# Keller-Box Analysis of Buongiorno Model with Brownian and Thermophoretic Diffusion for Casson Nanofluid over an Inclined Surface

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

## 3. Results and Discussion

## 4. Conclusions

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- The temperature profile increases near the boundary layer by improving the Biot number.
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- The velocity and temperature profiles drop by improving the nonlinear power index.
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- The heat exchange improved upon improving the radiation parameter.
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- The velocity distribution retards by increasing the Casson parameter.
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- The Nuselt number decreases by increasing the Casson parameter.
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- The skin friction declines by improving the Biot number.
- ➢
- The velocity profile shows an inverse relation with the inclination factor.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Contrast of the reduced Nusselt number $-\theta {}^{\prime}\left(0\right)$ and the reduced Sherwood number $-\varphi {}^{\prime}\left(0\right)$ against $\gamma $ = 90°, ${\gamma}_{1}\to \infty $, $\beta \to \infty $, $M,N,\lambda ,\delta =0$, with $m=1$, and $Pr=Le=10$.

Nb | Nt | Khan and Pop [45] | 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 reduced Nusselt number $-\theta {}^{\prime}\left(0\right)$, the reduced Sherwood number $-\varphi {}^{\prime}\left(0\right)$ and the skin friction coefficient ${C}_{fx}\left(0\right)$.

Nb | Nt | Pr | Le | M | N | $\mathit{\beta}$ | λ | δ | ${\mathit{\gamma}}_{1}$ | m | $\mathit{\gamma}$ | $-\mathit{\theta}{}^{\prime}\left(0\right)$ | $-\mathit{\varphi}{}^{\prime}\left(0\right)$ | ${\mathit{C}}_{\mathit{f}\mathit{x}}\left(0\right)$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | 0.1 | 6.5 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.9 | 0.1 | 0.5 | 45° | 0.0936 | 1.6159 | 0.5417 |

0.5 | 0.1 | 6.5 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.9 | 0.1 | 0.5 | 45° | 0.0447 | 1.6541 | 0.5449 |

0.1 | 0.13 | 6.5 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.9 | 0.1 | 0.5 | 45° | 0.0961 | 1.6038 | 0.5406 |

0.1 | 0.1 | 10.0 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.9 | 0.1 | 0.5 | 45° | 0.0563 | 1.7133 | 0.6235 |

0.1 | 0.1 | 6.5 | 10.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.9 | 0.1 | 0.5 | 45° | 0.0919 | 2.3622 | 0.5785 |

0.1 | 0.1 | 6.5 | 5.0 | 0.3 | 1.0 | 1.0 | 0.1 | 0.9 | 0.1 | 0.5 | 45° | 0.0957 | 1.5958 | 0.6332 |

0.1 | 0.1 | 6.5 | 5.0 | 0.1 | 5.0 | 1.0 | 0.1 | 0.9 | 0.1 | 0.5 | 45° | 0.0527 | 1.6374 | 0.5421 |

0.1 | 0.1 | 6.5 | 5.0 | 0.1 | 1.0 | 5.0 | 0.1 | 0.9 | 0.1 | 0.5 | 45° | 0.0916 | 1.5833 | 0.6563 |

0.1 | 0.1 | 6.5 | 5.0 | 0.1 | 1.0 | 1.0 | 1.0 | 0.9 | 0.1 | 0.5 | 45° | 0.0952 | 1.6184 | 0.5250 |

0.1 | 0.1 | 6.5 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 2.0 | 0.1 | 0.5 | 45° | 0.0949 | 1.6376 | 0.3809 |

0.1 | 0.1 | 6.5 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.9 | 0.2 | 0.5 | 45° | 0.1779 | 1.5828 | 0.5363 |

0.1 | 0.1 | 6.5 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.9 | 0.1 | 1.5 | 45° | 0.1253 | 1.5763 | 0.6968 |

0.1 | 0.1 | 6.5 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.9 | 0.1 | 0.5 | 60° | 0.0968 | 1.6098 | 0.5818 |

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**MDPI and ACS Style**

Rafique, K.; Imran Anwar, M.; Misiran, M.; Khan, I.; Alharbi, S.O.; Thounthong, P.; Nisar, K.S.
Keller-Box Analysis of Buongiorno Model with Brownian and Thermophoretic Diffusion for Casson Nanofluid over an Inclined Surface. *Symmetry* **2019**, *11*, 1370.
https://doi.org/10.3390/sym11111370

**AMA Style**

Rafique K, Imran Anwar M, Misiran M, Khan I, Alharbi SO, Thounthong P, Nisar KS.
Keller-Box Analysis of Buongiorno Model with Brownian and Thermophoretic Diffusion for Casson Nanofluid over an Inclined Surface. *Symmetry*. 2019; 11(11):1370.
https://doi.org/10.3390/sym11111370

**Chicago/Turabian Style**

Rafique, Khuram, Muhammad Imran Anwar, Masnita Misiran, Ilyas Khan, Sayer O. Alharbi, Phatiphat Thounthong, and Kottakkaran Sooppy Nisar.
2019. "Keller-Box Analysis of Buongiorno Model with Brownian and Thermophoretic Diffusion for Casson Nanofluid over an Inclined Surface" *Symmetry* 11, no. 11: 1370.
https://doi.org/10.3390/sym11111370