# Lorentz Forces Effects on the Interactions of Nanoparticles in Emerging Mechanisms with Innovative Approach

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## Abstract

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## 1. Introduction

## 2. Methods

**B**= [0, B${}_{0}$, 0] is applied perpendicularly to the heat and mass transfer flow. It is assumed that the strength of electric charge and magnetic field are maximum. The schematic diagram of the problem is shown in Figure 1.

## 3. Entropy Generation

## 4. Results and Discussion

#### Solution Validation

## 5. Conclusions

- (1)
- Velocity decreases with increasing the parameters M, f${}_{w}$, $\gamma $ and increases with increasing the parameter $\lambda $.
- (2)
- Temperature decreases with increasing the parameters Pr, $\lambda $, f${}_{w}$.
- (3)
- Nanoparticles concentration decreases with increasing the parameter f${}_{w}$ and increases with increasing the parameter Sc.
- (4)
- Entropy generation increases with increasing the parameters Re and Br.
- (5)
- Both the velocity components decrease with the Hall effect parameter m.
- (6)
- Streamlines show that the trapping increases at the left side of the surface for the parameter m.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 5.**Representation of the influence of a magnetic field parameter M = 1.00, 1.50, 2.00 on velocity f${}^{\prime}\left(\zeta \right)$.

**Figure 6.**Representation of the influence of a stretching/shrinking $\lambda $ = 1.00, 1.50, 2.00 on velocity f${}^{\prime}\left(\zeta \right)$.

**Figure 7.**Representation of the influence of a Stefan blowing parameter f${}_{w}$ = 0.001, 0.001, 0.002 on velocity f${}^{\prime}\left(\zeta \right)$.

**Figure 8.**Representation of the influence of an accretion/ablation parameter $\gamma $ = $\frac{\pi}{2}$, $\frac{\pi}{3}$, $\frac{\pi}{4}$ on velocity f${}^{\prime}\left(\zeta \right)$.

**Figure 9.**Representation of the influence of a Prandtl number Pr = 6.20, 8.20, 10.20 on temperature $\theta $($\zeta $).

**Figure 10.**Representation of the influence of a stretching/shrinking parameter $\lambda $ = 1.00, 1.50, 2.00 on temperature $\theta $($\zeta $).

**Figure 11.**Representation of the influence of a Stefan blowing parameter f${}_{w}$ = 1.00, 1.50, 2.00 on temperature $\theta $($\zeta $).

**Figure 12.**Representation of the influence of a Schmidt number Sc = 1.00, 5.00, 9.00 on nanoparticles concentration $\phi $($\zeta $).

**Figure 13.**Representation of the influence of a Stefan blowing parameter f${}_{w}$ = 1.00, 5.00, 9.00 on nanoparticles concentration $\phi $($\zeta $).

**Figure 14.**Representation of the influence of a Reynolds number Re = 1.00, 1.50, 2.00 on entropy generation rate N${}_{G}$($\zeta $).

**Figure 15.**Representation of the influence of a Brinkman number Br = 1.00, 1.50, 2.00 on entropy generation rate N${}_{G}$($\zeta $).

**Figure 16.**Representation of the influence of a Hall parameter m = 1.00, 1.50, 2.00 on velocity f${}^{\prime}\left(\zeta \right)$.

**Figure 17.**Representation of the influence of a Hall parameter m = 1.00, 1.50, 2.00 on velocity g$\left(\zeta \right)$.

**Figure 18.**Representation of the influence of a Hall parameter m = 0.60, ${\varphi}_{1}$ = 0.03, ${\varphi}_{2}$ = 0.04 on streamlines.

Thermophysical Properties | H${}_{2}$O | TiO${}_{2}$ (Titania) | GO (Graphene Oxide) |
---|---|---|---|

$\rho $ (Density) (kg/m${}^{3}$) | ${\rho}_{f}$ = 997.1 | ${\rho}_{{s}_{1}}$ = 4250 | ${\rho}_{{s}_{2}}$ = 1800 |

c${}_{P}$ (Heat capacity) (J/kg K) | (c${}_{P}$)${}_{f}$ = 4179 | (c${}_{P}$)${}_{{s}_{1}}$ = 686.2 | (c${}_{P}$)${}_{{s}_{2}}$ = 717 |

k (Thermal conductivity) (W/m K) | k${}_{f}$ = 0.613 | k${}_{{s}_{1}}$ = 8.9538 | k${}_{{s}_{2}}$ = 5000 |

$\sigma $ (Electrical conductivity) (Um)${}^{-1}$ | ${\sigma}_{f}$ = 0.05 | ${\sigma}_{{s}_{1}}$ = 2.38 × 10${}^{6}$ | ${\sigma}_{{s}_{2}}$ = 1.1 × 10${}^{-5}$ |

Properties | Nanofluid (TiO${}_{2}$/H${}_{2}$O) |
---|---|

Density ($\rho $) | ${\rho}_{nf}$ = (1 − $\varphi $)${\rho}_{f}$ + $\varphi $${\rho}_{s}$ |

Heat capacity ($\rho $c${}_{P}$) | ($\rho $c${}_{P}$)${}_{nf}$ = (1 − $\varphi $)($\rho $c${}_{P}$)${}_{f}$ + $\varphi $($\rho $c${}_{P}$)${}_{s}$ |

Dynamic viscosity ($\mu $) | $\frac{{\mu}_{nf}}{{\mu}_{f}}$ = $\frac{1}{{(1-\varphi )}^{2.5}}$ |

Thermal conductivity (k) | $\frac{{k}_{nf}}{{k}_{f}}$ = $\frac{{k}_{s}+2{k}_{f}-2\varphi ({k}_{f}-{k}_{s})}{{k}_{s}+2{k}_{f}+\varphi ({k}_{f}-{k}_{s})}$ |

Electrical conductivity ($\sigma $) | $\frac{{\sigma}_{nf}}{{\sigma}_{f}}$ = 1 + $\frac{3(\sigma -1)\varphi}{(\sigma +2)-(\sigma -1)\varphi}$, where $\sigma $ = $\frac{{\sigma}_{s}}{{\sigma}_{f}}$ |

Properties | Hybrid nanofluid (GO-TiO${}_{2}$/H${}_{2}$O) |

Density (${\rho}_{hnf}$) | ${\rho}_{hnf}$ = (1 − (${\varphi}_{1}$ + ${\varphi}_{2}$))${\rho}_{f}$ + ${\varphi}_{1}$${\rho}_{{s}_{1}}$ + ${\varphi}_{2}$${\rho}_{{s}_{2}}$ |

Heat capacity ($\rho $c${}_{P}$)${}_{hnf}$ | ($\rho $c${}_{P}$)${}_{hnf}$ = (1 − (${\varphi}_{1}$ + ${\varphi}_{2}$))($\rho $c${}_{P}$)${}_{f}$ + ${\varphi}_{1}$($\rho $c${}_{P}$)${}_{{s}_{1}}$ + ${\varphi}_{2}$($\rho $c${}_{P}$)${}_{{s}_{2}}$ |

Dynamic viscosity (${\mu}_{hnf}$) | $\frac{{\mu}_{hnf}}{{\mu}_{f}}$ = $\frac{1}{{\left[\begin{array}{c}1-({\varphi}_{1}+{\varphi}_{2})\end{array}\right]}^{2.5}}$ |

Thermal conductivity (k${}_{{}_{hnf}}$) | $\frac{{k}_{hnf}}{{k}_{f}}$ = $\frac{{\varphi}_{1}{k}_{{s}_{1}}+{\varphi}_{2}{k}_{{s}_{2}}+2\varphi {k}_{f}+2({\varphi}_{1}{k}_{{s}_{1}}+{\varphi}_{2}{k}_{{s}_{2}})-2{\varphi}^{2}{k}_{f}}{{\varphi}_{1}{k}_{{s}_{1}}+{\varphi}_{2}{k}_{{s}_{2}}+2\varphi {k}_{f}-({\varphi}_{1}{k}_{{s}_{1}}+{\varphi}_{2}{k}_{{s}_{2}})+{\varphi}^{2}{k}_{f}}$ |

Electrical conductivity (${\sigma}_{hnf}$) | $\frac{{\sigma}_{hnf}}{{\sigma}_{f}}$ = 1 + $\frac{3\left[\begin{array}{c}\frac{{\sigma}_{1}{\varphi}_{1}+{\sigma}_{2}{\varphi}_{2}}{{\sigma}_{f}}-({\varphi}_{1}+{\varphi}_{2})\end{array}\right]}{2+\left(\frac{{\sigma}_{1}{\varphi}_{1}+{\sigma}_{2}{\varphi}_{2}}{({\varphi}_{1}+{\varphi}_{2}){\sigma}_{f}}\right)-\left[\begin{array}{c}\frac{{\sigma}_{1}{\varphi}_{1}+{\sigma}_{2}{\varphi}_{2}}{{\sigma}_{f}}-({\varphi}_{1}+{\varphi}_{2})\end{array}\right]}$ |

$\mathit{\gamma}$ | Published Paper [19] | Present Work | Error |
---|---|---|---|

0${}^{0}$ | 5.6418 × 10${}^{-1}$ | 5.6417 × 10${}^{-1}$ | 0.0001 × 10${}^{-1}$ |

(15/2)${}^{0}$ | 5.7501 × 10${}^{-1}$ | 5.7500 × 10${}^{-1}$ | 0.0001 × 10${}^{-1}$ |

15${}^{0}$ | 5.8072 × 10${}^{-1}$ | 5.8071 × 10${}^{-1}$ | 0.0001 × 10${}^{-1}$ |

30${}^{0}$ | 5.7700 × 10${}^{-1}$ | 5.7700 × 10${}^{-1}$ | 0.0000 × 10${}^{-1}$ |

45${}^{0}$ | 5.52287 × 10${}^{-1}$ | 5.52285 × 10${}^{-1}$ | 0.0002 × 10${}^{-1}$ |

60${}^{0}$ | 5.0721 × 10${}^{-1}$ | 5.0720 × 10${}^{-1}$ | 0.0001 × 10${}^{-1}$ |

75${}^{0}$ | 4.3686 × 10${}^{-1}$ | 4.3684 × 10${}^{-1}$ | 0.0002 × 10${}^{-1}$ |

(165/2)${}^{0}$ | 3.8999 × 10${}^{-1}$ | 3.8998 × 10${}^{-1}$ | 0.0001 × 10${}^{-1}$ |

90${}^{0}$ | 3.3205 × 10${}^{-1}$ | 3.3205 × 10${}^{-1}$ | 0.0000 × 10${}^{-1}$ |

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**MDPI and ACS Style**

Khan, N.S.; Shah, Q.; Sohail, A.; Kumam, P.; Thounthong, P.; Bhaumik, A.; Ullah, Z.
Lorentz Forces Effects on the Interactions of Nanoparticles in Emerging Mechanisms with Innovative Approach. *Symmetry* **2020**, *12*, 1700.
https://doi.org/10.3390/sym12101700

**AMA Style**

Khan NS, Shah Q, Sohail A, Kumam P, Thounthong P, Bhaumik A, Ullah Z.
Lorentz Forces Effects on the Interactions of Nanoparticles in Emerging Mechanisms with Innovative Approach. *Symmetry*. 2020; 12(10):1700.
https://doi.org/10.3390/sym12101700

**Chicago/Turabian Style**

Khan, Noor Saeed, Qayyum Shah, Arif Sohail, Poom Kumam, Phatiphat Thounthong, Amiya Bhaumik, and Zafar Ullah.
2020. "Lorentz Forces Effects on the Interactions of Nanoparticles in Emerging Mechanisms with Innovative Approach" *Symmetry* 12, no. 10: 1700.
https://doi.org/10.3390/sym12101700