# Heat Transfer Analysis of a Magneto-Bio-Fluid Transport with Variable Thermal Viscosity Through a Vertical Ciliated Channel

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## Abstract

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## 1. Introduction

## 2. Statement of the Problem

## 3. Solution Methodology

## 4. Results and Discussions

#### 4.1. Velocity Distribution

#### 4.2. Pressure Distribution

#### 4.3. Volume Flow Rate

#### 4.4. Trapping Phenomenon

## 5. Application: Physiological Transport of Semen

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 2.**$u(y)$ for (

**a**) $M$ (

**b**) $\alpha $ (

**c**) ${g}_{t}$ (

**d**) ${S}_{t}$. The other parameters are fixed as $\delta =0.4$, $\beta =0.4$, $\epsilon =0.3$, $Q=0.9$, $M=2$, $\alpha =0.05$, ${g}_{t}=5$, ${S}_{t}=5$.

**Figure 3.**$dp/dx$ for (

**a**) $M$ (

**b**) $\alpha $ (

**c**) ${g}_{t}$ (

**d**) ${S}_{t}$. The other parameters are fixed as $\delta =0.4$, $\beta =0.4$, $\epsilon =0.3$, $Q=0.9$, $M=2$, $\alpha =0.05$, ${g}_{t}=5$, ${S}_{t}=5$.

**Figure 4.**Pressure difference ($\Delta p$) versus the flow rate ($Q$) for (

**a**) $M$ (

**b**) $\alpha $ (

**c**) ${g}_{t}$ (

**d**) ${S}_{t}$. The other parameters are fixed as $\delta =0.4$, $\beta =0.4$, $\epsilon =0.3$, $M=2$, $\alpha =0.05$, ${g}_{t}=5$, ${S}_{t}=5$.

**Figure 5.**Volume flow rate ($Q$) versus $\epsilon $ for (

**a**) $M$, (

**b**) $\alpha $, (

**c**) ${g}_{t}$, (

**d**) ${S}_{t}$. The other parameters are fixed as $\delta =0.4$, $\beta =0.4$, $\Delta p=-3$, $M=0.5$, $\alpha =0.05$, ${g}_{t}=3$, ${S}_{t}=3$.

**Figure 6.**Volume flow rate ($Q$) versus $\delta $ for (

**a**) $M$ (

**b**) $\epsilon $. The other parameters are fixed as $\epsilon =0.3$, $\beta =0.4$, $\Delta p=-3$, $M=0.5$, $\alpha =0.05$, ${g}_{t}=3$, ${S}_{t}=3$.

**Figure 7.**Streamlines for (

**a**) $M=1.5$, (

**b**) $M=2.5$, (

**c**) $M=3.5$. The other parameters are fixed as $\delta =0.2$, $\beta =0.2$, $\epsilon =0.3$, $Q=0.9$, $M=1.5$, $\alpha =0.03$, ${g}_{t}=5$, ${S}_{t}=5$.

**Figure 8.**Streamlines for (

**a**) $\alpha =0.0$, (

**b**) $\alpha =0.03$, (

**c**) $\alpha =0.05$. The other parameters are fixed as $\delta =0.2$, $\beta =0.2$, $\epsilon =0.3$, $Q=0.9$, $M=1.5$, ${g}_{t}=5$, ${S}_{t}=5$.

**Figure 9.**Streamlines for (

**a**) ${g}_{t}=0$, (

**b**) ${g}_{t}=3$, (

**c**) ${g}_{t}=5$. The other parameters are fixed as $\delta =0.2$, $\beta =0.2$, $\epsilon =0.3$, $Q=0.9$, $M=1.5$, $\alpha =0.03$, ${g}_{t}=3$, ${S}_{t}=3$.

**Figure 10.**Streamlines for (

**a**) ${S}_{t}=0$, (

**b**) ${S}_{t}=3$, (

**c**) ${S}_{t}=5$. The other parameters are fixed as $\delta =0.2$, $\beta =0.2$, $\epsilon =0.3$, $Q=0.9$, $M=1.5$, $\alpha =0.03$, ${g}_{t}=3$, ${S}_{t}=3$.

**Table 1.**Numerical results of $Q$ and ${Q}^{*}=(Q\times 2\pi {d}^{2}c)$, when $\delta =0.1$, $\epsilon =0.1$, $\beta =1.0$, and $\Delta p=0$.

$M$ | $\alpha $ | ${g}_{t}$ | ${S}_{t}$ | $Q$ | ${Q}^{*}$$\left(\mathrm{mL}/\mathrm{h}\right)$ |
---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0.02204 | 0.00012 |

0.01 | 0.1 | 0.1 | 0.05957 | 0.00032 | |

0.02 | 0.1 | 0.1 | 0.06443 | 0.00035 | |

0.03 | 0.1 | 0.1 | 0.07071 | 0.00038 | |

0.1 | 0.01 | 0.1 | 0.1 | 0.05932 | 0.00032 |

0.02 | 0.1 | 0.1 | 0.06411 | 0.00034 | |

0.03 | 0.1 | 0.1 | 0.07029 | 0.00031 | |

0.01 | 0.3 | 0.1 | 0.13398 | 0.00073 | |

0.01 | 0.5 | 0.1 | 0.20864 | 0.00113 | |

0.01 | 0.1 | 0.3 | 0.06249 | 0.00034 | |

0.01 | 0.1 | 0.5 | 0.06569 | 0.00035 | |

0.3 | 0.01 | 0.1 | 0.1 | 0.05716 | 0.00031 |

0.5 | 0.01 | 0.1 | 0.1 | 0.05208 | 0.00028 |

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**MDPI and ACS Style**

Ahmad Farooq, A.; Shah, Z.; Alzahrani, E.O.
Heat Transfer Analysis of a Magneto-Bio-Fluid Transport with Variable Thermal Viscosity Through a Vertical Ciliated Channel. *Symmetry* **2019**, *11*, 1240.
https://doi.org/10.3390/sym11101240

**AMA Style**

Ahmad Farooq A, Shah Z, Alzahrani EO.
Heat Transfer Analysis of a Magneto-Bio-Fluid Transport with Variable Thermal Viscosity Through a Vertical Ciliated Channel. *Symmetry*. 2019; 11(10):1240.
https://doi.org/10.3390/sym11101240

**Chicago/Turabian Style**

Ahmad Farooq, Ali, Zahir Shah, and Ebraheem O. Alzahrani.
2019. "Heat Transfer Analysis of a Magneto-Bio-Fluid Transport with Variable Thermal Viscosity Through a Vertical Ciliated Channel" *Symmetry* 11, no. 10: 1240.
https://doi.org/10.3390/sym11101240