# Finite Element Study of Magnetohydrodynamics (MHD) and Activation Energy in Darcy–Forchheimer Rotating Flow of Casson Carreau Nanofluid

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## Abstract

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## 1. Introduction

## 2. Statement of The Problem

## 3. Governing Equations

## 4. Finite Element Method Solutions

#### 4.1. Variational Formulations

#### 4.2. Finite Element Formulations

## 5. Results and Discussion

## 6. Conclusions

- The progressing values of Casson fluid parameter $\beta $, magnetic parameter M, porosity $kp$, Forchheimer number (Fr), and Weissenberg number (We) reduced the magnitude of secondary velocity $h(\xi ,\eta )$ and the primary velocity ${f}^{\prime}(\xi ,\eta )$ but concentration $\varphi $ and temperature $\theta $ are incremented.
- The concentration and temperature are incremented along with rising values of $\lambda $. The both components of velocity diminishes near the surface when $\lambda $ is incremented.
- Improvement in temperature is reported when Nb, Nt, and ${Q}_{s}$ are increased but inverse behaviour is noted for power-law index ($\u03f5$).
- The higher values of Nb, power-law index ($\u03f5$), and chemical reaction rate parameter $\mathsf{\Omega}$ recede the nanoparticle concentration but it increases against higher values of Nt and EE.
- The Nusselt number attains lower values for higher values of Casson fluid parameter $\beta $, Brownian (Nb), thermophoresis (Nt), magnetic (M), heat source ${Q}_{s}$, and rotating ($\lambda $).
- The Sherwood number attains higher values for higher values of Lewis number (Le), Nb, and $Nt$ but the reduced Sherwood number exhibits an opposite trend for magnetic, rotational, porosity $kp$, Forchheimer number (Fr), and Casson fluid parameter $\beta $ parameters.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Fluctuation of ${\tilde{f}}^{\prime}(\xi ,\eta )$, $\tilde{h}(\xi ,\eta )$, $\tilde{\theta}(\xi ,\eta )$, and $\tilde{\varphi}(\xi ,\eta )$ along with kp and Fr.

**Figure 4.**Fluctuation of ${\tilde{f}}^{\prime}(\xi ,\eta )$, $\tilde{h}(\xi ,\eta )$, $\tilde{\theta}(\xi ,\eta )$, and $\tilde{\varphi}(\xi ,\eta )$ along with M and $\beta $.

**Figure 5.**Fluctuation of ${\tilde{f}}^{\prime}(\xi ,\eta )$, $\tilde{h}(\xi ,\eta )$, $\tilde{\theta}(\xi ,\eta )$, and $\tilde{\varphi}(\xi ,\eta )$ along with $\lambda $ and $\tau $.

**Figure 6.**Fluctuation of ${\tilde{f}}^{\prime}(\xi ,\eta )$, $\tilde{h}(\xi ,\eta )$, $\tilde{\theta}(\xi ,\eta )$, and $\tilde{\varphi}(\xi ,\eta )$ along with We and $\u03f5$.

**Figure 7.**Fluctuation of $\tilde{\theta}(\xi ,\eta )$ along with Nb, Nt, ${\gamma}_{T}$, and ${Q}_{s}$.

**Figure 8.**Fluctuation of $\tilde{\varphi}(\xi ,\eta )$ along with $EE$, $\mathsf{\Omega}$, Nt, and Nb.

**Figure 9.**Fluctuation of $C{f}_{x}R{e}_{x}^{1/2}$ and $C{f}_{y}R{e}_{y}^{1/2}$ along with $Kp$ and Fr.

**Figure 10.**Fluctuation of $C{f}_{x}R{e}_{x}^{1/2}$ and $C{f}_{y}R{e}_{y}^{1/2}$ along with M and $\beta $.

**Figure 11.**Fluctuation of $N{u}_{x}R{e}_{x}^{1/2}$ and $S{h}_{x}R{e}_{x}^{1/2}$ along with Nb, Nt, M, and $\beta $.

**Figure 12.**Fluctuation of $N{u}_{x}R{e}_{x}^{1/2}$ and $S{h}_{x}R{e}_{x}^{1/2}$ along with Nb, Nt, $\lambda $, and ${Q}_{s}$.

**Figure 13.**Fluctuation of $S{h}_{x}R{e}_{x}^{1/2}$ along with Kp, Fr, EE, Le, and $\mathsf{\Omega}$.

**Table 1.**Comparison of skin friction coefficients for different values of kp, Fr, and $\lambda $ when $M=We=0$, $\beta \to \infty $ at $\xi =1$.

kp | Fr | $\mathit{\lambda}$ | Hayat et al. [14] | FEM (Our Results) | ||
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${\mathit{Cf}}_{\mathit{x}}{\mathit{Re}}_{\mathit{x}}^{\mathbf{1}/\mathbf{2}}$ | ${\mathit{Cf}}_{\mathit{y}}{\mathit{Re}}_{\mathit{y}}^{\mathbf{1}/\mathbf{2}}$ | ${\mathit{Cf}}_{\mathit{x}}{\mathit{Re}}_{\mathit{x}}^{\mathbf{1}/\mathbf{2}}$ | ${\mathit{Cf}}_{\mathit{y}}{\mathit{Re}}_{\mathit{y}}^{\mathbf{1}/\mathbf{2}}$ | |||

0.0 | 0.1 | 0.2 | −1.06329 | −0.23769 | −1.063285 | −0.237676 |

0.1 | −1.10532 | −0.22319 | −1.105309 | −0.223185 | ||

0.2 | −1.14675 | −0.21087 | −1.146743 | −0.210873 | ||

0.2 | 0.1 | 0.2 | −1.14675 | −0.21087 | −1.146743 | −0.210873 |

0.2 | −1.17424 | −0.20994 | −1.174241 | −0.209930 | ||

0.3 | −1.20121 | −0.20905 | −1.201216 | −0.209058 | ||

0.2 | 0.1 | 0.05 | −1.12589 | −0.05425 | −1.125878 | −0.054246 |

0.1 | −1.13024 | −0.10786 | −1.130254 | −0.107849 | ||

0.2 | −1.14675 | −0.21087 | −1.146759 | −0.210873 |

**Table 2.**Comparison of Nusselt number $-{\theta}^{\prime}\left(0\right)$ for kp, $\lambda $, Fr, and Pr at $\xi =1$ when M = We = 0, Nt = Nb = 0, ${\gamma}_{T}={Q}_{s}=0$, and $\beta \to \infty $.

kp | $\mathit{\lambda}$ | Fr | Pr | Rashid et al. [10] | FEM (Our Results) |
---|---|---|---|---|---|

0.0 | 0.5 | 1.0 | 1.0 | 0.508972 | 0.508969 |

1.0 | 0.485852 | 0.485847 | |||

2.0 | 0.457520 | 0.457459 | |||

0.2 | 0.1 | 1.0 | 1.0 | 0.542198 | 0.542187 |

0.5 | 0.506965 | 0.506956 | |||

0.9 | 0.467794 | 0.467790 | |||

0.2 | 0.5 | 0.0 | 2.0 | 0.844615 | 0.844608 |

2.0 | 0.783173 | 0.783165 | |||

3.0 | 0.734272 | 0.734277 | |||

0.2 | 0.5 | 1.0 | 2.0 | 0.811336 | 0.811327 |

3.0 | 1.064610 | 1.064582 | |||

4.0 | 1.279790 | 1.279804 |

**Table 3.**Comparison of Nusselt number $-{\theta}^{\prime}\left(0\right)$ for $\lambda $ and M at $\xi =1,Pr=2.0$ when other parameters are ignore.

$\mathit{\lambda}$ | Abbas et al. [34] | FEM (Our results) | ||||
---|---|---|---|---|---|---|

M = 0.5 | M = 1.0 | M = 2.0 | M = 0.5 | M = 1.0 | M = 2.0 | |

0.0 | 0.886 | 0.823 | 0.668 | 0.8862 | 0.8230 | 0.6682 |

0.5 | 0.841 | 0.800 | 0.663 | 0.8408 | 0.8003 | 0.6627 |

1.0 | 0.768 | 0.750 | 0.648 | 0.7684 | 0.7501 | 0.6483 |

2.0 | 0.641 | 0.643 | 0.603 | 0.6411 | 0.6429 | 0.6030 |

5.0 | 0.447 | 0.449 | 0.461 | 0.4467 | 0.4494 | 0.4612 |

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**MDPI and ACS Style**

Ali, B.; Rasool, G.; Hussain, S.; Baleanu, D.; Bano, S.
Finite Element Study of Magnetohydrodynamics (MHD) and Activation Energy in Darcy–Forchheimer Rotating Flow of Casson Carreau Nanofluid. *Processes* **2020**, *8*, 1185.
https://doi.org/10.3390/pr8091185

**AMA Style**

Ali B, Rasool G, Hussain S, Baleanu D, Bano S.
Finite Element Study of Magnetohydrodynamics (MHD) and Activation Energy in Darcy–Forchheimer Rotating Flow of Casson Carreau Nanofluid. *Processes*. 2020; 8(9):1185.
https://doi.org/10.3390/pr8091185

**Chicago/Turabian Style**

Ali, Bagh, Ghulam Rasool, Sajjad Hussain, Dumitru Baleanu, and Sehrish Bano.
2020. "Finite Element Study of Magnetohydrodynamics (MHD) and Activation Energy in Darcy–Forchheimer Rotating Flow of Casson Carreau Nanofluid" *Processes* 8, no. 9: 1185.
https://doi.org/10.3390/pr8091185