# The Dynamics of Water-Based Nanofluid Subject to the Nanoparticle’s Radius with a Significant Magnetic Field: The Case of Rotating Micropolar Fluid

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

## 3. Solution Procedure

- ${s}_{1}^{\prime}={s}_{2}$,
- ${s}_{2}^{\prime}={s}_{3}$,
- ${s}_{3}^{\prime}=\frac{(-1)}{(\frac{{\lambda}_{1}}{{\lambda}_{2}}+\nabla )}[{s}_{1}{s}_{3}+2\Gamma {s}_{4}-{K}_{p}{s}_{2}-(1+Fr){s}_{2}^{2}-\frac{{\lambda}_{3}}{{\lambda}_{2}}M{s}_{2}]$,
- ${s}_{4}^{\prime}={s}_{5}$,
- ${s}_{5}^{\prime}=\frac{(-1)}{(\frac{{\lambda}_{1}}{{\lambda}_{2}}+\nabla )}[{s}_{1}{s}_{5}-{s}_{2}{s}_{4}-2\Gamma {s}_{2}-Kp{s}_{4}-Fr{s}_{4}^{2}-\frac{{\lambda}_{3}}{{\lambda}_{2}}M{s}_{4}]$,
- ${s}_{6}^{\prime}={s}_{7}$,
- ${s}_{7}^{\prime}=\frac{(-1)}{(\frac{{\lambda}_{1}}{{\lambda}_{2}}+\frac{\nabla}{2})}[{s}_{1}{s}_{7}-\frac{\nabla}{{\lambda}_{2}}(2{s}_{6}-{s}_{3})]$,
- ${s}_{8}^{\prime}={s}_{9}$,
- ${s}_{9}^{\prime}=-\frac{{\lambda}_{5}}{{\lambda}_{4}}Pr{s}_{1}{s}_{9}$.

## 4. Results and Discussion

## 5. Conclusions

- The increase in the nanoparticle radius ${D}_{p}$ increased the velocity ${F}_{1}^{\prime}$ and microrotation H$\left(\eta \right)$ and
- Decreased the secondary velocity ${F}_{2}$.
- Decreased the Cu-nanofluid’s temperature.
- Increased the skin friction factor.
- Increased the Nusselt number.

- The magnetic parameter M reduced the component of velocity ${F}_{1}^{\prime}$, ${F}_{2}$ and
- Increased the microrotation of nanoparticles.
- Increased the temperature of the non-Newtonian fluid.
- Increased the skin friction coefficient $C{f}_{y}{\left(Re\right)}^{0.5}$ but lessened the $C{f}_{x}{\left(Re\right)}^{0.5}$.
- Reduced the Nusselt number.

- The boundary concentration parameter $\beta $ increased the microrotation distribution.
- The rotational parameter $\Gamma $ lowered the ${F}_{1}^{\prime}$, ${F}_{2}$, and H$\left(\eta \right)$ and
- Enhanced the temperature profile.
- Decrease the skin friction coefficients and Nusselt number.

- The higher input of the Forchheimer number $\left(Fr\right)$ decreased the velocity ${F}_{1}^{\prime}$, ${F}_{2}$, and microrotation $H\left(\eta \right)$ and
- Increased the temperature of the fluid.
- Increase the $C{f}_{y}{\left(Re\right)}^{0.5}$ but reduced the $C{f}_{x}{\left(Re\right)}^{0.5}$ and Nusselt number.

- The material parameter ∇ reduced the component of velocity ${F}_{2}$, microrotation, and temperature and
- Enlarged the velocity component ${F}_{2}$.
- Enlarged the $C{f}_{y}{\left(Re\right)}^{0.5}$ but reduced the $C{f}_{x}{\left(Re\right)}^{0.5}$.
- Enlarged the Nusselt number.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Variation in ${{F}_{1}}^{\prime}\left(\eta \right)$ and ${F}_{2}\left(\eta \right)$ along $\Gamma $ and ${D}_{p}$.

**Figure 3.**Variation in ${{F}_{1}}^{\prime}\left(\eta \right)$ and ${F}_{2}\left(\eta \right)$ along M and ${D}_{p}$.

**Figure 5.**Variation in ${{F}_{1}}^{\prime}\left(\eta \right)$ and ${F}_{2}\left(\eta \right)$ along ∇ and ${D}_{p}$.

**Figure 6.**Variation in ${{F}_{1}}^{\prime}\left(\eta \right)$ and ${F}_{2}\left(\eta \right)$ along $\Gamma $, M, and ${D}_{p}$.

**Figure 7.**Variation in ${{F}_{1}}^{\prime}\left(\eta \right)$ and ${F}_{2}\left(\eta \right)$ along ∇, $\beta $, and ${D}_{p}$.

**Figure 10.**Variation in $C{f}_{x}{\left(Re\right)}^{0.5}$ and $C{f}_{y}{\left(Re\right)}^{0.5}$ along ${D}_{p}$, M, and ∇.

**Figure 11.**Variation in $C{f}_{x}{\left(Re\right)}^{0.5}$ and $C{f}_{y}{\left(Re\right)}^{0.5}$ along ${D}_{p}$, $\Gamma $, and Fr.

**Table 1.**Attributes of the nanoparticles and base fluid [36].

Physical Features | Density ($\mathit{\rho}$) | Specific Heat (${\mathit{C}}_{\mathit{p}}$) | Thermal Conductivity ($\mathit{\kappa}$) |
---|---|---|---|

H${}_{2}$O | 0991.1 | 4179.0 | 00.613 |

Cu | 8933.0 | 0385.0 | 0401.0 |

Properties | Nanofluid |
---|---|

Viscosity $\left(\mu \right)$ | $\frac{{\mu}_{{n}_{f}}}{{\mu}_{{b}_{f}}}=1+2.5\Phi +4.5\left[\frac{1}{\frac{h}{{D}_{p}}(2+\frac{h}{{D}_{p}}){(1+\frac{h}{{D}_{p}})}^{2}}\right]$ |

Density $\left(\rho \right)$ | ${\rho}_{{n}_{f}}={\rho}_{f}(1-\Phi )+\Phi {\rho}_{s}$ |

Heat capacity $\left(\rho {C}_{p}\right)$ | ${\left(\rho {C}_{p}\right)}_{nf}={\left(\rho {C}_{p}\right)}_{f}(1-\Phi )+\Phi \frac{{\left(\rho {C}_{p}\right)}_{s}}{{\left(\rho {C}_{p}\right)}_{f}}$ |

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

Electrical conductivity $\left(\sigma \right)$ | $\frac{{\sigma}_{{n}_{f}}}{{\sigma}_{f}}=\left[1+\frac{3\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1\right)\Phi}{\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}+2\right)-\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1\right)\Phi}\right]$ |

**Table 3.**Comparison of the ${F}_{1}^{\u2033}\left(0\right)$ and ${F}_{2}^{\prime}\left(0\right)$ values with various values of $\Gamma $ ignoring other involved parameters.

$\Gamma $ | Wang et al. [43] | Ali et al. [38] | Current Results | |||
---|---|---|---|---|---|---|

$-{\mathit{F}}_{\mathbf{1}}^{\u2033}\left(\mathbf{0}\right)$ | $-{\mathit{F}}_{\mathbf{2}}^{\prime}\left(\mathbf{0}\right)$ | $-{\mathit{F}}_{\mathbf{1}}^{\u2033}\left(\mathbf{0}\right)$ | $-{\mathit{F}}_{\mathbf{2}}^{\prime}\left(\mathbf{0}\right)$ | $-{\mathit{F}}_{\mathbf{1}}^{\u2033}\left(\mathbf{0}\right)$ | $-{\mathit{F}}_{\mathbf{2}}^{\prime}\left(\mathbf{0}\right)$ | |

0 | 1.000 | 0.000 | 1.0000 | 0.0000 | 1.000009 | 0.000000 |

1 | 1.325 | 0.837 | 1.3250 | 0.8371 | 1.325019 | 0.837199 |

2 | 1.652 | 1.287 | 1.6523 | 1.2873 | 1.6523251 | 1.287359 |

**Table 4.**Comparison of the values of ${\Theta}^{\prime}\left(0\right)$ with various values of $\Gamma $ and Pr.

$\Gamma $ | Ali et al. [45] | Adnan et al. [44] | Current Results | |||
---|---|---|---|---|---|---|

$\mathit{Pr}=\mathbf{2}.\mathbf{0}$ | $\mathit{Pr}=\mathbf{7}.\mathbf{0}$ | $\mathit{Pr}=\mathbf{2}.\mathbf{0}$ | $\mathit{Pr}=\mathbf{7}.\mathbf{0}$ | $\mathit{Pr}=\mathbf{2}.\mathbf{0}$ | $\mathit{Pr}=\mathbf{7}.\mathbf{0}$ | |

0 | 0.9108 | 1.8944 | 0.9113 | 1.8944 | 0.911353 | 1.895401 |

0.5 | 0.8525 | 1.8500 | 0.8534 | 1.8500 | 0.852437 | 1.850177 |

1 | 0.7703 | 1.7877 | 0.7703 | 1.7877 | 0.770331 | 1.787625 |

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**MDPI and ACS Style**

Ali, B.; Ahammad, N.A.; Awan, A.U.; Oke, A.S.; Tag-ElDin, E.M.; Shah, F.A.; Majeed, S.
The Dynamics of Water-Based Nanofluid Subject to the Nanoparticle’s Radius with a Significant Magnetic Field: The Case of Rotating Micropolar Fluid. *Sustainability* **2022**, *14*, 10474.
https://doi.org/10.3390/su141710474

**AMA Style**

Ali B, Ahammad NA, Awan AU, Oke AS, Tag-ElDin EM, Shah FA, Majeed S.
The Dynamics of Water-Based Nanofluid Subject to the Nanoparticle’s Radius with a Significant Magnetic Field: The Case of Rotating Micropolar Fluid. *Sustainability*. 2022; 14(17):10474.
https://doi.org/10.3390/su141710474

**Chicago/Turabian Style**

Ali, Bagh, N. Ameer Ahammad, Aziz Ullah Awan, Abayomi S. Oke, ElSayed M. Tag-ElDin, Farooq Ahmed Shah, and Sonia Majeed.
2022. "The Dynamics of Water-Based Nanofluid Subject to the Nanoparticle’s Radius with a Significant Magnetic Field: The Case of Rotating Micropolar Fluid" *Sustainability* 14, no. 17: 10474.
https://doi.org/10.3390/su141710474