# Serious Solutions for Unsteady Axisymmetric Flow over a Rotating Stretchable Disk with Deceleration

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

## 3. Homotopy Analysis Method

#### 3.1. Zeroth-Order Deformation Equation

#### 3.2. Mth-Order Deformation

## 4. Convergence of the HAM Solution

## 5. Results and Discussion

## 6. Conclusions

- Results obtained by homotopy analysis method are in good agreement with existing numerical results;
- All the velocity profiles decrease with an increase in unsteadiness parameter $S$;
- Radial and axial velocity of the flow increases with the increase in disk stretching parameter $a$, whereas tangential velocity shows a decreasing trend with an increase in $a$;
- Variation trend decays with faster velocity to the ambient for fast deceleration as compared to the slow deceleration of the disk.

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Comparison of convergence of $g\left(\eta \right)\text{}\mathrm{when}\hslash =-0.333,a=1\mathrm{and}S=-1$ (line: 10th-order, dots: 5th-order).

**Figure 5.**Comparison of convergence of $f\left(\eta \right)\text{}\mathrm{when}\hslash =-0.333,a=1\mathrm{and}S=-1$ (line: 10th-order, dots: 5th-order).

**Figure 6.**For $S=-1/2$ solid line: $a=1$, Dashed line: $a=2$ 10th order HAM approximation for ${f}^{\prime}\left(\eta \right)$.

**Figure 8.**Variation of ${f}^{\prime}\left(\eta \right)$ for different values of unsteadiness parameter for $a=1$.

**Figure 9.**Variation of $g\left(\eta \right)$ for different values of unsteadiness parameter for $a=1$.

**Table 1.**Comparison of the numerical result [5] with homotopy analysis method (HAM) convergent result when $a=1,S=\text{}-1\text{}\mathrm{and}\text{}{\hslash}_{f}=-1/3,\text{}{\hslash}_{g}=-1/4$.

Order | ${\mathit{f}}^{\u2033}\left(0\right)$ | ${\mathit{g}}^{\prime}\left(0\right)$ | $\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\mathit{f}$ | $\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\mathit{g}$ |
---|---|---|---|---|

2nd | −0.7673 | −1.196 | 0.311 | 0.044 |

4th | −0.6945 | −1.246 | 0.021 | 0.0067 |

6th | −0.6681 | −1.264 | 0.0046 | 0.0021 |

8th | −0.6581 | −1.270 | 0.00091 | 0.000715 |

10th | −0.6543 | −1.271 | 0.00014 | 0.00023 |

**Table 2.**Comparison of the numerical result [5] with HAM convergent result when $a=1,S=-1/10\mathrm{and}{\hslash}_{f}=-1/3,{\hslash}_{g}=-1/4$.

Order | ${\mathit{f}}^{\u2033}\left(0\right)$ | ${\mathit{g}}^{\prime}\left(0\right)$ | $\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\mathit{f}$ | $\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\mathit{g}$ |
---|---|---|---|---|

2nd | −0.9642 | −1.3321 | 0.031 | 0.379 |

4th | −0.9374 | −1.4162 | 0.0047 | 0.0055 |

6th | −0.9262 | −1.446 | 0.00075 | 0.00091 |

8th | −0.9217 | −1.458 | 0.00012 | 0.00016 |

10th | −0.9200 | −1.4627 | 0.000018 | 0.000033 |

**Table 3.**Comparison of the numerical result [5] with HAM convergent result when $a=2,S=-1/10\mathrm{and}{\hslash}_{g}={\hslash}_{f}=-1/5$.

Order | ${\mathit{f}}^{\u2033}\left(0\right)$ | ${\mathit{g}}^{\prime}\left(0\right)$ | $\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\mathit{f}$ | $\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\mathit{g}$ |
---|---|---|---|---|

2nd | −2.779 | −1.658 | 0.408 | 0.543 |

4th | −2.9729 | −1.847 | 0.0876 | 0.136 |

6th | −3.044 | −1.924 | 0.0234 | 0.0437 |

8th | −3.072 | −1.953 | 0.0071 | 0.018 |

10th | −3.082 | −1.958 | 0.0024 | 0.012 |

**Table 4.**Comparison of the numerical result [5] with HAM convergent result when $a=1,S=-1/2\mathrm{and}{\hslash}_{g}={\hslash}_{f}=-1/4$.

Order | ${\mathit{f}}^{\u2033}\left(0\right)$ | ${\mathit{g}}^{\prime}\left(0\right)$ | $\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\mathit{f}$ | $\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\mathit{g}$ |
---|---|---|---|---|

2nd | −0.9062 | −1.2760 | 0.1051 | 0.0221 |

4th | −0.8592 | −1.3424 | 0.0283 | 0.0037 |

6th | −0.8319 | −1.3654 | 0.0077 | 0.00082 |

8th | −0.8172 | −1.3741 | 0.0021 | 0.00021 |

10th | −0.8093 | −1.3774 | 0.00058 | 0.00006 |

**Table 5.**The Convergence analysis of ${f}^{\u2033}\left(0\right)$ for different $\hslash $ when $a=1,$ and $S=-1/2$.

Order | $\mathit{\hslash}\mathit{f}=-1/4,\mathit{\hslash}\mathit{g}=-1/5$ | Err | $\mathit{\hslash}\mathit{f}=-1/5,\mathit{\hslash}\mathit{g}=-1/4$ | Err |
---|---|---|---|---|

2nd | −0.9007 | 0.1062 | −0.9243 | 0.1405 |

4th | −0.8535 | 0.0283 | −0.8826 | 0.0514 |

6th | −0.8282 | 0.0076 | −0.8658 | 0.0312 |

8th | −0.8149 | 0.0021 | −0.8325 | 0.0069 |

10th | −0.8080 | 0.0005 | −0.8201 | 0.0025 |

**Table 6.**The Convergence analysis of ${g}^{\prime}\left(0\right)$ for different $\hslash $ when $a=1,$ and $S=-1/2$.

Order | $\mathit{\hslash}\mathit{f}=-1/4,\mathit{\hslash}\mathit{g}=-1/5$ | Err | $\mathit{\hslash}\mathit{f}=-1/5,\mathit{\hslash}\mathit{g}=-1/4$ | Err |
---|---|---|---|---|

2nd | −1.2345 | 0.03718 | −1.2786 | 0.022 |

4th | −1.3157 | 0.0087 | −1.3445 | 0.0036 |

6th | −1.3502 | 0.0024 | −1.3578 | 0.0016 |

8th | −1.366 | 0.0007 | −1.3737 | 0.00021 |

10th | −1.3734 | 0.0002 | −1.3768 | 0.00006 |

**Table 7.**Comparison of the numerical result [5] with HAM convergent result for special case when analysis $=1,$ $S=0,\mathrm{and}\hslash f=-28/100,\hslash g=-1/3$.

Order | ${\mathit{f}}^{\u2033}$ | ${\mathit{g}}^{\prime}$ | $\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\mathit{f}$ | $\mathit{e}\mathit{r}\mathit{r}\mathit{o}\mathit{r}\mathit{g}$ |
---|---|---|---|---|

5th | −1.1785 | −1.44639 | 0.00044518 | 0.00019581 |

10th | −1.1751 | −1.45359 | 0.00001437 | 8.13 × 10^{−7} |

15th | −1.1739 | −1.45402 | 3.86 × 10^{−7} | 1.15 × 10^{−8} |

20th | −1.1737 | −1.45406 | 2.02 × 10^{−8} | 6.42 × 10^{−10} |

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**MDPI and ACS Style**

Sadiq, M.A.
Serious Solutions for Unsteady Axisymmetric Flow over a Rotating Stretchable Disk with Deceleration. *Symmetry* **2020**, *12*, 96.
https://doi.org/10.3390/sym12010096

**AMA Style**

Sadiq MA.
Serious Solutions for Unsteady Axisymmetric Flow over a Rotating Stretchable Disk with Deceleration. *Symmetry*. 2020; 12(1):96.
https://doi.org/10.3390/sym12010096

**Chicago/Turabian Style**

Sadiq, Muhammad Adil.
2020. "Serious Solutions for Unsteady Axisymmetric Flow over a Rotating Stretchable Disk with Deceleration" *Symmetry* 12, no. 1: 96.
https://doi.org/10.3390/sym12010096