{V}}}^2\right)+\tilde{\mathrm{D}}\mathrm{a}\left[{2\left({\displaystyle \frac{\partial \tilde{\mathrm{U}}}{\partial \tilde{\mathrm{X}}}}\right)}^2+{2\left({\displaystyle \frac{\partial \tilde{\mathrm{V}}}{\partial \tilde{\mathrm{Y}}}}\right)}^2+{\left({\displaystyle \frac{\partial \tilde{\mathrm{U}}}{\partial \tilde{\mathrm{Y}}}}+{\displaystyle \frac{\partial \tilde{\mathrm{V}}}{\partial \tilde{\mathrm{X}}}}\right)}^2\right]\right\}$$

(16)

$${\tilde{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n},\tilde{\mathrm{H}}\mathrm{a}}=\tilde{\chi }{\displaystyle \frac{{\tilde{\mathsf{\nu }}}_{\mathrm{f}}}{{\tilde{\mathsf{\nu }}}_{\mathrm{n}\mathrm{f}}}}{\tilde{\mathrm{H}}\mathrm{a}}^2{\tilde{\mathrm{V}}}^2$$

(17)

where $\tilde{\chi }={\displaystyle \frac{{{\tilde{\mathrm{T}}}_{\mathrm{o}}\tilde{\mathsf{\mu }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathrm{k}}}_{\mathrm{f}}}}{\left({\displaystyle \frac{{\tilde{\mathsf{\alpha }}}_{\mathrm{f}}}{\sqrt{\tilde{\mathrm{K}}}\Delta \tilde{\mathrm{T}}}}\right)}^2$, ${\tilde{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{T}}$ is the entropy generated by the heat flow (Equation (15)), ${\tilde{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathsf{\mu }}$ is the entropy generated by viscous dissipation (Equation (16)), and ${\tilde{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n},\tilde{\mathrm{H}}\mathrm{a}}$ is the entropy generated by a magnetic field (Equation (17)). The total entropy generated within the cavity is set by integrating the local entropy generation sources (Equation (14)) over the entire studied area.

$${\dot{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n}}=\int {\tilde{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n}}\mathrm{d}\tilde{\mathrm{A}}$$

(18)

Another factor derived from entropy generation in the cavity, which can be used to address irreversibility, is the Bejan number. This can be defined as follows:

$${\tilde{\mathrm{B}}}_{\mathrm{e}}={\displaystyle \frac{{\dot{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n},\mathrm{T}}}{{\dot{\mathrm{S}}}_{\mathrm{g}\mathrm{e}\mathrm{n}}}}$$

(19)

When ${\tilde{\mathrm{B}}}_{\mathrm{e}}$ close to approach 1, the process is predominated by heat transfer irreversibility. In contrast, when the value of $\tilde{\mathrm{B}}$e is less than 1/2, the irreversibility due to influences of viscous and magnetic field controls the processes. When ${\tilde{\mathrm{B}}}_{\mathrm{e}}$ = 1/2, the entropy generation due to the viscous dissipation and magnetic field is equal to the heat transfer effect [27].

3.5. Boundary Conditions

The boundary conditions are derived from the physical case study shown in Figure 1, and they are summarized below. These boundary conditions have been incorporated into the governing equations (Equations (5)–(8), (11) and (13)).

At the horizontal base wall

$$\begin{gathered} \tilde{\mathrm{Y}}=0,\hspace{1em}\hspace{1em}0\le \tilde{\mathrm{X}}\le {\mathrm{b}}_{\mathrm{w}},\hspace{1em}\hspace{1em}{\mathrm{b}}_{\mathrm{w}}+\tilde{\mathsf{\epsilon }}\le \tilde{\mathrm{X}}\le {\mathrm{b}}_{\mathrm{w}}+\tilde{\mathsf{\epsilon }}+{\mathrm{b}}_{\mathrm{s}},\hspace{1em}\hspace{1em}{\mathrm{b}}_{\mathrm{w}}+2\tilde{\mathsf{\epsilon }}+{\mathrm{b}}_{\mathrm{s}}\le \tilde{\mathrm{X}}\le 1, \\ \tilde{\mathrm{U}}=\tilde{\mathrm{V}}=0,{\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{Y}}}}=0,\tilde{\mathsf{\psi }}=0 \\ \tilde{\mathrm{Y}}=0,{\mathrm{b}}_{\mathrm{w}}\le \tilde{\mathrm{X}}\le {\mathrm{b}}_{\mathrm{w}}+\tilde{\mathsf{\epsilon }},{\mathrm{b}}_{\mathrm{w}}+\tilde{\mathsf{\epsilon }}+{\mathrm{b}}_{\mathrm{s}}\le \tilde{\mathrm{X}}\le {\mathrm{b}}_{\mathrm{w}}+2\tilde{\mathsf{\epsilon }}+{\mathrm{b}}_{\mathrm{s}}, \\ \tilde{\mathrm{U}}=\tilde{\mathrm{V}}=0,{\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{Y}}}}=-{\displaystyle \frac{{\tilde{\mathrm{k}}}_{\mathrm{f}}}{{\tilde{\mathrm{k}}}_{\mathrm{n}\mathrm{f}}}},\tilde{\mathsf{\psi }}=0 \end{gathered}$$

At the vertical walls

$$\tilde{\mathrm{X}}=1\hspace{1em}\mathrm{o}\mathrm{r}\hspace{1em}\tilde{\mathrm{X}}=0,\hspace{1em}\hspace{1em}0\le \tilde{\mathrm{Y}}\le 1\hspace{1em}\hspace{1em}\tilde{\mathrm{U}}=0,\tilde{\mathrm{V}}=\pm 1,\tilde{\mathsf{\theta }}=0,\tilde{\mathsf{\psi }}=0$$

At the corrugated upper wall

$$0\le \tilde{\mathrm{X}}\le 1,\tilde{\mathrm{Y}}=1+{\mathrm{A}}_m\sin \left(13\mathsf{\pi }\tilde{\mathrm{X}}\right), {\tilde{\mathrm{A}}}_m={\displaystyle \frac{\tilde{\mathrm{W}}}{50}}\hspace{1em}\tilde{\mathrm{U}}=\tilde{\mathrm{V}}=0,{\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{n}}}}=0,\tilde{\mathsf{\psi }}=0$$

Determining the boundary conditions for the heatfunctions can be challenging and complex. The starting points of these boundaries is derived from Equation (12). Neumann boundary condition is considered for isothermal walls, and it is derived from Equation (12) as follows: $\tilde{\mathrm{n}}$. Δ$\tilde{\mathsf{\Pi }}$ = 0. The Dirichlet boundary condition represents the insulated wall, and it is obtained from Equation (12), which is simplified into ∂$\tilde{\mathsf{\Pi }}$/∂$\tilde{\mathrm{n}}$ = 0 [1,30].

The first step to determine the boundary conditions of the heatfunction is by assuming a reference value of the heat function. The reference value of $\tilde{\mathsf{\Pi }}$ is 0 at $\tilde{\mathrm{X}}$ = 0; $\tilde{\mathrm{Y}}$ = 1, and therefore $\tilde{\mathsf{\Pi }}$ = 0 for $\tilde{\mathrm{Y}}$ = 1; 0 ≤ $\tilde{\mathrm{X}}$ ≤ 1 as this represents the heat flow at an adiabatic wall [30]. The heatfunctions values at the junction points are determined by integrating Equation (12) along boundaries from the reference point until the junction points that are required. Here are the procedures that have been considered to determine the heatfunctions boundary condition as follows:

Consider $\tilde{\mathrm{X}}=0,\tilde{\mathrm{Y}}=1,\tilde{\mathsf{\Pi }}\left(0,1\right)=0$, which is the starting point (reference value) for the boundary condition.

$$\begin{array}{l}\mathrm{a}\mathrm{t} \tilde{\mathrm{X}}=0, \tilde{\mathrm{Y}}=0, \tilde{\mathsf{\Pi }}\left(0,0\right)=\tilde{\mathsf{\Pi }}\left(0,1\right)-{\int }_0^1{\displaystyle \frac{\partial \tilde{\mathsf{\Pi }}}{\partial \tilde{\mathrm{Y}}}}\mathrm{d}\tilde{\mathrm{Y}}\hspace{1em}\hspace{1em}\tilde{\mathsf{\Pi }}\left(0,0\right)=\tilde{\mathsf{\Pi }}\left(0,1\right)+{\int }_0^1{\displaystyle \frac{{\tilde{\mathsf{\alpha }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\alpha }}}_{\mathrm{f}}}}{\displaystyle \frac{1}{\tilde{\mathrm{R}}\mathrm{e}\tilde{\mathrm{P}}\mathrm{r}}}{\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{X}}}}\mathrm{d}\tilde{\mathrm{Y}}\\ \mathrm{a}\mathrm{t} \tilde{\mathrm{X}}={\mathrm{b}}_{\mathrm{w}}, \tilde{\mathrm{Y}}=0, \tilde{\mathsf{\Pi }}\left({\mathrm{b}}_{\mathrm{w}},0\right)=\tilde{\mathsf{\Pi }}\left(0,0\right)+{\int }_0^{{\mathrm{b}}_{\mathrm{w}}}{\displaystyle \frac{\partial \tilde{\mathsf{\Pi }}}{\partial \tilde{\mathrm{X}}}}\mathrm{d}\tilde{\mathrm{X}}=\tilde{\mathsf{\Pi }}\left(0,0\right)+{\int }_0^{{\mathrm{b}}_{\mathrm{w}}}{\displaystyle \frac{{\tilde{\mathsf{\alpha }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\alpha }}}_{\mathrm{f}}}}{\displaystyle \frac{1}{\tilde{\mathrm{R}}\mathrm{e}\tilde{\mathrm{P}}\mathrm{r}}}{\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{Y}}}}\mathrm{d}\tilde{\mathrm{X}}\\ \mathrm{a}\mathrm{t} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t} \tilde{\mathrm{X}}={\mathrm{b}}_{\mathrm{w}}+\tilde{\epsilon },\hspace{1em}\hspace{1em}\tilde{\mathrm{Y}}=0,\\ \tilde{\mathsf{\Pi }}\left({\mathrm{b}}_{\mathrm{w}}+\tilde{\epsilon },0\right)=\tilde{\mathsf{\Pi }}\left({\mathrm{b}}_{\mathrm{w}},0\right)+{\int }_{{\mathrm{b}}_{\mathrm{w}}}^{{\mathrm{b}}_{\mathrm{w}}+\tilde{\epsilon }}{\displaystyle \frac{\partial \tilde{\mathsf{\Pi }}}{\partial \tilde{\mathrm{X}}}}\mathrm{d}\tilde{\mathrm{X}}=\tilde{\mathsf{\Pi }}\left({\mathrm{b}}_{\mathrm{w}},0\right)+{\int }_{{\mathrm{b}}_{\mathrm{w}}}^{{\mathrm{b}}_{\mathrm{w}}+\tilde{\epsilon }}{\displaystyle \frac{{\tilde{\mathsf{\alpha }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\alpha }}}_{\mathrm{f}}}}{\displaystyle \frac{1}{\tilde{\mathrm{R}}\mathrm{e}\tilde{\mathrm{P}}\mathrm{r}}}{\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{Y}}}}\mathrm{d}\tilde{\mathrm{X}}\\ \mathrm{a}\mathrm{t} \tilde{\mathrm{X}}={\mathrm{b}}_{\mathrm{w}}+\tilde{\epsilon }+{\mathrm{b}}_{\mathrm{s}},\tilde{\mathrm{Y}}=0,\\ \tilde{\mathsf{\Pi }}\left({\mathrm{b}}_{\mathrm{w}}+\tilde{\epsilon }+{\mathrm{b}}_{\mathrm{s}},0\right)=\tilde{\mathsf{\Pi }}\left({\mathrm{b}}_{\mathrm{w}}+\tilde{\epsilon },0\right)+{\int }_{{\mathrm{b}}_{\mathrm{w}}}^{{\mathrm{b}}_{\mathrm{w}}+\tilde{\epsilon }+{\mathrm{b}}_{\mathrm{s}}}{\displaystyle \frac{\partial \tilde{\mathsf{\Pi }}}{\partial \tilde{\mathrm{X}}}}\mathrm{d}\tilde{\mathrm{X}}=\tilde{\mathsf{\Pi }}\left({\mathrm{b}}_{\mathrm{w}}+\tilde{\epsilon },0\right)+{\int }_{{\mathrm{b}}_{\mathrm{w}}}^{{\mathrm{b}}_{\mathrm{w}}+\tilde{\epsilon }+{\mathrm{b}}_{\mathrm{s}}}{\displaystyle \frac{{\tilde{\mathsf{\alpha }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\alpha }}}_{\mathrm{f}}}}{\displaystyle \frac{1}{\tilde{\mathrm{R}}\mathrm{e}\tilde{\mathrm{P}}\mathrm{r}}}{\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{Y}}}}\mathrm{d}\tilde{\mathrm{X}}\\ \mathrm{a}\mathrm{t} \tilde{\mathrm{X}}=1, \tilde{\mathrm{Y}}=1, \tilde{\mathsf{\Pi }}\left(1,1\right)=0 \mathrm{a}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r} \mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l} \mathrm{i}\mathrm{s} \mathrm{i}\mathrm{n}\mathrm{s}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\\ \mathrm{a}\mathrm{t} \tilde{\mathrm{X}}=1, \tilde{\mathrm{Y}}=0, \tilde{\mathsf{\Pi }}\left(1,0\right)=\tilde{\mathsf{\Pi }}\left(1,1\right)-{\int }_0^1{\displaystyle \frac{\partial \tilde{\mathsf{\Pi }}}{\partial \tilde{\mathrm{Y}}}}\mathrm{d}\tilde{\mathrm{Y}}=\tilde{\mathsf{\Pi }}\left(1,1\right)+{\int }_0^1{\displaystyle \frac{{\tilde{\mathsf{\alpha }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\alpha }}}_{\mathrm{f}}}}{\displaystyle \frac{1}{\tilde{\mathrm{R}}\mathrm{e}\tilde{\mathrm{P}}\mathrm{r}}}{\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{X}}}}\mathrm{d}\tilde{\mathrm{Y}}\\ \mathrm{a}\mathrm{t} \tilde{\mathrm{X}}={\mathrm{b}}_{\mathrm{w}}+2\tilde{\epsilon }+{\mathrm{b}}_{\mathrm{s}}, \tilde{\mathrm{Y}}=0,\\ \tilde{\mathsf{\Pi }}\left({\mathrm{b}}_{\mathrm{w}}+2\tilde{\epsilon }+{\mathrm{b}}_{\mathrm{s}},0\right)=\tilde{\mathsf{\Pi }}\left(1,0\right)+{\int }_1^{{\mathrm{b}}_{\mathrm{w}}+2\tilde{\epsilon }+{\mathrm{b}}_{\mathrm{s}}}{\displaystyle \frac{\partial \tilde{\mathsf{\Pi }}}{\partial \tilde{\mathrm{X}}}}\mathrm{d}\tilde{\mathrm{X}}=\tilde{\mathsf{\Pi }}\left(1,0\right)-{\int }_{{\mathrm{b}}_{\mathrm{w}}+2\tilde{\epsilon }+{\mathrm{b}}_{\mathrm{s}}}^1{\displaystyle \frac{{\tilde{\mathsf{\alpha }}}_{\mathrm{n}\mathrm{f}}}{{\tilde{\mathsf{\alpha }}}_{\mathrm{f}}}}{\displaystyle \frac{1}{\tilde{\mathrm{R}}\mathrm{e}\tilde{\mathrm{P}}\mathrm{r}}}{\displaystyle \frac{\partial \tilde{\mathsf{\theta }}}{\partial \tilde{\mathrm{Y}}}}\mathrm{d}\tilde{\mathrm{X}}\end{array}$$

4. Numerical Method and Validation

A finite element technique based on the variational method has been employed to discretize the governing equations (Equations (5)–(8), (11) and (13)).

A finite elements technique has been employed based on the variation method to discretize the governor equations. Non-uniform triangular elements are used to evaluate the boundary conditions area efficiently. A homemade C++ model using an Intel Core Ultra 7 165H 1.40 GHz processor (by Lenovo, Beijing, China) has been developed to handle the finite elements equations. A LU decomposition method is used to solve the matrices. To calculate the convergence steady-state value of velocity, temperature, and pressure, a projection method algorithm developed by Rannacher has been used [1]. This method consists of several steps, from computing an intermediate velocity using the equation of momentum. This intermediate velocity does not meet the continuity condition, so a projection step is performed to bring the pressure back and obtain the correct velocity [1].

The present work’s convergence criteria for streamfunction and streamlines.

The convergence limits that have been considered in the following approach are ${\displaystyle \frac{\sum _{i=1}^N\left|{\tilde{\mathsf{\psi }}}_i^{m+1}-{\tilde{\mathsf{\psi }}}_i^m\right|}{\sum _{i=1}^N\left|{\tilde{\mathsf{\psi }}}_i^{m+1}\right|}} \le {10}^{-7}$ and ${\displaystyle \frac{\sum _{i=1}^N\left|{\tilde{\mathsf{\theta }}}_i^{m+1}-{\tilde{\mathsf{\theta }}}_i^m\right|}{\sum _{i=1}^N\left|{\tilde{\mathsf{\theta }}}_i^{m+1}\right|}} \le {10}^{-7}$.

Accuracy testing is necessary to ensure that the solutions are not dependent on the grid. A series of trial calculations were carried out using different non-uniform grids.

Table 2. Grid sensitivity and accuracy based on nanofluid TiO₂-water with $\tilde{\mathsf{\Phi }}$ = 0.1, $\tilde{\mathrm{R}}$i = 1, $\tilde{\mathrm{H}}$a = 0, $\tilde{\mathrm{D}}$a = 0.01, and $\tilde{\mathrm{G}}$r = 10⁴.

Elements	Grid	|$\tilde{\mathsf{\psi }}$max|	$\tilde{\mathsf{\theta }}$max	${\overline{\tilde{\mathbf{N}}\mathbf{u}}}_{\mathbf{h}}$
2052	30 × 30	0.056116	0.065968	17.1827
3809	40 × 40	0.056445	0.065017	17.4335
6023	50 × 50	0.056495	0.064776	17.5001
8930	60 × 60	0.056508	0.064404	17.5406
12,226	70 × 70	0.056511	0.064304	17.5570
15,993	80 × 80	0.056510	0.064300	17.5590

shows the results of these trials. The data indicates that there is a slight difference between the results obtained from the 70 × 70 grid and the 80 × 80 grid. Therefore, it is advisable to consider a 70 × 70 non-uniform triangular element grid (Figure 2) in the current analysis to optimize computational efficiency.

The numerical model’s accuracy was validated against a published numerical result by Mittal et al. [26]. The results for the average $\tilde{\mathrm{N}}$u number was compared with Mittal’s results for all tested cases, and they showed excellent agreement, as presented in

Table 3. Comparison between the present study and [26] for average $\tilde{\mathrm{N}}$u number for nanofluid (Al₂O₃-water) and porous media at $\tilde{\mathrm{R}}$i = 100.

$\tilde{\mathbf{D}}$a	$\tilde{\mathbf{G}}$r = 100				$\tilde{\mathbf{G}}$r = 10,000
	Ref.	Present	Ref.	Present	Ref.	Present
	$\tilde{\mathsf{\Phi }}$ = 0%	$\tilde{\mathsf{\Phi }}$ = 0%	$\tilde{\mathsf{\Phi }}$ = 5%	$\tilde{\mathsf{\Phi }}$ = 5%	$\tilde{\mathsf{\Phi }}$ = 2%	$\tilde{\mathsf{\Phi }}$ = 2%
0.001	1.0020	1.0017	1.1520	1.1520	1.4300	1.4000
0.01	1.0060	1.0055	1.1550	1.1560	2.5140	2.4000
0.1	1.0060	1.0059	1.1580	1.1590	2.8500	2.8000

.

The data from of the present study were compared with those of a previously published study by Khorasanizadeh et al. [27] to validate the accuracy of the entropy generation analysis. The comparative data has been organized in

Table 4. Comparison between the results of the present study [27] for rectangular enclosures filled with Water-Cu nanofluid.

$\tilde{\mathbf{R}}$a	$\tilde{\mathbf{R}}$e	$\tilde{\mathsf{\Phi }}$	$\tilde{\mathsf{\chi }}$	${\dot{\mathbf{S}}}_{\mathbf{g}\mathbf{e}\mathbf{n}}$
				Ref.	Present
105	1	0	0.0001	5.200	5.100
105	100	0.05	0.0001	9.700	9.560
104	10	0.05	0.0001	4.000	3.950

to support its accuracy.

A comparison has been made between the current study and a study conducted by Al-Khaleel et al. [30]. The comparison is presented in

Table 5. Comparison between the present study and [30] for average maximum streamfunction ($\tilde{\mathsf{\psi }}$_max) at $\tilde{\mathrm{D}}$a = 10⁻² and $\tilde{\mathrm{H}}$a = 0.

$\tilde{\mathbf{R}}$e	$\tilde{\mathbf{G}}$r	Ref.	Present
50	104	0.110	0.115
100	103	0.097	0.102
250	104	0.108	0.113
500	105	0.135	0.139

and shows the maximum streamfunction at various Reynolds and Grashoff numbers. The results indicate an acceptable difference between the two studies.

A comparison with a recent experimental study [31,32] to validate the current program results. The results determine the program’s accuracy.

5. Results and Discussion

The following section discusses the resulting data for different values of $\tilde{\mathrm{R}}$i ($\tilde{\mathrm{R}}$i = 1, 10, and 100), $\tilde{\mathrm{H}}$a (0 ≤ $\tilde{\mathrm{H}}$a ≤ 75 step 25), and solid volume fraction (0 ≤ $\tilde{\mathsf{\Phi }}$ ≤ 1.5 step 0.05). The value of $\tilde{\mathrm{G}}$r = 10⁴ and $\tilde{\mathrm{D}}$a = 10⁻³. The cavity is assumed to be a square cavity (W = H) with $\tilde{\mathsf{\epsilon }}$ = b_w = b_s = 0.2. Three different cases are discussed in this study:

• Case-1, where the isothermal vertical walls move from top to bottom ($\tilde{\mathrm{V}}$ = −1).
• Case-2, where the isothermal vertical walls move oppositely, the right wall moving from top to bottom ($\tilde{\mathrm{V}}$ = −1), and the left wall moving from bottom to top ($\tilde{\mathrm{V}}$ = 1).
• Case-3, where the isothermal vertical walls move from bottom to top ($\tilde{\mathrm{V}}$ = 1).

For all scenarios, the program results are presented in terms of streamfunction, isotherms, heatfunction, Nusselt number, entropy generation, and Bejan number.

5.1. Streamfunction

Case-1: regardless of the value of $\tilde{\mathrm{R}}\mathrm{i}$, the walls’ movement creates two symmetric elliptical vortices in opposite directions, controlling the flow and heat inside the enclosure.

The difference in density along the Y-direction causes the flow to move upward, where it interacts with the upper wall and then changes its direction. This process creates vortices. Additionally, the symmetrical boundary conditions along the Y-axis result in the formation of two symmetrical vortices. As presented in Figure 3, most of the flow is concentrated near the sliding walls due to the walls’ movement, making the center of the elliptical streamfunction move and located near these walls. As $\tilde{\mathrm{H}}$a increases, the core center of the elliptical streamfunction moves towards the moving isothermal walls for all $\tilde{\mathrm{R}}$i values. The shape of the streamfunctions does not change significantly as $\tilde{\mathsf{\Phi }}$ increases for various values of $\tilde{\mathrm{H}}$a, but the intensity of the flow increases due to the increase in the conduction effect. The values of the streamfunction depend strongly on the values of $\tilde{\mathrm{H}}$a and $\tilde{\mathsf{\Phi }}$, as the effect of adding nanoparticles significantly depends on the value of $\tilde{\mathrm{H}}$a and $\tilde{\mathrm{R}}\mathrm{i}$. At a Rayleigh number ($\tilde{\mathrm{R}}\mathrm{i}$) of 1, adding nanoparticles ($\tilde{\mathsf{\Phi }}$) between 0.05 and 0.1 will cause the maximum streamfunction to decrease. This is due to an increase in the buoyancy force, which in turn causes a decrease in fluid movement. Additionally, the maximum value of the streamfunction will decline as the Hartmann number ($\tilde{\mathrm{H}}$a) decreases. This is because there is an increase in the transverse movement, accompanied by a decrease in fluid movement along the Y-axis (buoyancy force). At a certain $\tilde{\mathrm{H}}$a value (non-zero), increasing $\tilde{\mathsf{\Phi }}$ from 0.1 to 0.15 will drop the maximum streamfunction. This is due to an increase in the movement of fluid in the transverse direction, coupled with a low fluid velocity ($\tilde{\mathrm{R}}\mathrm{i}$ = 1). As $\tilde{\mathrm{H}}$a increases, this increase in the streamfunction becomes more noticeable (

Table 6. Comparison of average Nusselt value between the present study and at $\tilde{P}$r = 0.71, $\tilde{R}$e = 1201.72 and (W = H).

$\tilde{\mathbf{R}}$i	Ref.	Present
1	2	2.11
10	2.875	2.92

). At $\tilde{\mathrm{R}}\mathrm{i}$ values of 10 and 100, the maximum streamfunction decreases with an increase in $\tilde{\mathrm{H}}$a for all $\tilde{\mathsf{\Phi }}$ values due to high convection effects. The impact of $\tilde{\mathrm{H}}$a is more significant at a low value $\tilde{\mathsf{\Phi }}$ and high $\tilde{\mathrm{R}}\mathrm{i}$ values. The strength of these circulations increases as the $\tilde{\mathrm{R}}\mathrm{i}$ number increases due to an increase in the buoyancy force inside the cavity. The percentage reduction in the maximum streamfunction due to an increase in $\tilde{\mathsf{\Phi }}$ at a specific $\tilde{\mathrm{H}}$a value depends mainly on the $\tilde{\mathrm{R}}\mathrm{i}$ value. It increases as $\tilde{\mathsf{\Phi }}$ increases at high $\tilde{\mathrm{R}}\mathrm{i}$ values and decreases at low $\tilde{\mathrm{R}}\mathrm{i}$ values ($\tilde{\mathrm{R}}\mathrm{i}$ = 1).

Table 7. The percentage change in maximum streamfunction, case-1.

${(\left|{\tilde{\mathsf{\psi }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }},\tilde{\mathrm{H}}\mathrm{a}}-{\left|{\tilde{\mathsf{\psi }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}})\times 100/{\left|{\tilde{\mathsf{\psi }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}}$
		$\tilde{\mathsf{\Phi }}$ = 0.05	$\tilde{\mathsf{\Phi }}$ = 0.1	$\tilde{\mathsf{\Phi }}$ = 0.15
$\tilde{\mathrm{R}}\mathrm{i}$ = 1	$\tilde{\mathrm{H}}$a = 0	−0.0377	−0.0683	−0.0919
	$\tilde{\mathrm{H}}$a = 25	−0.3975	−0.3625	0.0413
	$\tilde{\mathrm{H}}$a = 50	−0.7548	−0.6533	0.1814
	$\tilde{\mathrm{H}}$a = 75	−0.9180	−0.7908	0.2335
$\tilde{\mathrm{R}}\mathrm{i}$ = 100	$\tilde{\mathrm{H}}$a = 0	−1.9115	−3.3081	−4.3558
	$\tilde{\mathrm{H}}$a = 25	−1.8495	−2.8556	−3.1980

presents the percentage change in the maximum streamfunction for some values of $\tilde{\mathsf{\Phi }}$ and $\tilde{\mathrm{H}}$a in certain cases.

Case-2: when the vertical walls move in opposite directions, two elliptical shapes of streamfunction are created. These shapes are bounded by lines to form a ∞ shape. At high values of $\tilde{\mathrm{H}}$a = 75, the elliptical shape of streamfunction reduces and disappears regardless of the values of $\tilde{\mathrm{R}}\mathrm{i}$ and $\tilde{\mathsf{\Phi }}$. When $\tilde{\mathrm{R}}\mathrm{i}$ = 1, the shape of the streamfunction is approximately symmetrical due to the low convection effect. However, when $\tilde{\mathrm{R}}\mathrm{i}$ = 100, the streamfunction is not symmetrical due to the non-symmetrical boundary conditions resulting from the different movements of the vertical walls. At $\tilde{\mathrm{R}}\mathrm{i}$ = 1 and 10, the same effect as in Case-1 is observed, where the fluid movement in the vertical left wall reduces the fluid movement. At $\tilde{\mathrm{R}}\mathrm{i}$ = 10, the effect of $\tilde{\mathrm{H}}$a ≥ 50 at $\tilde{\mathsf{\Phi }}$ = 0.15 enhances the fluid movement due to an increase in fluid density with low $\tilde{\mathrm{R}}\mathrm{i}$.

Table 8. The percentage change in maximum streamfunction, case-2.

${(\left|{\tilde{\mathsf{\psi }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }},\tilde{\mathrm{H}}\mathrm{a}}-{\left|\tilde{\mathsf{\psi }}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}})\times 100/{\left|{\tilde{\mathsf{\psi }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}}$
		$\tilde{\mathsf{\Phi }}$ = 0.05	$\tilde{\mathsf{\Phi }}$ = 0.1	$\tilde{\mathsf{\Phi }}$ = 0.15
$\tilde{\mathrm{R}}\mathrm{i}=$ 1	$\tilde{\mathrm{H}}$a = 25	−0.3475	−0.2911	0.1095
	$\tilde{\mathrm{H}}$a = 75	−0.9306	−0.7790	0.3023
$\tilde{\mathrm{R}}\mathrm{i}=$ 10	$\tilde{\mathrm{H}}$a = 0	−0.1130	−0.2203	−0.3216
	$\tilde{\mathrm{H}}$a = 50	−0.8070	−0.7646	1.2492
$\tilde{\mathrm{R}}\mathrm{i}=$ 100	$\tilde{\mathrm{H}}$a = 0	−1.4046	−2.4334	−3.2035
	$\tilde{\mathrm{H}}$a = 75	−1.2811	−1.3874	−0.5031

shows the percentage reduction in maximum streamfunction for some values of $\tilde{\mathsf{\Phi }}$ and $\tilde{\mathrm{H}}$a.

Case-3: In this scenario, two symmetrical streamfunctions are generated within the cavity. These streamfunctions are governed by symmetrical boundary conditions around the vertical center. The shape of the streamfunctions in this case is similar to the shape of the streamfunctions in case-1. However, the directions of the streamfunctions are different (as illustrated in Figure 2). The behavior of the streamfunctions remains the same for various values of $\tilde{\mathsf{\Phi }}$ and $\tilde{\mathrm{R}}\mathrm{i}$ as in case 1, but the values of $\tilde{\mathsf{\Phi }}$ and $\tilde{\mathrm{R}}\mathrm{i}$ are different.

Table 9. The percentage change in maximum streamfunction, case-3.

${(\left|{\widetilde{\stackrel{~}{\mathsf{\psi }}}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }},\tilde{\mathrm{H}}\mathrm{a}}-{\left|{\tilde{\mathsf{\psi }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}})\times 100/{\left|{\tilde{\mathsf{\psi }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}}$
		$\tilde{\mathsf{\Phi }}$ = 0.05	$\tilde{\mathsf{\Phi }}$ = 0.1	$\tilde{\mathsf{\Phi }}$ = 0.15
$\tilde{\mathrm{R}}\mathrm{i}=$ 1	$\tilde{\mathrm{H}}$a = 0	−0.0076	−0.0105	−0.0093
	$\tilde{\mathrm{H}}$a = 75	−0.8915	−0.7368	0.3146
$\tilde{\mathrm{R}}\mathrm{i}=$ 100	$\tilde{\mathrm{H}}$a = 0	−0.3534	−0.6851	−0.9711
	$\tilde{\mathrm{H}}$a = 50	−0.5563	−0.2997	−0.6740

displays the percentage change in the maximum streamfunction due to an increase in $\tilde{\mathsf{\Phi }}$ for different values of $\tilde{\mathrm{R}}\mathrm{i}$.

5.2. Isotherms

The isotherms tend to cluster around the heat sources at the bottom wall in all cases. As the nanoparticle volume fraction increases at all values of $\tilde{\mathrm{H}}$a and $\tilde{\mathrm{R}}\mathrm{i}$, the maximum temperature decreases. This is because of increasing the effect of conduction, leading to a decrease in temperature inside the cavity. When $\tilde{\mathrm{H}}$a increases, the maximum temperature for all values of $\tilde{\mathsf{\Phi }}$ and $\tilde{\mathrm{R}}\mathrm{i}$ increases too. This happens because there is a decrease in heat transfer from the heat sources at the bottom of the cavity to the upper region and an increase in heat transfer in the transverse direction. The maximum temperature also increases as $\tilde{\mathrm{R}}\mathrm{i}$ increases.

Case-1: Figure 4 shows the isotherms for different cases. The dense isotherms are located near the heat sources and at the cavity center. The symmetrical isotherms about the vertical line at the cavity center are due to the symmetrical boundary conditions in the vertical direction. The effect of $\tilde{\mathsf{\Phi }}$ on the isotherms’ shape decreases as $\tilde{\mathrm{H}}$a increases (as shown in Figure 4). However, the impact of $\tilde{\mathrm{H}}$a on the isotherms is noticeable for all values of $\tilde{\mathrm{R}}\mathrm{i}$. An increase in $\tilde{\mathrm{H}}$a tries to spread the heat (isotherms) in the transverse direction, thereby reducing the concentration of isotherms in the center region. At higher values of $\tilde{\mathrm{R}}\mathrm{i}$, the $\tilde{\mathrm{H}}$a effect attempts to create isotherms in a parabolic shape around the base wall, which results in a reduced temperature distribution in the transverse direction due to increased heat transfer in that direction. As $\tilde{\mathsf{\Phi }}$ increases for all values of $\tilde{\mathrm{H}}$a and $\tilde{\mathrm{R}}\mathrm{i}$, the percentage of reduction in maximum temperature inside the cavity also increases. The effect of $\tilde{\mathsf{\Phi }}$ becomes more remarkable as $\tilde{\mathrm{R}}\mathrm{i}$ increases. At lower $\tilde{\mathsf{\Phi }}$ values, the percentage of decrease in maximum temperature value due to adding more nanoparticles is greater than at higher $\tilde{\mathsf{\Phi }}$ values, as shown in the following equation:

$${\mathrm{R}\mathrm{D}=(\left|{\tilde{\mathsf{\psi }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}1}-{\left|{\tilde{\mathsf{\psi }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}=0})/{(\left|{\tilde{\mathsf{\psi }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}2}-{\left|{\tilde{\mathsf{\psi }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}=0})$$

For example, at $\tilde{\mathsf{\Phi }}$₁ = 0.05 and $\tilde{\mathsf{\Phi }}$₂ = 0.1, with $\tilde{\mathrm{R}}\mathrm{i}$ = 100 and $\tilde{\mathrm{H}}$a = 25, RD = 1.93.

Table 10. The percentage change in maximum isotherms, case-1.

${(\left|{\tilde{\mathsf{\theta }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }},\tilde{\mathrm{H}}\mathrm{a}}-{\left|{\tilde{\mathsf{\theta }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}})\times 100/{\left|{\tilde{\mathsf{\theta }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}}$
		$\tilde{\mathsf{\Phi }}$ = 0.05	$\tilde{\mathsf{\Phi }}$ = 0.1	$\tilde{\mathsf{\Phi }}$ = 0.15
$\tilde{\mathrm{R}}\mathrm{i}$ = 1	$\tilde{\mathrm{H}}$a = 25	−7.0116	−13.415	−19.2531
	$\tilde{\mathrm{H}}$a = 75	−6.1149	−12.1202	−18.0556
$\tilde{\mathrm{R}}\mathrm{i}$ = 10	$\tilde{\mathrm{H}}$a = 0	−6.9715	−13.613	−19.96
	$\tilde{\mathrm{H}}$a = 75	−8.9113	−17.393	−25.285
$\tilde{\mathrm{R}}\mathrm{i}$ = 100	$\tilde{\mathrm{H}}$a = 25	−9.823	−18.902	−27.109
	$\tilde{\mathrm{H}}$a = 50	−10.602	−20.017	−28.379

shows the percentage change in maximum isotherms due to an increase in $\tilde{\mathsf{\Phi }}$ for different values of $\tilde{\mathrm{R}}\mathrm{i}$ and $\tilde{\mathrm{H}}$a.

Case-2: The same effect of case-1 is noted here for all values of $\tilde{\mathrm{R}}\mathrm{i}$ and $\tilde{\mathsf{\Phi }}$. However, when $\tilde{\mathrm{R}}\mathrm{i}$ is large (i.e., $\tilde{\mathrm{R}}\mathrm{i}$ = 100), increasing $\tilde{\mathrm{H}}$a from 0 to 75 does not have a noticeable impact on the isotherm values. The isotherm shapes are non-symmetrical due to the non-symmetrical boundary conditions with Y-axis as shown in Figure 4. As mentioned, the movement of the right-moving wall in the negative direction helps to spread the isotherms in the Y-direction more than the left-moving wall. The non-symmetrical isotherms are more remarkable at low values of $\tilde{\mathrm{R}}\mathrm{i}$. At $\tilde{\mathrm{R}}\mathrm{i}$ = 100 and $\tilde{\mathrm{H}}$a = 75, the isotherms tend to be symmetrical due to increased fluid movement and $\tilde{\mathrm{H}}$a tends to distribute the heat in the transverse direction. The effect of $\tilde{\mathrm{H}}$a on the isotherm shape is significant for all values of $\tilde{\mathrm{R}}\mathrm{i}$ and $\tilde{\mathsf{\Phi }}$ (Figure 4). At $\tilde{\mathrm{R}}\mathrm{i}$ = 1, the isotherms are concentrated near the right wall and the heat sources, resulting in an unsymmetrical shape. However, a symmetrical shape of the isotherms is observed at high values of $\tilde{\mathrm{H}}$a = 75 and $\tilde{\mathrm{R}}\mathrm{i}$ = 100.

Table 11. The percentage change in maximum isotherms, case-2.

${(\left|{\tilde{\mathsf{\theta }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }},\tilde{\mathrm{H}}\mathrm{a}}-{\left|{\tilde{\mathsf{\theta }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}})\times 100/{\left|{\tilde{\mathsf{\theta }}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}}$
		$\tilde{\mathsf{\Phi }}$ = 0.05	$\tilde{\mathsf{\Phi }}$ = 0.1	$\tilde{\mathsf{\Phi }}$ = 0.15
$\tilde{\mathrm{R}}\mathrm{i}=$ 1	$\tilde{\mathrm{H}}$a = 25	−6.3891	−12.227	−17.634
	$\tilde{\mathrm{H}}$a = 75	−5.5019	−11.008	−16.57
$\tilde{\mathrm{R}}\mathrm{i}=$ 10	$\tilde{\mathrm{H}}$a = 0	−7.8314	−15.135	−22.155
	$\tilde{\mathrm{H}}$a = 75	−9.1976	−17.813	−25.79
$\tilde{\mathrm{R}}\mathrm{i}=$ 100	$\tilde{\mathrm{H}}$a = 0	−11.144	−20.828	−29.298
	$\tilde{\mathrm{H}}$a = 50	−11.314	−21.105	−29.637

points out the percentage change in maximum isotherms for some cases due to an increase in $\tilde{\mathsf{\Phi }}$ and $\tilde{\mathrm{H}}$a.

Case-3: The same effect as case-1 is spotted here. As the convection effect ($\tilde{\mathrm{R}}\mathrm{i}$) increases, the reduction in isotherms percentage also increases. The denser isotherms tend to be located around the lower heat sources. The $\tilde{\mathrm{H}}$a effect tries to spread the isotherms in the transverse direction. The impact of $\tilde{\mathrm{H}}$a on isotherms is significantly more noticeable than the effect of $\tilde{\mathsf{\Phi }}$, particularly at low $\tilde{\mathrm{R}}\mathrm{i}$ (as shown in Figure 4). Due to the symmetric boundary conditions around the Y-axis, the shape of isotherms is symmetric. The percentage change in the maximum temperature for some cases is tabulated in

Table 12. The percentage change in maximum isotherms, case-3.

${(\left|{\mathsf{\theta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }},\tilde{\mathrm{H}}\mathrm{a}}-{\left|{\mathsf{\theta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}})\times 100/{\left|{\mathsf{\theta }}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}}$
		$\tilde{\mathsf{\Phi }}$ = 0.05	$\tilde{\mathsf{\Phi }}$ = 0.1	$\tilde{\mathsf{\Phi }}$ = 0.15
$\tilde{\mathrm{R}}\mathrm{i}=$ 1	$\tilde{\mathrm{H}}$a = 0	−6.8709	−13.0376	−18.6837
	$\tilde{\mathrm{H}}$a = 75	−5.6269	−11.1926	−16.7592
$\tilde{\mathrm{R}}\mathrm{i}=$ 10	$\tilde{\mathrm{H}}$a = 25	−7.2647	−14.1153	−20.6392
	$\tilde{\mathrm{H}}$a = 75	−8.7446	−16.9719	−24.6290
$\tilde{\mathrm{R}}\mathrm{i}=$ 100	$\tilde{\mathrm{H}}$a = 0	−10.4928	−19.6325	−27.6950
	$\tilde{\mathrm{H}}$a = 50	−10.8242	−20.2358	−28.4800

.

5.3. Heatfunction

Heat transfer within a cavity occurs through two main methods: conduction and convection (Figure 5). For any Da and at low values of $\tilde{\mathrm{R}}\mathrm{i}$ and Gr, conduction dominates. However, at high values of $\tilde{\mathrm{R}}\mathrm{i}$ and Gr, the convection becomes the dominant. Recently, heatlines have been considered a main function to illustrate the heat transfer within the cavity [30], and still, most of the CFD-ready packages don’t include it. As a fact, the heat transfers from the hot surface towards the end on the cold surface, so the heatlines will follow these rules. Heatlines are perpendicular to isothermal walls and parallel to insulated walls. The maximum value of heatlines directly depends on the maximum value of the isotherms. The positive value of the heat function represents anti-clockwise heat flow, while the negative value represents clockwise heat flow. Thermal boundary conditions have a greater impact on heat transfer than velocity gradient; therefore, dense heatlines are located near heat sources.

Case-1: At low values of $\tilde{\mathrm{R}}\mathrm{i}$, where $\tilde{\mathrm{R}}\mathrm{i}$ = 1; heat transfer primarily occurs through conduction from the heat sources to the lower portions of the vertical isothermal moving wall, which is more than 90% of the total heat. The circular heatlines in the upper portion of the enclosure have an opposite direction to the heatlines in the lower portion of the same half of the enclosure.

For example, when $\tilde{\mathrm{H}}$a = 0 and $\tilde{\mathsf{\Phi }}$ = 0, the heatlines transfer to 0–25% of the vertical wall length (lower portion), and this percentage increases to 50% at $\tilde{\mathrm{H}}$a = 75 and $\tilde{\mathsf{\Phi }}$ = 0.15. An increase in $\tilde{\mathrm{H}}$a and $\tilde{\mathsf{\Phi }}$ causes a decrease in the circular heatlines in the upper portion. When $\tilde{\mathrm{R}}\mathrm{i}$ = 10, the circular heatlines appear, and the straight heatlines transfer directly to 30% of the vertical walls’ length at $\tilde{\mathrm{H}}$a = 0 and $\tilde{\mathsf{\Phi }}$ = 0. The circular heatlines vanish at $\tilde{\mathrm{H}}$a = 75 and $\tilde{\mathsf{\Phi }}$ = 0.15, and only straight heatlines demonstrate the heat transfer. The magnetic field tends to distribute the heat uniformly in the transverse direction. At high values of $\tilde{\mathrm{R}}\mathrm{i}$, $\tilde{\mathrm{R}}\mathrm{i}$ = 100, circular heatlines disappear. The heatlines try to fill the entire enclosure as $\tilde{\mathrm{H}}$a and $\tilde{\mathsf{\Phi }}$ increase due to the increase in fluid movement. The effect of an increase in $\tilde{\mathsf{\Phi }}$ on the heatlines at high $\tilde{\mathrm{H}}$a = 75 is not noticeable, as shown in Figure 5.

Case-2: The asymmetrical heatlines, in this case, are caused by nonsymmetrical boundary conditions with respect to Y-direction. The heat distribution is enhanced along the moving wall on the right side, while the left side wall receives less heat in the bottom portion. At $\tilde{\mathrm{R}}\mathrm{i}$ = 1, circular heatlines appear due to low fluid movement, which reduces as $\tilde{\mathrm{H}}$a and $\tilde{\mathsf{\Phi }}$ increase. The circular heatlines vanish at $\tilde{\mathrm{R}}\mathrm{i}$ = 100. The left vertical wall’s movement creates circular heatlines near the lower part of the wall, which increases the path of the non-circular heatlines from the heat sources to the cold left wall by changing the heatlines’ path around these towards the left wall. These circular heatlines vanish as $\tilde{\mathrm{R}}\mathrm{i}$ and $\tilde{\mathrm{H}}$a increase, and the path of the heatlines becomes shorter and easier, as shown in Figure 5.

Case-3: The heatlines in this scenario are symmetric due to the symmetrical boundary conditions. When fluid movement is low ($\tilde{\mathrm{R}}\mathrm{i}$ = 1) (Figure 5), circular heatlines can be observed in the lower portions of the vertical walls. This is because the vertical wall proceeds in the adverse direction of the fluid movement. As mentioned in Case-2 for the left wall, these circular heatlines decrease as $\tilde{\mathrm{H}}$a and $\tilde{\mathsf{\Phi }}$ increase. The circular heatlines vanish when $\tilde{\mathrm{R}}\mathrm{i}$ is high, and the heatlines’ path reduces as $\tilde{\mathrm{R}}\mathrm{i}$ increases.

5.4. Entropy Generation

The contours of total entropy generation are shown in Figure 6 for different studied scenarios. The entropy generation occurs due to the difference in temperature, velocity, and magnetic field. However, the entropy generated due to temperature difference is greater than that generated due to velocity and magnetic field. As a result, the dense entropy distribution contour is located near the heat sources (due to high temperature and high thermal boundary layers) and the moving walls (due to the velocity effect and high hydrodynamic boundary layer).

Case-1: the contours of the total entropy generation seem symmetrical because of symmetrical boundary conditions with the Y-axis. The intensity of entropy generation is generally located near the vertical walls and the heat sources, especially at low $\tilde{\mathrm{R}}\mathrm{i}$ and $\tilde{\mathrm{H}}$a. The effect of $\tilde{\mathrm{H}}$a tends to distribute the entropy in the transverse axis due to increased heat transfer. At high values of $\tilde{\mathrm{R}}\mathrm{i}$ = 100 and $\tilde{\mathrm{H}}$a = 75, the entropy generation lines move horizontally (parallel to the base wall) in the middle of the cavity (Figure 6). There is no noticeable remarkable entropy generation near the upper wall due to zero wall velocity and thermally insulated conditions.

Case-2: the entropy generation appears nonsymmetrical due to nonsymmetrical boundary conditions. However, when the values of $\tilde{\mathrm{R}}\mathrm{i}$ and $\tilde{\mathrm{H}}$a are increased to 100, the entropy generation change and becomes symmetrical. The intensity of entropy generation contours is denser near the right wall than near the left wall. This is due to the movement of the right wall, which distributes heat flow along the wall (Figure 6).

Case-3: the entropy generation contour is dense near the base wall and the lower part of the vertical walls. This is because the movement of the vertical walls is opposite to the fluid movement, resulting in the dense entropy generation near the base wall (Figure 6).

5.5. Average Nusselt Number at Hot Source

Figure 7 presents the average $\tilde{\mathrm{N}}\mathrm{u}$ number along the heat sources for all cases. It can be noted that the average $\tilde{\mathrm{N}}\mathrm{u}$ number decreases as the $\tilde{\mathrm{R}}\mathrm{i}$ number increases. This is because the temperature near the heat sources increases with an increase in $\tilde{\mathrm{R}}\mathrm{i}$, resulting in a decrease in $\tilde{\mathrm{N}}\mathrm{u}$ (as given in Equation (9)). At lower values of $\tilde{\mathrm{R}}\mathrm{i}$ number, a higher $\tilde{\mathrm{N}}\mathrm{u}$ number is observed. The average $\tilde{\mathrm{N}}\mathrm{u}$ number also decreases as $\tilde{\mathrm{H}}$a increases. This is due to the rise in heat dissipation in the transverse direction, leading to an increase in the average temperature above the heat sources. It is clear that introducing the nanoparticles into the porous medium leads to an increase in the average Nusselt number. The rate of increment depends on the value of $\tilde{\mathrm{R}}\mathrm{i}$. At $\tilde{\mathrm{R}}\mathrm{i}$ = 1, the effect of increasing $\tilde{\mathsf{\Phi }}$ on the average $\tilde{\mathrm{N}}\mathrm{u}$ number reduces as $\tilde{\mathrm{H}}$a increases due to low fluid movement (low convection). However, for high values of $\tilde{\mathrm{R}}\mathrm{i}$ = 10 and 100, the effect of increasing $\tilde{\mathsf{\Phi }}$ on the average $\tilde{\mathrm{N}}\mathrm{u}$ number is roughly the same for all values of $\tilde{\mathrm{H}}$a. The percentage increase in the average $\tilde{\mathrm{N}}\mathrm{u}$ number due to added $\tilde{\mathsf{\Phi }}$ depends on $\tilde{\mathrm{R}}\mathrm{i}$ and $\tilde{\mathrm{H}}$a. For example, at $\tilde{\mathrm{R}}\mathrm{i}$ = 1, $\tilde{\mathrm{H}}$a = 0, and $\tilde{\mathsf{\Phi }}$ = 0.15, the average $\tilde{\mathrm{N}}\mathrm{u}$ number increases by approximately 18.45% for case-1, 19.74% for case-2-R, 21.04% for case-2-L and 21.48% for case-3.

Table 13. The percentage change in $\tilde{\mathrm{N}}\mathrm{u}{(\left|{\tilde{\mathrm{N}}\mathrm{u}}_{\mathrm{h}}\right|}_{\tilde{\mathsf{\Phi }},\tilde{\mathrm{H}}\mathrm{a}}-{\left|{\tilde{\mathrm{N}}\mathrm{u}}_{\mathrm{h}}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}})\times 100/{\left|{\tilde{\mathrm{N}}\mathrm{u}}_{\mathrm{h}}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}}$.

		$\tilde{\mathsf{\Phi }}$ = 0.05	$\tilde{\mathsf{\Phi }}$ = 0.1	$\tilde{\mathsf{\Phi }}$ = 0.15		$\tilde{\mathsf{\Phi }}$ = 0.05	$\tilde{\mathsf{\Phi }}$ = 0.1	$\tilde{\mathsf{\Phi }}$ = 0.15
Case-1					Case-2—R
$\tilde{\mathrm{R}}\mathrm{i}=$ 1	$\tilde{\mathrm{H}}$a = 0	6.0314	12.1656	18.4543	$\tilde{\mathrm{H}}$a = 50	5.7124	11.5635	17.6364
	$\tilde{\mathrm{H}}$a = 75	3.9877	8.7239	14.3719	$\tilde{\mathrm{H}}$a = 75	4.8852	10.0608	15.6804
$\tilde{\mathrm{R}}\mathrm{i}=$ 10	$\tilde{\mathrm{H}}$a = 25	5.6283	12.2966	20.1092	$\tilde{\mathrm{H}}$a = 0	5.6416	11.7158	18.4518
	$\tilde{\mathrm{H}}$a = 75	8.2621	18.1148	29.5096	$\tilde{\mathrm{H}}$a = 75	6.2869	13.9295	22.9128
$\tilde{\mathrm{R}}\mathrm{i}=$ 100	$\tilde{\mathrm{H}}$a = 0	8.7163	19.1010	31.0973	$\tilde{\mathrm{H}}$a = 0	8.4530	18.3633	29.6857
	$\tilde{\mathrm{H}}$a = 75	11.6315	24.5817	38.8383	$\tilde{\mathrm{H}}$a = 25	8.9660	19.3598	31.1549
Case-2—L					Case-3
$\tilde{\mathrm{R}}\mathrm{i}=$ 1	$\tilde{\mathrm{H}}$a = 25	6.3846	12.9623	19.8239	$\tilde{\mathrm{H}}$a = 0	6.8837	14.0251	21.4829
	$\tilde{\mathrm{H}}$a = 75	5.1647	11.0236	17.7445	$\tilde{\mathrm{H}}$a = 25	6.4682	13.2405	20.3778
$\tilde{\mathrm{R}}\mathrm{i}=$ 10	$\tilde{\mathrm{H}}$a = 0	8.1944	17.2683	27.6383	$\tilde{\mathrm{H}}$a = 50	4.6370	13.8467	29.4897
	$\tilde{\mathrm{H}}$a = 50	8.1840	18.4074	34.2044	$\tilde{\mathrm{H}}$a = 75	9.6157	20.5602	32.9092
$\tilde{\mathrm{R}}\mathrm{i}=$ 100	$\tilde{\mathrm{H}}$a = 0	12.6280	26.5325	41.8510	$\tilde{\mathrm{H}}$a = 25	11.9096	24.9062	39.1550
	$\tilde{\mathrm{H}}$a = 75	12.7955	26.8032	42.1463	$\tilde{\mathrm{H}}$a = 75	12.2510	25.6390	40.3154

shows the percentage change in $\tilde{\mathrm{N}}\mathrm{u}$ number versus $\tilde{\mathrm{H}}$a and $\tilde{\mathsf{\Phi }}$.

5.6. Local Nusselt Number

The local Nusselt number is proportional inversely to dimensionless temperature along the heat sources. Figure 8 shows the local $\tilde{\mathrm{N}}\mathrm{u}$ number for different cases. The local Nusselt number increases as $\tilde{\mathrm{H}}$a increases, while it reduces as $\tilde{\mathsf{\Phi }}$ and $\tilde{\mathrm{R}}\mathrm{i}$ increase due to the same effect that been addressed in average Nusselt number. At high values of $\tilde{\mathrm{R}}\mathrm{i}$ = 100, the effect of an increase in $\tilde{\mathsf{\Phi }}$ on the local Nusselt number diminishes.

Case-1: the minimum local Nusselt number (indicating maximum temperature) along the heat flux is located near the end of the heat source towards the center of the enclosure. The local Nusselt sketch in Figure 8 is symmetrical due to symmetrical boundary conditions.

Case-2: the local Nusselt number values along the heat sources are not symmetrical because of non-symmetrical boundary conditions. The minimum local Nusselt number over the right heat source is located near the end of the heat source towards the center of the cavity. This is because the right wall movement enhances the heat transfer. On the other hand, the minimum local heat transfer over the left heat source moves from the right end toward the left end of the heat source as $\tilde{\mathrm{R}}\mathrm{i}$ increases. This is because the left wall movement does not enhance the heat transfer.

Case-3: the uniform local Nusselt number distribution above the heat sources is symmetrical due to symmetrical boundary conditions. The minimum local Nusselt number over the heat source is situated near the end of the heat source toward the vertical walls. This location moves towards the center of the enclosure as $\tilde{\mathrm{R}}\mathrm{i}$ rises.

5.7. Bejan Number

The variation of $\tilde{\mathrm{B}}$e number versus $\tilde{\mathrm{R}}\mathrm{i}$, $\tilde{\mathsf{\Phi }}$, and $\tilde{\mathrm{H}}$a is plotted in Figure 9 for different cases. In all cases, the $\tilde{\mathrm{B}}$e number decreases as $\tilde{\mathsf{\Phi }}$ and $\tilde{\mathrm{H}}$a rise. This is because the system becomes more stable as the buoyancy force decreases. While $\tilde{\mathrm{H}}$a increases, the entropy generation due to the magnetic effect also rises, leading to an increase in total entropy generation and, ultimately, a decrease in the $\tilde{\mathrm{B}}$e number. The effect of $\tilde{\mathsf{\Phi }}$ on the $\tilde{\mathrm{B}}$e number is most remarkable at high values of $\tilde{\mathrm{R}}\mathrm{i}$ and $\tilde{\mathrm{H}}$a for all cases. The percentage reduction in $\tilde{\mathrm{B}}$e number also increases as $\tilde{\mathrm{H}}$a increases for all values of $\tilde{\mathsf{\Phi }}$. Based on the data, the minimum $\tilde{\mathrm{B}}$e number (minimal entropy generation) is recorded at high $\tilde{\mathrm{H}}$a and $\tilde{\mathsf{\Phi }}$ for all values of $\tilde{\mathrm{R}}\mathrm{i}$. In Figure 8, the $\tilde{\mathrm{B}}$e numbers increase as $\tilde{\mathrm{R}}\mathrm{i}$ increases, this is because the entropy generation due to heat transfer increases with $\tilde{\mathrm{R}}\mathrm{i}$. The direction of the wall’s movement (positive or negative) has little impact on the $\tilde{\mathrm{B}}$e number.

Table 14. The percentage change in $\tilde{\mathrm{B}}$e number ${(\left|\tilde{\mathrm{B}}\mathrm{e}\right|}_{\tilde{\mathsf{\Phi }},\tilde{\mathrm{H}}\mathrm{a}}-{\left|\tilde{\mathrm{B}}\mathrm{e}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}})\times 100/{\left|\tilde{\mathrm{B}}\mathrm{e}\right|}_{\tilde{\mathsf{\Phi }}=0,\tilde{\mathrm{H}}\mathrm{a}}$.

	Case-1				Case-2				Case-3
	$\tilde{\mathbf{H}}$a	$\tilde{\mathsf{\Phi }}=$ 0.05	$\tilde{\mathsf{\Phi }}=$ 0.1	$\tilde{\mathsf{\Phi }}=$ 0.15	$\tilde{\mathbf{H}}$a	$\tilde{\mathsf{\Phi }}=$ 0.05	$\tilde{\mathsf{\Phi }}=$ 0.1	$\tilde{\mathsf{\Phi }}=$ 0.15	$\tilde{\mathbf{H}}$a	$\tilde{\mathsf{\Phi }}=$ 0.05	$\tilde{\mathsf{\Phi }}=$ 0.1	$\tilde{\mathsf{\Phi }}=$ 0.15
$\tilde{\mathrm{R}}\mathrm{i}=$ 1	0	−0.8992	−1.8001	−2.7056	25	−1.2219	−2.3261	−3.3334	0	−0.9124	−1.8318	−2.7604
	75	−1.5221	−2.8467	−4.0660	50	−1.5953	−2.9474	−4.0988	25	−1.2233	−2.3478	−3.3861
$\tilde{\mathrm{R}}\mathrm{i}=$ 10	0	−0.5726	−1.1814	−1.8679	25	−0.8619	−1.7249	−2.6050	50	−0.5618	−1.9784	−4.6275
	50	−0.2965	−1.6199	−5.0097	75	−1.8668	−3.6994	−5.4666	75	−2.0629	−4.0507	−5.9542
$\tilde{\mathrm{R}}\mathrm{i}=$ 100	25	−0.9887	−2.0359	−3.1259	0	−0.7114	−1.4946	−2.3495	0	−0.7597	−1.5718	−2.4445
	75	−2.2338	−4.3877	−6.4092	25	−1.0315	−2.0949	−3.1847	75	−2.3306	−4.5329	−6.6013

shows the percentage change in $\tilde{\mathrm{B}}$e number for some cases against $\tilde{\mathrm{H}}$a and $\tilde{\mathsf{\Phi }}$.

6. Conclusions

This study is a continuous study that investigates the influence of magnetic fields on heat transfer, fluid flow, and entropy generation in a corrugated lid-driven enclosure with a constant heat flux source. The study employs the finite element method to discrete the governor equations. It presents the results in graphical and tabular formats for various parameters and discusses them. The numerical results describe heat via isotherms and heatlines, fluid via streamlines, and irreversibility via entropy lines and Bejan number. Based on the study, the following conclusions can be drawn:

• Case-3 is the most efficient scenario among others with similar physical parameters.
• The maximum streamfunction value decreases as $\tilde{\mathrm{H}}$a and $\tilde{\mathsf{\Phi }}$ increase, except for low $\tilde{\mathrm{R}}\mathrm{i}$ where the maximum streamfunction increases as $\tilde{\mathsf{\Phi }}$ increases ($\tilde{\mathsf{\Phi }}$ = 0.15).
• The increase in nanoparticle volume fraction will reduce the maximum temperature, and this reduction increases as $\tilde{\mathrm{H}}$a and $\tilde{\mathrm{R}}\mathrm{i}$ increase.
• At low $\tilde{\mathrm{R}}\mathrm{i}$, circular heatlines appear at the top portion near the vertical wall, which has a negative sign (V = −1), and at the lower portion near the vertical wall, which has a positive sign (V = +1). These circular lines disappear at high $\tilde{\mathrm{R}}\mathrm{i}$.
• The dense entropy generation is located in the center and near the heat sources. At high values of $\tilde{\mathrm{R}}\mathrm{i}$ = 100 and $\tilde{\mathrm{H}}$a, the entropy generation lines are distributed horizontally (parallel to the base wall) in the middle of the cavity.
• For all cases, the average $\tilde{\mathrm{N}}\mathrm{u}$ number decreases as $\tilde{\mathrm{R}}\mathrm{i}$ and $\tilde{\mathrm{H}}$a increase. The increase in the nanoparticle volume fraction inside the porous medium causes an increase in the average $\tilde{\mathrm{N}}\mathrm{u}$ number and the rate of increment depends on the value of the $\tilde{\mathrm{R}}\mathrm{i}$.
• The local Nusselt number increases as $\tilde{\mathsf{\Phi }}$ and $\tilde{\mathrm{R}}\mathrm{i}$ decrease, and $\tilde{\mathrm{H}}$a increases. The effect of $\tilde{\mathsf{\Phi }}$ on the local Nusselt number above the heat sources decreases at a high value of $\tilde{\mathrm{R}}\mathrm{i}$ = 100.
• The $\tilde{\mathrm{B}}$e number decreases as $\tilde{\mathsf{\Phi }}$ and $\tilde{\mathrm{H}}$a increase. The effect of $\tilde{\mathsf{\Phi }}$ on the $\tilde{\mathrm{B}}$e increases significantly at high values of $\tilde{\mathrm{R}}\mathrm{i}$ and $\tilde{\mathrm{H}}$a for all cases. The direction of the vertical wall movement (positive or negative) does not noticeably affect the $\tilde{\mathrm{B}}$e number. The variation of the average $\tilde{\mathrm{N}}\mathrm{u}$ Number and $\tilde{\mathrm{B}}$e Number is linear with $\tilde{\mathrm{H}}$a for all volume fractions of nanoparticles.