# Magnetohydrodynamic (MHD) Flow of Micropolar Fluid with Effects of Viscous Dissipation and Joule Heating Over an Exponential Shrinking Sheet: Triple Solutions and Stability Analysis

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## Abstract

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## 1. Introduction

- To extend the problem of Aurangzaib et al. [15] by considering the effect of magnetic, viscous, and Joule heating functions.
- To find the maximum number of the multiple solutions.
- To perform stability analysis of multiple solutions in order to determine a stable solution.

## 2. Mathematical Formulation

## 3. Stability Analysis

## 4. Results and Discussion

_{c}. At S = S

_{c}, both solutions meet each other. Beyond this critical value, no similarity solution exists as the boundary layer separates from the sheet. The variation of the shear stress with the parameter S is depicted in Figure 3 for various values of micropolar parameter K. As anticipated, in the first solution, the shear stress rises with all values of S and declines with K. This increment in the shear stress is due to the high values of suction since suction produces more resistance in the fluid flow. Yet, for the second and third solutions, the shear stress reduces with increasing S and has dual solutions at S

_{c1}= 2.1255 and S

_{c2}= 2.67528, respectively corresponding to K = 0.1 and K = 0.2. The variation of the couple stress coefficient with S is depicted in Figure 4 for different values of K. This variation is found to be negligible in the first solution, whereas the couple stress coefficient decreases (increases) with an increase in S (K) in the second solution. Physically, this is due to the fact that the material parameter supports the particles of the skew-symmetric of the fluid and therefore the couple stress coefficient increases in the second solution. In the third solution, the variation of the couple stress coefficient is totally different. It increases with S and decreases with K. Like shear stress, the dimensionless heat transfer coefficient shows the dual nature at the same critical points (Figure 5). All other parameters are kept constant. The first solution for the dimensionless heat transfer coefficient also shows the same nature as the first solution for shear stress. Moreover, heat transfer enhances in the first and second solutions for the high values of mass suctions.

## 5. Conclusions

- Two solutions exist for the case of Newtonian fluid.
- Three solutions exist for the case of no-Newtonian fluid in the specific values of the suction parameter.
- Ranges of single and multiple solutions are dependent on the suction parameter.
- Results of the stability analysis of solutions indicate that only the first solution is stable.
- The thickness of the momentum boundary layer enhances in all solutions when the slip parameter is increased.
- The thickness of the thermal boundary layer is directly proportional to the values of the Eckert number. As the Eckert number increases, the temperature of fluid also rises due to the high impact of the kinetic energy.
- The occurrence of a higher velocity of the fluid is possible for unstable solutions when the magnetic parameter increases.
- The thermal field has been noted as lower corresponding to a larger Prandtl number in all solutions.
- The angular velocity of the micropolar fluid increased in the first and third solutions for the higher values of material and slip parameters.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Comparison of skin friction coefficient ${f}^{\u2033}\left(0\right)$ at the various values of S for different values of K with Aurangzaib et al. [15].

**Figure 3.**The skin friction coefficient ${f}^{\u2033}\left(0\right)$ at various values of S for different values of K.

**Figure 4.**The couple stress coefficient ${g}^{\prime}\left(0\right)$ at various values of S for different values of K.

**Figure 5.**The heat transfer coefficient $-{\theta}^{\prime}\left(0\right)$ along S for different values of K.

**Table 1.**Smallest eigenvalues for different values of $K$ and $S$ when $M=0.5$, $Pr=1$, $m=0.5$, and $Ec=0.1$.

$\mathit{\epsilon}$ | ||||
---|---|---|---|---|

$\mathit{K}$ | $\mathit{S}$ | 1st Solution | 2nd Solution | 3rd Solution |

0.1 | 3 | 0.96439 | −0.78535 | −0.84652 |

2.5 | 0.86425 | −0.60258 | −0.64035 | |

0.2 | 3 | 0.88162 | −0.69025 | −0.72184 |

3.7 | 0.61582 | −0.46325 | −0.52287 |

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**MDPI and ACS Style**

Lund, L.A.; Omar, Z.; Khan, I.; Raza, J.; Sherif, E.-S.M.; Seikh, A.H.
Magnetohydrodynamic (MHD) Flow of Micropolar Fluid with Effects of Viscous Dissipation and Joule Heating Over an Exponential Shrinking Sheet: Triple Solutions and Stability Analysis. *Symmetry* **2020**, *12*, 142.
https://doi.org/10.3390/sym12010142

**AMA Style**

Lund LA, Omar Z, Khan I, Raza J, Sherif E-SM, Seikh AH.
Magnetohydrodynamic (MHD) Flow of Micropolar Fluid with Effects of Viscous Dissipation and Joule Heating Over an Exponential Shrinking Sheet: Triple Solutions and Stability Analysis. *Symmetry*. 2020; 12(1):142.
https://doi.org/10.3390/sym12010142

**Chicago/Turabian Style**

Lund, Liaquat Ali, Zurni Omar, Ilyas Khan, Jawad Raza, El-Sayed M. Sherif, and Asiful H. Seikh.
2020. "Magnetohydrodynamic (MHD) Flow of Micropolar Fluid with Effects of Viscous Dissipation and Joule Heating Over an Exponential Shrinking Sheet: Triple Solutions and Stability Analysis" *Symmetry* 12, no. 1: 142.
https://doi.org/10.3390/sym12010142