# Micromagnetorotation of MHD Micropolar Flows

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## Abstract

**:**

## 1. Introduction

## 2. Model and Governing Equations

#### 2.1. Statement of the Problem

- No-slip and no-penetration conditions for the velocity field, i.e., ${\mathit{\upsilon}|}_{\overline{x}=-L}=0$ and ${\mathit{\upsilon}|}_{\overline{x}=L}={\upsilon}_{0}$.
- Condiff–Dahler conditions for the microrotation field and the angular velocity, i.e., $\mathit{W}=\delta \mathit{w}$, where $\delta $ is called wall coefficient. In this study $\delta =0$, which means that the microelements close to the wall are unable to rotate [34].

#### 2.2. Governing Equations

_{m}approximation that is incorporated in several studies in the relative literature, such as [30,31]. This approach allows the neglecting of the solution of the magnetic induction equation leading to a reduction of the equations that need to be solved.

#### 2.3. Solution of the Governing Equations

## 3. Results

#### 3.1. Effect of Magnetization for Various Values of the Micropolar Effect Parameter, $\epsilon $

#### 3.2. Effect of Magnetization for Various Values of the Size Effect Parameter, $\lambda $

#### 3.3. Effect of Magnetization for Various Values of the Hartmann Number, $Ha$

#### 3.4. Effect of Magnetization for Various Values of the Electric Effect Parameter, ζ

#### 3.5. Effect of Magnetization on Skin Friction, ${C}_{f}$

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 2.**Effect of σ

_{m}on velocity (left) and microrotation (right) for λ = 5, Ha = 1, ζ = 0 and ε equal to (

**a**) 0.1 (top), (

**b**) 0.5 (middle), and (

**c**) 0.9 (bottom).

**Figure 3.**Difference in (

**a**) velocity, $\Delta \upsilon $, and (

**b**) microrotation, $\Delta \Omega $, via considering ${\sigma}_{m}=1$ or ${\sigma}_{m}=0$ for ε = 0.1, 0.5 and 0.6.

**Figure 4.**Effect of ${\sigma}_{m}$ on velocity (left) and microrotation (right), for ε = 0.5, Ha = 1, ζ = 0 and λ equal to: (

**a**) 5 (top), (

**b**) 10 (middle), and (

**c**) 20 (bottom).

**Figure 5.**Difference in (

**a**) velocity, $\Delta \upsilon $, and (

**b**) microrotation, $\Delta \Omega $, via considering ${\sigma}_{m}=1$ or ${\sigma}_{m}=0$ for λ = 5, 10 and 20.

**Figure 6.**Effect of σ

_{m}on velocity (left) and microrotation (right) for ε = 0.5, λ = 5, ζ = 0 and Ha equal to: (

**a**) 1 (up), (

**b**) 3 (middle), and (

**c**) 18 (bottom).

**Figure 7.**Difference in (

**a**) velocity, $\Delta \upsilon $, and (

**b**) microrotation, $\Delta \Omega $, via considering ${\sigma}_{m}=1$ or ${\sigma}_{m}=0$ for ε = 0.5, λ = 5, ζ = 0 and Ha = 0.1, 1, 3 and 18.

**Figure 8.**Effect of σ

_{m}on the velocity (left) and microrotation (right) for ε = 0.5, λ = 5, Ha = 1 and ζ equal to (

**a**) 0.1 (top), (

**b**) 1 (second row), (

**c**) 3 (third row) and (

**d**) 15 (bottom).

**Figure 9.**Difference in (

**a**) velocity, $\Delta \upsilon $, and (

**b**) microrotation, $\Delta \Omega $, via considering ${\sigma}_{m}=1$ or ${\sigma}_{m}=0$ for ε = 0.5, λ = 5, Ha = 1 and ζ = 0.1, 1, 5 and 15.

**Table 1.**Range of $Ha$ complex number solution for various $\lambda ,\epsilon ,{\sigma}_{m}$, and $H\alpha $.

$\mathit{\lambda}$ | $\mathit{\epsilon}$ | ${\mathit{\sigma}}_{\mathit{m}}$ | $\mathit{H}\mathit{a}$ |
---|---|---|---|

5 | 0.2 | 0 | 2.8989–6.8990 |

5 | 0.2 | 1 | 4.9282–8.9283 |

5 | 0.9 | 0 | 4.0766–22.0767 |

5 | 0.9 | 1 | 9.4932–9.4933 |

20 | 0.2 | 0 | 11.5959–27.5960 |

20 | 0.2 | 1 | 19.7128–35.7129 |

20 | 0.9 | 0 | 16.3067–88.3068 |

20 | 0.9 | 1 | 37.9729–109.9730 |

$\mathit{R}\mathit{e}$ | ${\mathit{C}}_{\mathit{f}}$ | |||||
---|---|---|---|---|---|---|

Newtonian | MHD | Micropolar | ${\mathit{\sigma}}_{\mathit{m}}=0$ | ${\mathit{\sigma}}_{\mathit{m}}=1$ | $\mathit{\delta}{\mathit{\sigma}}_{\mathit{m}}$ | |

10 | 0.1000 | 0.4149 | 0.0459 | 0.0729 | 0.0802 | 0.73 |

50 | 0.0200 | 0.0830 | 0.0092 | 0.0146 | 0.0160 | 0.14 |

100 | 0.0100 | 0.0415 | 0.0046 | 0.0073 | 0.0080 | 0.07 |

200 | 0.0050 | 0.0207 | 0.0023 | 0.0036 | 0.0040 | 0.04 |

300 | 0.0033 | 0.0138 | 0.0015 | 0.0024 | 0.0027 | 0.03 |

400 | 0.0025 | 0.0104 | 0.0011 | 0.0018 | 0.0020 | 0.02 |

500 | 0.0020 | 0.0083 | 0.0009 | 0.0015 | 0.0016 | 0.01 |

600 | 0.0017 | 0.0069 | 0.0008 | 0.0012 | 0.0013 | 0.01 |

700 | 0.0014 | 0.0059 | 0.0007 | 0.0010 | 0.0011 | 0.01 |

800 | 0.0013 | 0.0052 | 0.0006 | 0.0009 | 0.0010 | 0.01 |

900 | 0.0011 | 0.0046 | 0.0005 | 0.0008 | 0.0009 | 0.01 |

1000 | 0.0010 | 0.0041 | 0.0004 | 0.0007 | 0.0008 | 0.01 |

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Aslani, K.-E.; Benos, L.; Tzirtzilakis, E.; Sarris, I.E.
Micromagnetorotation of MHD Micropolar Flows. *Symmetry* **2020**, *12*, 148.
https://doi.org/10.3390/sym12010148

**AMA Style**

Aslani K-E, Benos L, Tzirtzilakis E, Sarris IE.
Micromagnetorotation of MHD Micropolar Flows. *Symmetry*. 2020; 12(1):148.
https://doi.org/10.3390/sym12010148

**Chicago/Turabian Style**

Aslani, Kyriaki-Evangelia, Lefteris Benos, Efstratios Tzirtzilakis, and Ioannis E. Sarris.
2020. "Micromagnetorotation of MHD Micropolar Flows" *Symmetry* 12, no. 1: 148.
https://doi.org/10.3390/sym12010148