# Numerical Analysis with Keller-Box Scheme for Stagnation Point Effect on Flow of Micropolar Nanofluid over an Inclined Surface

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

## 3. Results and Discussion

_{fx}(0) against changed magnitudes of concerned physical factors are portrayed in Table 2. From Table 2, it can be found that $-\theta {}^{\prime}(0)$ diminishes with the growth of $Nb,Nt,\mathrm{Pr},Le,M,R,{\lambda}_{1}$ and $\zeta $, whereas it boosts on improving the magnitudes of $K,\lambda ,\delta $ and $\gamma $. Moreover, $-\varphi {}^{\prime}(0)$ increases on the growing of $Nb,Nt,\mathrm{Pr},Le,K,{\lambda}_{1},\lambda ,\delta ,R$ and $\gamma $, while it decreases against the cumulative magnitudes of $M$ and $\zeta $. Physically, by increasing the Brownian motion effect, the thermal boundary layer thickness increases and it affects a large amount of the liquid. Besides, the Sherwood number increases and the Nusselt number declines as we increase the thermophoresis effect. It is due to the fact that the thermal boundary layer turns thicker due to deeper dispersal penetration into the liquid. In sum, ${C}_{fx}\left(0\right)$ improves on the growth of $Nb,Le,M,K,R,{\lambda}_{1}$ and $\zeta $. On the other hand, ${C}_{fx}\left(0\right)$ decreases with the increment in $Nt,\mathrm{Pr},\lambda ,\delta $ and $\gamma $. It is observed that the skin friction reduces on improving the values of stagnation parameter, and its negative values identify the presence of drag force (employs the stretching sheet) on the motion of the micropolar nanofluid. It is not shocking, because the boundary layer is developed due to the stretching.

#### 3.1. Velocity Profile

#### 3.2. Temperature Profile

#### 3.3. Concentration Profile

## 4. Conclusions

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- Energy and mass exchange enhance with the growth of the stagnation parameter.
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- The chemical reaction diminishes the concentration field with higher values.
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- The stagnation parameter shows direct correspondence with the velocity profile.
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- The heat generation or absorption factor declines the energy transport rate, whereas it improves the mass flux rate.
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- The velocity profile shows an opposite behavior for $\gamma <1$, and $\gamma >1$.
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- The velocity profile shows direct relation against growing magnitudes of bouncy impacts for $\gamma <1$, and $\gamma >1$.
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- This study can be utilized in the building envelop applications because of the heat transfer.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations Table

$C$ | Fluid Concentration | ${R}^{\ast}$ | Dimensionless Reaction Rate | ${\lambda}_{1}$ | Heat Generation Parameter |

${C}_{f}$ | Skin friction coefficient | $R$ | Chemical reaction parameter | ${\mathrm{Re}}_{x}$ | Reynolds number |

${C}_{\infty}$ | Ambient nanoparticle volume fraction | $Le$ | Lewis number | $Sh$ | Sherwood number |

${C}_{w}$ | Surface volume fraction | $Nb$ | Brownian motion parameter | $T$ | Fluid temperature |

${c}_{p}$ | Specific heat at constant pressure | $Nt$ | Thermophoretic parameter | ${T}_{w}$ | Wall temperature |

${D}_{B}$ | Brownian diffusion coefficient | $Nu$ | Nusselt number | ${T}_{\infty}$ | Ambient temperature |

${D}_{T}$ | Thermophoretic diffusion coefficient | $\mathrm{Pr}$ | Prandtle number | $U$ | Composite velocity |

$f$ | Similarity function for velocity | ${Q}_{0}$ | Dimensionless heat generation | ${U}_{\infty}$ | Wall velocity |

$\rho {c}_{p}$ | Volume heat capacity | $\mu $ | Kinematic viscosity | $\nu $ | Dynamic viscosity |

$\varphi $ | Dimensionless solid volume fraction | $w$ | Condition at the wall | $\infty $ | Ambient condition |

$\delta $ | Solutal buoyancy parameter | ${\beta}_{t}$ | Thermal expansion coefficient | ${\beta}_{c}$ | Concentration expansion coefficient |

$\sigma $ | Electric conductivity | ${\gamma}^{\ast}$ | Spin gradient viscosity | ${k}_{1}^{\ast}$ | Vertex viscosity |

${j}^{\ast}$ | Micro inertia per unit mass | $\zeta $ | Inclination parameter | ${}^{{}^{\prime}}$ | Differentiation with respect to $\eta $ |

$u$ | Velocity in $x$ direction | $v$ | Velocity in $y$ direction | $x$ | Cartesian coordinate |

$\theta $ | Dimensionless temperature | $\gamma $ | Velocity ratio parameter | $k$ | Thermal conductivity |

$\rho $ | Fluid density | $\lambda $ | Bouncy parameter | ${B}_{0}$ | Uniform magnetic field strength |

$K$ | Material parameter | $\eta $ | Similarity independent variable | $\alpha $ | Thermal diffusivity |

${N}^{\ast}$ | Non-dimensional angular velocity | $g$ | Gravitational acceleration | $-\theta {}^{\prime}(0)$ | Reduced Nusselt number |

$-\varphi {}^{\prime}(0)$ | Reduced Sherwood number |

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**Table 1.**Contrast of $-\theta {}^{\prime}(0)$ and $-\varphi {}^{\prime}(0)$ against $M,R,{\lambda}_{1},\gamma ,\lambda ,\delta =0$ with $Le,\mathrm{Pr}=10$ and $\zeta ={90}^{0}$.

$\mathit{N}\mathit{b}$ | $\mathit{N}\mathit{t}$ | Khan and Pop [42] | Present Results | ||
---|---|---|---|---|---|

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

0.1 | 0.1 | 0.9524 | 2.1294 | 0.9524 | 2.1294 |

0.2 | 0.2 | 0.3654 | 2.5152 | 0.3654 | 2.5152 |

0.3 | 0.3 | 0.1355 | 2.6088 | 0.1355 | 2.6088 |

0.4 | 0.4 | 0.0495 | 2.6038 | 0.0495 | 2.6038 |

0.5 | 0.5 | 0.0179 | 2.5731 | 0.0179 | 2.5731 |

**Table 2.**Values of $-\theta {}^{\prime}(0)$, $-\varphi {}^{\prime}(0)$ and ${C}_{fx}\left(0\right)$.

$\mathit{N}\mathit{b}$ | $\mathit{N}\mathit{t}$ | $\mathbf{Pr}$ | $\mathit{L}\mathit{e}$ | $\mathit{M}$ | $\mathit{K}$ | $\mathit{R}$ | ${\mathit{\lambda}}_{1}$ | $\mathit{\lambda}$ | $\mathit{\delta}$ | $\mathit{\gamma}$ | $\mathit{\zeta}$ | $-\mathit{\theta}{}^{\prime}(0)$ | $-\mathit{\varphi}{}^{\prime}(0)$ | ${\mathit{C}}_{\mathit{f}\mathit{x}}\left(0\right)$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.1 | 0.1 | 7.0 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.5 | 45^{0} | 0.8966 | 2.8563 | 0.6967 |

0.5 | 0.1 | 7.0 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.5 | 45^{0} | 0.0640 | 2.8806 | 0.7234 |

0.1 | 0.3 | 7.0 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.5 | 45^{0} | 0.4931 | 3.6809 | 0.6509 |

0.1 | 0.1 | 10.0 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.5 | 45^{0} | 0.8322 | 2.9741 | 0.6955 |

0.1 | 0.1 | 7.0 | 10.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.5 | 45^{0} | 0.8100 | 4.1661 | 0.7506 |

0.1 | 0.1 | 7.0 | 5.0 | 0.5 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.5 | 45^{0} | 0.8885 | 2.8524 | 0.8200 |

0.1 | 0.1 | 7.0 | 5.0 | 0.1 | 3.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.5 | 45^{0} | 0.9120 | 2.8642 | 0.9514 |

0.1 | 0.1 | 7.0 | 5.0 | 0.1 | 1.0 | 3.0 | 0.1 | 0.1 | 0.9 | 0.5 | 45^{0} | 0.8122 | 4.4944 | 0.7482 |

0.1 | 0.1 | 7.0 | 5.0 | 0.1 | 1.0 | 1.0 | 0.5 | 0.1 | 0.9 | 0.5 | 45^{0} | 0.1890 | 3.3436 | 0.7051 |

0.1 | 0.1 | 7.0 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.5 | 0.9 | 0.5 | 45^{0} | 0.9007 | 2.8579 | 0.6064 |

0.1 | 0.1 | 7.0 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.1 | 2.0 | 0.5 | 45^{0} | 0.9071 | 2.8607 | 0.4742 |

0.1 | 0.1 | 7.0 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 1.5 | 45^{0} | 1.0243 | 2.9339 | −1.4412 |

0.1 | 0.1 | 7.0 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.5 | 60^{0} | 0.8937 | 2.8552 | 0.7570 |

0.1 | 0.1 | 7.0 | 5.0 | 0.1 | 1.0 | 1.0 | 0.1 | 0.1 | 0.9 | 0.5 | 90^{0} | 0.8867 | 2.8523 | 0.9033 |

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**MDPI and ACS Style**

Rafique, K.; Anwar, M.I.; Misiran, M.; Khan, I.; Seikh, A.H.; Sherif, E.-S.M.; Nisar, K.S.
Numerical Analysis with Keller-Box Scheme for Stagnation Point Effect on Flow of Micropolar Nanofluid over an Inclined Surface. *Symmetry* **2019**, *11*, 1379.
https://doi.org/10.3390/sym11111379

**AMA Style**

Rafique K, Anwar MI, Misiran M, Khan I, Seikh AH, Sherif E-SM, Nisar KS.
Numerical Analysis with Keller-Box Scheme for Stagnation Point Effect on Flow of Micropolar Nanofluid over an Inclined Surface. *Symmetry*. 2019; 11(11):1379.
https://doi.org/10.3390/sym11111379

**Chicago/Turabian Style**

Rafique, Khuram, Muhammad Imran Anwar, Masnita Misiran, Ilyas Khan, Asiful H. Seikh, El-Sayed M. Sherif, and Kottakkaran Sooppy Nisar.
2019. "Numerical Analysis with Keller-Box Scheme for Stagnation Point Effect on Flow of Micropolar Nanofluid over an Inclined Surface" *Symmetry* 11, no. 11: 1379.
https://doi.org/10.3390/sym11111379