# Numerical Solution of Non-Newtonian Fluid Flow Due to Rotatory Rigid Disk

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## Abstract

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## 1. Introduction

## 2. Problem Formulation

## 3. Computational Outline

## 4. Analysis

## 5. Closing Remarks

**Both the**Brownian and thermophoresis aspects are entertained by incorporating nanoparticles. The flow characteristics are reported numerically with the support of computational algorithm. The summary is as follows:

- CF velocities which includes $[G(\xi ),\text{}{F}^{\prime}(\xi )]$ reflects decline trend towards $\beta $.
- CF velocities are decreasing function of $\lambda $ and $\gamma $.
- CFT $[T(\xi )]$ admits inciting nature towards both ${N}_{T}$ and ${N}_{B}$ but opposite trend is observed for Pr.
- CFC $[C(\xi )]$ shows decline values for both Le, and ${N}_{B}$.
- CFC $[C(\xi )]$ reflect inciting trend for ${N}_{T}$.
- Comparative values of HTR and MTR are provided for involved flow controlling parameters.

## Author Contributions

## Funding

## Acknowledgment

## Conflicts of Interest

## Nomenclature

$V=(\overline{u},\overline{v},\overline{w})$ | Velocity field |

($\overline{r}$, $\overline{\varphi}$, $\overline{z}$) | Polar coordinates |

$\nu $ | Kinematic viscosity |

$\lambda $ | Casson fluid parameter |

${\rho}_{f}$ | Fluid density |

$\sigma $ | Electrical conductivity |

${B}_{0}$ | Uniform applied magnetic field |

$\alpha $ | Thermal diffusivity |

${D}_{B}$ | Brownian diffusion coefficient |

${D}_{T}$ | Thermophoretic diffusion coefficient |

${\overline{T}}_{\infty}$ | Ambient temperature |

$L$ | Velocity slip parameter |

${\overline{T}}_{w}$ | Surface temperature |

${\overline{C}}_{w}$ | Surface concentration |

$\overline{C}$ | Concentration |

${F}^{\prime}(\xi ),\text{}G(\xi )$ | Dimensionless velocities |

$T(\xi )$ | Dimensionless temperature |

$C\left(\mathsf{\xi}\right)$ | Dimensionless concentration |

$\gamma $ | Magnetic field parameter |

$\mathrm{Pr}$ | Prandtl number |

${N}_{B}$ | Brownian motion parameter |

${N}_{T}$ | Thermophoresis parameter |

$Le$ | Lewis number |

$\beta $ | Velocity slip parameter |

${\mathrm{Re}}_{r}$ | Reynolds number |

## References

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**Table 1.**Local Nusselt number comparison with Hayat et al. [32].

$\frac{Nu}{\sqrt{{\mathrm{Re}}_{\overline{r}}}}=-\frac{dT(0)}{d\xi}$ | |||||||

$\beta $ | $\gamma $ | ${N}_{T}$ | $Le$ | $\mathrm{Pr}$ | ${N}_{B}$ | Hayat et al. [32] | Present values |

0.2 | - | - | - | - | - | 0.32655 | 0.326600 |

0.5 | - | - | - | - | - | 0.30360 | 0.30363 |

0.8 | - | - | - | - | - | 0.28715 | 0.28724 |

- | 0.0 | - | - | - | - | 0.30494 | 0.30502 |

- | 0.7 | - | - | - | - | 0.24421 | 0.24434 |

- | 1.4 | - | - | - | - | 0.17566 | 0.17575 |

- | - | 0.5 | - | - | - | 0.25913 | 0.25916 |

- | - | 0.7 | - | - | - | 0.23865 | 0.23879 |

- | - | 1.0 | - | - | - | 0.21010 | 0.21025 |

- | - | 0.5 | - | - | 0.29633 | 0.29642 | |

- | - | 1.0 | - | - | 0.28954 | 0.28963 | |

- | - | - | 1.5 | - | - | 0.28395 | 0.28398 |

- | - | - | - | 0.5 | - | 0.24989 | 0.24999 |

- | - | - | - | 1.0 | - | 0.29211 | 0.29224 |

- | - | - | - | 1.5 | - | 0.32286 | 0.32294 |

- | - | - | - | - | 0.5 | 0.26341 | 0.26358 |

- | - | - | - | - | 0.7 | 0.23677 | 0.23687 |

- | - | - | - | - | 1.0 | 0.20056 | 0.20068 |

**Table 2.**Local Sherwood number comparison with Hayat et al. [32].

$\frac{Sh}{\sqrt{{\mathrm{Re}}_{\overline{r}}}}=-\frac{dC(0)}{d\xi}$ | |||||||

$\beta $ | $\gamma $ | ${N}_{T}$ | $Le$ | $\mathrm{Pr}$ | ${N}_{B}$ | Hayat et al. [32] | Present values |

0.2 | - | - | - | - | - | 0.27583 | 0.27593 |

0.5 | - | - | - | - | - | 0.26933 | 0.26945 |

0.8 | - | - | - | - | - | 0.26493 | 0.26498 |

- | 0.0 | - | - | - | - | 0.27000 | 0.27012 |

- | 0.7 | - | - | - | - | 0.25387 | 0.25394 |

- | 1.4 | - | - | - | - | 0.23722 | 0.23735 |

- | - | 0.5 | - | - | - | 0.22206 | 0.22215 |

- | - | 0.7 | - | - | - | 0.22539 | 0.22564 |

- | - | 1.0 | - | - | - | 0.22285 | 0.22288 |

- | - | 0.5 | - | - | 0.21373 | 0.21380 | |

- | - | 1.0 | - | - | 0.30132 | 0.30145 | |

- | - | - | 1.5 | - | - | 0.38690 | 0.38696 |

- | - | - | - | 0.5 | - | 0.22934 | 0.22944 |

- | - | - | - | 1.0 | - | 0.26624 | 0.26636 |

- | - | - | - | 1.5 | - | 0.31262 | 0.31276 |

- | - | - | - | - | 0.5 | 0.30338 | 0.30342 |

- | - | - | - | - | 0.7 | 0..31875 | 0..31887 |

- | - | - | - | - | 1.0 | 0.32959 | 0.32978 |

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**MDPI and ACS Style**

Rehman, K.U.; Malik, M.Y.; Khan, W.A.; Khan, I.; Alharbi, S.O.
Numerical Solution of Non-Newtonian Fluid Flow Due to Rotatory Rigid Disk. *Symmetry* **2019**, *11*, 699.
https://doi.org/10.3390/sym11050699

**AMA Style**

Rehman KU, Malik MY, Khan WA, Khan I, Alharbi SO.
Numerical Solution of Non-Newtonian Fluid Flow Due to Rotatory Rigid Disk. *Symmetry*. 2019; 11(5):699.
https://doi.org/10.3390/sym11050699

**Chicago/Turabian Style**

Rehman, Khalil Ur, M. Y. Malik, Waqar A Khan, Ilyas Khan, and S. O. Alharbi.
2019. "Numerical Solution of Non-Newtonian Fluid Flow Due to Rotatory Rigid Disk" *Symmetry* 11, no. 5: 699.
https://doi.org/10.3390/sym11050699