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Mathematical Entropy Analysis of Natural Convection of MWCNT—Fe_{3}O_{4}/Water Hybrid Nanofluid with Parallel Magnetic Field via Galerkin Finite Element Process

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## Abstract

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^{®}software computer package. According to the results, there is a difference in the occurrence of the magnetic parameter and an increase in heat transmission when the right wall is recessed inward. The heat transmission is also significantly reduced when the right wall is exposed to the outside. The number of Nusselt grows directly proportional to the number of nanofluids in the environment. In contrast, all porous media with low Darcy and Hartmann numbers, high porosity, and low volume fraction have high Nusselt numbers. It is found that double streamlines for the hot side and single cooling for Darcy, Rayleigh, and Hartmann numbers. A cold isotherm at various physical parameters is needed in the top cavity. Rayleigh’s number and a solid volume fraction raise Darcy’s number, increasing heat transmission inside the cavity and thermal entropy determines entropy components.

## 1. Introduction

_{2}O

_{3}–Cu water-based hybrid nanofluid mixed convection in a wave-shaped canal with a spherical cylinder was inspected by Hussain et al. [13]. The Galerkin finite element method was adopted to discretize a two-dimensional system of partial differential equations. In fluid flow and heat transfer studies, the Reynolds number is crucial. The fluid’s speed changes when the Reynolds number is increased. The velocity rises with each hump, and the local Nusselt number on the wavy lowermost wall is immensely enriched. By Fares et al., triangular cavities were used to study the convection of a TiO

_{2}–SiO

_{2}/water-based hybrid nanofluid [14]. When the Richardson number $Ri$ is more significant than 1, the importance of mixed convection flow behavior is demonstrated.

_{2}O

_{3}/Water hybrid nanofluid were studied numerically by Mebarek-Oudina et al. [19]. The results discovered that increasing the volume percentage of the hybrid nano-liquid is essential when free convection is poor. It is due to the internal heat generation rate increases; conductive and convective thermal transfer are both enhanced. A spongy compartment occupied with Fe

_{3}O

_{4}-CuO hybrid nanofluid was the setting for Qin’s [20] investigation of the influence of the Lorentz force. The MHD and porous medium source terms were combined, and the vorticity formulation was used to simplify the equations. There is evidence that MHD leads to less convection and more noticeable conduction, which reduces heat transfer. As a result, transporting nanomaterials becomes more accessible and faster when greater buoyant force is taken into account. Due to an increase in convection, the isothermal fluctuation rises. There is a rise in entropy-generating levels as the Rayleigh number upsurges. Due to the reduction in nanoparticle velocity, the Hartmann number escalations while the temperature gradient diminutions.

_{2}O

_{3}–Cu/H

_{2}O nano-liquid rectangular cavity with various configurations of a grooved directing dense were studied by Tayebi and Chamkha [22]. With the finite volume discretization approach, governing equations were numerically solved with nanoparticle Brownian motion incorporated in the computations. Abu-Libdeh et al. [23] implemented the finite element approach to solve the heat transference response of Ag-MgO H

_{2}O-based hybrid nanoliquids in a spongy chamber under continuous magnetic stresses. The results suggest that the magnetic field may be utilized as a heat transference regulator, as the heat transition rate reduces while the Ha number increases. Thermal conductance was examined by Brahimi et al. [24] in an insertion dominated with an Ag-MgO water-based nano liquid beneath a constant magnetic field. Because the box’s construction sources the flow to ramble above the cliff bars, this exceptional liquid undertaking decelerates the flowing at the surrounding, allowing the molecules to transmit heat more effectively. In addition, the enclosure’s restrictions are based on fluidity. Every aspect of this undertaking had a considerable impact. The fluidity of the meandering box is enhanced by increasing $Ra$, $Da$, or porosity, whereas nanoparticle volume fraction and $Ha$ tend to resist it.

_{3}O

_{4}/water hybrid nanofluid to explore MHD-free convective utilizing the dual MRT lattice Boltzmann technique. It is thought that MWCNTs, Fe

_{3}O

_{4}nano-solids, and water are all Newtonian working fluids. The Rayleigh number was increased by adding MWCNT and Fe

_{3}O

_{4}nanoparticles, whereas increasing the magnetic field had the opposite effect, as demonstrated by the results. Several nanofluids, including MWCNT-Fe

_{3}O

_{4}hybrid nanofluids, were compared in a study by Saghir et al. [29]. The hybrid performed better in all temperatures, according to the data. Other nanofluids were also examined, and the MWCNT-Fe

_{3}O

_{4}combination was shown to perform better than the others. The inspiration of magnetics on the performance of MWCNT/Fe

_{3}O

_{4}hybrid nanofluid by Shin et al. [30] was studied in order to advance photo-thermal energy alteration presentation. Investigators are using an MWCNT/Fe

_{3}O

_{4}hybrid nanofluid with varying magnetic strengths to experiment with the impact of external magnetic forces on photo-thermal energy conversion proficiency. Thermal conductivity and photo-thermal energy alteration were studied using Fe

_{3}O

_{4}and MWCNT nanofluids [31]. Zafar et al. employed a multilayer perceptron coupled to an artificial neural network to investigate the thermophysical characteristics and stability of H

_{2}O-based hybrid nanofluids covered with multi-walled carbon nanotubes. It was able to accurately replicate the circumstances of the experiments, regardless of the temperature or concentration. MWCNT/Fe

_{3}O

_{4}nanofluids were tested as a plate heat exchanger coolant by Alklaibi et al. [32]. Several parameters showed that MWCNT + Fe

_{3}O

_{4}/water nanofluids improved plate heat exchanger performance. Increases in coolant flow velocity and nanoparticle concentration both contribute to amplification.

_{3}O

_{4}/H

_{2}O). The effect of various control variables such as $Ra$, $Da$, $Ha$, $\varphi ,$ and porous amounts on the flowing characteristics and heat transition was investigated. The equations for energy, motion, and continuity were solved using the application of the COMSOL Multiphysics

^{®}software computer package.

## 2. Problem Description

- ▪ The flow is incompressible, and NF is power-law non-Newtonian.
- ▪ Relative movement between fluid and Fe
_{3}O_{4}, MWCNT particles is zero, and thermal equilibrium exists between them. - ▪ The temperature and velocity fields are laminar, steady state, and 2D.
- ▪ The effects of radiation and viscous dissipation are neglected.

_{3}O

_{4}/H

_{2}O) nanofluid fills the inside of the domain. Additionally, it is assumed that the uniform magnetic field $\left(B={B}_{x}{e}_{x}+{B}_{y}{e}_{y}\right)$ of constant magnitude $(B=\sqrt{{B}_{x}^{2}+{B}_{y}^{2}})$ is applied, where ${\overrightarrow{e}}_{x}$ and ${\overrightarrow{e}}_{y}$ are unit vectors in the Cartesian coordinate system. The electric current $J$ and the electromagnetic force $F$ are defined by $J=\sigma (\overrightarrow{V}\times \overrightarrow{B})$ and $F=\sigma (\overrightarrow{V}\times \overrightarrow{B})\times \overrightarrow{B}$, respectively. Figure 1 illustrates a graphical issue with the enclosure’s fundamental geometric characteristics. Semi-cylinders are adiabatic, with the left wall being hot and the right wall being cool. Table 1 demonstrates the main thermos-physical possessions of the essential liquid (H

_{2}O) and the nano molecules.

## 3. Mathematical Model

- The hot wall:

- The cold wall:

- The insolated walls:

#### Non-Dimensional Entropy Generation

## 4. Authentication and Grid Independence Evaluation

## 5. Results and Discussion

_{3}O

_{4}) are illustrated in Table 1. The fallouts are classified into the subsequent four subdivisions. The values of all parameters used are based on previous research [40,43].

#### 5.1. Effect of Rayleigh Number

_{3}O

_{4}/water are shown on streamlines and isotherm. In Rayleigh’s initial values. Laminar currents flashed and faded on the hot side end to end with the upstream currents. The latter drew thick lines at the first bend, accompanied by a moderate field at the lower end. It is noticeable when the moment of the Rayleigh number increases, a fluctuation in the stream state. This phenomenon leads to the occurrence of the middle contour. In addition, it is seen that the currents are driven toward the scorching end of the solid. It proves that the flow phases were observed at higher Rayleigh values in the transition to turbulence. The heated isotherm extends to the edge of the cooled wall. At the same time, the coolant isotherm turns to the other end. It leads to a diminution in density due to the heat in the cavity fluid.

_{avg}over Ra pour ($Da=0.01$, $Ha=0$, $\epsilon =0.4$). For all values of $Da$, $N{u}_{avg}$ improves with $Ra$. However, the heat transfer and Nusselt number are enhanced for greater $Ra$ values. The result of the conductive heat will not affect the lower values of $Ra$. In contrast, small bands show apparent escalations (enlarged by 23$\%$). Additionally, $N{u}_{avg}$ improved Rayleigh’s number to less than 10

^{3}. In the lower ranges, the improvement of fluid movement is of greater importance. The porous parameter $\epsilon $ is exposed in Figure 7. The Nusselt number improves to a higher value and keeps improving. The flow and diffusion of nanoparticles are less accessible, facilitating the heat transfer process. $\epsilon $ enhanced with the improvement of $Ra$ does not affect the improvement of $N{u}_{avg}$. The isothermal behavior is due to the convergence between convection and thermal conductivity. At high values of $Ra$, we notice a significant improvement in heat transfer rates.

#### 5.2. Effect of Hartmann Number

#### 5.3. Effect of Darcy’s Number

#### 5.4. Effect of Nanofluid Loading

#### 5.5. Effect of Porosity

#### 5.6. Effect of General Entropy of Different Numbers

_{3}O

_{4}/H

_{2}O for collective amounts of $Ra$. For lower amounts, the flow claims to be laminar; under these conditions, the entropy formation was at the emergent stage, which can be found mainly near the lower corrugated wall. As it propagates into a turbulent phase of higher Rayleigh number ($Ra$) values, entropy formation is also augmented from significant parts of the enclosure. In particular, the diagonal dominance trend can be visualized for the more significant Rayleigh number ($Ra$).

_{3}O

_{4}/H

_{2}O for snowballing amounts of $Ra$. For lower quantities, the flow claims to be laminar. Under these conditions, the entropy formation was at the emergent stage, which can be found mainly near the bottom wall. This result showed that $Ra$ disturbed the entropy system in the flow. Figure 19, Figure 20 and Figure 21 present the variation of entropy on Rayleigh function for varied $Da$, notable that the entropy varied proportionally by $Ra$ as well as the descending $Da$ and volume fraction and Hartmann number to diminish.

_{3}O

_{4}/H

_{2}O for accumulative values of the Hartman number $Ha$. For higher values, the flow pretends to be laminar. Under these conditions, the entropy formation was augmented in the porous medium and lessened in the non-porous medium.

_{3}O

_{4}/H

_{2}O for accumulative amounts of $Da$. For lower amounts, the flow claims to be laminar. Under these conditions, entropy formation dwindled in the porous medium and rose in the nanofluid medium.

_{3}O

_{4}/H

_{2}O for increasing values of volume fraction $\varphi $. For lower values, the flow pretends to be laminar; consequently, the propagation of heating and randomness in the porousness surface by diminishing volume fraction $\varphi $ and stabilizing so that the volume fraction $\varphi $ intensifications in the opposite way for the no-porous medium.

_{3}O

_{4}/H

_{2}O for aggregate values of porosity $\epsilon .$ For lower values, the flow pretends to be laminar. Therefore, the propagation of heat and randomness in a porous medium decreases porosity and stabilizes; hence, porosity increases in the opposite way for the non-porous medium.

## 6. Conclusions

_{3}O

_{4}/water packed in the porous cavity medium in layers and under a magnetic field. Then, the problem studied is solved using the finite element method. The validation is conducted by comparing with existing results. The results will be benefits technological applications. Thus, the numerical results are extracted from the following.

- Activation of double streamlines for the hot side and single cooling for smaller amounts of Darcy numbers, Rayleigh numbers, and more significant amounts of Hartmann numbers.
- In the upper part of the cavity, a cold isotherm is required below it at varying values of physical parameters.
- At low values of Darcy and Hartmann number, Rayleigh number, high porosity, and lower volume fraction dominated by high $N{u}_{avg}$ in all porous mediums.
- The high heat transfer rate inside the cavity is due to a rise in Darcy’s number with $Ra$ and a solid volume fraction. These results are reflected when Hartmann is elevated.
- Thermal entropy production determines the entropy components.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$b$ | Length of porous aria ($m$) | ${B}_{0}$ | The intensity of the magnetic field | |

a,c | Length of non-porous aria ($m$) | $Fc$ | Forchheimer coefficient | |

$Ha$ | Hartmann number | $k$ | Thermal conductivity ($W\xb7{m}^{-1}\xb7{K}^{-1}$) | |

$K$ | Permeability (${m}^{2}$) | $L$ | Height of cavity ($m$) | |

$Nu$ | Nusselt number | $P$ | Pressure ($Pa$) | |

$Pr$ | Prandtl number | $Ra$ | Rayleigh number | |

$Da$ | Darcy number | $u,v$ | Velocity components ($m/s$) | |

$U,V$ | Dimensional velocity components | $T$ | Temperature ($K$) | |

$x,y$ | Coordinates ($m$) | $S$ | Entropy | |

Greek symbols | ||||

$\alpha $ | Thermal diffusivity (${m}^{2}/s$) | $\beta $ | Thermal expansion coefficient ${K}^{-1}$ | |

$\epsilon $ | Porosity | $\lambda $ | The length of the Baffle ($m$) | |

$\mu $ | Dynamic viscosity ($W\xb7{m}^{-1}\xb7{K}^{-1}$) | $\nu $ | Kinematic diffusivity (${m}^{2}\xb7{s}^{-1}$) | |

$\theta $ | Adimentional temperature | $\rho $ | Density ($kg\xb7{m}^{-3}$) | |

$\sigma $ | Electrical conductivity ($\mathsf{\Omega}.\mathrm{M}$) | $\varphi $ | Solid volume fraction | |

Subscripts | ||||

$avg$ | Average | $c$ | Cold | |

$f$ | Fluid | $h$ | Hot | |

$loc$ | Local | $hnf$ | Hybrid nanofluid | |

$tot$ | Total | bf | Base fluid |

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**Figure 2.**Comparing among streamlines (left) and isotherms (right): (

**a**) the current study and (

**b**) Corcione et al. [40] for $Ra={10}^{5}$, and $\varphi =0.04$. (

**c**). Comparison of vertical velocity profile (right) and temperature (left) at a specific Y coordinate among the current outcomes and the experimental findings by [41] ($Ra={10}^{5}$, $Ha=0,$ and $Da=0.01$ ).

**Figure 3.**Streamlines and isotherms with divers Rayleigh quantity $Ra$ at $Da=0.01$, $Ha=0$, $\varphi =0.02$, and $\epsilon =0.4$.

**Figure 4.**Variation of $N{u}_{avg}$ with $Ra$ for different $Ha$ at $Da=0.01$, $\varphi =0.02$, and $\epsilon =0.4$.

**Figure 5.**Variation of $N{u}_{avg}$ with $Ra$ for different $Da$ at $Ha=0$, $\varphi =0.02$, and $\epsilon =0.4$.

**Figure 6.**Variation of $N{u}_{avg}$ with $Ra$ for different $\varphi $ at $Da=0.01$, $Ha=0$, and $\epsilon =0.4$.

**Figure 7.**Variation of $N{u}_{avg}$ with $Ra$ for different $\epsilon $ at $Ha=0$, $Da=0.01$, and $\varphi =0.02$.

**Figure 8.**Streamlines and isotherms with diverse Hartmann quantities $Ha$ at $Da=0.01$, $Ra={10}^{5}$, $\varphi =0.02$, and $\epsilon =0$.

**Figure 9.**Variation of $N{u}_{avg}$ with $Ha$ for different $Da$ at $Ra={10}^{5}$, $\varphi =0.02$, and $\epsilon =0$.

**Figure 10.**Variation of $N{u}_{avg}$ with $Ha$ for different $\varphi $ at $Ra={10}^{5}$, $Da=0.01$, and $\epsilon =0$.

**Figure 11.**Variation of $N{u}_{avg}$ with $Ha$ for diverse $\epsilon $ at $Ha=0$, $Da=0.01$, and $\varphi =0.02$.

**Figure 12.**Streamlines and isotherms with diverse Darcy quantities $Da$ at $Ha=0$, $Ra={10}^{5}$, $\varphi =0.02$, and $\epsilon =0$.

**Figure 13.**Variation of $N{u}_{avg}$ with $Da$ for diverse $\varphi $ at $Ra={10}^{5}$, $Ha=0$, and $\epsilon =0.4$.

**Figure 14.**Variation of $N{u}_{avg}$ with $Da$ for diverse $\epsilon $ at $Ra={10}^{5}$, $Ha=0$, and $\varphi =0.02$.

**Figure 15.**Streamlines and isotherms with varied $\varphi $ at $Da=0.01$, $Ra={10}^{5}$, $Ha=0$, and $\epsilon =0$.

**Figure 16.**Discrepancy of $N{u}_{avg}$ with $\varphi $ for varied $\epsilon $ at $Ra={10}^{5}$, $Ha=0$, and $Da=0.01$.

**Figure 17.**Streamlines and isotherms with diverse porosity at $Da=0.01$, $Ra={10}^{5}$, $Ha=0$, and $\varphi =0.02$.

**Figure 18.**Change of the entropy with diverse Ra when $Da=0.01$, $Ha=0$, $\varphi =0.02$, and $\epsilon =0.2$.

**Figure 19.**Changes in ${S}_{tot}$ with $Ra$ for diverse $Ha$ when $Da=0.01$, $\varphi =0.02$, and $\epsilon =0.2$.

**Figure 20.**Changes in ${S}_{tot}$ with $Ra$ for varied $\varphi $ when $Da=0.01$, $Ha=0$, and $\epsilon =0.2$.

**Figure 21.**Changes in ${S}_{tot}$ with $Ra$ for varied $Da$ when $Ha=0$, $\varphi =0.02$, and $\epsilon =0.2$.

**Figure 22.**Change of the entropy with varied $Ha$ when $Da=0.01$, $Ra={10}^{5}$, $\varphi =0.02,$ and $\epsilon =0.2$.

**Figure 23.**Changes in ${S}_{tot}$ with $Ha$ for diverse $Da$ when $Ra={10}^{5}$, $\varphi =0.02$, and $\epsilon =0.2$.

**Figure 24.**Changes in ${S}_{tot}$ with $Ha$ for diverse $\varphi $ when $Ra={10}^{5}$, $Da=0.01$, and $\epsilon =0.2$.

**Figure 25.**Change of the entropy with varied $Da$ when $Ha=0$, $Ra={10}^{5}$, $\varphi =0.02,$ and $\epsilon =0.2$.

**Figure 26.**Changes in ${S}_{tot}$ with $Da$ for varied $\varphi $ when $Ra={10}^{5}$, $Ha=0$, $Da=0.01$, and $\epsilon =0.2$.

**Figure 27.**Entropy changes with diverse $\varphi $ when $Ha=0$, $Ra={10}^{5}$, $Da=0.01$, and $\epsilon =0.2$.

**Figure 28.**Changes in ${S}_{tot}$ with $Da$ for diverse $\epsilon $ when $Ra={10}^{5}$, $Ha=0$, $Da=0.01$, and $\varphi =0.02$.

**Figure 29.**Entropy changes with diverse $\epsilon $ when $Ha=0$, $Ra={10}^{5}$, $Da=0.01$, and $\varphi =0.02$.

Material | ${C}_{p}\left(J/kg\xb7K\right)$ | $k\left(W/m\xb7k\right)$ | $\rho \left(kg/{m}^{3}\right)$ | $\beta \times {10}^{-5}\left({K}^{-1}\right)$ | $\sigma \left(s/m\right)$ |

Water | 4179 | 0.613 | 997.1 | 21 | 5.5 × 10^{−6} |

Fe_{3}O_{4} | 670 | 6 | 5810 | 1.3 | 2.5 × 10^{4} |

MWCNT | 711 | 3000 | 2100 | 4.2 | 1.9 × 10^{−4} |

**Table 2.**Variation of $N{u}_{avg},{\psi}_{max}$, Max (${S}_{tot}$ ) with mesh elements number at $Ra={10}^{5}$, $\varphi =0.02$, $Ha=0$, $Da=0.01$, and $\epsilon =0.4$.

Number of elements | 604 | 1092 | 1642 | 2660 | 5852 | 15,886 | 26,098 |

$N{u}_{avg}$ | 6.219 | 6.320 | 6.564 | 6.427 | 6.564 | 6.667 | 6.668 |

${\psi}_{max}$ | 0.471 | 0.471 | 0.457 | 0.462 | 0.457 | 0.455 | 0.453 |

$\mathrm{Max}\left({S}_{tot}\right)$ in the porous medium | 1.644 | 1.644 | 2.042 | 1.993 | 2.042 | 2.219 | 2.220 |

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## Share and Cite

**MDPI and ACS Style**

Ghali, D.; Redouane, F.; Abdelhak, R.; Belhadj Mahammed, A.; Zineb, C.D.; Jamshed, W.; Eid, M.R.; Eldin, S.M.; Musa, A.; Mohd Nasir, N.A.A.
Mathematical Entropy Analysis of Natural Convection of MWCNT—Fe_{3}O_{4}/Water Hybrid Nanofluid with Parallel Magnetic Field via Galerkin Finite Element Process. *Symmetry* **2022**, *14*, 2312.
https://doi.org/10.3390/sym14112312

**AMA Style**

Ghali D, Redouane F, Abdelhak R, Belhadj Mahammed A, Zineb CD, Jamshed W, Eid MR, Eldin SM, Musa A, Mohd Nasir NAA.
Mathematical Entropy Analysis of Natural Convection of MWCNT—Fe_{3}O_{4}/Water Hybrid Nanofluid with Parallel Magnetic Field via Galerkin Finite Element Process. *Symmetry*. 2022; 14(11):2312.
https://doi.org/10.3390/sym14112312

**Chicago/Turabian Style**

Ghali, Djellouli, Fares Redouane, Roubi Abdelhak, Amine Belhadj Mahammed, Chikr Djaoutsi Zineb, Wasim Jamshed, Mohamed R. Eid, Sayed M. Eldin, Awad Musa, and Nor Ain Azeany Mohd Nasir.
2022. "Mathematical Entropy Analysis of Natural Convection of MWCNT—Fe_{3}O_{4}/Water Hybrid Nanofluid with Parallel Magnetic Field via Galerkin Finite Element Process" *Symmetry* 14, no. 11: 2312.
https://doi.org/10.3390/sym14112312