# Pulsating Flow of an Ostwald—De Waele Fluid between Parallel Plates

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## Abstract

**:**

## 1. Introduction

## 2. Analytical Solution

#### 2.1. Flow Enhancement

#### 2.2. Dispersion Coefficient

## 3. Numerical Solution of the Velocity Distribution

^{−4}, 10

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^{−6}.

## 4. Results

#### 4.1. Wall Shear Stress

#### 4.1.1. Wall Shear Stress for $\mathsf{\Omega}\gg 1$ and $Re~1$

#### 4.1.2. Wall Shear Stress for $\mathsf{\Omega}\ll 1$ and $Re~1$

#### 4.1.3. Wall Shear Stress for $Re\gg 1$ and $\mathsf{\Omega}~1$

#### 4.1.4. Wall Shear Stress for $Re\ll 1$ and $\mathsf{\Omega}~1$

#### 4.2. Discharge

#### 4.2.1. Discharge for $\mathsf{\Omega}\gg 1$ and $Re~1$

#### 4.2.2. Discharge for $\mathsf{\Omega}\ll 1$ and $Re~1$

#### 4.2.3. Discharge for $\mathsf{\Omega}~1$ and $Re\gg 1$

#### 4.2.4. Discharge for $\mathsf{\Omega}~1$ and $Re\ll 1$

#### 4.3. Flow Enhancement

#### 4.4. Dispersion Enhancement

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Effects of the adjustment parameter $\delta $ on the velocity profiles. $n=0.5,\text{}Re=1,\text{}\mathsf{\Omega}=1$, and $\epsilon =0.1$.

**Figure 2.**Comparison between analytical (lines) and numerical (symbols) velocity profiles. $n=0.5,Re=1,$ $\mathsf{\Omega}=1$, and $\epsilon =0.1$. The inset shows the pulsation of the maximum velocity (reached at $y=1$). The vertical lines in the inset are indicative of the times at which the velocity profiles were plotted in the main figure.

**Figure 11.**Dependency of the flow enhancement coefficient with dimensionless frequency and Reynolds number for $n=0.75$ and $\epsilon =0.1$.

**Figure 12.**Dependency of the flow enhancement coefficient with dimensionless frequency and flow index for $Re=1$ and $\epsilon =0.1$.

**Figure 13.**Dependency of the dispersion enhancement coefficient with dimensionless frequency and Reynolds numbers for $n=0.75$ and $\epsilon =0.1$.

**Figure 14.**Dependency of the dispersion enhancement coefficient with dimensionless frequency and flow index for $Re=1$ and $\epsilon =0.1$.

$\mathit{t}=0.25$ | $\mathit{t}=0.50$ | $\mathit{t}=0.75$ | $\mathit{t}=1.00$ | |
---|---|---|---|---|

$Re=1,\text{}\mathsf{\Omega}=1$ | −1.6% | −1.8% | −2.0% | −1.9 |

$Re=100,\text{}\mathsf{\Omega}=1$ | −0.4% | −0.4% | −0.4% | −0.4% |

$Re=1,\text{}\mathsf{\Omega}=100$ | −0.7% | −0.7% | −0.7% | −0.7% |

$\mathit{t}=0.25$ | $\mathit{t}=0.50$ | $\mathit{t}=0.75$ | $\mathit{t}=1.00$ | |
---|---|---|---|---|

$Re=1,\text{}\mathsf{\Omega}=1$ | 0.4% | −0.1% | 0.6% | −0.1% |

$Re=100,\text{}\mathsf{\Omega}=1$ | 0.6% | 0.6% | −0.3% | −0.3% |

$Re=1,\text{}\mathsf{\Omega}=100$ | Less than 0.1% | Less than 0.1% | Less than 0.1% | Less than 0.1% |

$\mathit{t}=0.25$ | $\mathit{t}=0.50$ | $\mathit{t}=0.75$ | $\mathit{t}=1.00$ | |
---|---|---|---|---|

Discharge | −14.9% | −19.7% | −84.9% | −42.8% |

Wall shear stress | 6.7% | −1.9% | 30.9% | −3.5% |

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**MDPI and ACS Style**

González, R.; Tamburrino, A.; Vacca, A.; Iervolino, M.
Pulsating Flow of an Ostwald—De Waele Fluid between Parallel Plates. *Water* **2020**, *12*, 932.
https://doi.org/10.3390/w12040932

**AMA Style**

González R, Tamburrino A, Vacca A, Iervolino M.
Pulsating Flow of an Ostwald—De Waele Fluid between Parallel Plates. *Water*. 2020; 12(4):932.
https://doi.org/10.3390/w12040932

**Chicago/Turabian Style**

González, Rodrigo, Aldo Tamburrino, Andrea Vacca, and Michele Iervolino.
2020. "Pulsating Flow of an Ostwald—De Waele Fluid between Parallel Plates" *Water* 12, no. 4: 932.
https://doi.org/10.3390/w12040932