# Convective Effect on Magnetohydrodynamic (MHD) Stagnation Point Flow of Casson Fluid over a Vertical Exponentially Stretching/Shrinking Surface: Triple Solutions

^{1}

^{2}

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^{4}

^{5}

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## Abstract

**:**

_{1}= 0 where λ

_{1}is a mixed convection parameter and A > 0.1, and a single solution exists when λ

_{1}> 0. Moreover, the effects of numerous applied parameters on velocity, temperature distributions, skin friction, and local Nusselt number are examined and given through tables and graphs for both shrinking and stretching surfaces.

## 1. Introduction

## 2. Mathematical Description of the Problem

## 3. Stability Analysis

## 4. Three-Stage Lobatto IIIA Formula

## 5. Discussion

## 6. Conclusions

- Triple solutions for the coefficient of skin friction, the gradient of temperature, velocity, and temperature profiles occur for specific values of the applied quantity examined in the current examination.
- The critical value and the range of the first and second solutions for the coefficient of skin friction rise with a higher magnitude of the Casson parameter.
- For the stable solution, the velocity of the Casson fluid reduces (as expected) over both surfaces for the strong field of the Lorentz force.
- For the shrinking surface, additional mass suction is required for the occurrence of single and multiple solutions for non-Newtonian Casson fluid (${S}_{c1},$ ${S}_{c2}$ corresponding to $\beta =1.5,5$) compared to Newtonian fluid (${S}_{c3}$ when $\beta =\infty $).
- The magnitude of $-\theta \prime \left(0\right)$ increases (corresponding critical points of $S$ for Biot number $Bi=5,10,\infty $ are ${S}_{c1}=2.6152,$ ${S}_{c2}=2.6141,$ and ${S}_{c3}=2.6107$) for the advanced values of convective parameter.
- Fluid temperature reduces in all solutions and both surfaces when the effect of $Pr$ increases.
- The only first solution is stable from triple solutions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Table 1.**Values of $-\left(1+\frac{1}{\beta}\right){f}^{\u2033}\left(0\right)$ for different $\beta $ and $M$ values where $A=0,M={M}^{2},\lambda =1,{\lambda}_{1}=0$ and $S=0$.

$\mathit{\beta}$ | $\mathit{M}$ | Hussain et al. [34] | Present Results |
---|---|---|---|

$-\left(1+\frac{1}{\mathit{\beta}}\right){\mathit{f}}^{\u2033}\left(0\right)$ | |||

0.7 | 0.5 | 2.146677 | 2.146676800 |

1.2 | 0.5 | 1.865142 | 1.865142292 |

1.2 | 0.0 | 1.735577 | 1.735580976 |

1.2 | 0.4 | 1.819679 | 1.819679224 |

1.2 | 0.7 | 1.980908 | 1.980908405 |

**Table 2.**The values of the smallest eigenvalue ${\gamma}_{1}$ for numerous ${\lambda}_{1}$ and $\lambda $ values where $=1.5,$ $M=0.25,A=0.1,Pr=1,Bi=5,S=3$.

${\mathit{\lambda}}_{1}$ | $\mathit{\lambda}$ | ${\mathit{\gamma}}_{1}$ | ||
---|---|---|---|---|

1st Solution | 2nd Solution | 3rd Solution | ||

−0.2 | −1 | 0.28165 | −0.40384 | −1.52864 |

−0.5 | 0.75239 | −1.042571 | −1.87263 | |

0.5 | 1.17384 | −1.92538 | −2.162901 | |

1 | 1.57243 | −2.35820 | −2.82736 | |

0 | −1 | 0.42962 | −0.5386 | --- |

1 | 2.00518 | −0.79284 | --- | |

0.2 | −1 | 1.26739 | --- | --- |

1 | 3.17427 | --- | --- |

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**MDPI and ACS Style**

Lund, L.A.; Omar, Z.; Khan, I.; Baleanu, D.; Nisar, K.S.
Convective Effect on Magnetohydrodynamic (MHD) Stagnation Point Flow of Casson Fluid over a Vertical Exponentially Stretching/Shrinking Surface: Triple Solutions. *Symmetry* **2020**, *12*, 1238.
https://doi.org/10.3390/sym12081238

**AMA Style**

Lund LA, Omar Z, Khan I, Baleanu D, Nisar KS.
Convective Effect on Magnetohydrodynamic (MHD) Stagnation Point Flow of Casson Fluid over a Vertical Exponentially Stretching/Shrinking Surface: Triple Solutions. *Symmetry*. 2020; 12(8):1238.
https://doi.org/10.3390/sym12081238

**Chicago/Turabian Style**

Lund, Liaquat Ali, Zurni Omar, Ilyas Khan, Dumitru Baleanu, and Kottakkaran Sooppy Nisar.
2020. "Convective Effect on Magnetohydrodynamic (MHD) Stagnation Point Flow of Casson Fluid over a Vertical Exponentially Stretching/Shrinking Surface: Triple Solutions" *Symmetry* 12, no. 8: 1238.
https://doi.org/10.3390/sym12081238