# The Solutions of Non-Integer Order Burgers’ Fluid Flowing through a Round Channel with Semi Analytical Technique

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Imposing Condition and Geometry

#### 3.1. Velocity Field Due to Rotating Circular Pipe

#### 3.2. Shear Stress Due to Rotating Circular Cylinder

## 4. Results and Discussions

## 5. Conclusions

- The ordinary fluid have less velocity as compared to fractional order derivative fluid models. This result can be verify from the graph of fractional parameter $\alpha $ available in Figure 12, which has decreasing altitude when the velocity is increasing.
- The velocity of the fluid increases for Burgers’ fluid model as fluid becomes more thick in this model.
- The graph of the parameters $\beta $, ${\lambda}_{3}$, $\nu $, t, r, and $\mu $ showed an increase/upward in behaviour with increase in velocity and stress function.
- The parameters ${\lambda}_{1}$, ${\lambda}_{2}$ and $\alpha $ are behaving opposite to the influence of velocity and shear stress.
- The fractional Burgers’ fluid is flowing faster than the Maxwell and Newtonian fluids.
- In future, authors will try to study the fluid motion by considering the effects of temperature and magnetic field.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The aspect of velocity $F(\mathrm{r},t)$ for various values of t and fixed values for $\mathsf{\Omega}=1,R=1,p=1,s=1,\alpha =0.3,\beta =0.7,\nu =0.053,n=12,{\lambda}_{1}=5,{\lambda}_{2}=20$ and ${\lambda}_{3}=55$.

**Figure 3.**The aspect of stress $Q(\mathrm{r},t)$ for various values of t and fixed values for $\mu =0.010,\mathsf{\Omega}=2.5,R=1,p=1,s=1,\alpha =0.3,\beta =0.7,\nu =0.053,n=12,{\lambda}_{1}=5,{\lambda}_{2}=20$ and ${\lambda}_{3}=55$.

**Figure 4.**The aspect of velocity $F(\mathrm{r},t)$ for various values of $\mathrm{r}$ and fixed values for $\mathsf{\Omega}=1,R=1,p=1,s=1,\alpha =0.3,\beta =0.7,\nu =0.053,n=12,{\lambda}_{1}=5,{\lambda}_{2}=20$ and ${\lambda}_{3}=55$.

**Figure 5.**The aspect of stress $Q(\mathrm{r},t)$ for various values of $\mathrm{r}$ and fixed values for $\mu =0.09,\mathsf{\Omega}=1.5,R=1,p=1,s=1,\alpha =0.3,\beta =0.7,\nu =0.053,n=12,{\lambda}_{1}=5,{\lambda}_{2}=20$ and ${\lambda}_{3}=55$.

**Figure 6.**The aspect of velocity $F(\mathrm{r},t)$ for various values of ${\lambda}_{1}$ and fixed values for $\mathsf{\Omega}=1,R=1,p=1,s=1,\alpha =0.3,\beta =0.8,\nu =0.013,n=12,\mathrm{r}=0.8,{\lambda}_{2}=20$ and ${\lambda}_{3}=55$.

**Figure 7.**The aspect of stress $Q(\mathrm{r},t)$ for various values of ${\lambda}_{1}$ and fixed values for $\mu =0.09,\mathsf{\Omega}=2.5,R=1,p=1,s=1,\alpha =0.3,\beta =0.8,\nu =0.013,n=12,\mathrm{r}=0.8,{\lambda}_{2}=20$ and ${\lambda}_{3}=55$.

**Figure 8.**The aspect of velocity $F(\mathrm{r},t)$ for various values of ${\lambda}_{2}$ and fixed values for $\mathsf{\Omega}=1,R=1,p=1,s=1,\alpha =0.2,\beta =0.7,\nu =0.013,n=12,\mathrm{r}=0.8,{\lambda}_{1}=5$ and ${\lambda}_{3}=55$.

**Figure 9.**The aspect of stress $Q(\mathrm{r},t)$ for various values of ${\lambda}_{2}$ and fixed values for $\mu =0.09,\mathsf{\Omega}=1.2,R=1,p=1,s=1,\alpha =0.2,\beta =0.7,\nu =0.013,n=12,\mathrm{r}=0.8,{\lambda}_{1}=5$ and ${\lambda}_{3}=55$.

**Figure 10.**The aspect of velocity $F(\mathrm{r},t)$ for various values of ${\lambda}_{3}$ and fixed values for $\mathsf{\Omega}=1,R=1,p=1,s=1,\alpha =0.2,\beta =0.8,\nu =0.013,n=12,\mathrm{r}=0.8,{\lambda}_{1}=5$ and ${\lambda}_{2}=20$.

**Figure 11.**The aspect of stress $Q(\mathrm{r},t)$ for various values of ${\lambda}_{3}$ and fixed values for $\mu =0.09,\mathsf{\Omega}=1.2,R=1,p=1,s=1,\alpha =0.2,\beta =0.8,\nu =0.013,n=12,\mathrm{r}=0.8,{\lambda}_{1}=5$ and ${\lambda}_{2}=20$.

**Figure 12.**The aspect of velocity $F(\mathrm{r},t)$ for various values of $\alpha $ and fixed values for $\mathsf{\Omega}=1,R=1,p=1,s=1,\beta =0.7,\nu =0.013,n=12,\mathrm{r}=0.8,{\lambda}_{1}=5,{\lambda}_{2}=20$ and ${\lambda}_{3}=55$.

**Figure 13.**The aspect of stress $Q(\mathrm{r},t)$ for various values of $\alpha $ and fixed values for $\mu =0.09,\mathsf{\Omega}=1.2,R=1,p=1,s=1,\beta =0.7,\nu =0.013,n=12,\mathrm{r}=0.8,{\lambda}_{1}=5,{\lambda}_{2}=20$ and ${\lambda}_{3}=55$.

**Figure 14.**The aspect of velocity $F(\mathrm{r},t)$ for various values of $\beta $ and fixed values for $\mathsf{\Omega}=1,R=1,p=1,s=1,\alpha =0.3,\nu =0.013,n=12,\mathrm{r}=0.8,{\lambda}_{1}=5,{\lambda}_{2}=20$ and ${\lambda}_{3}=55$.

**Figure 15.**The aspect of stress $Q(\mathrm{r},t)$ for various values of $\beta $ and fixed values for $\mu =0.09,\mathsf{\Omega}=1.2,R=1,p=1,s=1,\alpha =0.3,\nu =0.013,n=12,\mathrm{r}=0.8,{\lambda}_{1}=5,{\lambda}_{2}=20$ and ${\lambda}_{3}=55$.

**Figure 16.**The aspect of velocity $F(\mathrm{r},t)$ for various values of $\nu $ and fixed values for $\mathsf{\Omega}=1,R=1,p=1,s=1,\alpha =0.3,\beta =0.8,n=12,\mathrm{r}=0.8,{\lambda}_{1}=5,{\lambda}_{2}=20$ and ${\lambda}_{3}=55$.

**Figure 17.**The aspect of stress $Q(\mathrm{r},t)$ for various values of $\mu $ and fixed values for $\mathsf{\Omega}=2.5,R=1,p=1,s=1,\alpha =0.3,\beta =0.8,\nu =0.013,n=12,\mathrm{r}=0.8,{\lambda}_{1}=5,{\lambda}_{2}=20$ and ${\lambda}_{3}=55$.

**Table 1.**Comparison of exact solutions [30] and numerical solutions (obtained from “Stehfest’s algorithm”) for the fractional order derivative model of the Maxwell fluid.

r | Exact w(r,t) [30] | Numerical V(r,t) | Error |
---|---|---|---|

0 | 0.000 | 0.000 | 0.000 |

0.02 | 0.029 | 0.029 | 0.000 |

0.04 | 0.058 | 0.058 | 0.000 |

0.06 | 0.087 | 0.087 | 0.000 |

0.08 | 0.115 | 0.116 | −0.001 |

0.10 | 0.145 | 0.145 | 0.000 |

0.12 | 0.174 | 0.174 | 0.000 |

0.14 | 0.203 | 0.203 | 0.000 |

0.16 | 0.233 | 0.233 | 0.000 |

0.18 | 0.262 | 0.263 | −0.001 |

0.20 | 0.292 | 0.292 | 0.000 |

0.22 | 0.322 | 0.322 | 0.000 |

0.24 | 0.352 | 0.352 | 0.000 |

0.26 | 0.383 | 0.381 | 0.002 |

0.28 | 0.414 | 0.411 | 0.003 |

0.30 | 0.444 | 0.441 | 0.003 |

0.32 | 0.475 | 0.473 | 0.002 |

0.34 | 0.506 | 0.503 | 0.003 |

0.36 | 0.538 | 0.534 | 0.004 |

0.38 | 0.569 | 0.565 | 0.004 |

0.40 | 0.601 | 0.596 | 0.005 |

0.42 | 0.632 | 0.628 | 0.004 |

0.44 | 0.664 | 0.661 | 0.003 |

0.46 | 0.695 | 0.694 | 0.001 |

0.48 | 0.727 | 0.725 | 0.002 |

..... | ..... | ..... | ..... |

**Table 2.**Comparison of exact solutions [29] and numerical solutions (obtained from “Stehfest’s algorithm”) for the fractional order derivative model of the Newtonian fluid.

r | Exact w(r,t) [29] | Numerical V(r,t) | Error |
---|---|---|---|

0 | 0.000 | 0.000 | 0.000 |

0.02 | 0.023 | 0.023 | 0.000 |

0.04 | 0.046 | 0.046 | 0.000 |

0.06 | 0.069 | 0.069 | 0.000 |

0.08 | 0.093 | 0.092 | 0.001 |

0.10 | 0.116 | 0.115 | 0.001 |

0.12 | 0.139 | 0.138 | 0.001 |

0.14 | 0.162 | 0.161 | 0.001 |

0.16 | 0.185 | 0.185 | 0.000 |

0.18 | 0.208 | 0.208 | 0.000 |

0.20 | 0.232 | 0.232 | 0.000 |

0.22 | 0.255 | 0.256 | −0.001 |

0.24 | 0.279 | 0.280 | −0.001 |

0.26 | 0.304 | 0.304 | 0.000 |

0.28 | 0.329 | 0.329 | 0.000 |

0.30 | 0.354 | 0.354 | 0.000 |

0.32 | 0.379 | 0.379 | 0.000 |

0.34 | 0.405 | 0.404 | 0.001 |

0.36 | 0.431 | 0.430 | 0.001 |

0.38 | 0.458 | 0.456 | 0.002 |

0.40 | 0.484 | 0.482 | 0.002 |

0.42 | 0.511 | 0.509 | 0.002 |

0.44 | 0.538 | 0.536 | 0.002 |

0.46 | 0.564 | 0.564 | 0.000 |

0.48 | 0.591 | 0.592 | −0.001 |

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**MDPI and ACS Style**

Imran, M.; Ching, D.L.C.; Safdar, R.; Khan, I.; Imran, M.A.; Nisar, K.S.
The Solutions of Non-Integer Order Burgers’ Fluid Flowing through a Round Channel with Semi Analytical Technique. *Symmetry* **2019**, *11*, 962.
https://doi.org/10.3390/sym11080962

**AMA Style**

Imran M, Ching DLC, Safdar R, Khan I, Imran MA, Nisar KS.
The Solutions of Non-Integer Order Burgers’ Fluid Flowing through a Round Channel with Semi Analytical Technique. *Symmetry*. 2019; 11(8):962.
https://doi.org/10.3390/sym11080962

**Chicago/Turabian Style**

Imran, M., D.L.C. Ching, Rabia Safdar, Ilyas Khan, M. A. Imran, and K. S. Nisar.
2019. "The Solutions of Non-Integer Order Burgers’ Fluid Flowing through a Round Channel with Semi Analytical Technique" *Symmetry* 11, no. 8: 962.
https://doi.org/10.3390/sym11080962