# The Impact of Nanoparticles Due to Applied Magnetic Dipole in Micropolar Fluid Flow Using the Finite Element Method

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## Abstract

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## 1. Introduction

## 2. Problem Description

## 3. Implementation of Method

#### 3.1. Variational Formulations

#### 3.2. Finite Element Formulation

## 4. Results and Discussion

## 5. Concluding Remarks

- With increasing values of thermal stratification ${S}_{1}$ corresponds in decreasing of velocity and temperature profiles, Further, the heat transfer rate increases by increasing values of parameter ${S}_{1}$.
- The velocity, temperature, and micro-rotational velocity is higher in the micropolar ferromagnetic fluid as compare to the ferrimagnetic fluid.
- Thermal conduction of nanoparticles enhances with the inconsistency of volume fraction.
- The effect of K on the velocity profile and the micro-rotational velocity is increasing whereas it is declining in the thermal boundary layer.
- The velocity profile reduces with increasing values of suction/injection parameter ${f}_{w}$ and ferromagnetic parameter $\beta $ in the presence of magnetic dipole while the temperature field increases.
- The velocity profile is a decreasing function of slip parameter ${S}_{f}$ while also an increasing function of temperature profile and relative boundary layer of nanofluids.
- In the presence of magnetic dipole reducing the rate of heat transfer has been perceived.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$R{e}_{x}$ | Local Reynold number |

$Rd$ | Thermal radiation parameter |

${S}_{1}$ | Thermal Stratification |

${\alpha}_{nf}$ | Normal anxiety moduli |

$\rho {}_{n}f$ | Fluid density |

$\mu nf$ | Viscosity of fluid |

${K}_{1}$ | pyromagnetic coefficient |

M | Magnetic penetrability |

${\mu}_{0}$ | Magnetic field |

${({\rho}_{cp})}_{nf}$ | Thermal capability of nano-fluid |

${\gamma}_{nf}$ | Spin gradient viscosity |

$qr$ | Rosseland eradicative heat flux |

${\sigma}^{*}$ | Stefan-Boltzmann number |

${\kappa}^{*}$ | Mean assimilation coefficient |

$\u03f5$ | Curie temperature |

T | Non-dimensional temperature |

${T}_{w}$ | Temperature at surface |

${T}_{\infty}$ | Temperature away from the surface |

m | Micro-rotation parameter |

${u}_{w}$ | Velocity of sheet |

$u,v$ | Velocity components |

$\beta $ | Ferromagnetic parameter |

$\lambda $ | Viscous dissipation |

${\alpha}_{nf}$ | Normal anxiety moduli |

$Pr$ | Prandtl number |

$\delta $ | boundary parameter |

${S}_{f}$ | Slip parameter |

K | micro-rotation parameter |

$\varphi $ | Solid volume fraction |

${\kappa}_{s}$ | thermal conductivity of nanoparticles |

${\kappa}_{f}$ | thermal conductivity of base fluid |

${\rho}_{s}$ | density of the nanoparticles |

${\rho}_{f}$ | density of base fluid |

$\gamma $ | Strength of magnetic field |

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**Table 1.**Physical properties of base fluids (water $60\%$ + ethylene glycol $40\%$) and nanoparticles.

Property | Base Fluid [39] | Paramagnetic (Ta) [40] | Diamagnetic (Cu) [40] | Ferromagnetic (Fe) [40] |
---|---|---|---|---|

${C}_{p}(J$·$(kg$·${K)}^{-1})$ | 3752 | 140 | 385 | 447 |

$\rho (kg$·${m}^{-3})$ | 1054 | 16,600 | 8933 | 7870 |

$\kappa (W$·$(m$·${K)}^{-1})$ | 0.416 | 57.5 | 401 | 80.2 |

**Table 2.**Finite Element Method (FEM) convergence results of $\tilde{h}(\eta )$, $\tilde{g}(\eta )$, ${\tilde{\theta}}_{1}(\eta )$, and ${\tilde{\theta}}_{2}(\eta )$ for various number of elements when $Pr=2$, $K=0.1$, $\lambda =0.01$, ${S}_{f}=0.2$, $\beta =0.5$, $E=2$, $\delta =0.5$, ${f}_{w}=0.2$, $N=0.5$.

Number of Elements | $\tilde{\mathit{h}}(3)$ | $\tilde{\mathit{g}}(3)$ | ${\tilde{\mathit{\theta}}}_{1}(3)$ | ${\tilde{\mathit{\theta}}}_{2}(3)$ |
---|---|---|---|---|

40 | 0.007067 | 0.006707 | 0.143487 | 0.000010 |

100 | 0.007279 | 0.006842 | 0.142996 | 0.000011 |

200 | 0.007309 | 0.006861 | 0.142925 | 0.000012 |

340 | 0.007315 | 0.006865 | 0.142909 | 0.000012 |

500 | 0.007317 | 0.006866 | 0.142905 | 0.000012 |

700 | 0.007318 | 0.006867 | 0.142903 | 0.000012 |

**Table 3.**Comparison of $-{\tilde{\theta}}^{\prime}(0)$ for various values of Prandtl number $Pr$, when all others parameters are zero.

Pr | Sohaib et al. [35] | Liaqat et al. [48] | Bagh et al. [49] | Majeed et al. [50] | Bachok et al. [51] | FEM (Current Results) |
---|---|---|---|---|---|---|

0.72 | 0.808633 | 0.808634 | 0.808634 | 0.808640 | 0.8086 | 0.808633 |

1.00 | 1.000008 | 1.000001 | 1.000001 | 1.000000 | 1.0000 | 1.000009 |

3.00 | 1.923677 | 1.923678 | 1.923683 | 1.923609 | 1.9237 | 1.923680 |

10.0 | 3.720668 | 3.720668 | 3.720674 | 3.720580 | 3.7207 | 3.720669 |

**Table 4.**Comparison of skin friction for various values of K and $\delta $ when all others parameters are zero.

K | $\mathit{\delta}$ | Qasim et al. [52] | Abid Hussanan et al. [53] | Kumar [54] | FEM (Current Results) |
---|---|---|---|---|---|

0.0 | 0.5 | −1.000000 | −1.0000000 | - | −1.0000089 |

1.0 | −1.224741 | −1.2247448 | - | −1.2248199 | |

2.0 | −1.414218 | −1.4142135 | - | −1.4144797 | |

4.0 | −1.732052 | −1.7320508 | - | −1.7332924 | |

0.0 | 0.0 | −1.000000 | - | −1.000008 | −1.0000089 |

1.0 | −1.367872 | - | −1.367996 | −1.3679971 | |

2.0 | −1.621225 | - | −1.621575 | −1.6215754 | |

4.0 | −2.004133 | - | −2.005420 | −2.0054211 |

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**MDPI and ACS Style**

Ali, L.; Liu, X.; Ali, B.; Mujeed, S.; Abdal, S.; Mutahir, A.
The Impact of Nanoparticles Due to Applied Magnetic Dipole in Micropolar Fluid Flow Using the Finite Element Method. *Symmetry* **2020**, *12*, 520.
https://doi.org/10.3390/sym12040520

**AMA Style**

Ali L, Liu X, Ali B, Mujeed S, Abdal S, Mutahir A.
The Impact of Nanoparticles Due to Applied Magnetic Dipole in Micropolar Fluid Flow Using the Finite Element Method. *Symmetry*. 2020; 12(4):520.
https://doi.org/10.3390/sym12040520

**Chicago/Turabian Style**

Ali, Liaqat, Xiaomin Liu, Bagh Ali, Saima Mujeed, Sohaib Abdal, and Ali Mutahir.
2020. "The Impact of Nanoparticles Due to Applied Magnetic Dipole in Micropolar Fluid Flow Using the Finite Element Method" *Symmetry* 12, no. 4: 520.
https://doi.org/10.3390/sym12040520