# Impact of Nonlinear Thermal Radiation on the Time-Dependent Flow of Non-Newtonian Nanoliquid over a Permeable Shrinking Surface

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## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

## 3. Numerical Procedure

^{−5}) were acquired, in order to fulfill the convergence criterion.

## 4. Results and Discussion

## 5. Conclusions

- Multiple results have been obtained for decelerating flow only, and for precise values of $S$.
- The liquid velocity declines in the first result and increases in the second due to $K$. The concentration and temperature fields rise in the first result and diminish in the second result.
- The thicknesses of the concentration and thermal boundary layers increase due to $\gamma $ in both results.
- The liquid temperature and nanomaterial concentration decrease due to thermal radiation in both results.
- The nanoparticles distribution can be controlled through the mechanism of Brownian motion and the thermophoresis effect.
- The time-dependent and non-Newtonian parameters delay the separation of the boundary.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 15.**The skin friction ${C}_{f}{\mathrm{Re}}_{x}^{1/2}$ versus $S$ for different values of $A$.

**Figure 16.**The Nusselt number $N{u}_{x}{\mathrm{Re}}_{x}^{-1/2}$ versus $S$ for different values of $A$.

**Figure 17.**The Sherwood number $S{h}_{x}{\mathrm{Re}}_{x}^{-1/2}$ versus $S$ for different values of $A$.

**Figure 18.**The skin friction ${C}_{f}{\mathrm{Re}}_{x}^{1/2}$ versus $A$ for different values of $K$.

**Figure 19.**The Nusselt number $N{u}_{x}{\mathrm{Re}}_{x}^{-1/2}$ versus $A$ for different values of $K$.

**Figure 20.**The Sherwood number, $S{h}_{x}{\mathrm{Re}}_{x}^{-1/2}$, versus $A$ for different values of $K$.

**Table 1.**Values of skin friction, Nusselt number, and Sherwood number versus $S$ for different values of $A$ when $K=1,Nb=1,Nt=1.5,Le=1,{Q}_{w}=1.5,{R}_{d}=2,\gamma =0.3$ are fixed.

S | A | ${\mathit{C}}_{\mathit{f}}{\mathbf{Re}}_{\mathit{x}}^{1/2}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{-1/2}$ | $\mathit{S}{\mathit{h}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{-1/2}$ | |||
---|---|---|---|---|---|---|---|

First Solution | Second Solution | First Solution | Second Solution | First Solution | Second Solution | ||

2.8 | −3 | 0.9627 | −1.7424 | 1.0894 | 1.0855 | 2.7589 | 2.5893 |

−2 | 1.1099 | −1.5024 | 1.0863 | 1.0804 | 2.6085 | 2.4075 | |

−1 | 1.2370 | −1.1420 | 1.0816 | 1.0713 | 2.1800 | 2.0495 | |

2.6 | −3 | 0.7457 | −1.4673 | 1.0867 | 1.0831 | 2.5777 | 2.4429 |

−2 | 0.9505 | −1.2430 | 1.0828 | 1.0770 | 2.4220 | 2.2570 | |

−1 | 1.1154 | −0.8527 | 1.0766 | 1.0659 | 2.2332 | 2.0209 | |

2.4 | −3 | 0.3372 | −1.0314 | 1.0831 | 1.0807 | 2.3910 | 2.3136 |

−2 | 0.7068 | −0.8738 | 1.0782 | 1.0735 | 2.2324 | 2.1203 | |

−1 | 0.9524 | −0.4982 | 1.0699 | 1.0602 | 2.0347 | 1.8766 |

**Table 2.**Critical values of ${S}_{c}$ for different values of $A$ when $K=1,Nb=1,Nt=1.5,Le=1$, ${Q}_{w}=1.5,{R}_{d}=2,\gamma =0.3$ are fixed.

A | S_{c} |
---|---|

−3 | 2.3079 |

−2 | 2.2338 |

−1 | 2.1665 |

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**MDPI and ACS Style**

Zaib, A.; Khan, U.; Khan, I.; M. Sherif, E.-S.; Nisar, K.S.; Seikh, A.H.
Impact of Nonlinear Thermal Radiation on the Time-Dependent Flow of Non-Newtonian Nanoliquid over a Permeable Shrinking Surface. *Symmetry* **2020**, *12*, 195.
https://doi.org/10.3390/sym12020195

**AMA Style**

Zaib A, Khan U, Khan I, M. Sherif E-S, Nisar KS, Seikh AH.
Impact of Nonlinear Thermal Radiation on the Time-Dependent Flow of Non-Newtonian Nanoliquid over a Permeable Shrinking Surface. *Symmetry*. 2020; 12(2):195.
https://doi.org/10.3390/sym12020195

**Chicago/Turabian Style**

Zaib, A., Umair Khan, Ilyas Khan, El-Sayed M. Sherif, Kottakkaran Sooppy Nisar, and Asiful H. Seikh.
2020. "Impact of Nonlinear Thermal Radiation on the Time-Dependent Flow of Non-Newtonian Nanoliquid over a Permeable Shrinking Surface" *Symmetry* 12, no. 2: 195.
https://doi.org/10.3390/sym12020195