# Thermo-Diffusion and Multi-Slip Effect on an Axisymmetric Casson Flow over a Unsteady Radially Stretching Sheet in the Presence of Chemical Reaction

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

## 3. Results and Discussion

## 4. Conclusions

- The velocity profiles are observed to be decreased with increasing values of the Casson, unsteadiness, magnetic, Prandtl number, Dufour, Soret number, chemical, Schmidt number and slips parameters, but the effect of increasing buoyancy parameter values and injection parameters is the opposite.
- Increments in unsteadiness, magnetic field, buoyancy, Prandtl number, Soret, thermal slip, Dufour, and chemical parameters decline the fluid temperature. However, the opposite effect is observed with increasing values of Schmidt number, magnetic, Casson, suction/injection, and hydrodynamic slip parameters.
- The concentration profle are found to be reduced with increasing values of the unsteadiness, buoyancy, Soret, Schmidt number, thermal slip, Prandtl number, chemical reaction, and suction parameters. But the concentration profile are enhanced by increment in the magnetic field, Casson, Prandtl number, Dufour, injection, and hydrodynamic slip parameters.
- The obtained results are presented in graphical and tabular formats. An excellent agreement of our numerical results is obtained with the existing literature which assists with the authenticity of proposed study.
- Destructive chemical reactions are favorable in order to enhance the mass transfer rate.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Influence of $\beta $ on velocity profile ${f}^{\prime}$, temperature profile $\theta $, and concentration profile $\varphi $ against $\eta $.

**Figure 3.**Influence of M on velocity profile ${f}^{\prime}$, temperature profile $\theta $, and concentration profile $\varphi $ against $\eta $.

**Figure 4.**Influence of $\alpha $ on velocity profile ${f}^{\prime}$, temperature profile $\theta $, and concentration profile $\varphi $ against $\eta $.

**Figure 5.**Influence of $\lambda $ on velocity profile ${f}^{\prime}$, temperature profile $\theta $, and concentration profile $\varphi $ against $\eta $.

**Figure 6.**Influence of $Pr$ on velocity profile ${f}^{\prime}$, temperature profile $\theta $, and concentration profile $\varphi $ against $\eta $.

**Figure 7.**Influence of ${D}_{s}$ on velocity profile ${f}^{\prime}$, temperature profile $\theta $, and concentration profile $\varphi $ against $\eta $.

**Figure 8.**Influence of $Sc$ on velocity profile ${f}^{\prime}$, temperature profile $\theta $, and concentration profile $\varphi $ against $\eta $.

**Figure 9.**Influence of ${D}_{f}$ on velocity profile ${f}^{\prime}$, temperature profile $\theta $, and concentration profile $\varphi $ against $\eta $.

**Figure 10.**Influence of ${R}_{o}$ on velocity profile ${f}^{\prime}$, temperature profile $\theta $, and concentration profile $\varphi $ against $\eta $.

**Figure 11.**Influence of ${\delta}_{1}$ on velocity profile ${f}^{\prime}$, temperature profile $\theta $, and concentration profile $\varphi $ against $\eta $.

**Figure 12.**Influence of ${\delta}_{2}$ on velocity profile ${f}^{\prime}$, temperature profile $\theta $, and concentration profile $\varphi $ against $\eta $.

**Figure 13.**Influence of S on velocity profile ${f}^{\prime}$, temperature profile $\theta $, and concentration profile $\varphi $ against $\eta $.

**Figure 14.**Influence of $-S$ on velocity profile ${f}^{\prime}$, temperature profile $\theta $, and concentration profile $\varphi $ against $\eta $.

**Table 1.**Present results are Compared with [29] of $-{f}^{\u2033}(0)$ and $-{\theta}^{\prime}(0)$ for various values of $\alpha $, S and $Pr$.

$\mathit{\alpha}$ | S | Pr | Azeem et al. [13] | KBM (Present Results) | ||
---|---|---|---|---|---|---|

$-{\mathit{f}}^{\u2033}(0)$ | $-{\mathit{\theta}}^{\prime}(0)$ | $-{\mathit{f}}^{\u2033}(0)$ | $-{\mathit{\theta}}^{\prime}(0)$ | |||

0.5 | −1.0 | 1.0 | 0.620400 | 0.620400 | 0.620436 | 0.620436 |

−0.5 | 0.887200 | 0.887200 | 0.887247 | 0.887247 | ||

0.0 | 1.308999 | 1.308999 | 1.308670 | 1.308670 | ||

0.5 | 1.907999 | 1.907999 | 1.907973 | 1.907973 | ||

1.0 | 2.655999 | 2.655999 | 2.655591 | 2.655591 | ||

0.0 | 0.5 | 1.0 | 1.798999 | 1.798999 | 1.798668 | 1.798668 |

0.5 | 1.907999 | 1.907999 | 1.907973 | 1.907973 | ||

1.0 | 2.016999 | 2.016999 | 2.016665 | 2.016665 | ||

0.5 | 0.5 | 0.5 | 1.907999 | 1.119999 | 1.907973 | 1.118889 |

0.7 | 1.907999 | 1.450000 | 1.907973 | 1.467003 | ||

1.0 | 1.907999 | 1.907999 | 1.907973 | 1.907973 |

**Table 2.**Influence of some parameters on $-{f}^{\u2033}(0)$, $-{\theta}^{\prime}(0)$, and $-{\varphi}^{\prime}(0)$ when $Sc=10,{D}_{s}={D}_{f}=0.5,{R}_{o}=1.0,{\delta}_{1}={\delta}_{2}=0.2,S=0$.

$\mathit{\alpha}$ | $\mathit{\beta}$ | M | $\mathit{P}\mathit{r}$ | $\mathit{\lambda}$ | ${\mathit{f}}^{\u2033}(0)$ | $-{\mathit{\theta}}^{\prime}(0)$ | $-{\mathit{\varphi}}^{\prime}(0)$ |
---|---|---|---|---|---|---|---|

0.5 | 0.3 | 1.0 | 10.0 | 1.0 | −2.93927 | 2.51714 | 0.97907 |

1.0 | −3.11489 | 2.65722 | 1.15877 | ||||

1.5 | −3.27767 | 2.77597 | 1.32801 | ||||

0.5 | 1.0 | −1.87877 | 2.48897 | 0.88058 | |||

3.0 | −1.48554 | 2.48759 | 0.81457 | ||||

5.0 | −1.40157 | 2.48426 | 0.79476 | ||||

0.3 | 2.0 | −3.36540 | 2.50392 | 0.94527 | |||

4.0 | −4.02817 | 2.48569 | 0.89369 | ||||

6.0 | −4.54380 | 2.47292 | 0.85671 | ||||

1.0 | 15.0 | −2.94933 | 2.56412 | 0.88783 | |||

20.0 | −2.95470 | 2.58853 | 0.84059 | ||||

25.0 | −2.95780 | 2.60340 | 0.81173 | ||||

10.0 | 2.0 | −3.08296 | 3.12406 | 1.14151 | |||

3.0 | −3.08036 | 3.51020 | 1.32529 | ||||

3.5 | −3.07908 | 3.86004 | 1.48219 |

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**MDPI and ACS Style**

Faraz, F.; Imran, S.M.; Ali, B.; Haider, S.
Thermo-Diffusion and Multi-Slip Effect on an Axisymmetric Casson Flow over a Unsteady Radially Stretching Sheet in the Presence of Chemical Reaction. *Processes* **2019**, *7*, 851.
https://doi.org/10.3390/pr7110851

**AMA Style**

Faraz F, Imran SM, Ali B, Haider S.
Thermo-Diffusion and Multi-Slip Effect on an Axisymmetric Casson Flow over a Unsteady Radially Stretching Sheet in the Presence of Chemical Reaction. *Processes*. 2019; 7(11):851.
https://doi.org/10.3390/pr7110851

**Chicago/Turabian Style**

Faraz, Faraz, Syed Muhammad Imran, Bagh Ali, and Sajjad Haider.
2019. "Thermo-Diffusion and Multi-Slip Effect on an Axisymmetric Casson Flow over a Unsteady Radially Stretching Sheet in the Presence of Chemical Reaction" *Processes* 7, no. 11: 851.
https://doi.org/10.3390/pr7110851