# Stability Analysis and Dual Solutions of Micropolar Nanofluid over the Inclined Stretching/Shrinking Surface with Convective Boundary Condition

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

## 3. Stability Analysis

## 4. Numerical Method

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Comparison of ${f}^{\u2033}\left(0\right)$ with results of Bhattacharyya et al., when $m=0.5,\lambda =-1,\xi ={90}^{\xb0}.$

**Figure 4.**${f}^{\u2033}\left(0\right)$ versus the stretching/shrinking parameter $\lambda $ for varying angle $\xi $.

**Figure 6.**${g}^{\prime}\left(0\right)$ versus the stretching/shrinking parameter $\lambda $ for varying angle $\xi $.

**Figure 7.**$-{\theta}^{\prime}\left(0\right)$ versus the suction S for varying material parameter $K$.

**Figure 8.**$-{\theta}^{\prime}\left(0\right)$ versus the stretching/shrinking parameter $\lambda $ for varying angle $\xi $.

**Figure 9.**$-{\varnothing}^{\prime}\left(0\right)$ versus the suction S for varying material parameter $K$.

**Figure 10.**$-{\varnothing}^{\prime}\left(0\right)$ versus the stretching/shrinking parameter $\lambda $ for varying angle $\xi $.

**Table 1.**Comparison of results for the reduced Nusselt and Sherwood numbers when $S=0,\lambda =1,Bi=\infty ,\xi ={90}^{\xb0},K=0,Sc=10,Pr=$ 10.

${\mathit{N}}_{\mathit{b}}$ | ${\mathit{N}}_{\mathit{t}}$ | Khan and | Pop [56] | Present | Results |
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$-{\mathit{\theta}}^{\prime}\left(0\right)$ | $-{\varnothing}^{\prime}\left(0\right)$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ | $-{\varnothing}^{\prime}\left(0\right)$ | ||

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

0.3 | 0.5201 | 2.5286 | 0.52005 | 2.5285 | |

0.5 | 0.3211 | 3.0351 | 0.3212 | 3.0351 | |

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

0.3 | 0.2731 | 2.6555 | 0.2731 | 2.6555 | |

0.5 | 0.1681 | 2.8883 | 0.1681 | 2.8883 |

K | $\mathit{S}$ | ${\mathit{\epsilon}}_{1}$ | |
---|---|---|---|

1st Solution | 2nd Solution | ||

0 | 3 | 1.53376 | −1.51742 |

2.5 | 1.08913 | −1.20287 | |

2 | 0.97563 | −0.8101 | |

0.5 | 3 | 0.86261 | −0.96571 |

2.5 | 0.54185 | −0.67231 | |

2 | 0.05935 | −0.03765 | |

1 | 3 | 0.45512 | −0.52843 |

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**MDPI and ACS Style**

Lund, L.A.; Omar, Z.; Khan, U.; Khan, I.; Baleanu, D.; Nisar, K.S.
Stability Analysis and Dual Solutions of Micropolar Nanofluid over the Inclined Stretching/Shrinking Surface with Convective Boundary Condition. *Symmetry* **2020**, *12*, 74.
https://doi.org/10.3390/sym12010074

**AMA Style**

Lund LA, Omar Z, Khan U, Khan I, Baleanu D, Nisar KS.
Stability Analysis and Dual Solutions of Micropolar Nanofluid over the Inclined Stretching/Shrinking Surface with Convective Boundary Condition. *Symmetry*. 2020; 12(1):74.
https://doi.org/10.3390/sym12010074

**Chicago/Turabian Style**

Lund, Liaquat Ali, Zurni Omar, Umair Khan, Ilyas Khan, Dumitru Baleanu, and Kottakkaran Sooppy Nisar.
2020. "Stability Analysis and Dual Solutions of Micropolar Nanofluid over the Inclined Stretching/Shrinking Surface with Convective Boundary Condition" *Symmetry* 12, no. 1: 74.
https://doi.org/10.3390/sym12010074