# Molecular Interaction and Magnetic Dipole Effects on Fully Developed Nanofluid Flowing via a Vertical Duct Applying Finite Volume Methodology

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## Abstract

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## 1. Introduction

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_{3,}water–Ag) together with pure water. The laminar forced mixed convection, heat transport, and entropy generation in rectangular three-dimensional ducts were explored numerically in [10], where the duct of finite length was heated symmetrically with constant wall temperature. The impacts of buoyancy force, entropy generation, and Magnetohydrodynamic (MHD) mixed convection on nanofluids flow in a duct of an angled step were discussed by Atashafrooz [11]. The results indicated that the Hartmann number has a significant impact on the friction coefficient, Nusselt number, and entropy generation patterns along the lower wall.

_{2}O

_{3}–water) flow, occurring inside a lid-driven quadrilateral cavity under the influence of magnetic force for different conductivity structures by employing the finite element method, has been numerically studied by Selimefendigil and Öztop [21]. Zanchini [22] examined analytically the mixed convection for laminar and steady flow with a thermal viscosity in an annular duct. The findings designated that the cumulative impacts of variable flow rate and buoyancy forces on the averaged friction Fanning factor cross-section significantly affected the flow and that the lower values of such quantities are possible. El. Hasadi et al. [23] scrutinized a laminar mixed convection in a horizontal duct (semicircular) along with the flat wall by employing a finite control volume approach to solve the governing energy and momentum equations numerically. Barletta and Nield [24] analyzed forced mixed convection in a rectangular lid-driven cavity for buoyant laminar flow in view of uniform motion, where the standing sides were heated at various temperatures. Barletta et al. [25] examined a fully developed flow within an axial region having electrical wire immersed in a porous medium. The effects of MHD power, buoyancy, and heat generation were considered due to Joule heating and viscous dissipation. The mixed convection seepage flow was evaluated according to Boussinesq approximation and Darcy law, while the buoyancy effect was modeled by wall temperature. The effect of mixed convection and laminar heat transport enhancement for Lewis number, taking a tube having fixed concentration at the fluid–solid interface and constant heat flux, was discussed by Orfi and Galanis [26]. Barletta [27] analytically investigated mixed convection for a rectangular duct fully developed flow by using Fourier transforms. The temperature field, the velocity field, the friction factor, and the Nusselt number have been evaluated. In a further study of Barletta [28], the Boussinesq approximation was employed to assess the appropriate condition for horizontal mixed convection in a tilted duct and considered a sample case regarding an inclined plane to explain this condition.

_{2}O

_{3}) over a thermally driven cavity. Numerical simulations were performed by Zhang and Zhang [33] to study the thermal transmission and magnetic nanofluids’ flow properties under magnetic field intensities, magnetic field directions, and volume fractions. According to their findings, the effect of the thermal performance was small in a poor magnetic field but increased significantly in a powerful magnetic field. Ekiciler [34] numerically analyzed flow characteristics and heat transfer by incorporating four dissimilar fluids and modifying the position, length, and altitude of a rib mounted in a two-dimensional channel through turbulent fluid properties and forced convection. Umavathi et al. [35] examined free convection (double diffusive) of nanofluids inside a confined rectangular channel. The equations for momentum and energy were written as difference equations and solved numerically. The influencing factors of the magnetic field, as well as nanofluid over forced convection flow and thermal irreversibility in a conduit with an immediate contraction, were inspected by Atashafrooz et al. [36]. Mayeli et al. [37] used numerical analysis to interpret the entropy generation and convective heat transfer of the laminar nanofluids’ flow across a channel with wavy surfaces under a magnetic field. According to the calculations, skin friction, total entropy generation, and Nusselt number increased as the strength of a magnetic field increased. The micropolar flow within a channel having permeable walls, with the effect of magnetohydrodynamic, was interpreted by Ahmad et al. [38]. Not only skin friction and Nusselt number increased with the effect of the imposed magnetic field but micro-rotational also increased with its effect. To determine the impact of a hybrid methodology on fluid motion and heat transfer, a numerical investigation is carried out by Aidaoui et al. [39]. The authors considered a correlation between multiple promoting approaches to strengthen transfer phenomena in chaotic flow, nanofluid flow, and magnetic induction. The thermal characteristics of copper–water and aluminum oxide–water hybrid nanofluid flow inside a complex non-Darcian porous wavy enclosure was elucidated by Mandal et al. [40]. The left enclosure wall was assumed to be heated and wavy. They noted that the Darcy number caused a reduction in the flow intensity. The same phenomenon was investigated by Biswas et al. [41] and Manna et al. [42], subject to magnetohydrodynamic effects. An analysis of hybrid nanofluid-filled enclosures with thermo-fluidic transport and magnetic field effects was presented by Mondal et al. [43]. Further recent work is elaborated in the refs. [44,45,46].

## 2. Governing Equations

## 3. Numerical Approach: Finite Volume Method

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

List of symbols | |

$H$ | magnetic field |

$\overline{H}$ | magnetic field intensity |

$H$ | $\mathrm{magnetic}\mathrm{field}\mathrm{strength}[T]$ |

$\overline{B}$ | magnetic field induction |

${\overline{H}}_{0}$ | magnetic field intensity at the origin |

${x}_{E}$ | ordinate of eastward located neighboring point of point P |

${x}_{P}$ | ordinate of general point P |

${x}_{W}$ | ordinate of westward located neighboring point of point P |

${x}_{N}$ | ordinate of northward located neighboring point of point P |

${u}_{E}$ | velocity at the eastward located point E |

${u}_{P}$ | velocity of general point P |

${u}_{W}$ | velocity at the westward located point W |

${u}_{N}$ | velocity at the northward located point N |

${y}_{P}$ | abscissa of general point P |

${y}_{S}$ | abscissa of the ordinate of southward located neighboring point of point P |

${y}_{N}$ | abscissa of the ordinate of northward located neighboring point of point P |

$k$ | grid space along vertical direction |

$h$ | grid space along horizontal direction |

${f}_{P}$ | value of the function at point |

${a}_{n,m}$ | coefficient for the spectral expression for u |

${b}_{n,m}$ | coefficient for the spectral expression for u |

$M$ | magnetic parameter |

${N}_{u}$ | Nusselt number |

$Re$ | Reynolds number |

$T$ | $\mathrm{temperature}[K]$ |

${T}_{0}$ | reference temperature |

$D$ | hydraulic diameter of the duct |

${\dot{q}}_{nf}$ | constant peripherally averaged wall heat flux |

Greek letters | |

$\theta $ | dimensionless temperature |

$\varphi $ | nanoparticle volume fraction |

$\rho $ | $\mathrm{density}[kg.{m}^{-3}]$ |

${\rho}_{s}$ | density for the solid particles |

${\rho}_{f}$ | density of the base fluids |

${\rho}_{nf}$ | density of the nanofluids |

${\sigma}_{nf}$ | electrical diffusivity of the nanofluid |

$\epsilon $ | corresponding location of the dipole |

$\sigma $ | aspect ratio of the duct |

$\lambda $ | pressure gradient |

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**Figure 2.**(

**a**) Control volume domain at a specific mesh point P. (

**b**) Comparison of our numerical results for the Nusselt number Nu with literature Ali et al. [17], for the limiting case. (

**c**) Comparison of our numerical results along the line x = 0.5 with the literature Ali et al. [17], for the limiting case. (

**d**) Comparison of our numerical results along the line y = 0.5 with the literature Ali et al. [17], for the limiting case.

**Figure 3.**(

**a**) Variation in the Nusselt number with different $\varphi \mathrm{and}M$. (

**b**) Variation in the Nusselt number with different $\lambda \mathrm{and}\sigma $. (

**c**) Variation in the Nusselt number with different ratio $\raisebox{1ex}{$Gr$}\!\left/ \!\raisebox{-1ex}{$Re$}\right.\mathrm{and}\epsilon $.

**Figure 4.**(

**a**) Velocity distribution in the duct for the fixed $M=0,Gr/Re={10}^{4},\sigma =1,$ $\varphi =0.02,\epsilon =0.2,\mathrm{and}\lambda =10$ (

**b**) Temperature distribution in the duct for the fixed $M=0,Gr/Re={10}^{4},\sigma =1,\varphi =0.02,{\epsilon}_{0}=0.2$, and $\lambda =10$.

**Figure 5.**(

**a**) Velocity distribution in the duct for the fixed $M=100,Gr/Re={10}^{4},\sigma =1,\varphi =0.02,$ $\epsilon =0.2$, and $\lambda =10$. (

**b**) Temperature distribution in the duct for the fixed $M=100,Gr/Re={10}^{4},\sigma =1,\varphi =0.02,$ $\epsilon =0.2$, and $\lambda =10$.

**Figure 6.**(

**a**) Velocity distribution in the duct for the fixed $Gr/Re={10}^{4},\sigma =1,\varphi =0.02,$$\epsilon =0.2$, and $\lambda =10$. (

**b**) Temperature distribution in the duct for the fixed $Gr/Re={10}^{4},\sigma =1,$ $\varphi =0.02,\epsilon =0.2,$ and$\lambda =10$.

**Table 1.**Comparison of present results for Nu in the limiting case when $\sigma =1,\lambda =1,\varphi =0.0$, and $M=0$.

Gr/Re | Nu (Present Method) | Nu (Spectral Method) |
---|---|---|

1 | 1.1746 × 10^{4} | 1.1745 × 10^{4} |

10 | 1.2292 × 10^{4} | 1.2291 × 10^{4} |

100 | 1.8436 × 10^{4} | 1.8433 × 10^{4} |

1000 | 1.4612 × 10^{4} | 1.4608 × 10^{4} |

5000 | 1.9351 × 10^{4} | 1.9343 × 10^{4} |

Thermo-Physical | $\mathit{\rho}\left(\mathbf{k}\mathbf{g}{\mathbf{m}}^{-3}\right)$ | ${\mathit{C}}_{\mathit{p}}\left(\mathbf{J}\mathbf{k}{\mathbf{g}}^{-1}{\mathbf{K}}^{-1}\right)$ | $\mathit{k}\left(\mathbf{W}{\mathbf{m}}^{-1}{\mathbf{K}}^{-1}\right)$ | $\mathit{\beta}\left({\mathbf{K}}^{-1}\right)$ |
---|---|---|---|---|

Water (H_{2}O) | 997.1 | 4179 | 0.613 | 21 × 10^{−5} |

Silver (Ag) | 10500 | 235 | 429 | 1.89 × 10^{−5} |

**Table 3.**Nusselt number variation with $\sigma $ and $M$ for the fixed $\lambda =200,\frac{Gr}{Re=0},\varphi =0.02$, and $\epsilon =0.2$.

$\mathit{\sigma}$ | Nu | ||||
---|---|---|---|---|---|

M = 0 | M = 50 | M = 100 | M = 150 | M = 200 | |

0.2 | 18.7016 | 19.8204 | 20.9379 | 22.0550 | 23.1728 |

0.4 | 1.2432 | 1.4441 | 1.6535 | 1.8710 | 2.0967 |

0.6 | 0.3056 | 0.3824 | 0.4654 | 0.5546 | 0.6497 |

0.8 | 0.1342 | 0.1753 | 0.2207 | 0.2701 | 0.3235 |

1.0 | 0.0808 | 0.1073 | 0.1367 | 0.1688 | 0.2037 |

**Table 4.**Variation in $Nu$ with $\sigma $ and $M$ for the fixed $\lambda =200,Gr/Re=100,\varphi =0.02$, and $\epsilon =0.2$.

σ | Nu | ||||
---|---|---|---|---|---|

M = 0 | M = 50 | M = 100 | M = 150 | M = 200 | |

0.2 | 1.2608 | 1.3143 | 1.3677 | 1.4208 | 1.4737 |

0.4 | 0.5254 | 0.5767 | 0.6289 | 0.6819 | 0.7359 |

0.6 | 0.2800 | 0.3181 | 0.3576 | 0.3984 | 0.4406 |

0.8 | 0.1804 | 0.2088 | 0.2384 | 0.2693 | 0.3014 |

1.0 | 0.1318 | 0.1537 | 0.1766 | 0.2005 | 0.2254 |

**Table 5.**Variation in $Nu$ with Gr/Re and $M$ for the fixed $\lambda =200,\sigma =0.5,\varphi =0.02$, and $\epsilon =0.2$.

$\raisebox{1ex}{$\mathit{G}\mathit{r}$}\!\left/ \!\raisebox{-1ex}{$\mathit{R}\mathit{e}$}\right.$ | Nu | ||||
---|---|---|---|---|---|

M = 0 | M = 50 | M = 100 | M = 150 | M = 200 | |

−100 | 2.3919 | 4.6080 | 7.4516 | 10.8838 | 14.8753 |

−50 | 1.1507 | 1.5449 | 1.9836 | 2.4648 | 2.9868 |

0 | 0.5583 | 0.6756 | 0.8004 | 0.9325 | 1.0717 |

50 | 0.4289 | 0.4940 | 0.5619 | 0.6323 | 0.7053 |

100 | 0.3729 | 0.4173 | 0.4630 | 0.5099 | 0.5581 |

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**MDPI and ACS Style**

Ali, K.; Ahmad, S.; Ahmad, S.; Jamshed, W.; Hussain, S.M.; Tag El Din, E.S.M.
Molecular Interaction and Magnetic Dipole Effects on Fully Developed Nanofluid Flowing via a Vertical Duct Applying Finite Volume Methodology. *Symmetry* **2022**, *14*, 2007.
https://doi.org/10.3390/sym14102007

**AMA Style**

Ali K, Ahmad S, Ahmad S, Jamshed W, Hussain SM, Tag El Din ESM.
Molecular Interaction and Magnetic Dipole Effects on Fully Developed Nanofluid Flowing via a Vertical Duct Applying Finite Volume Methodology. *Symmetry*. 2022; 14(10):2007.
https://doi.org/10.3390/sym14102007

**Chicago/Turabian Style**

Ali, Kashif, Shabbir Ahmad, Sohail Ahmad, Wasim Jamshed, Syed M. Hussain, and El Sayed M. Tag El Din.
2022. "Molecular Interaction and Magnetic Dipole Effects on Fully Developed Nanofluid Flowing via a Vertical Duct Applying Finite Volume Methodology" *Symmetry* 14, no. 10: 2007.
https://doi.org/10.3390/sym14102007