# Integer and Non-Integer Order Study of the GO-W/GO-EG Nanofluids Flow by Means of Marangoni Convection

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## Abstract

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## 1. Introduction

_{2}SO

_{4}, potassium permanganate KMnO

_{4}, and sodium nitrate NaNO

_{3}.

_{2}O nanofluid flow under the impact of Marangoni convection. The numerical approach to find the solution of a different type of problem was previously discussed [22,23,24,25,26,27]. The numerical scheme of Runge Kutta method of order 4 (RK-4) was used in their study to determine the impact of the physical parameters and numerical outputs.

## 2. Problem Formulation

## 3. Preliminaries on the Caputo Fractional Derivatives

#### 3.1. Definition 1

#### 3.2. Property 1

#### 3.3. Property 2

## 4. Solution Methodology

## 5. Results and Discussions

## 6. Conclusions

- The rising values of $\varphi $ lead to the linear enhancement of the velocity field, which was observed more clearly in the non-integer case compared with the classical model.
- The increasing values of the magnetic parameter increase the temperature field and decrease the Nusselt number. This effect is somewhat better in the fractional case compared to the integer model.
- Due to the rising values of $\varphi $, the thermal boundary layer increases and this effect is somewhat better in the GO-EG nanofluid rather than the GO-W nanofluid.
- The cooling efficiency and heat transfer of the GO-EG nanofluid is far better than that of the GO-W nanofluid.
- With the Lorentz force, resistance arises in the transport phenomenon. This particular phenomenon controls the GO-W and GO-EG nanofluid velocities. Also, this effect is more visible in GO-EG than in GO-W.
- Due to the fractional order $\alpha =1,0.95,0.90,0.85,$ the heat transfer rate enhances in growing increments and this effect is far better in the GO-W nanofluid compared with the GO-EG nanofluid.

## 7. Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**The impact of $M$ versus classical $\mathsf{\Theta}(\eta )$, when $\varphi =0.2,\mathrm{Pr}=6.7$.

**Figure 7.**The impact of $M$ versus fractional $\mathsf{\Theta}(\eta )$, when $\varphi =0.2,\mathrm{Pr}=6.7$.

**Figure 8.**The impact of $\varphi $ versus the integer order of $\mathsf{\Theta}(\eta )$, when $M=0.1,\mathrm{Pr}=6.7$.

**Figure 9.**The impact of $\varphi $ versus fractional $\mathsf{\Theta}(\eta )$, when $M=0.1,\mathrm{Pr}=6.7$.

**Table 1.**The experimental values (thermophysical properties) of water, ethylene glycol, and graphene oxide nanoparticles.

Model | $\mathit{\rho}\left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^{3}\right)$ | ${\mathit{C}}_{\mathit{p}}\left(\mathbf{k}{\mathbf{g}}^{-1}/{\mathbf{k}}^{-1}\right)$ | $\mathit{k}\left(\mathbf{W}{\mathbf{m}}^{-1}{\mathbf{k}}^{-1}\right)$ |
---|---|---|---|

Water (W) | 997.1 | 4179 | 0.613 |

Graphene oxide (GO) | 1800 | 717 | 5000 |

Ethylene glycol (EG) | 1.115 | 0.58 | 0.1490 |

$\begin{array}{l}\mathit{\alpha}=1\\ \mathit{\eta}.\end{array}$ | $\begin{array}{l}{\mathsf{\Theta}}^{\prime}(0)\\ \mathit{G}\mathit{O}-\mathit{W}\end{array}$ | $\begin{array}{l}{\mathsf{\Theta}}^{\prime}(0)\\ \mathit{G}\mathit{O}-\mathit{E}\mathit{G}\end{array}$ | $\begin{array}{l}\mathit{\alpha}=0.95\\ \mathit{\eta}.\end{array}$ | $\begin{array}{l}{\mathsf{\Theta}}^{\prime}(0)\\ \mathit{G}\mathit{O}-\mathit{W}\end{array}$ | $\begin{array}{l}{\mathsf{\Theta}}^{\prime}(0)\\ \mathit{G}\mathit{O}-\mathit{E}\mathit{G}\end{array}$ | $\begin{array}{l}\mathit{\alpha}=0.9\\ \mathit{\eta}.\end{array}$ | $\begin{array}{l}{\mathsf{\Theta}}^{\prime}(0)\\ \mathit{G}\mathit{O}-\mathit{W}\end{array}$ | $\begin{array}{l}{\mathsf{\Theta}}^{\prime}(0)\\ \mathit{G}\mathit{O}-\mathit{E}\mathit{G}\end{array}$ | $\begin{array}{l}\mathit{\alpha}=0.85\\ \mathit{\eta}.\end{array}$ | $\begin{array}{l}{\mathsf{\Theta}}^{\prime}(0)\\ \mathit{G}\mathit{O}-\mathit{W}\end{array}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | 1.0921 | 1.0903 | 0.1 | 1.1039 | 1.1014 | 0.1 | 1.1167 | 1.1133 | 0.1 | 1.1304 | 1.1259 |

0.2 | 1.1708 | 1.1639 | 0.2 | 1.1845 | 1.1758 | 0.2 | 1.1983 | 1.1875 | 0.2 | 1.2121 | 1.1986 |

0.3 | 1.2380 | 1.2234 | 0.3 | 1.2504 | 1.2328 | 0.3 | 1.2624 | 1.2413 | 0.3 | 1.2736 | 1.2486 |

0.4 | 1.2953 | 1.2706 | 0.4 | 1.3051 | 1.2764 | 0.4 | 1.3141 | 1.2809 | 0.4 | 1.3221 | 1.2840 |

0.5 | 1.3442 | 1.3076 | 0.5 | 1.3509 | 1.3095 | 0.5 | 1.3567 | 1.3101 | 0.5 | 1.3617 | 1.3096 |

0.6 | 1.3862 | 1.3361 | 0.6 | 1.3899 | 1.3344 | 0.6 | 1.3928 | 1.3318 | 0.6 | 1.3952 | 1.3288 |

0.7 | 1.4225 | 1.3578 | 0.7 | 1.4235 | 1.3532 | 0.7 | 1.4242 | 1.3484 | 0.7 | 1.4250 | 1.3441 |

0.8 | 1.4544 | 1.3741 | 0.8 | 1.4534 | 1.3676 | 0.8 | 1.4527 | 1.3619 | 0.8 | 1.4527 | 1.3577 |

0.9 | 1.4832 | 1.3866 | 0.9 | 1.4809 | 1.3794 | 0.9 | 1.4796 | 1.3741 | 0.9 | 1.4800 | 1.3715 |

1.0 | 1.5098 | 1.3966 | 1.0 | 1.5072 | 1.3900 | 1. | 1.5064 | 1.3866 | 1.0 | 1.5083 | 1.3872 |

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**MDPI and ACS Style**

Gul, T.; Anwar, H.; Khan, M.A.; Khan, I.; Kumam, P.
Integer and Non-Integer Order Study of the GO-W/GO-EG Nanofluids Flow by Means of Marangoni Convection. *Symmetry* **2019**, *11*, 640.
https://doi.org/10.3390/sym11050640

**AMA Style**

Gul T, Anwar H, Khan MA, Khan I, Kumam P.
Integer and Non-Integer Order Study of the GO-W/GO-EG Nanofluids Flow by Means of Marangoni Convection. *Symmetry*. 2019; 11(5):640.
https://doi.org/10.3390/sym11050640

**Chicago/Turabian Style**

Gul, Taza, Haris Anwar, Muhammad Altaf Khan, Ilyas Khan, and Poom Kumam.
2019. "Integer and Non-Integer Order Study of the GO-W/GO-EG Nanofluids Flow by Means of Marangoni Convection" *Symmetry* 11, no. 5: 640.
https://doi.org/10.3390/sym11050640