# Numerical Solutions of Unsteady Boundary Layer Flow with a Time-Space Fractional Constitutive Relationship

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. The Numerical Technique

#### 3.1. Discretization Method

#### 3.2. Solvability of the Implicit Difference Scheme

## 4. Validation of the Numerical Discretization Method

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Term | Definition |

$u$ | velocity component in the x-direction |

$v$ | velocity component along $y$-direction |

$a$ | constant parameter |

$n$ | stretching index |

${\tau}_{xy}$ | shear stress |

$\rho $ | density of fluid |

$\alpha $ | stress fractional parameter |

$\gamma $ | space fractional parameter |

$\lambda $ | relaxation parameter |

$\tilde{\mathrm{R}}\mathrm{e}$ | Reynolds number |

${\mu}_{\gamma}$ | fractional dynamic viscosity |

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$\alpha =0.1$ $\gamma =0.98$ | ${L}_{2}$Error | Order | ${L}_{\infty}$Error | Order |

1/8 | 3.9055e-05 | - | 8.4866e-05 | - |

1/16 | 1.9529e-05 | 0.9989 | 4.2644e-05 | 0.9282 |

1/32 | 9.7050e-06 | 1.0088 | 2.1337e-05 | 0.9901 |

1/64 | 4.8460e-06 | 1.0019 | 1.0655e-05 | 1.0018 |

$\alpha =0.1$ $\gamma =0.95$ | ${L}_{2}$Error | Order | ${L}_{\infty}$Error | Order |

1/8 | 4.0576e-05 | - | 8.9865e-05 | - |

1/16 | 2.0305e-05 | 0.9988 | 4.4954e-05 | 0.9993 |

1/32 | 1.0087e-05 | 1.0093 | 2.2335e-05 | 1.0092 |

1/64 | 5.0183e-06 | 1.0071 | 1.1127e-05 | 1.0052 |

$\alpha =0.3$ $\gamma =0.8$ | ${L}_{2}$Error | Order | ${L}_{\infty}$Error | Order |

1/8 | 5.7726e-05 | - | 1.3641e-04 | - |

1/16 | 2.6869e-05 | 1.1033 | 6.3664e-05 | 1.0993 |

1/32 | 1.2973e-05 | 1.0504 | 3.0797e-05 | 1.0477 |

1/64 | 6.3905e-06 | 1.0215 | 1.5257e-05 | 1.0133 |

$\alpha =0.4$ $\gamma =0.8$ | ${L}_{2}$Error | Order | ${L}_{\infty}$Error | Order |

1/8 | 5.6212e-05 | - | 1.3151e-04 | - |

1/16 | 2.6040e-05 | 1.1101 | 6.1069e-05 | 1.1067 |

1/32 | 1.2539e-05 | 1.0543 | 2.9888e-05 | 1.0309 |

1/64 | 6.1704e-06 | 1.0230 | 1.4809e-05 | 1.0131 |

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**MDPI and ACS Style**

Yang, W.; Chen, X.; Meng, Y.; Zhang, X.; Mi, S.
Numerical Solutions of Unsteady Boundary Layer Flow with a Time-Space Fractional Constitutive Relationship. *Symmetry* **2020**, *12*, 1446.
https://doi.org/10.3390/sym12091446

**AMA Style**

Yang W, Chen X, Meng Y, Zhang X, Mi S.
Numerical Solutions of Unsteady Boundary Layer Flow with a Time-Space Fractional Constitutive Relationship. *Symmetry*. 2020; 12(9):1446.
https://doi.org/10.3390/sym12091446

**Chicago/Turabian Style**

Yang, Weidong, Xuehui Chen, Yuan Meng, Xinru Zhang, and Shiyun Mi.
2020. "Numerical Solutions of Unsteady Boundary Layer Flow with a Time-Space Fractional Constitutive Relationship" *Symmetry* 12, no. 9: 1446.
https://doi.org/10.3390/sym12091446