# Symmetric MHD Channel Flow of Nonlocal Fractional Model of BTF Containing Hybrid Nanoparticles

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## Abstract

**:**

_{2}) nanoparticles were dissolved in base fluid water (H

_{2}O) to form a hybrid nanofluid. The MHD free convection flow of the nanofluid (Ag-TiO

_{2}-H

_{2}O) was considered in a microchannel (flow with a bounded domain). The BTF model was generalized using a nonlocal Caputo-Fabrizio fractional operator (CFFO) without a singular kernel of order $\alpha $ with effective thermophysical properties. The governing equations of the model were subjected to physical initial and boundary conditions. The exact solutions for the nonlocal fractional model without a singular kernel were developed via the fractional Laplace transform technique. The fractional solutions were reduced to local solutions by limiting $\alpha \to 1$. To understand the rheological behavior of the fluid, the obtained solutions were numerically computed and plotted on various graphs. Finally, the influence of pertinent parameters was physically studied. It was found that the solutions were general, reliable, realistic and fixable. For the fractional parameter, the velocity and temperature profiles showed a decreasing trend for a constant time. By setting the values of the fractional parameter, excellent agreement between the theoretical and experimental results could be attained.

## 1. Introduction

## 2. Description of the Problem

## 3. Thermophysical Properties of Hybrid Nanofluid

## 4. Problem Solutions and Dimensionless Analysis

#### 4.1. Solutions of the Energy Equation

#### 4.2. Solution of Momentum Equation

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Material | Base fluid | Nanoparticles | |
---|---|---|---|

${\mathit{H}}_{2}\mathit{O}$ | $\mathit{T}\mathit{i}{\mathit{O}}_{2}$ | $\mathit{A}\mathit{g}$ | |

$\rho \left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ | 997.1 | 425 | 10500 |

${C}_{p}\left(\mathrm{J}/\mathrm{kg}\text{}\mathrm{K}\right)$ | 4179 | 6862 | 235 |

$K\left(\mathrm{W}/\mathrm{m}\text{}\mathrm{K}\right)$ | 0.613 | 8.9538 | 429 |

${\beta}_{T}\times {10}^{-5}({K}^{-1})$ | 21 | 0.9 | 1.89 |

$\sigma $ | 0.05 | $1\times {10}^{-12}$ | $3.60\times {10}^{7}$ |

Pr | 6.2 | - | - |

**Table 2.**Expressions of nanofluid and hybrid nanofluid [42].

Thermophysical Properties | Nanofluid | Hybrid Nanofluid |
---|---|---|

Density | ${\rho}_{nf}=\left(1-\varphi \right){\rho}_{f}+\varphi {\rho}_{s},$ | ${\rho}_{hnf}=\left(1-{\varphi}_{hnf}\right){\rho}_{f}+{\varphi}_{Ag}{\rho}_{Ag}+{\varphi}_{Ti{O}_{2}}{\rho}_{Ti{O}_{2}},$ |

Dynamic viscosity | ${\mu}_{nf}=\frac{{\mu}_{f}}{{\left(1-\varphi \right)}^{2.5}},$ | ${\mu}_{hnf}=\frac{{\mu}_{f}}{{\left\{1-\left({\varphi}_{Ag}+{\varphi}_{Ti{O}_{2}}\right)\right\}}^{2.5}}$ |

Thermal expansion | ${\left({\beta}_{T}\rho \right)}_{nf}=\left(1-\varphi \right){\left({\beta}_{T}\rho \right)}_{f}+\varphi {\left({\beta}_{T}\rho \right)}_{s}$ | ${\left(\rho {\beta}_{T}\right)}_{hnf}=\left(1-{\varphi}_{hnf}\right){\left(\rho {\beta}_{T}\right)}_{f}+{\varphi}_{Ag}{\left(\rho {\beta}_{t}\right)}_{Ag}+{\varphi}_{Ti{O}_{2}}{\left(\rho {\beta}_{T}\right)}_{Ti{O}_{2}}$ |

Heat Capacitance | ${\left(\rho {C}_{p}\right)}_{nf}=\left(1-\varphi \right){\left(\rho {C}_{p}\right)}_{f}+\varphi {\left(\rho {C}_{p}\right)}_{s}$ | ${\left(\rho Cp\right)}_{hnf}=\left(1-{\varphi}_{hnf}\right){\left(\rho Cp\right)}_{f}+{\varphi}_{Ag}{\left(\rho Cp\right)}_{Ag}+{\varphi}_{Ti{O}_{2}}{\left(\rho Cp\right)}_{Ti{O}_{2}}$ |

Electrical conductivity | $\frac{{\sigma}_{nf}}{{\sigma}_{f}}=1+\frac{3\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1\right)\varphi}{\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}+2\right)-\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1\right)\varphi}$ | $\frac{{\sigma}_{hnf}}{{\sigma}_{f}}=1+\frac{3\left(\frac{+{\varphi}_{Ag}{\sigma}_{Ag}+{\varphi}_{Ti{O}_{2}}{\sigma}_{Ti{O}_{2}}}{{\sigma}_{f}}-\varphi \right)}{\left(\frac{{\varphi}_{Ag}{\sigma}_{Ag}+{\varphi}_{Ti{O}_{2}}{\sigma}_{Ti{O}_{2}}}{\varphi {\sigma}_{f}}+2\right)-\left(\frac{{\varphi}_{Ag}{\sigma}_{Ag}+{\varphi}_{Ti{O}_{2}}{\sigma}_{Ti{O}_{2}}}{{\sigma}_{f}}-\varphi \right)}$ |

Thermal conductivity | $\frac{{K}_{nf}}{{K}_{f}}=\frac{{k}_{s}+2{k}_{f}-2\varphi \left({k}_{s}-{k}_{f}\right)}{{k}_{s}+2{k}_{f}+\varphi \left({k}_{s}-{k}_{f}\right)},$ | $\frac{{k}_{hnf}}{{k}_{f}}=\frac{\frac{{\varphi}_{Ag}{k}_{Ag}+{\varphi}_{Ti{O}_{2}}{k}_{Ti{O}_{2}}}{{\varphi}_{hnf}}+2{k}_{f}+2\left({\varphi}_{Ag}{k}_{Ag}+{\varphi}_{Ti{O}_{2}}{k}_{{}_{Ti{O}_{2}}}\right)-2{k}_{f}{\varphi}_{hnf}}{\frac{{\varphi}_{Ag}{k}_{Ag}+{\varphi}_{Ti{O}_{2}}{k}_{Ti{O}_{2}}}{{\varphi}_{hnf}}+2{k}_{f}+\left({\varphi}_{Ag}{k}_{Ag}+{\varphi}_{Ti{O}_{2}}{k}_{{}_{Ti{O}_{2}}}\right)-{k}_{f}{\varphi}_{hnf}}$ |

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**MDPI and ACS Style**

Saqib, M.; Shafie, S.; Khan, I.; Chu, Y.-M.; Nisar, K.S.
Symmetric MHD Channel Flow of Nonlocal Fractional Model of BTF Containing Hybrid Nanoparticles. *Symmetry* **2020**, *12*, 663.
https://doi.org/10.3390/sym12040663

**AMA Style**

Saqib M, Shafie S, Khan I, Chu Y-M, Nisar KS.
Symmetric MHD Channel Flow of Nonlocal Fractional Model of BTF Containing Hybrid Nanoparticles. *Symmetry*. 2020; 12(4):663.
https://doi.org/10.3390/sym12040663

**Chicago/Turabian Style**

Saqib, Muhammad, Sharidan Shafie, Ilyas Khan, Yu-Ming Chu, and Kottakkaran Sooppy Nisar.
2020. "Symmetric MHD Channel Flow of Nonlocal Fractional Model of BTF Containing Hybrid Nanoparticles" *Symmetry* 12, no. 4: 663.
https://doi.org/10.3390/sym12040663