# MHD Nanofluids in a Permeable Channel with Porosity

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## Abstract

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## 1. Introduction

## 2. Problem Description

## 3. Nusselt Number

## 4. Skin-Friction

## 5. Results and Discussion

## 6. Conclusions

- It was found that the velocity of nanofluids increases with an increase of the volume fraction, radiation, and permeability parameter in the case of suction whereas an opposite behavior was noted in the case of injection.
- The velocity of $Ag$ nanofluids decreases with an increase of the magnetic parameter while the opposite behavior was noted in the case of injection.
- The temperature of $Ag$ nanofluids was found to decrease with an increase of $\varphi $ for the extraction of fluid from the walls whereas a very small change was observed in the case of injection.
- Finally, it was noticed that different types of nanoparticles have different effects on the velocity and temperature due to suction and injection.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${H}_{2}O$ | Water |

${C}_{2}{H}_{6}{O}_{2}$ | Ethelyn glycol |

$v\left(y,t\right)$ | Velocity component in the $x-$ direction |

$T\left(y,t\right)$ | Temperature |

${v}_{\omega}>0$ | Suction |

${v}_{\omega}<0$ | Injection |

${\rho}_{nf}$ | Density of nanofluid |

$M$ | Magnetic parameter |

$Pe$ | Peclet number |

${\mu}_{nf}$ | Dynamic viscosity of nanofluid |

${\left(\rho \beta \right)}_{nf}$ | thermal expansion coefficient |

$g$ | Acceleration due to gravity |

${\left(\rho {c}_{p}\right)}_{nf}$ | Heat capacitance of nanofluids |

${k}_{nf}$ | The thermal conductivity of nanofluid |

$\alpha $ | Mean radiation absorption coefficient |

$Re$ | Reynolds’ number |

$Gr$ | Grashof number |

$N$ | Radiation parameter |

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**Figure 2.**Velocity profiles for different values of $\varphi $ of $Ag$ in water based nanofluids when $Gr=0.1,$ $N=0.1,$ ${r}_{c}=20\mathrm{nm},$ $Pe=0.1,$ $\lambda =1,$ $M=1,$ $K=0.3,$ ${v}_{0}=2,$ $t=5,$ $\omega =0.2.$

**Figure 3.**Velocity profiles for different values of $\varphi $ of $Ag$ in water based nanofluids when $Gr=0.1,$ $N=0.1,$ ${r}_{c}=20\mathrm{nm},$ $Pe=0.1,$ $\lambda =1,$ $M=1,$ $K=0.3,$ ${v}_{0}=-0.01,$ $t=5,$ $\omega =0.2.$

**Figure 4.**Velocity profiles for different values of $\varphi $ of $Ag$ in EG based nanofluids when $Gr=0.1,$ $N=0.1,$ ${r}_{c}=20\mathrm{nm},$ $Pe=0.1,$ $\lambda =1,$ $M=2,$ $K=3,$ ${v}_{0}=4,$ $t=5,$ $\omega =0.2.$

**Figure 5.**Velocity profiles for different values of $\varphi $ of $Ag$ in EG based nanofluids when $Gr=0.1,$ $N=0.1,$ ${r}_{c}=20\mathrm{nm},$ $Pe=0.1,$ $\lambda =1,$ $M=2,$ $K=3,$ $t=5,$ ${v}_{0}=-0.01,$ $\omega =0.2.$

**Figure 6.**Velocity profiles for different values of $Gr$ of $Ag$ in water based nanofluids when $N=0.1,$ $Pe=0.1,$ ${r}_{c}=20\mathrm{nm},$ $\varphi =0.04,$ $\lambda =1,$ $M=2,$ $K=3,$ ${v}_{0}=10,$ $t=5,$ $\omega =0.2.$

**Figure 7.**Velocity profiles for different values of $Gr$ of $Ag$ in water based nanofluids when $N=0.1,$ $Pe=0.1,$ ${r}_{c}=20\mathrm{nm},$ $\varphi =0.04,$ $Re=0.1,$ $\lambda =1,$ $M=2,$ $K=3,$ ${v}_{0}=-1,$ $t=5,$ $\omega =0.2.$

**Figure 8.**Velocity profiles for different values of $K$ of $Ag$ in water based nanofluids when $Gr=0.1,$ $N=0.1,$ $Pe=0.1,$ ${r}_{c}=20\mathrm{nm},$ $\varphi =0.04,$ $\lambda =1,$ $M=2,$ ${v}_{0}=6,$ $t=10,$ $\omega =0.2.$

**Figure 9.**Velocity profiles for different values of $K$ of $Ag$ in water based nanofluids when $Gr=0.1,$ $N=0.1,$ $Pe=0.1,$ ${r}_{c}=20\mathrm{nm},$ $\varphi =0.04,$ $\lambda =1,$ $M=2,$ ${v}_{0}=-0.01,$ $t=10,$ $\omega =0.2.$

**Figure 10.**Velocity profiles for different values of $M$ of $Ag$ in water based nanofluids when $Gr=0.1,$ $N=0.1,$ $Pe=0.1,$ ${r}_{c}=20\mathrm{nm},$ $\varphi =0.04,$ $\lambda =1,$ $K=0.3$ ${v}_{0}=5,$ $t=10,$ $\omega =0.2.$

**Figure 11.**Velocity profiles for different values of $M$ of $Ag$ in water based nanofluids when $Gr=0.1,$ $N=0.1,$ $Pe=0.1,$ ${r}_{c}=20\mathrm{nm},$ $\varphi =0.04,$ $\lambda =1,$ $K=0.3$ ${v}_{0}=-0.01$ $t=10,$ $\omega =0.2.$

**Figure 12.**Velocity profiles for different values of $N$ of $Ag$ in water based nanofluids when $Gr=0.1,$ $Pe=0.1,$ ${r}_{c}=20\mathrm{nm},$ $\varphi =0.04,$ $\lambda =1,$ $M=1,$ $K=1,$ ${v}_{0}=7,$ $t=2,$ $\omega =0.2.$

**Figure 13.**Velocity profiles for different values of $N$ of $Ag$ in water based nanofluids when $Gr=0.1,$ $Pe=0.1,$ ${r}_{c}=20\mathrm{nm},$ $\varphi =0.04,$ $\lambda =1,$ $M=1,$ $K=1,$ ${v}_{0}=-1,$ $t=2,$ $\omega =0.2.$

**Figure 14.**Velocity profiles for different types of nanoparticles in water based nanofluids when $Gr=0.1,$ $N=0.1,$ $Pe=0.1,$ ${r}_{c}=20\mathrm{nm},$ $\lambda =1,$ $M=2,$ $K=3,$ $t=5,$ $\omega =0.2.$

**Figure 15.**Temperature profiles for different values of $\varphi $ of $Ag$ in water based nanofluids when ${r}_{c}=20\mathrm{nm},$ $N=1,$ $t=1,$ ${v}_{0}=10,$ $\omega =0.2.$

**Figure 16.**Temperature profiles for different values of $\varphi $ of $Ag$ in water based nanofluids when ${r}_{c}=20\mathrm{nm},$ $N=1,$ $t=1,$ ${v}_{0}=-1,$ $\omega =0.2.$

**Figure 17.**Temperature profiles for different values of $N$ of $Ag$ in water based nanofluids when ${r}_{c}=20\mathrm{nm},$ $\varphi =0.04,$ $t=1,$ ${v}_{0}=-1,$ $\omega =0.2.$

**Figure 18.**Temperature profiles for different values of $N$ of $Ag$ in water based nanofluids when ${r}_{c}=20\mathrm{nm},$ $\varphi =0.04,$ $t=1,$ ${v}_{0}=10,$ $\omega =0.2.$

**Figure 19.**Temperature profiles for different types of nanoparticles in water based nanofluids when ${r}_{c}=20\mathrm{nm},$ $N=1,$ $t=1,$ ${v}_{0}=-1,$ $\omega =0.2.$

Model | C_{P} (kg^{−1} K^{−1}) | $\mathit{\rho}\mathbf{(}\mathbf{kg}{\mathbf{m}}^{\mathbf{-}\mathbf{3}}\mathbf{)}$ | k (Wm^{−1} K^{−1}) | β x 10^{−5} (K^{−1}) | $\mathit{\sigma}\text{}\mathbf{(}\mathbf{S}\mathbf{/}\mathbf{m})$ |
---|---|---|---|---|---|

Water $({H}_{2}O)$ | 4179 | 997.1 | 0.613 | 21 | $5.5\times {10}^{-6}$ |

EG $({C}_{2}{H}_{6}{O}_{2})$ | 0.58 | 1.115 | 0.1490 | 6.5 | $1.07\times {10}^{-6}$ |

Alumina $(A{l}_{2}{O}_{3})$ | 756 | 3970 | 40 | 0.85 | $1.07\times {10}^{-6}$ |

Silver $(Ag)$ | 235 | 10,500 | 429 | 1.89 | $6.30\times {10}^{7}$ |

Copper $(Cu)$ | 385 | 8933 | 401 | 1.67 | $59.6\times {10}^{6}$ |

Titanium Dioxide $(Ti{O}_{2})$ | 686.2 | 4250 | 8.9528 | 0.9 | $2.6\times {10}^{6}$ |

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**MDPI and ACS Style**

Khan, I.; Alqahtani, A.M.
MHD Nanofluids in a Permeable Channel with Porosity. *Symmetry* **2019**, *11*, 378.
https://doi.org/10.3390/sym11030378

**AMA Style**

Khan I, Alqahtani AM.
MHD Nanofluids in a Permeable Channel with Porosity. *Symmetry*. 2019; 11(3):378.
https://doi.org/10.3390/sym11030378

**Chicago/Turabian Style**

Khan, Ilyas, and Aisha M. Alqahtani.
2019. "MHD Nanofluids in a Permeable Channel with Porosity" *Symmetry* 11, no. 3: 378.
https://doi.org/10.3390/sym11030378