# Hydromagnetic Flow of Micropolar Nanofluid

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

^{1}. Hence, we assume that (see [34,35,36,37])

## 3. Results and Discussion

## 4. Conclusions

- ⮚
- The temperature profile falls by improving ${\lambda}_{1}$.
- ⮚
- The Nusselt number enhances with the increase of the chemical reaction effect.
- ⮚
- The Nusselt number improves the higher the magnitude of inclination.
- ⮚
- The Sherwood number upturns by improving the inclination parameter.
- ⮚
- The Sherwood number decreases by improving the chemical reaction impact.
- ⮚
- The Sherwood number reduces the larger the value of ${\lambda}_{1}$.
- ⮚
- The skin friction decreases for higher values of ${\lambda}_{1}$.
- ⮚
- Skin friction decreases with the enhancement of the chemical reaction parameter.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$C$ | Fluid concentration | ${\lambda}_{1}$ | Heat generation or absorption | $R$ | Chemical reaction |

${C}_{f}$ | Skin friction coefficient | $\alpha $ | Thermal diffusivity | $R{e}_{x}$ | Reynolds number |

${C}_{\infty}$ | Ambient nanoparticle volume fraction | $Le$ | Lewis number | $Sh$ | Sherwood number |

${C}_{w}$ | Surface volume fraction | $Nb$ | Brownian motion parameter | $T$ | Fluid temperature |

${C}_{p}$ | Specific heat at constant pressure | $Nt$ | Thermophoretic parameter | ${T}_{w}$ | Wall temperature |

${D}_{B}$ | Brownian diffusion coefficient | $Nu$ | Nusselt number | ${T}_{\infty}$ | Ambient temperature |

${D}_{T}$ | Thermophoretic diffusion coefficient | $Pr$ | Prandtle number | ${u}_{w}$ | wall velocity |

$f$ | Similarity function for velocity | $g$ | Gravitational acceleration | ${u}_{\infty}$ | ambient velocity |

$\rho {c}_{p}$ | Volume heat capacity | $\mu $ | Kinematic viscosity | $\nu $ | Dynamic viscosity |

$\varphi $ | Dimensionless solid volume fraction | $w$ | Condition at the wall | $\infty $ | Ambient condition |

$Gr$ | Local Grashof number | ${\beta}_{t}$ | Thermal expansion coefficient | ${\beta}_{c}$ | Concentration expansion coefficient |

$\sigma $ | Electric conductivity | ${\gamma}^{*}$ | Spin gradient viscosity | ${k}_{1}^{*}$ | Vertex viscosity |

${j}^{*}$ | Micro inertia per unit mass | $\gamma $ | Inclination parameter | $\prime $ | Differentiation with respect to $\eta $ |

$u$ | Velocity in $x$ direction | $v$ | Velocity in $y$ direction | $x$ | Cartesian coordinate |

$\theta $ | Dimensionless temperature | $S$ | Suction or injection parameter | $k$ | Thermal conductivity |

$\rho $ | Fluid density | $Gc$ | Bouncy parameter | ${B}_{0}$ | Uniform magnetic field strength |

$K$ | Material parameter | $\eta $ | Similarity independent variable | ||

${N}^{*}$ | Non-dimensional angular velocity | $a$ | Stretching rate |

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**Table 1.**Contrast of the reduced Nusselt number $-\theta \prime \left(0\right)$ and the reduced Sherwood number $-\varphi \prime \left(0\right)$ when, $K,{\lambda}_{1},R,M,Gr,Gc,S=0$ and $Le=Pr=10$ with $\gamma =90\xb0$.

$\mathit{N}\mathit{b}$ | $\mathit{N}\mathit{t}$ | Khan and Pop [38] | Present Results | ||
---|---|---|---|---|---|

$-\mathit{\theta}\prime \left(0\right)$ | $-\mathit{\varphi}\prime \left(0\right)$ | $-\mathit{\theta}\prime \left(0\right)$ | $-\mathit{\varphi}\prime \left(0\right)$ | ||

0.1 | 0.1 | 0.9524 | 2.1294 | 0.9524 | 2.1294 |

0.2 | 0.2 | 0.3654 | 2.5152 | 0.3654 | 2.5152 |

0.3 | 0.3 | 0.1355 | 2.6088 | 0.1355 | 2.6088 |

0.4 | 0.4 | 0.0495 | 2.6038 | 0.0495 | 2.6038 |

0.5 | 0.5 | 0.0179 | 2.5731 | 0.0179 | 2.5731 |

**Table 2.**Values of the $-{\theta}^{\prime}\left(0\right),-\varphi \text{\u2019}\left(0\right)$ and ${C}_{fx}\left(0\right)$.

Nb | Nt | Pr | Le | M | K | R | λ_{1} | $\mathit{G}\mathit{r}$ | $\mathit{G}\mathit{c}$ | S | $\mathit{\gamma}$ | $-\mathit{\theta}\prime \left(0\right)$ | $\mathit{\varphi}\prime \left(0\right)$ | ${\mathit{C}}_{\mathit{f}\mathit{x}}\left(0\right)$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | 0.1 | 7.0 | 5.0 | 0.5 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.1 | 45° | 1.1123 | 2.9146 | 1.5227 |

0.5 | 0.1 | 7.0 | 5.0 | 0.5 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.1 | 45° | 0.0978 | 3.1368 | 1.5550 |

0.1 | 0.5 | 7.0 | 5.0 | 0.5 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.1 | 45° | 0.3667 | 4.5619 | 1.4355 |

0.1 | 0.1 | 10.0 | 5.0 | 0.5 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.1 | 45° | 1.1212 | 2.9771 | 1.5206 |

0.1 | 0.1 | 7.0 | 10.0 | 0.5 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.1 | 45° | 0.9984 | 4.5817 | 1.5818 |

0.1 | 0.1 | 7.0 | 5.0 | 1.5 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.1 | 45° | 1.0760 | 2.9014 | 2.0284 |

0.1 | 0.1 | 7.0 | 5.0 | 0.5 | 3.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.1 | 45° | 1.1457 | 2.9288 | 2.0779 |

0.1 | 0.1 | 7.0 | 5.0 | 0.5 | 1.0 | 2.0 | 0.1 | 0.1 | 0.9 | 0.1 | 45° | 1.0466 | 3.8926 | 1.5556 |

0.1 | 0.1 | 7.0 | 5.0 | 0.5 | 1.0 | 1.0 | 0.5 | 0.1 | 0.9 | 0.1 | 45° | 0.3832 | 3.4477 | 1.5301 |

0.1 | 0.1 | 7.0 | 5.0 | 0.5 | 1.0 | 1.0 | 0.1 | 1.0 | 0.9 | 0.1 | 45° | 1.1218 | 2.9175 | 1.3303 |

0.1 | 0.1 | 7.0 | 5.0 | 0.5 | 1.0 | 1.0 | 0.1 | 0.1 | 2.0 | 0.1 | 45° | 1.1237 | 2.9185 | 1.2989 |

0.1 | 0.1 | 7.0 | 5.0 | 0.5 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.5 | 45° | 2.4776 | 3.0066 | 1.7661 |

0.1 | 0.1 | 7.0 | 5.0 | 0.5 | 1.0 | 1.0 | 0.1 | 0.1 | 1.0 | 0.0 | 45° | 0.8317 | 2.8251 | 1.4666 |

0.1 | 0.1 | 7.0 | 5.0 | 0.5 | 1.0 | 1.0 | 0.1 | 0.1 | 1.0 | −0.5 | 45° | −0.0228 | 1.8648 | 1.2083 |

0.1 | 0.1 | 7.0 | 5.0 | 0.5 | 1.0 | 1.0 | 0.1 | 0.1 | 1.0 | 0.1 | 60° | 1.1092 | 2.9136 | 1.5830 |

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**MDPI and ACS Style**

Rafique, K.; Anwar, M.I.; Misiran, M.; Khan, I.; Baleanu, D.; Nisar, K.S.; Sherif, E.-S.M.; Seikh, A.H.
Hydromagnetic Flow of Micropolar Nanofluid. *Symmetry* **2020**, *12*, 251.
https://doi.org/10.3390/sym12020251

**AMA Style**

Rafique K, Anwar MI, Misiran M, Khan I, Baleanu D, Nisar KS, Sherif E-SM, Seikh AH.
Hydromagnetic Flow of Micropolar Nanofluid. *Symmetry*. 2020; 12(2):251.
https://doi.org/10.3390/sym12020251

**Chicago/Turabian Style**

Rafique, Khuram, Muhammad Imran Anwar, Masnita Misiran, Ilyas Khan, Dumitru Baleanu, Kottakkaran Sooppy Nisar, El-Sayed M. Sherif, and Asiful H. Seikh.
2020. "Hydromagnetic Flow of Micropolar Nanofluid" *Symmetry* 12, no. 2: 251.
https://doi.org/10.3390/sym12020251