# Stability Analysis of the Magnetized Casson Nanofluid Propagating through an Exponentially Shrinking/Stretching Plate: Dual Solutions

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Stability Analysis

## 4. Numerical Methods

#### 4.1. Shooting Method

#### 4.2. Three-Stage Lobatto IIIA Formula

## 5. Result and Discussion

## 6. Conclusion Remarks

- For both surfaces, there are dual solutions;
- For the stable solution, the velocity profile decreases for higher values of Casson parameter $\beta $ and the magnetic field $M$;
- With a rise in the intensity of the convection parameter $A$, the thermal boundary layer is enhanced. Yet, the Prandtl number $Pr$ has the inverse relationship with the temperature profile in both solutions.
- Stability analysis reveals that an unstable (/stable) solution is a second (/first) solution.
- Dual solutions vary, and no solution depends on the parameters involved.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

u, v | Velocity components | $A$ | Biot number |

T | Temperature | ${f}^{\prime}$ | Dimensionless velocity |

${T}_{0}$ | Reference temperature | $R{e}_{x}$ | Local Reynolds number |

${T}_{f}$ | Temperature of the hot fluid below the surface | ${C}_{f}$ | Skin friction coefficient |

${T}_{\infty}$ | Ambient temperature | ${N}_{u}$ | Local Nusselt number |

$\beta $ | Casson parameter | ${f}_{w}$ | Injunction/suction parameter |

C | Concentration | ${h}_{f}$ | Convective heat transfer coefficient |

${C}_{0}$ | Reference concentration | ${u}_{w}$ | Velocity of surface |

${C}_{\infty}$ | Ambient concentration | γ_{1} | Smallest eigen value |

B | Magnetic field | $\tau $ | Stability transformed variable |

M | Hartmann number | $\lambda $ | Stretching/shrinking parameter |

Pr | Prandtl number | $\psi $ | Stream function |

${D}_{B}$ | Brownian diffusion | ${U}_{0}$ | A constant |

${D}_{T}$ | Thermophoretic diffusion | $\delta $ | Velocity slip condition |

${v}_{w}$ | Suction/injection velocity | ${\sigma}^{*}$ | Electrical conductivity |

${S}_{h}$ | Local Sherwood number | Thermal buoyancy parameter | |

${k}_{f}$ | Thermal conductivity of the nanofluid | ||

${N}_{b}$ | Brownian motion parameter | $\epsilon $ | Unknown eigenvalue |

${N}_{t}$ | Thermophoresis parameter | $\eta $ | Transformed variable |

$Sc$ | Schmidt number | $\alpha $ | Thermal diffusivity |

${A}^{*}$ | Velocity slip factor | ${A}_{1}^{*}$ | Slip factor of velocity |

$N$ | Concentration condition | ${N}_{1}$ | Slip factor of concentration |

${C}_{w}$ | Variable concentration at the sheet | $\theta $ | Dimensionless temperature |

${\tau}_{w}$ | Heat capacity of the nanofluid and the effective heat capacity of the nanoparticle material | $\varnothing $ | Dimensionless concentration |

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**Figure 8.**$-{\varnothing}^{\prime}\left(0\right)$ for different values of ${f}_{w}$ by varying $\beta $.

**Figure 15.**$\varnothing \left(\eta \right)$ for different values of ${N}_{b}$ by varying ${\delta}_{c}$.

**Figure 16.**$\varnothing \left(\eta \right)$ for different values of ${N}_{t}$ by varying ${\delta}_{c}$.

**Table 1.**Compression of $-{f}^{\u2033}\left(0\right)$, $-{\theta}^{\prime}\left(0\right)$ and $-{\varnothing}^{\prime}\left(0\right)$.

${\mathit{N}}_{\mathit{t}}$ | [38]${\mathit{N}}_{\mathit{b}}=0.1$ | [37] ${\mathit{N}}_{\mathit{b}}=0.1$ | Present Results ${\mathit{N}}_{\mathit{b}}=0.1$ | ||||||
---|---|---|---|---|---|---|---|---|---|

$-{\mathit{f}}^{\u2033}\left(0\right)$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ | $-{\varnothing}^{\prime}\left(0\right)$ | $-{\mathit{f}}^{\u2033}\left(0\right)$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ | $-{\varnothing}^{\prime}\left(0\right)$ | $-{\mathit{f}}^{\u2033}\left(0\right)$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ | $-{\varnothing}^{\prime}\left(0\right)$ | |

0.1 | 1.28181 | 0.25374 | 0.37525 | 1.28180857 | 0.25373483 | 0.37525393 | 1.28180857 | 0.25373483 | 0.37525393 |

0.2 | 1.28181 | 0.25192 | 0.20423 | 1.28180857 | 0.25191722 | 0.20422841 | 1.28180857 | 0.25191726 | 0.20422841 |

0.3 | 1.28181 | 0.25008 | 0.03662 | 1.28180857 | 0.25007701 | 0.03660736 | 1.28180857 | 0.25007701 | 0.03660735 |

0.4 | 1.28181 | 0.24821 | −0.12757 | 1.28180857 | 0.24821431 | −0.12757046 | 1.28180857 | 0.24821430 | −0.12757045 |

0.5 | 1.28181 | 0.24633 | −0.28826 | 1.28180857 | 0.24632922 | −0.28826784 | 1.28180857 | 0.24632921 | −0.28826784 |

**Table 2.**Values of ${\epsilon}_{1}$ for different values of ${f}_{w}$ and $\lambda $ by keeping $\beta =2.5$.

$\mathit{\lambda}$ | ${\mathit{f}}_{\mathit{w}}$ | ${\mathit{\epsilon}}_{1}$ | |
---|---|---|---|

1^{st} Solution | 2^{nd} Solution | ||

−1 | 2.1 | 0.17016 | −0.15422 |

−1 | 2.05 | 0.10934 | −0.04286 |

−1 | 2.048 | 0.03401 | −0.00966 |

1 | 2.1 | 1.83592 | −2.63523 |

−1.03 | 2.1 | 0.07409 | −0.08139 |

−1.035 | 2.1 | 0.02781 | −0.03149 |

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**MDPI and ACS Style**

Lund, L.A.; Omar, Z.; Khan, I.; Sherif, E.-S.M.; Abdo, H.S.
Stability Analysis of the Magnetized Casson Nanofluid Propagating through an Exponentially Shrinking/Stretching Plate: Dual Solutions. *Symmetry* **2020**, *12*, 1162.
https://doi.org/10.3390/sym12071162

**AMA Style**

Lund LA, Omar Z, Khan I, Sherif E-SM, Abdo HS.
Stability Analysis of the Magnetized Casson Nanofluid Propagating through an Exponentially Shrinking/Stretching Plate: Dual Solutions. *Symmetry*. 2020; 12(7):1162.
https://doi.org/10.3390/sym12071162

**Chicago/Turabian Style**

Lund, Liaquat Ali, Zurni Omar, Ilyas Khan, El-Sayed M. Sherif, and Hany S. Abdo.
2020. "Stability Analysis of the Magnetized Casson Nanofluid Propagating through an Exponentially Shrinking/Stretching Plate: Dual Solutions" *Symmetry* 12, no. 7: 1162.
https://doi.org/10.3390/sym12071162