# Developments of Electro-Osmotic Two-Phase Flows of Fourth-Grade Fluid through Convergent and Divergent Channels

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## Abstract

**:**

## 1. Introduction

## 2. Development of a Mathematical Model of Multiphase Flow of Non-Newtonian Fluid with Electro-Osmotic Phenomena

**Geometry 1:**

**Geometry 2:**

#### 2.1. Flow Equations for Fluid Phase

#### 2.2. Governing Equations (Particle Phase)

## 3. Dimensionalization of the Problem

## 4. Perturbation Solution

## 5. Comparative Analysis

## 6. Results and Discussion

## 7. Concluding Remarks

- ❖
- An increase in the behavior of both particle and fluid phase velocities is viewed in convergent and divergent geometries when enhancement is made in the fourth-grade parameter;
- ❖
- A remarkable decrease in the velocity profiles of fluid and particle phases in both channels is noted when the value of the electro-osmotic parameter is enhanced;
- ❖
- The credible incline is measured in the velocity profile of both phases in the divergent channel when the value of particle concentration is increased, and a very dubious decline has been seen in the velocities of both phases in the convergent channel;
- ❖
- When the volumetric flow rate upraised in both channels the velocity profile of fluid and particle phases improved as the volumetric flow rate more in velocities.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Perturbation Solution | Numerical Solution | Absolute Error | ||||
---|---|---|---|---|---|---|

$\mathit{M}$ | ${\mathit{u}}_{\mathit{v}\mathit{f}}$ | ${\mathit{u}}_{\mathit{v}\mathit{p}}$ | ${\mathit{u}}_{\mathit{v}\mathit{f}}$ | ${\mathit{u}}_{\mathit{v}\mathit{p}}$ | ${\mathit{u}}_{\mathit{v}\mathit{f}}$ | ${\mathit{u}}_{\mathit{v}\mathit{p}}$ |

1.0 | 1.45064 | 1.45089 | 1.45198 | 1.45001 | $0.134\%$ | $0.088\%$ |

2.0 | 1.35730 | 1.35744 | 1.35598 | 1.35671 | $0.132\%$ | $0.073\%$ |

3.0 | 1.21666 | 1.21680 | 1.21549 | 1.21612 | $0.117\%$ | $0.068\%$ |

4.0 | 1.04626 | 1.04638 | 1.04519 | 1.04590 | $0.107\%$ | $0.048\%$ |

5.0 | 0.86436 | 0.86446 | 0.86332 | 0.86399 | $0.104\%$ | $0.047\%$ |

Pseudo-Spectral Collocation Method | Shooting Method | |||
---|---|---|---|---|

$\mathit{C}$ | ${\mathit{u}}_{\mathit{v}\mathit{f}}$ | ${\mathit{u}}_{\mathit{v}\mathit{p}}$ | ${\mathit{u}}_{\mathit{v}\mathit{f}}$ | ${\mathit{u}}_{\mathit{v}\mathit{p}}$ |

0.1 | 1.41085 | 1.41110 | 1.41090 | 1.41001 |

0.2 | 1.33491 | 1.33506 | 1.33231 | 1.33325 |

0.3 | 1.23917 | 1.23932 | 1.23523 | 1.23567 |

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**MDPI and ACS Style**

Fatima, N.; Nazeer, M.; Lashin, M.M.A.; Ghafar, M.M.; Gorji, M.R.; Hameed, M.K.
Developments of Electro-Osmotic Two-Phase Flows of Fourth-Grade Fluid through Convergent and Divergent Channels. *Mathematics* **2023**, *11*, 1832.
https://doi.org/10.3390/math11081832

**AMA Style**

Fatima N, Nazeer M, Lashin MMA, Ghafar MM, Gorji MR, Hameed MK.
Developments of Electro-Osmotic Two-Phase Flows of Fourth-Grade Fluid through Convergent and Divergent Channels. *Mathematics*. 2023; 11(8):1832.
https://doi.org/10.3390/math11081832

**Chicago/Turabian Style**

Fatima, Nahid, Mubbashar Nazeer, Maha M. A. Lashin, M. M. Ghafar, M. R. Gorji, and M. K. Hameed.
2023. "Developments of Electro-Osmotic Two-Phase Flows of Fourth-Grade Fluid through Convergent and Divergent Channels" *Mathematics* 11, no. 8: 1832.
https://doi.org/10.3390/math11081832