# Fractional Order Forced Convection Carbon Nanotube Nanofluid Flow Passing Over a Thin Needle

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Preliminaries on the Caputo Fractional Derivatives

**Definition**

**1.**

**Property**

**1.**

**Property**

**2.**

## 4. Solution Methodology

## 5. Results and Discussion

## 6. Conclusions

- Greater values of $\mathrm{Pr}$ cause decreases in the thickness of the thermal boundary layer when using the classical model, but by means of the fractional model for the same values of the Prandtl number, the thermal boundary layer near the needle surface increases and decreases after the critical point.
- Lower values of n lead to a decrease in the temperature profile using the classical model values, and this effect is upturned for the fractional order values $\alpha =0.95,0.90$ near the wall and change to an upsurge in the thermal boundary layer after the point of inflection.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Physical Properties | $\mathbf{Density}\text{}\mathit{\rho}(\mathit{K}\mathit{g}/{\mathit{m}}^{3})$ | $\mathbf{Thermal}\text{}\mathbf{Conduct}\text{}\mathit{k}(\mathit{W}{\mathit{m}}^{-1}/{\mathit{k}}^{-1})$ | $\mathbf{Specific}\text{}\mathbf{Heat}{\mathit{c}}_{\mathit{p}}(\mathit{K}{\mathit{g}}^{-1}/{\mathit{k}}^{-1})$ | |
---|---|---|---|---|

Base fluid Water | 997 | 0.613 | 4179 | |

Nanoparticles | SWCNT | 2600 | 6600 | 425 |

MWCNT | 1600 | 3000 | 796 |

**Table 2.**The classical and fractional order comparison for the skin fraction comprising (SWCNTs/MWCNTs). When $\alpha =1,0.95,0.90,\mathrm{Pr}=0.0005,\varphi =0.0001,m=0.01,n=0.$

$\mathit{\alpha}=1,\mathit{\eta}.$ | $\begin{array}{l}{\mathit{f}}^{\u2033}(\mathit{a})\\ \mathit{S}\mathit{W}\mathit{C}\mathit{N}\mathit{T}\mathit{s}\end{array}$ | $\begin{array}{l}{\mathit{f}}^{\u2033}(\mathit{a})\\ \mathit{M}\mathit{W}\mathit{C}\mathit{N}\mathit{T}\mathit{s}\end{array}$ | $\mathit{\alpha}=0.95,\mathit{\eta}.$ | $\begin{array}{l}{\mathit{f}}^{\u2033}(\mathit{a})\\ \mathit{S}\mathit{W}\mathit{C}\mathit{N}\mathit{T}\mathit{s}\end{array}$ | $\begin{array}{l}{\mathit{f}}^{\u2033}(\mathit{a})\\ \mathit{M}\mathit{W}\mathit{C}\mathit{N}\mathit{T}\mathit{s}\end{array}$ | $\mathit{\alpha}=0.90,\mathit{\eta}.$ | $\begin{array}{l}{\mathit{f}}^{\u2033}(\mathit{a})\\ \mathit{S}\mathit{W}\mathit{C}\mathit{N}\mathit{T}\mathit{s}\end{array}$ | $\begin{array}{l}{\mathit{f}}^{\u2033}(\mathit{a})\\ \mathit{M}\mathit{W}\mathit{C}\mathit{N}\mathit{T}\mathit{s}\end{array}$ |
---|---|---|---|---|---|---|---|---|

0.1 | 0.0988 | 0.0988 | 0.1 | 0.0985 | 0.0985 | 0.1 | 0.0982 | 0.0982 |

0.2 | 0.0967 | 0.0967 | 0.2 | 0.0960 | 0.0961 | 0.2 | 0.0953 | 0.0954 |

0.3 | 0.0938 | 0.0939 | 0.3 | 0.0927 | 0.0928 | 0.3 | 0.0915 | 0.0916 |

0.4 | 0.0901 | 0.0903 | 0.4 | 0.0886 | 0.0888 | 0.4 | 0.0869 | 0.0871 |

0.5 | 0.0856 | 0.0859 | 0.5 | 0.0837 | 0.0839 | 0.5 | 0.0815 | 0.0818 |

0.6 | 0.0804 | 0.0807 | 0.6 | 0.0781 | 0.0784 | 0.6 | 0.0755 | 0.0758 |

0.7 | 0.0745 | 0.0749 | 0.7 | 0.0717 | 0.0722 | 0.7 | 0.0688 | 0.0693 |

0.8 | 0.0679 | 0.0683 | 0.8 | 0.0648 | 0.0653 | 0.8 | 0.0615 | 0.0621 |

0.9 | 0.0606 | 0.0612 | 0.9 | 0.0572 | 0.0579 | 0.9 | 0.0538 | 0.0545 |

1.0 | 0.0527 | 0.0534 | 1.0 | 0.0492 | 0.0499 | 1.0 | 0.0456 | 0.0464 |

**Table 3.**The classical and fractional order comparison for the Nusselt number (SWCNTs/MWCNTs). When $\alpha =1,0.95,0.90,\mathrm{Pr}=0.0005,\varphi =0.0001,m=0.0,n=1.$

$\begin{array}{l}\mathit{\alpha}=1,\\ \mathit{\eta}.\end{array}$ | $\begin{array}{l}{\mathbf{\Theta}}^{\prime}(\mathit{a})\\ \mathit{S}\mathit{W}\mathit{C}\mathit{N}\mathit{T}\mathit{s}\end{array}$ | $\begin{array}{l}{\mathbf{\Theta}}^{\prime}(\mathit{a})\\ \mathit{M}\mathit{W}\mathit{C}\mathit{N}\mathit{T}\mathit{s}\end{array}$ | $\begin{array}{l}\mathit{\alpha}=0.95,\\ \mathit{\eta}.\end{array}$ | $\begin{array}{l}{\mathbf{\Theta}}^{\prime}(\mathit{a})\\ \mathit{S}\mathit{W}\mathit{C}\mathit{N}\mathit{T}\mathit{s}\end{array}$ | $\begin{array}{l}{\mathbf{\Theta}}^{\prime}(\mathit{a})\\ \mathit{M}\mathit{W}\mathit{C}\mathit{N}\mathit{T}\mathit{s}\end{array}$ | $\begin{array}{l}\mathit{\alpha}=0.90,\\ \mathit{\eta}.\end{array}$ | $\begin{array}{l}{\mathbf{\Theta}}^{\prime}(\mathit{a})\\ \mathit{S}\mathit{W}\mathit{C}\mathit{N}\mathit{T}\mathit{s}\end{array}$ | $\begin{array}{l}{\mathbf{\Theta}}^{\prime}(\mathit{a})\\ \mathit{M}\mathit{W}\mathit{C}\mathit{N}\mathit{T}\mathit{s}\end{array}$ |
---|---|---|---|---|---|---|---|---|

0.1 | 0.3000 | 0.3000 | 0.1 | 0.3000 | 0.3000 | 0.1 | 0.3000 | 0.3000 |

0.2 | 0.3000 | 0.2999 | 0.2 | 0.3000 | 0.2999 | 0.2 | 0.3000 | 0.2999 |

0.3 | 0.3000 | 0.2999 | 0.3 | 0.3000 | 0.2999 | 0.3 | 0.2999 | 0.2999 |

0.4 | 0.2999 | 0.2998 | 0.4 | 0.2999 | 0.2998 | 0.4 | 0.2999 | 0.2998 |

0.5 | 0.2999 | 0.2998 | 0.5 | 0.2999 | 0.2997 | 0.5 | 0.2999 | 0.2997 |

0.6 | 0.2999 | 0.2997 | 0.6 | 0.2999 | 0.2997 | 0.6 | 0.2998 | 0.2996 |

0.7 | 0.2998 | 0.2996 | 0.7 | 0.2998 | 0.2996 | 0.7 | 0.2998 | 0.2995 |

0.8 | 0.2998 | 0.2995 | 0.8 | 0.2998 | 0.2995 | 0.8 | 0.2998 | 0.2994 |

0.9 | 0.2998 | 0.2994 | 0.9 | 0.2997 | 0.2994 | 0.9 | 0.2997 | 0.2993 |

1.0 | 0.2997 | 0.2993 | 1.0 | 0.2997 | 0.2992 | 1.0 | 0.2997 | 0.2992 |

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**MDPI and ACS Style**

Gul, T.; Khan, M.A.; Noman, W.; Khan, I.; Abdullah Alkanhal, T.; Tlili, I.
Fractional Order Forced Convection Carbon Nanotube Nanofluid Flow Passing Over a Thin Needle. *Symmetry* **2019**, *11*, 312.
https://doi.org/10.3390/sym11030312

**AMA Style**

Gul T, Khan MA, Noman W, Khan I, Abdullah Alkanhal T, Tlili I.
Fractional Order Forced Convection Carbon Nanotube Nanofluid Flow Passing Over a Thin Needle. *Symmetry*. 2019; 11(3):312.
https://doi.org/10.3390/sym11030312

**Chicago/Turabian Style**

Gul, Taza, Muhammad Altaf Khan, Waqas Noman, Ilyas Khan, Tawfeeq Abdullah Alkanhal, and Iskander Tlili.
2019. "Fractional Order Forced Convection Carbon Nanotube Nanofluid Flow Passing Over a Thin Needle" *Symmetry* 11, no. 3: 312.
https://doi.org/10.3390/sym11030312