# Impact of Thermal Radiation and Heat Source/Sink on Eyring–Powell Fluid Flow over an Unsteady Oscillatory Porous Stretching Surface

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## Abstract

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## 1. Introduction

## 2. Formulation of the Problem

#### Physical Quantities of Interest

## 3. Solution by HAM

## 4. HAM Solution Convergence

## 5. Results and Discussion

#### Table Discussion

## 6. Conclusions

- The amplitude of the velocity decreased with an increase in $A$ and porosity $\kappa $, while it increased with an increase in the dimensionless fluid parameters $K$ and $\lambda .$
- The temperature increased with an increase in $A$, the radiation parameter $Rd$, and the heat source/sink $\gamma $, while it decreased with an increase in the Prandtl number $Pr$ and the ratio of the oscillation frequency of the sheet to its stretching rate $A$.
- The local Nusselt number increased with an increase in the Prandtl number $Pr$, the heat source/sink $\gamma $, the dimensionless fluid parameter $K$ and radiation parameter $Rd$, while it decreased with an increase in the porosity $\kappa $ and the dimensionless fluid parameter $\lambda $.

## Author Contributions

## Conflicts of Interest

## Nomenclature

$P$ | pressure (Pa) |

c | constant |

X,Y | topological space |

x,y | coordinates |

$\overrightarrow{u}$, $\overrightarrow{v}$ | velocity components $\left({\mathrm{ms}}^{-1}\right)$ |

${C}_{p}$ | specific heat $\left(\frac{\mathrm{J}}{\mathrm{kgK}}\right)$ |

$\mathsf{\Psi}$, $\mathsf{\Upsilon}$ | fluid materials |

${Q}_{0}$ | heat source/sink |

$k$ | thermal conductivity (${\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1}$) |

${Q}_{rad}$ | radiative heat flux $\left({\mathrm{Wm}}^{-2}\right)$ |

${k}^{\prime}$ | absorption coefficient |

$K$ | fluid parameter |

$A$ | ratio of the oscillation frequency |

$Rd$ | radiation parameter |

$Pr$ | Prandtl number |

${C}_{f}$ | skin fraction coefficient |

$N{u}_{x}$ | local Nusselt number |

Greek Letters | |

$\mu $ | dynamic viscosity $\left(\mathrm{mPa}\right)$ |

$\upsilon $ | constant |

$\upsilon $ | kinematic viscosity (m^{2}/s) |

$\rho $ | density (kg/m^{3}) |

${\sigma}^{\prime}$ | Stefan–Boltzmann constant |

$\kappa $ | porosity term |

$\gamma $ | heat source/sink |

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**Figure 2.**The $\hslash $-curve graph of velocity profile, when $Pr=0.5,\text{}K=0.5,\text{}Rd=0.5,\text{}\lambda =0.5,\text{}\kappa =0.5,\text{}\gamma =0.5,\text{}\mathrm{Sin}\tau =1.0,\text{}A=0.5$.

**Figure 3.**The $\hslash $-curve graph of temperature profile, when $Pr=0.5,\text{}K=0.5,\text{}Rd=0.5,\text{}\lambda =0.5,\text{}\kappa =0.5,\text{}\gamma =0.5,\text{}\mathrm{Sin}\tau =1.0,\text{}A=0.5$.

**Figure 4.**Impact of $A$ on velocity profile ${F}^{\prime}(\mathsf{\Gamma})$ when $Pr=0.5,\text{}K=0.5,\text{}Rd=0.5,\text{}\lambda =0.5,\text{}\kappa =0.5,\text{}\gamma =0.5,\text{}\mathrm{Sin}\tau =1.0$.

**Figure 5.**Impact of $A$ on temperature profile $G(\mathsf{\Gamma})$, when $Pr=0.5,\text{}K=0.5,\text{}Rd=0.5,\text{}\lambda =0.5,\text{}\kappa =0.5,\gamma =0.5,\text{}\mathrm{Sin}\tau =1.0$.

**Figure 6.**Impact of $Pr$ on temperature profile $G(\mathsf{\Gamma})$, when $A=0.5,\text{}K=0.5,\text{}Rd=0.5,\text{}\lambda =0.5,\text{}\kappa =0.5,\text{}\gamma =0.5,\text{}\mathrm{Sin}\tau =1.0$.

**Figure 7.**Impact of $Rd$ on temperature profile $G(\mathsf{\Gamma})$, when $K=0.5,\text{}Pr=0.5,\text{}\lambda =0.5,\text{}\kappa =0.5,\text{}\gamma =0.5,\text{}\mathrm{Sin}\tau =1.0,\text{}A=0.5$.

**Figure 8.**Impact of $K$ on velocity profile ${F}^{\prime}(\mathsf{\Gamma})$, when $Pr=0.5,\text{}Rd=0.5,\text{}\lambda =0.5,\text{}\kappa =0.5,\text{}\gamma =0.5,\text{}\mathrm{Sin}\tau =1.0,\text{}A=0.5$.

**Figure 9.**Impact of $\kappa $ on velocity profile ${F}^{\prime}(\mathsf{\Gamma})$, when $Pr=0.5,\text{}Rd=0.5,\text{}\lambda =0.5,\text{}\kappa =0.5,\text{}\gamma =0.5,\text{}\mathrm{Sin}\tau =1.0,\text{}A=0.5$.

**Figure 10.**Impact of $\lambda $ on velocity profile ${F}^{\prime}(\mathsf{\Gamma})$ when $Pr=0.5,\text{}Rd=0.5,\text{}K=0.5,\text{}\kappa =0.5,\text{}\gamma =0.5,\text{}\mathrm{Sin}\tau =1.0,\text{}A=0.5$.

**Figure 11.**Impact of $\gamma $ on temperature profile $G(\mathsf{\Gamma})$, when $Pr=0.5,\text{}Rd=0.5,\text{}K=0.5,\text{}\kappa =0.5,\text{}\lambda =0.5,\text{}\mathrm{Sin}\tau =1.0,\text{}A=0.5$.

**Figure 12.**Impact of $\tau $ on velocity profile ${F}^{\prime}(\mathsf{\Gamma})$, when $Pr=0.5,\text{}Rd=0.5,\text{}K=0.5,\text{}\kappa =0.5,\text{}\gamma =0.5,\text{}\lambda =0.5,\text{}A=0.5$.

**Figure 13.**Impact of $\tau $ on temperature profile $G(\mathsf{\Gamma})$, when $Pr=0.5,\text{}Rd=0.5,\text{}K=0.5,\text{}\kappa =0.5,\text{}\gamma =0.5,\text{}\lambda =0.5,\text{}A=0.5$.

**Table 1.**The numerical values of skin fraction $(1+K){F}^{\u2033}(0)-\frac{K}{3}\mathsf{\Psi}({F}^{\u2033}(0)),$ when $A=0.5,\text{}\mathsf{\Psi}=1$ at time instant $\tau =\pi /2$.

$\mathit{\kappa}$ | $\mathit{K}$ | $\mathit{\lambda}$ | ${\mathit{C}}_{\mathit{f}}$ |
---|---|---|---|

0.5 | - | - | −1.29447 |

0.7 | - | - | −1.39927 |

0.9 | - | - | −1.50053 |

1.1 | 1.0 | - | −1.59913 |

- | 1.3 | - | −1.68426 |

- | 1.5 | - | −1.73747 |

- | 1.7 | 0.5 | −1.78810 |

- | - | 0.6 | −1.81316 |

- | - | 0.7 | −1.84069 |

**Table 2.**The numerical values of heat flux $\left(1+\frac{4}{3}Rd\right){G}^{\prime}(0),$ when $A=0.5,\text{}\mathsf{\Psi}=1$ at time instant $\tau =\pi /2$.

$\mathit{P}\mathit{r}$ | $\mathit{\gamma}$ | $\mathit{Rd}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}$ |
---|---|---|---|

1.0 | - | - | 1.82770 |

1.2 | - | - | 1.80574 |

1.4 | - | - | 1.78404 |

1.6 | 2.5 | - | 1.76259 |

- | 2.6 | - | 1.79740 |

- | 2.7 | - | 1.83133 |

- | 2.8 | 0.3 | 1.85965 |

- | - | 0.4 | 1.97571 |

- | - | 0.5 | 2.08693 |

**Table 3.**The association between HAM and numerical solution for ${F}^{\prime}(\mathsf{\Gamma}),$ when $K=0,\text{}A=0.2,\text{}\kappa =0.5,\text{}Rd=1.0,\text{}\mathrm{Sin}\tau =1.0,\text{}\lambda =Pr=\gamma =0.6$.

$\mathsf{\Gamma}$ | HAM Solution ${\mathit{F}}^{\mathbf{\prime}}(\mathsf{\Gamma})$ | Numerical Solution ${\mathit{F}}^{\mathbf{\prime}}(\mathsf{\Gamma})$ | Absolute Error AE |
---|---|---|---|

0.0 | 1.12757 × 10^{−17} | 0.000000 | 1.12757 × 10^{−17} |

0.5 | 0.378563 | 0.381439 | 0.002876 |

1.0 | 0.586888 | 0.596535 | 0.009647 |

1.5 | 0.699341 | 0.715829 | 0.016488 |

2.0 | 0.758826 | 0.779103 | 0.020276 |

2.5 | 0.789566 | 0.808656 | 0.019089 |

3.0 | 0.804988 | 0.816817 | 0.011828 |

3.5 | 0.812414 | 0.866641 | 0.054227 |

4.0 | 0.815772 | 0.873029 | 0.057257 |

4.5 | 0.817129 | 0.875971 | 0.058842 |

5.0 | 0.817552 | 0.876772 | 0.059220 |

**Table 4.**The association between HAM and numerical solution for $G(\mathsf{\Gamma})$, when $K=0,\text{}A=0.2,\text{}\kappa =0.5,\text{}Rd=1.0,\text{}\mathrm{Sin}\tau =1.0,\text{}\lambda =Pr=\gamma =0.6$.

$\mathsf{\Gamma}$ | HAM Solution $\mathit{G}(\mathsf{\Gamma})$ | Numerical Solution $\mathit{G}(\mathsf{\Gamma})$ | Absolute Error AE |
---|---|---|---|

0 | 1.000000 | 1.000000 | 0.000000 |

1.0 | 0.513778 | 0.543757 | 0.029978 |

2.0 | 0.266242 | 0.288424 | 0.022182 |

3.0 | 0.133781 | 0.152038 | 0.018257 |

4.0 | 0.065247 | 0.080017 | 0.014769 |

5.0 | 0.030998 | 0.042041 | 0.011042 |

6.0 | 0.014391 | 0.021973 | 0.007582 |

7.0 | 0.006548 | 0.011297 | 0.004748 |

8.0 | 0.002927 | 0.005503 | 0.002575 |

9.0 | 0.001288 | 0.002181 | 0.000892 |

10.0 | 0.000559 | 2.093 × 10^{−6} | 0.000557 |

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## Share and Cite

**MDPI and ACS Style**

Dawar, A.; Shah, Z.; Idrees, M.; Khan, W.; Islam, S.; Gul, T.
Impact of Thermal Radiation and Heat Source/Sink on Eyring–Powell Fluid Flow over an Unsteady Oscillatory Porous Stretching Surface. *Math. Comput. Appl.* **2018**, *23*, 20.
https://doi.org/10.3390/mca23020020

**AMA Style**

Dawar A, Shah Z, Idrees M, Khan W, Islam S, Gul T.
Impact of Thermal Radiation and Heat Source/Sink on Eyring–Powell Fluid Flow over an Unsteady Oscillatory Porous Stretching Surface. *Mathematical and Computational Applications*. 2018; 23(2):20.
https://doi.org/10.3390/mca23020020

**Chicago/Turabian Style**

Dawar, Abdullah, Zahir Shah, Muhammad Idrees, Waris Khan, Saeed Islam, and Taza Gul.
2018. "Impact of Thermal Radiation and Heat Source/Sink on Eyring–Powell Fluid Flow over an Unsteady Oscillatory Porous Stretching Surface" *Mathematical and Computational Applications* 23, no. 2: 20.
https://doi.org/10.3390/mca23020020