# Significance of Thermal Slip and Convective Boundary Conditions in Three Dimensional Rotating Darcy-Forchheimer Nanofluid Flow

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## Abstract

**:**

## 1. Introduction

_{2}O

_{3}-based nanofluids. Sarafraz et al. [11] discussed the potential of nanofluids and solar collectors in thermal energy production systems used in smart cities. Sarafraz et al. [12] reported some major findings and performed smart optimization using tube solar collectors and the response surface method (RSM) over heat pipes. Tlili et al. [13] discussed the impact of MHD in nanofluid flow across a cylindrical channel. Furthermore, Tlili et al. [14] reported some good results of entropy optimization and MHD in stagnation point nanofluid flow using a stretching sheet. Some recent and advanced studies on the effects of different nanoparticles in nanofluids’ flow can be seen in [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29] and the references cited therein.

## 2. Formulation

- (1)
- Skin friction:$$\begin{array}{cc}\hfill {Re}^{1/2}{C}_{fx}& =-{f}^{\u2033}\left(0\right),\hfill \\ \hfill {Re}^{1/2}{C}_{fy}& =-{g}^{\u2033}\left(0\right)\hfill \end{array}$$
- (2)
- Local Nusselt and Sherwood numbers:$$\begin{array}{cc}\hfill N{u}_{x}& =\frac{x{q}_{w}}{\widehat{k}\left({T}_{w}-{T}_{\infty}\right)}=-\left(1+\frac{4}{3}{R}_{1}\right){\theta}^{\prime}\left(0\right),\hfill \\ \hfill S{h}_{x}& =\frac{x{q}_{m}}{\widehat{k}\left(C-{C}_{\infty}\right)}=-{\varphi}^{\prime}\left(0\right).\hfill \end{array}$$$${q}_{w}=-k{\left.\frac{\partial T}{\partial z}\right|}_{z=0}+{q}_{r},\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{q}_{m}=-k{\left.\frac{\partial C}{\partial z}\right|}_{y=0}.$$

## 3. Methodology

## 4. Analysis of the Solutions

## 5. Concluding Remarks

- Lorentz force generated by the MHD resulted in reducing trend in both the axial and transverse velocity fields.
- Both the axial and transverse velocity fields greatly declined for larger values of the Forchheimer number.
- The thermal radiation parameter greatly raised the thermal state of the field.
- The chemical reaction part involved in the governing equations showed the opposite trend in the temperature profile for both the chemical reaction parameter and the Arrhenius activation energy parameter, respectively.
- Both the Brownian diffusion and thermophoresis were rising factors for the thermal distribution.
- The augmented Biot number resulted in a rise in the thermal field.
- The augmented thermal slip parameter enhanced the temperature field.
- Stronger Brownian diffusion resulted in a higher concentration of the nanoparticles.
- A declination was noticed for stronger thermophoresis.
- The Arrhenius activation energy gave rise to the concentration field.
- Both the Forchheimer number and porosity factor resulted in enhancement of the skin friction, while both slip parameters resulted in a decline of the skin friction.
- The activation energy enhanced heat flux with a clear reduction in mass flux. The thermal slip factor resulted in a decline of both the heat and mass flux rates.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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λ | ${\mathit{F}}_{\mathit{r}}$ | ${\mathit{M}}_{1}$ | ${\mathsf{\Omega}}_{1}$ | $\mathit{\alpha}$ | ${\mathit{\gamma}}_{1}$ | ${\mathit{\gamma}}_{2}$ | Skin Friction | |
---|---|---|---|---|---|---|---|---|

$-{\mathit{f}}^{\u2033}\left(\mathbf{0}\right)$ | $-{\mathit{g}}^{\u2033}\left(\mathbf{0}\right)$ | |||||||

0.0 | 0.5 | 0.5 | 0.1 | 0.5 | 0.5 | 0.5 | 0.694509 | 0.369163 |

0.2 | 0.727417 | 0.382899 | ||||||

0.4 | 0.747188 | 0.393225 | ||||||

0.0 | 0.692725 | 0.377525 | ||||||

0.5 | 0.727417 | 0.382899 | ||||||

1.0 | 0.757275 | 0.388372 | ||||||

0.0 | 0.685707 | 0.365622 | ||||||

0.5 | 0.727417 | 0.382899 | ||||||

1.0 | 0.870732 | 0.499395 | ||||||

0.0 | 0.743564 | 0.344124 | ||||||

0.4 | 0.326767 | 0.295244 | ||||||

0.8 | 0.233129 | 0.267431 | ||||||

0.1 | 0.0 | 0.725903 | 0.0539441 | |||||

0.5 | 0.727417 | 0.382899 | ||||||

1.0 | 0.729788 | 0.791255 | ||||||

0.0 | 1.34844 | 0.410502 | ||||||

0.5 | 0.727417 | 0.382899 | ||||||

1.0 | 0.510365 | 0.370739 | ||||||

0.0 | 0.728882 | 0.636562 | ||||||

0.5 | 0.727417 | 0.382899 | ||||||

1.0 | 0.726853 | 0.280113 |

$\mathit{\lambda}$ | ${\mathit{F}}_{\mathit{r}}$ | ${\mathit{R}}_{1}$ | Pr | ${\mathit{N}}_{\mathit{t}}$ | ${\mathit{N}}_{\mathit{b}}$ | ${\mathit{\lambda}}_{1}$ | ${\mathit{S}}_{\mathit{c}}$ | ${\mathit{K}}_{\mathit{r}}$ | ${\mathit{\sigma}}_{1}$ | ${\mathit{E}}_{1}$ | ${\mathit{\gamma}}_{3}$ | ${\mathit{\gamma}}_{4}$ | $-\left(1+\frac{4}{3}{\mathit{R}}_{1}\right){\mathit{\theta}}^{\prime}\left(0\right)$ | $-{\mathit{\varphi}}^{\prime}\left(0\right)$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.0 | 0.5 | 0.5 | 1.0 | 1.0 | 0.3 | 0.2 | 0.1 | 0.5 | 0.3 | 0.5 | 0.2 | 0.3 | 0.314518 | 0.764965 |

0.2 | 0.288237 | 0.773312 | ||||||||||||

0.4 | 0.227716 | 0.782004 | ||||||||||||

0.0 | 0.302424 | 0.76908 | ||||||||||||

0.5 | 0.288237 | 0.773312 | ||||||||||||

1.0 | 0.2758 | 0.776765 | ||||||||||||

0.0 | 0.204117 | 0.778736 | ||||||||||||

0.5 | 0.288237 | 0.773312 | ||||||||||||

1.0 | 0.344522 | 0.782106 | ||||||||||||

1.0 | 0.288237 | 0.773312 | ||||||||||||

2.0 | 0.350164 | 0.791577 | ||||||||||||

3.0 | 0.369261 | 0.851746 | ||||||||||||

1.0 | 0.288237 | 0.773312 | ||||||||||||

1.5 | 0.237309 | 0.815223 | ||||||||||||

2.0 | 0.198657 | 0.872646 | ||||||||||||

0.3 | 0.288237 | 0.773312 | ||||||||||||

0.6 | 0.268343 | 0.821967 | ||||||||||||

0.9 | 0.236591 | 0.840249 | ||||||||||||

0.0 | 0.394603 | 0.652187 | ||||||||||||

0.2 | 0.288237 | 0.773312 | ||||||||||||

0.4 | 0.195891 | 0.879378 | ||||||||||||

1.0 | 0.288237 | 0.773312 | ||||||||||||

1.5 | 0.306753 | 1.00232 | ||||||||||||

2.0 | 0.317076 | 1.19733 | ||||||||||||

0.2 | 0.343959 | 0.427825 | ||||||||||||

0.3 | 0.321331 | 0.566382 | ||||||||||||

0.5 | 0.288237 | 0.773312 | ||||||||||||

0.0 | 0.299917 | 0.696956 | ||||||||||||

0.3 | 0.288237 | 0.773312 | ||||||||||||

0.6 | 0.278423 | 0.83771 | ||||||||||||

0.2 | 0.5 | 0.5 | 1.0 | 1.0 | 0.3 | 0.2 | 0.1 | 0.5 | 0.3 | 0.0 | 0.2 | 0.3 | 0.258141 | 0.964634 |

0.3 | 0.276492 | 0.847455 | ||||||||||||

0.6 | 0.293967 | 0.737371 | ||||||||||||

0.1 | 0.425244 | 0.316425 | ||||||||||||

0.3 | 0.394744 | 0.773723 | ||||||||||||

0.6 | 0.301575 | 0.774264 | ||||||||||||

0.1 | 0.291202 | 0.773487 | ||||||||||||

0.3 | 0.288237 | 0.773312 | ||||||||||||

0.6 | 0.283906 | 0.773093 |

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**MDPI and ACS Style**

Shafiq, A.; Rasool, G.; Khalique, C.M.
Significance of Thermal Slip and Convective Boundary Conditions in Three Dimensional Rotating Darcy-Forchheimer Nanofluid Flow. *Symmetry* **2020**, *12*, 741.
https://doi.org/10.3390/sym12050741

**AMA Style**

Shafiq A, Rasool G, Khalique CM.
Significance of Thermal Slip and Convective Boundary Conditions in Three Dimensional Rotating Darcy-Forchheimer Nanofluid Flow. *Symmetry*. 2020; 12(5):741.
https://doi.org/10.3390/sym12050741

**Chicago/Turabian Style**

Shafiq, Anum, Ghulam Rasool, and Chaudry Masood Khalique.
2020. "Significance of Thermal Slip and Convective Boundary Conditions in Three Dimensional Rotating Darcy-Forchheimer Nanofluid Flow" *Symmetry* 12, no. 5: 741.
https://doi.org/10.3390/sym12050741