# Magnetohydrodynamics Stagnation-Point Flow of a Nanofluid Past a Stretching/Shrinking Sheet with Induced Magnetic Field: A Revised Model

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Stability Analysis

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Roman Letters | |

a | constant variable |

C | nanoparticle volume fraction |

${C}_{f}$ | skin friction coefficient |

${C}_{p}$ | specific heat capacity |

${D}_{B}$ | Brownian diffusion coefficient |

${D}_{T}$ | thermophoresis diffusion coefficient |

$f\left(\eta \right)$ | dimensionless stream function |

$h\left(\eta \right)$ | dimensionless induced magnetic field |

${H}_{0}\left(x\right)$ | applied magnetic field |

${H}_{e}\left(x\right)$ | magnetic field at the edge |

${H}_{1},{H}_{2}$ | induced magnetic field components along the x and y directions, respectively |

k | thermal conductivity |

$Le$ | Lewis number |

M | magnetic parameter |

$Nb$ | Brownian motion parameter |

$Nt$ | thermophoresis parameter |

$N{u}_{x}$ | local Nusselt number |

$Pr$ | Prandtl number |

${q}_{w}$ | surface heat flux |

$R{e}_{x}$ | local Reynolds number |

S | suction/injection parameter |

$S{h}_{x}$ | local Sherwood number |

t | time |

T | temperature of the nanofluid |

${u}_{e}\left(x\right)$ | velocity at the edge of the boundary layer |

$u,v$ | velocity components along the x and y directions, respectively |

${v}_{0}$ | constant mass velocity |

$x,y$ | Cartesian coordinates |

Greek Symbols | |

$\alpha $ | thermal diffusivity of the nanofluid |

$\gamma $ | eigenvalue |

${\gamma}_{1}$ | smallest eigenvalue |

$\u03f5$ | ratio of nanoparticle heat capacity to the base fluid heat capacity |

$\eta $ | similarity variable |

$\theta \left(\eta \right)$ | dimensionless temperature |

$\lambda $ | stretching/shrinking parameter |

$\mu $ | magnetic permeability |

${\mu}_{e}$ | magnetic diffusivity |

$\nu $ | kinematic viscosity |

$\rho $ | density |

$\tau $ | dimensionless time |

${\tau}_{w}$ | surface shear stress |

$\varphi \left(\eta \right)$ | dimensionless nanoparticle volume fraction |

$\chi $ | reciprocal of the magnetic Prandtl number |

Subscripts | |

w | condition at the surface |

∞ | condition outside of boundary layer |

c | critical value |

f | base fluid |

p | nanoparticle |

Superscripts | |

${}^{\prime}$ | differentiation with respect to $\eta $ |

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**Table 1.**Comparison values of the skin friction coefficient ${f}^{\u2033}\left(0\right)$ for some values of $\lambda $ when $M=0$.

$\mathit{\lambda}$ | Present Result | Aman et al. [55] | Bhattacharyya et al. [56] | |||
---|---|---|---|---|---|---|

Upper Branch | Lower Branch | Upper Branch | Lower Branch | Upper Branch | Lower Branch | |

−0.25 | 1.40224 | 1.4022 | 1.40224051 | |||

−0.3 | 1.42758 | 1.4276 | ||||

−0.4 | 1.46861 | 1.4686 | ||||

−0.5 | 1.49567 | 1.4957 | 1.49566948 | |||

−0.615 | 1.50724 | 1.5072 | 1.50724089 | |||

−0.75 | 1.48929 | 1.4893 | 1.48929834 | |||

−1.0 | 1.32881 | 0 | 1.3288 | 0 | 1.32881689 | 0 |

−1.15 | 1.08223 | 0.11670 | 1.0822 | 0.1167 | 1.08223164 | 0.11667340 |

−1.18 | 1.00045 | 0.17836 | 1.0004 | 0.1784 | ||

−1.2465 | 0.55430 | 0.55430 | 0.5543 | 0.5543 | 0.55428565 | 0.55428565 |

S | $\mathit{\lambda}$ | First Solution | Second Solution |
---|---|---|---|

1 | −1.8 | 0.40484 | −0.39610 |

−1.82 | 0.12550 | −0.12458 | |

−1.822 | 0.03052 | −0.03049 | |

2 | −2.8 | 0.73421 | −0.71159 |

−2.85 | 0.28333 | −0.27970 | |

−2.858 | 0.08134 | −0.08104 | |

3 | −4.3 | 0.42854 | −0.42058 |

−4.31 | 0.25712 | −0.25423 | |

−4.315 | 0.08807 | −0.08773 |

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**MDPI and ACS Style**

Junoh, M.M.; Md Ali, F.; Pop, I.
Magnetohydrodynamics Stagnation-Point Flow of a Nanofluid Past a Stretching/Shrinking Sheet with Induced Magnetic Field: A Revised Model. *Symmetry* **2019**, *11*, 1078.
https://doi.org/10.3390/sym11091078

**AMA Style**

Junoh MM, Md Ali F, Pop I.
Magnetohydrodynamics Stagnation-Point Flow of a Nanofluid Past a Stretching/Shrinking Sheet with Induced Magnetic Field: A Revised Model. *Symmetry*. 2019; 11(9):1078.
https://doi.org/10.3390/sym11091078

**Chicago/Turabian Style**

Junoh, Mohamad Mustaqim, Fadzilah Md Ali, and Ioan Pop.
2019. "Magnetohydrodynamics Stagnation-Point Flow of a Nanofluid Past a Stretching/Shrinking Sheet with Induced Magnetic Field: A Revised Model" *Symmetry* 11, no. 9: 1078.
https://doi.org/10.3390/sym11091078