# MHD Flow and Heat Transfer in Sodium Alginate Fluid with Thermal Radiation and Porosity Effects: Fractional Model of Atangana–Baleanu Derivative of Non-Local and Non-Singular Kernel

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Framing of the Problem

## 3. Problem Solution, Skin Friction, and Nusselt Number

## 4. Skin Friction and Nusselt Number

## 5. Discussion

## 6. Conclusions

- ➣
- Velocity rises for a large value of $Gr,\hspace{0.17em}\hspace{0.17em}\gamma \hspace{0.17em}$, and $t.$
- ➣
- Velocity reduces for a large value of $M,\hspace{0.17em}\mathrm{Pr},\hspace{0.17em}\hspace{0.17em}N$, and $K$.
- ➣
- Temperature is increased by increasing $t$ and $\alpha $, while decreasing with the increase of $\mathrm{Pr}$.
- ➣
- The temperature and velocity of the fractional fluid model converge faster compared to an ordinary fluid model.
- ➣
- The Atangana–Baleanu fractional model reduced the velocity profile up to 45.76% and temperature profile up to 13.74% compared to an ordinary model.

**Suggestions for future research work.**

- ➣
- The researchers extend this work for different kind of nanofluids.
- ➣
- The authors also can take this model in different geometries.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Plots of velocity for four different $\gamma $ when $K=0.5,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ $\mathrm{Pr}=7.2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ $M=1,$ $Gr=10,\hspace{0.17em}\hspace{0.17em}$ $N=0.5\hspace{0.17em}$, and $t=10$.

**Figure 3.**Plots of velocity for four different $Gr$ when $K=0.5,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ $\mathrm{Pr}=7.2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ $\gamma =0.3,\hspace{0.17em}\hspace{0.17em}$ $M=1,$ $N=0.5\hspace{0.17em}$, and $t=10$.

**Figure 4.**Plots of velocity for four different $K$ when $\mathrm{Pr}=7.2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ $\gamma =0.3,\hspace{0.17em}\hspace{0.17em}$ $M=1,\hspace{0.17em}\hspace{0.17em}Gr=10$.$N=0.5\hspace{0.17em}$, and $t=10$.

**Figure 5.**Plots of velocity for four different $M$ values when $K=0.5,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ $\mathrm{Pr}=7.2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ $\gamma =0.3,\hspace{0.17em}\hspace{0.17em}$ $\hspace{0.17em}Gr=10$, $N=0.5\hspace{0.17em}$, and $t=10$.

**Figure 6.**Plots of velocity for four different $N$ values when $K=0.5,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ $\mathrm{Pr}=7.2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ $\gamma =0.3,\hspace{0.17em}\hspace{0.17em}$ $M=1,\hspace{0.17em}\hspace{0.17em}Gr=10$, and $t=10$.

**Figure 7.**Plots of velocity for four different $\mathrm{Pr}$ values when $K=0.5,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ $\gamma =0.3,\hspace{0.17em}\hspace{0.17em}$ $M=1,\hspace{0.17em}\hspace{0.17em}Gr=10$, $N=0.5\hspace{0.17em}$, and $t=10$.

**Figure 8.**Plots of temperature for four different $\mathrm{Pr}$ values when $N=0.5,\hspace{0.17em}\hspace{0.17em}\alpha =0.1$ and $t=10$.

**Figure 9.**Plots of velocity for four different $t$ values when $K=0.5,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ $\mathrm{Pr}=7.2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ $\gamma =0.3,\hspace{0.17em}\hspace{0.17em}$ $M=1,\hspace{0.17em}\hspace{0.17em}Gr=10$, and $N=0.5$.

**Figure 10.**Plots of temperature for four different $t$ values when $\mathrm{Pr}=7.2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\alpha =0.1$, and $N=0.5\hspace{0.17em}$.

**Figure 11.**Plots of temperature for four different $\alpha $ values when $\mathrm{Pr}=7.2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}N=0.5$, and $t=10\hspace{0.17em}$.

**Figure 12.**Comparison of fractional SA fluid and ordinary SA fluid (velocity) when $K=0.5,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$ $\mathrm{Pr}=7.2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}$$\gamma =0.3,\hspace{0.17em}\hspace{0.17em}$$M=1,\hspace{0.17em}\hspace{0.17em}Gr=10$ $N=0.5\hspace{0.17em}$, and $t=10$.

**Figure 13.**Comparison of fractional SA fluid and ordinary SA fluid (temperature) when $\mathrm{Pr}=7.2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\alpha =0.1$, $N=0.5\hspace{0.17em}$, and $t=10$.

$\mathit{t}$ | $\mathbf{Pr}$ | $\mathit{N}$ | $\mathit{\alpha}$ | $\mathit{N}\mathit{u}$ |
---|---|---|---|---|

1 | 0.7 | 0.5 | 0.7 | 0.858 |

2 | 0.739 | |||

3 | 0.684 | |||

7 | 2.297 | |||

9 | 2.592 | |||

1.5 | 1.641 | |||

2.5 | 2.585 | |||

0.8 | 0.795 | |||

0.9 | 0.713 |

$\mathit{t}$ | $\mathit{\alpha}$ | $\mathit{M}$ | $\mathbf{Pr}$ | $\mathit{N}$ | $\mathit{\gamma}$ | $\mathit{G}\mathit{r}$ | $\mathit{K}$ | ${\mathit{C}}_{\mathit{f}}$ |
---|---|---|---|---|---|---|---|---|

0.3 | 0.5 | 0.4 | 0.7 | 0.5 | 0.5 | 0.1 | 1.5 | 0.366 |

1.3 | 0.510 | |||||||

1 | 0.476 | |||||||

0.7 | 0.411 | |||||||

0.9 | 0.488 | |||||||

8.4 | 0.275 | |||||||

19.4 | 0.218 | |||||||

7.2 | 0.451 | |||||||

9.2 | 0.916 | |||||||

2.5 | 0.245 | |||||||

3.5 | 0.168 | |||||||

1.5 | 0.159 | |||||||

2.5 | 0.123 | |||||||

0.3 | 1.097 | |||||||

0.5 | 1.829 | |||||||

4.5 | 0.268 | |||||||

10.5 | 0.194 |

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**MDPI and ACS Style**

Khan, A.; Khan, D.; Khan, I.; Taj, M.; Ullah, I.; Aldawsari, A.M.; Thounthong, P.; Sooppy Nisar, K.
MHD Flow and Heat Transfer in Sodium Alginate Fluid with Thermal Radiation and Porosity Effects: Fractional Model of Atangana–Baleanu Derivative of Non-Local and Non-Singular Kernel. *Symmetry* **2019**, *11*, 1295.
https://doi.org/10.3390/sym11101295

**AMA Style**

Khan A, Khan D, Khan I, Taj M, Ullah I, Aldawsari AM, Thounthong P, Sooppy Nisar K.
MHD Flow and Heat Transfer in Sodium Alginate Fluid with Thermal Radiation and Porosity Effects: Fractional Model of Atangana–Baleanu Derivative of Non-Local and Non-Singular Kernel. *Symmetry*. 2019; 11(10):1295.
https://doi.org/10.3390/sym11101295

**Chicago/Turabian Style**

Khan, Arshad, Dolat Khan, Ilyas Khan, Muhammad Taj, Imran Ullah, Abdullah Mohammed Aldawsari, Phatiphat Thounthong, and Kottakkaran Sooppy Nisar.
2019. "MHD Flow and Heat Transfer in Sodium Alginate Fluid with Thermal Radiation and Porosity Effects: Fractional Model of Atangana–Baleanu Derivative of Non-Local and Non-Singular Kernel" *Symmetry* 11, no. 10: 1295.
https://doi.org/10.3390/sym11101295