# Study of Entropy Generation with Multi-Slip Effects in MHD Unsteady Flow of Viscous Fluid Past an Exponentially Stretching Surface

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling

## 3. Entropy Analysis

## 4. Solution to the Problem

#### 4.1. Homotopy Analysis Method

#### 4.2. Numerical Method

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Effects of (

**a**) the magnetic field parameter $M$, (

**b**) the slip parameter $\gamma $, and (

**c**) the unsteadiness parameter $A$ on the velocity profile ${f}^{\prime}(\xi )$.

**Figure 3.**Effects of (

**a**) the magnetic field parameter $M$, (

**b**) the slip parameter $\gamma $, and (

**c**) the unsteadiness parameter $A$ on the temperature profile $\theta (\xi )$.

**Figure 4.**Effects of (

**a**) the temperature exponent ${A}_{o}$, (

**b**) the Prandtl number $\mathrm{Pr}$, (

**c**) the Eckert number $Ec$ on the temperature profile $\theta (\xi )$.

**Figure 6.**Effects of (

**a**) the magnetic field parameter $M$, (

**b**) the slip paramete $\gamma $, and (

**c**) the unsteadiness parameter $A$ on local entropy generation number ${N}_{G}$.

**Figure 7.**Effects of (

**a**) the temperature exponent ${A}_{o}$, (

**b**) the group parameter $Br/\Omega $, and (

**c**) the Reynolds number ${\mathrm{Re}}_{L}$ on the local entropy generation number ${N}_{G}$.

**Figure 8.**Effects of (

**a**) the magnetic field parameter $M$, (

**b**) the slip parameter $\gamma $

**,**and (

**c**) the unsteadiness parameter $A$ on the Bejan number $Be$.

**Figure 9.**Effects of (

**a**) the temperature exponent ${A}_{o}$, (

**b**) the group parameter $Br/\Omega $, (

**c**) the Reynolds number ${\mathrm{Re}}_{L}$ on the Bejan number.

**Table 1.**Optimal values of ${\hslash}_{f}$ and ${\hslash}_{\theta}$, as well as their corresponding averaged root mean squared residual errors ${E}_{m,f}$ and ${E}_{m,\theta}$ at different orders of approximation when $M=1.0,\text{}\gamma =0.5,\text{}A=1.0,{A}_{0}=0.5,$ $\mathrm{Pr}=1.0,\text{}Ec=0.2$.

$\mathit{O}\mathit{r}\mathit{d}\mathit{e}\mathit{r}\mathit{o}\mathit{f}\mathit{A}\mathit{p}\mathit{p}\mathit{r}\mathit{o}\mathit{x}\mathit{i}\mathit{m}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{m}$ | ${\mathit{\hslash}}_{\mathit{f}}$ | ${\mathit{\hslash}}_{\mathit{\theta}}$ | ${\mathit{E}}_{\mathit{m},\mathit{\theta}}$ | ${\mathit{E}}_{\mathit{m},\mathit{\theta}}$ |
---|---|---|---|---|

$3$ | $-0.464353$ | $-0.396157$ | $1.05905\times {10}^{-2}$ | $2.69482\times {10}^{-3}$ |

$6$ | $-0.405309$ | $-0.424927$ | $1.81301\times {10}^{-5}$ | $1.25275\times {10}^{-4}$ |

$9$ | $-0.382564$ | $-0.413877$ | $1.49644\times {10}^{-5}$ | $2.93299\times {10}^{-5}$ |

$12$ | $-0.369398$ | $-0.368290$ | $8.38391\times {10}^{-7}$ | $9.89133\times {10}^{-6}$ |

$15$ | $-0.314784$ | $-0.326304$ | $2.48338\times {10}^{-6}$ | $3.91650\times {10}^{-6}$ |

**Table 2.**Convergence table for the $[m/m]$ homotopy Padé approximation of $-{f}^{\u2033}(0)\text{}\mathrm{and}\text{}-{\theta}^{\prime}(0)$ when ${\hslash}_{f}=-0.314784,\text{}{\hslash}_{\theta}=-0.398461,$ $M=1.0,\text{}\gamma =0.5,\text{}A=1.0,{A}_{0}=0.5,$ $\mathrm{Pr}=1.0,\text{}Ec=0.2$.

$[\mathit{m}/\mathit{m}]$ | $-{\mathit{f}}^{\u2033}(0)$ | $-{\mathit{\theta}}^{\prime}(0)$ |
---|---|---|

$\left[5/5\right]$ | $0.95890$ | $1.93308$ |

$\left[10/10\right]$ | $0.95923$ | $1.92904$ |

$\left[15/15\right]$ | $0.95921$ | $1.92905$ |

$\left[20/20\right]$ | $0.95921$ | $1.92905$ |

$\left[25/25\right]$ | $0.95921$ | $1.92905$ |

$\left[30/30\right]$ | $0.95921$ | $1.92905$ |

$\left[35/35\right]$ | $0.95921$ | $1.92905$ |

$\mathit{M}$ | $\mathbf{Pr}$ | $-{\mathit{\theta}}^{\prime}(0)$ Magyari and Keller [6] | $-{\mathit{\theta}}^{\prime}(0)$ E1-Aziz [9] | $-{\mathit{\theta}}^{\prime}(0)$ Bidin and Nazar [10] | $-{\mathit{\theta}}^{\prime}(0)$ Anuar Ishak [11] | Present | |
---|---|---|---|---|---|---|---|

$-{\mathit{\theta}}^{\prime}(0)$ HAM | $-{\mathit{\theta}}^{\prime}(0)$ Numerical | ||||||

$\mathbf{0.0}$ | $\mathbf{1.0}$ | $\mathbf{0.954782}$ | $\mathbf{0.954785}$ | $\mathbf{0.9548}$ | $0.9548$ | $0.95478$ | $0.95478$ |

$2.0$ | $----$ | $----$ | $1.4714$ | $1.4715$ | $1.47146$ | $1.47146$ | |

$3.0$ | $1.869075$ | $1.869072$ | $1.8691$ | $1.86907$ | $1.86907$ | ||

$5.0$ | $2.500135$ | $2.500132$ | $2.5001$ | $2.50012$ | $2.50012$ | ||

$10.0$ | $3.660379$ | $3.660372$ | $3.6604$ | $3.66027$ | $3.66027$ | ||

$1.0$ | $1.0$ | $----$ | $----$ | $----$ | $0.8611$ | $0.86109$ | $0.86109$ |

**Table 4.**Numerical values of $-{f}^{\u2033}(0)$ that were obtained by the homotopy analysis method (HAM) and the shooting method for variation in the values of the slip parameter $\gamma $, the magnetic field parameter $M$, and the unsteadiness parameter $A$.

$\mathit{\gamma}$ | $\mathit{M}$ | $\mathit{A}$ | $-{\mathit{f}}^{\u2033}\left(0\right)$$\mathit{H}\mathit{A}\mathit{M}$ | $-{\mathit{f}}^{\u2033}\left(0\right)$$\mathit{N}\mathit{u}\mathit{m}\mathit{e}\mathit{r}\mathit{i}\mathit{c}\mathit{a}\mathit{l}$ |
---|---|---|---|---|

$0.5$ | $1.0$ | $0.2$ | $0.85691$ | $0.85691$ |

$0.5$ | $0.89946$ | $0.89946$ | ||

$1.0$ | $0.95920$ | $0.95920$ | ||

$1.5$ | $1.00841$ | $1.00841$ | ||

$0.5$ | $0.0$ | $1.0$ | $0.88048$ | $0.88048$ |

$1.0$ | $0.95920$ | $0.95920$ | ||

$2.0$ | $1.02039$ | $1.02039$ | ||

$3.0$ | $1.06998$ | $1.06998$ | ||

$0.0$ | $1.0$ | $1.0$ | $2.04499$ | $2.04499$ |

$0.5$ | $0.95920$ | $0.95920$ | ||

$1.0$ | $0.63912$ | $0.63912$ | ||

$2.0$ | $0.38676$ | $0.38676$ |

**Table 5.**Numerical values of $-{\theta}^{\prime}(0)$ that were obtained by the HAM and the shooting method for variation in the values of the slip parameter $\gamma $, the magnetic field parameter $M$, the unsteadiness parameter $A$, the temperature exponent ${A}_{o}$, the Prandtl number $\mathrm{Pr}$ and the Eckert number $Ec$.

$\mathit{\gamma}$ | $\mathit{M}$ | $\mathit{A}$ | ${\mathit{A}}_{0}$ | $\mathbf{Pr}$ | $\mathit{E}\mathit{c}$ | $-{\mathit{\theta}}^{\prime}\left(0\right)\mathit{H}\mathit{A}\mathit{M}$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$$\mathit{N}\mathit{u}\mathit{m}\mathit{e}\mathit{r}\mathit{i}\mathit{c}\mathit{a}\mathit{l}$ |
---|---|---|---|---|---|---|---|

$0.5$ | $1.0$ | $1.0$ | $0.5$ | $1.0$ | $0.2$ | $1.92905$ | $1.92905$ |

$0.5$ | $1.86944$ | $1.86944$ | |||||

$0.7$ | $1.82970$ | $1.82970$ | |||||

$0.9$ | $1.78996$ | $1.78996$ | |||||

$0.5$ | $1.0$ | $1.0$ | $0.5$ | $0.7$ | $0.2$ | $1.61368$ | $1.61368$ |

$1.0$ | $1.92905$ | $1.92905$ | |||||

$2.0$ | $2.72687$ | $2.72687$ | |||||

$3.0$ | $3.33743$ | $3.33743$ | |||||

$0.5$ | $1.0$ | $1.0$ | $0.5$ | $1.0$ | $0.2$ | $1.92905$ | $1.92905$ |

$1.0$ | $1.97510$ | $1.97510$ | |||||

$1.5$ | $2.02057$ | $2.02057$ | |||||

$2.0$ | $2.06548$ | $2.06548$ | |||||

$0.5$ | $1.0$ | $0.2$ | $0.5$ | $1.0$ | $0.2$ | $0.96247$ | $0.96247$ |

$0.5$ | $1.40692$ | $1.40692$ | |||||

$1.0$ | $1.92905$ | $1.92905$ | |||||

$1.5$ | $2.33731$ | $2.33731$ | |||||

$0.5$ | $0.0$ | $1.0$ | $0.5$ | $1.0$ | $0.2$ | $1.95329$ | $1.95329$ |

$1.0$ | $1.92905$ | $1.92905$ | |||||

$2.0$ | $1.91235$ | $1.91235$ | |||||

$3.0$ | $1.90015$ | $1.90015$ | |||||

$0.0$ | $1.0$ | $1.0$ | $0.5$ | $1.0$ | $0.2$ | $1.88525$ | $1.88525$ |

$0.5$ | $1.92905$ | $1.92905$ | |||||

$1.0$ | $1.92378$ | $1.92378$ | |||||

$2.0$ | $1.91240$ | 1.91240 |

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## Share and Cite

**MDPI and ACS Style**

Haider, S.; Saeed Butt, A.; Li, Y.-Z.; Imran, S.M.; Ahmad, B.; Tayyaba, A.
Study of Entropy Generation with Multi-Slip Effects in MHD Unsteady Flow of Viscous Fluid Past an Exponentially Stretching Surface. *Symmetry* **2020**, *12*, 426.
https://doi.org/10.3390/sym12030426

**AMA Style**

Haider S, Saeed Butt A, Li Y-Z, Imran SM, Ahmad B, Tayyaba A.
Study of Entropy Generation with Multi-Slip Effects in MHD Unsteady Flow of Viscous Fluid Past an Exponentially Stretching Surface. *Symmetry*. 2020; 12(3):426.
https://doi.org/10.3390/sym12030426

**Chicago/Turabian Style**

Haider, Sajjad, Adnan Saeed Butt, Yun-Zhang Li, Syed Muhammad Imran, Babar Ahmad, and Asia Tayyaba.
2020. "Study of Entropy Generation with Multi-Slip Effects in MHD Unsteady Flow of Viscous Fluid Past an Exponentially Stretching Surface" *Symmetry* 12, no. 3: 426.
https://doi.org/10.3390/sym12030426