# Two-Phase Flow of Eyring–Powell Fluid with Temperature Dependent Viscosity over a Vertical Stretching Sheet

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

**For fluid phase:**

**For dust phase:**

## 3. Results and Discussion

**Figure 2.**${f}^{\prime}\left(\eta \right)$ and ${F}^{\prime}\left(\eta \right)$ at $M=B=\lambda =0.5$, ${\gamma}_{1}=\alpha =0.1$ and $\beta =N=0.6$ for various values of $\mathrm{Pr}$.

**Figure 3.**$\theta \left(\eta \right)$ and ${\theta}_{p}\left(\eta \right)$ at $M=B=\lambda =0.5$, ${\gamma}_{1}=\alpha =0.1$ and $\beta =N=0.6$ for various values of $\mathrm{Pr}$.

**Figure 4.**${f}^{\prime}\left(\eta \right)$ and ${F}^{\prime}\left(\eta \right)$ at $M=B=\lambda =0.5,{\gamma}_{1}=0.1$, $\beta =N=0.6$ and $\mathrm{Pr}=10$ for various values of $\alpha $.

**Figure 5.**$\theta \left(\eta \right)$ and ${\theta}_{p}\left(\eta \right)$ at $M=B=\lambda =0.5,{\gamma}_{1}=0.1$, $\beta =N=0.6$ and $\mathrm{Pr}=10$ for various values of $\alpha $.

**Figure 6.**${f}^{\prime}\left(\eta \right)$ and ${F}^{\prime}\left(\eta \right)$ at $B=\lambda =0.5,\alpha ={\gamma}_{1}=0.1$, $\beta =N=0.6$ and $\mathrm{Pr}=10$ for various values of $M$.

**Figure 7.**$\theta \left(\eta \right)$ and ${\theta}_{p}\left(\eta \right)$ at $B=\lambda =0.5,\alpha ={\gamma}_{1}=0.1$, $\beta =N=0.6$ and $\mathrm{Pr}=10$ for various values of $M$.

**Figure 8.**${f}^{\prime}\left(\eta \right)$ and ${F}^{\prime}\left(\eta \right)$ at $M=\lambda =0.5,\alpha ={\gamma}_{1}=0.1$, $\beta =N=0.6$ and $\mathrm{Pr}=10$ for various values of $B$.

**Figure 9.**$\theta \left(\eta \right)$ and ${\theta}_{p}\left(\eta \right)$ at $M=\lambda =0.5,\alpha ={\gamma}_{1}=0.1$, $\beta =N=0.6$ and $\mathrm{Pr}=10$ for various values of $B$.

**Figure 10.**${f}^{\prime}\left(\eta \right)$ and ${F}^{\prime}\left(\eta \right)$ at $M=B=\lambda =0.5,\alpha =0.1$, $\beta =N=0.6$ and $\mathrm{Pr}=10$ for various values of ${\gamma}_{1}$.

**Figure 11.**$\theta \left(\eta \right)$ and ${\theta}_{p}\left(\eta \right)$ at $M=B=\lambda =0.5,\alpha =0.1$, $\beta =N=0.6$ and $\mathrm{Pr}=10$ for various values of ${\gamma}_{1}$.

**Figure 12.**${f}^{\prime}\left(\eta \right)$ and ${F}^{\prime}\left(\eta \right)$ at $M=B=0.5,{\gamma}_{1}=\alpha =0.1$, $\beta =N=0.6$ and $\mathrm{Pr}=10$ for various values of $\lambda $.

**Figure 13.**$\theta \left(\eta \right)$ and ${\theta}_{p}\left(\eta \right)$ at $M=B=0.5,{\gamma}_{1}=\alpha =0.1$, $\beta =N=0.6$ and $\mathrm{Pr}=10$ for various values of $\lambda $.

**Figure 14.**${f}^{\prime}\left(\eta \right)$ and ${F}^{\prime}\left(\eta \right)$ at $M=B=\lambda =0.5,N=0.6$, ${\gamma}_{1}=\alpha =0.1$ and $\mathrm{Pr}=10$ for various values of $\beta $.

**Figure 15.**$\theta \left(\eta \right)$ and ${\theta}_{p}\left(\eta \right)$ at $M=B=\lambda =0.5,N=0.6$, ${\gamma}_{1}=\alpha =0.1$ and $\mathrm{Pr}=10$ for various values of $\beta $.

**Figure 16.**${f}^{\prime}\left(\eta \right)$ and ${F}^{\prime}\left(\eta \right)$ at $M=B=\lambda =0.5,\beta =0.6$, ${\gamma}_{1}=\alpha =0.1$ and $\mathrm{Pr}=10$ for various values of $N$.

**Figure 17.**$\theta \left(\eta \right)$ and ${\theta}_{p}\left(\eta \right)$ at $M=B=\lambda =0.5,\beta =0.6$, ${\gamma}_{1}=\alpha =0.1$ and $\mathrm{Pr}=10$ for various values of $N$.

## 4. Conclusions and Future Work

- For certain applications, the fluid’s flow and heat transfer can be regulated by embedding the particles of fine dust.
- The process velocity (temperature) of fluid and dust have the opposite effects for buoyancy force parameter variability.
- In the mixed convection regime, the local shear stress increases and the local rate of heat transfer decreases as the value of buoyancy parameter increases for all values of the Prandtl number and the viscosity variation parameter.
- The fluid–particle interaction parameter variability is favourable for the thickness of the dust boundary layer. However, for the thickness of boundary layer of momentum, it is unfavourable.
- The velocity distribution was suppressed with the Prandtl number compared to temperature distribution.
- The velocity profiles increase and the viscosity of the fluid decrease near the surface of the plate owing to increase in the value of the viscosity variation parameter. The temperature profiles of both phases are enhanced by rising $\alpha $.
- The quantity of skin friction decreases with greater values of fluid parameters, mixed convection, conjugate parameter of heat transfer, mass concentration of particle phase and fluid–particle interaction parameter.
- Increase in the value of the viscosity variation parameter leads to increase in the local shear stress and to decrease in the local rate of heat transfer. Its effect on the increase of the rate of heat transfer is less than that of the local shear stress.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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$\mathbf{Pr}$ | [35] | [36] | [37] | Present |
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1 | 1.3333 | 1.3333 | 1.3333 | 1.3329 |

3 | 2.50970 | 2.50972 | 2.50972 | 2.50969 |

10 | 4.79690 | 4.79686 | 4.79687 | 4.79689 |

100 | 15.7120 | 15.7118 | 15.7120 | 15.7098 |

Existing Literature | Model of Problem | Boundary Condition | Limiting Cases | $\mathbf{Value}\text{}\mathbf{of}\text{}{\mathit{f}}^{\u2033}\left(0\right)$ |
---|---|---|---|---|

Exact Solution (15) | ${f}^{\u2033}=-{e}^{-\eta}$ | - | - | −1.0000 |

[38] | ${f}^{\u2034}-{{f}^{\prime}}^{2}+f{f}^{\u2033}-A\left({f}^{\prime}+\frac{1}{2}\eta {f}^{\u2033}\right)+\lambda \theta =0$ | $\begin{array}{l}f\left(0\right)=0\\ {f}^{\prime}\left(0\right)=1\\ {f}^{\prime}(\infty )=0\end{array}$ | $A=\lambda =0$ | −1.0000 |

[39] | ${f}^{\u2034}-{{f}^{\prime}}^{2}+f{f}^{\u2033}-{k}_{1}\left(2{f}^{\prime}{f}^{\u2034}-{{f}^{\u2033}}^{2}-f{f}^{\u2034}\right)=0$ | $\begin{array}{l}f\left(0\right)=0\\ {f}^{\prime}\left(0\right)=1\\ {f}^{\prime}(\infty )=0\end{array}$ | ${k}_{1}=0$ | −1.0000 |

[40] | $\begin{array}{l}\left(1+M\right)\left(1+2\eta \gamma \right){f}^{\u2034}-\alpha M{\left(1+2\eta \gamma \right)}^{2}{{f}^{\u2033}}^{2}{f}^{\u2034}\\ +2\gamma \left(1+M\right){f}^{\u2033}+f{f}^{\u2033}-\frac{4}{3}\alpha M\left(\gamma +2\eta {\gamma}^{2}\right){{f}^{\u2033}}^{3}\\ -{{f}^{\prime}}^{2}+\lambda \theta \mathrm{sin}\phi =0\end{array}$ | $\begin{array}{l}f\left(0\right)=0\\ {f}^{\prime}\left(0\right)=1\\ {f}^{\prime}(\infty )=0\end{array}$ | $\begin{array}{l}\alpha =\gamma =0\\ \lambda =M=0\end{array}$ | −1.0000 |

[41] | $\left(1+M\right){f}^{\u2034}-{{f}^{\prime}}^{2}+f{f}^{\u2033}-MB{{f}^{\u2033}}^{2}{f}^{\u2034}-H{f}^{\prime}=0$ | $\begin{array}{l}B=H=0,\\ M=0.0001\end{array}$ | −1.0000 | |

Present study | $\begin{array}{l}\left(1+M\right){f}^{\u2034}-{{f}^{\prime}}^{2}+f{f}^{\u2033}+\beta N\left({F}^{\prime}-{f}^{\prime}\right)\\ -BM{{f}^{\u2033}}^{2}{f}^{\u2034}-\alpha {f}^{\u2033}{\theta}^{\prime}-\alpha \theta {f}^{\u2034}+\lambda \theta =0,\end{array}$ | $\begin{array}{l}B=M=0\\ N=\beta =0\\ \alpha =\lambda =0\end{array}$ | −1.0015 |

**Table 3.**Numerical results of ${C}_{f}{\mathrm{Re}}_{x}{}^{1/2}$ and $N{u}_{x}{\mathrm{Re}}_{x}{}^{-1/2}$ for various values of $\mathrm{Pr},\alpha ,M,B,\lambda ,{\gamma}_{1},\beta $ and $N$.

$\mathbf{P}\mathbf{r}$ | $\mathit{\alpha}$ | $\mathit{M}$ | $\mathit{B}$ | $\mathit{\lambda}$ | ${\mathit{\gamma}}_{1}$ | $\mathit{\beta}$ | $\mathit{N}$ | ${\mathit{C}}_{\mathit{f}}{\mathbf{Re}}_{\mathit{x}}{}^{1/2}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}{}^{-1/2}$ |
---|---|---|---|---|---|---|---|---|---|

7 | 0.1 | 0.6 | 0.6 | 0.1 | 0.5 | 0.5 | 0.5 | −1.076579 | 0.105283 |

9 | −1.078272 | 0.104603 | |||||||

12 | −1.079770 | 0.103939 | |||||||

10 | 0.2 | −1.077788 | 0.104348 | ||||||

0.4 | −1.075612 | 0.104349 | |||||||

0.7 | −1.072307 | 0.104350 | |||||||

10 | 0.1 | 0.5 | −1.048326 | 0.104357 | |||||

0.9 | −1.173040 | 0.104324 | |||||||

1.5 | −1.351057 | 0.104292 | |||||||

10 | 0.1 | 0.6 | 0.1 | −1.133436 | 0.104328 | ||||

0.5 | −1.088912 | 0.104324 | |||||||

0.9 | −1.048210 | 0.104322 | |||||||

10 | 0.1 | 0.1 | 0.6 | 0.3 | −1.754138 | 0.342542 | |||

0.5 | −1.741401 | 0.630439 | |||||||

0.9 | −1.706099 | 1.433335 | |||||||

10 | 0.1 | 0.6 | 0.6 | 0.1 | 0.1 | −1.085300 | 0.104326 | ||

0.6 | −1.081074 | 0.104324 | |||||||

1.2 | −1.076098 | 0.104318 | |||||||

10 | 0.1 | 0.6 | 0.6 | 0.1 | 0.5 | 0.1 | −1.180434 | 0.104421 | |

0.4 | −1.114385 | 0.104359 | |||||||

0.9 | −1.044242 | 0.104282 | |||||||

10 | 0.1 | 0.6 | 0.6 | 0.1 | 0.5 | 0.5 | 0.1 | −1.182364 | 0.104415 |

0.4 | −1.125083 | 0.104364 | |||||||

0.9 | −1.019745 | 0.104280 |

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**MDPI and ACS Style**

Aljabali, A.; Mohd Kasim, A.R.; Arifin, N.S.; Ariffin, N.A.N.; Ling Chuan Ching, D.; Waini, I.; Khashi’ie, N.S.; Zainal, N.A.
Two-Phase Flow of Eyring–Powell Fluid with Temperature Dependent Viscosity over a Vertical Stretching Sheet. *Mathematics* **2022**, *10*, 3111.
https://doi.org/10.3390/math10173111

**AMA Style**

Aljabali A, Mohd Kasim AR, Arifin NS, Ariffin NAN, Ling Chuan Ching D, Waini I, Khashi’ie NS, Zainal NA.
Two-Phase Flow of Eyring–Powell Fluid with Temperature Dependent Viscosity over a Vertical Stretching Sheet. *Mathematics*. 2022; 10(17):3111.
https://doi.org/10.3390/math10173111

**Chicago/Turabian Style**

Aljabali, Ahlam, Abdul Rahman Mohd Kasim, Nur Syamilah Arifin, Noor Amalina Nisa Ariffin, Dennis Ling Chuan Ching, Iskandar Waini, Najiyah Safwa Khashi’ie, and Nurul Amira Zainal.
2022. "Two-Phase Flow of Eyring–Powell Fluid with Temperature Dependent Viscosity over a Vertical Stretching Sheet" *Mathematics* 10, no. 17: 3111.
https://doi.org/10.3390/math10173111