# How Fluid Particle Interaction Affects the Flow of Dusty Williamson Fluid

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

$\left(u,v\right)$ | velocities components of the fluid along $x$ and $y$ axes |

$\left({u}_{p},{v}_{p}\right)$ | velocities components of the particle along $x$ and $y$ axes |

$\mu $ | viscosity of the fluid |

$\rho $ | density of fluid |

${\rho}_{p}$ | density of dust |

${\alpha}_{1}$ | aligned angle |

${\tau}_{v}=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$k$}\right.$ | relaxation time of particles phase |

$k$ | Stoke’s resistance (drag force) |

${c}_{p}$ | specific heat of fluid |

${c}_{s}$ | specific heat of dust particle |

$T$ | temperature of fluid |

${T}_{p}$ | temperature of particle |

${\gamma}_{T}$ | thermal relaxation time |

${q}_{r}$ | radiative heat flux |

$a$ | positive constant |

${h}_{s}$ | heat transfer parameter |

${\sigma}^{\ast}$ | Stefan-Boltzmann constant |

${k}^{\ast}$ | Absorption parameter |

${T}^{4}$ | coefficient linear function of temperature |

$N=\raisebox{1ex}{${\rho}_{p}$}\!\left/ \!\raisebox{-1ex}{$\rho $}\right.$ | mass concentration of particle phase |

$M=\raisebox{1ex}{$\sigma {B}_{0}{}^{2}$}\!\left/ \!\raisebox{-1ex}{$\rho a$}\right.$ | magnetic field parameter |

$\beta =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$a{\tau}_{\nu}$}\right.$ | fluid-particle interaction |

$\mathrm{Pr}=\raisebox{1ex}{$\mu {c}_{p}$}\!\left/ \!\raisebox{-1ex}{$k$}\right.$ | Prandtl number |

$\gamma =\raisebox{1ex}{${c}_{s}$}\!\left/ \!\raisebox{-1ex}{${c}_{p}$}\right.$ | specific heat ratio of mixture |

$b=-{h}_{s}{\left(v/a\right)}^{1/2}$ | conjugate parameter for NH |

${\lambda}_{3}=\sqrt{2{a}^{3}/v}\Gamma x$ | Williamson parameter |

$R=-4{\sigma}^{\ast}{T}_{\infty}{}^{3}/k{k}^{\ast}$ | radiation parameter |

## 3. Numerical Procedure and Scope of Investigation

## 4. Results and Discussion

## 5. Conclusions

- The rising fluid particle interaction has decreased the velocity of fluid phase and increased the velocity of dust phase.
- The incremented fluid particle interaction has increased the temperature of the fluid phase and decreased the temperature of the dust phase.
- The skin friction coefficient has increased due to the fluid–particle interaction and Williamson parameter.
- The heat transfer has increased by reason of the rising fluid–particle interaction and decreased because of the Williamson parameter.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Comparative study of $\theta (0)$ when $M={\lambda}_{1}=\beta =N=0,$ $\gamma \to \infty $ and $b=1$.

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**MDPI and ACS Style**

Mohd Kasim, A.R.; Arifin, N.S.; Mohd Zokri, S.; Ariffin, N.A.N.; Shafie, S.
How Fluid Particle Interaction Affects the Flow of Dusty Williamson Fluid. *Symmetry* **2023**, *15*, 203.
https://doi.org/10.3390/sym15010203

**AMA Style**

Mohd Kasim AR, Arifin NS, Mohd Zokri S, Ariffin NAN, Shafie S.
How Fluid Particle Interaction Affects the Flow of Dusty Williamson Fluid. *Symmetry*. 2023; 15(1):203.
https://doi.org/10.3390/sym15010203

**Chicago/Turabian Style**

Mohd Kasim, Abdul Rahman, Nur Syamilah Arifin, Syazwani Mohd Zokri, Noor Amalina Nisa Ariffin, and Sharidan Shafie.
2023. "How Fluid Particle Interaction Affects the Flow of Dusty Williamson Fluid" *Symmetry* 15, no. 1: 203.
https://doi.org/10.3390/sym15010203