# Simple Particle Relaxation Modeling in One-Dimensional Oscillating Flows

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## Abstract

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## 1. Introduction

## 2. Method

#### 2.1. Drag Models

#### 2.2. Slip Velocity Amplitude

#### 2.3. Particle Relaxation

## 3. Result and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | displacement amplitude |

a | particle acceleration |

${C}_{D}$ | drag coefficient |

d | particle diameter |

F | force |

$Re$ | Reynolds number |

$Stk$ | oscillation Stokes number |

t | time |

u | velocity |

U | velocity amplitude |

$W{o}^{2}$ | Womersley number (frequency parameter) |

$\gamma $ | density ratio |

$\u03f5$ | amplitude parameter |

$\eta $ | dynamic viscosity |

$\rho $ | density |

$\tau $ | relaxation time |

$\varphi $ | phase shift |

$\omega $ | angular frequency |

Abbreviations | |

$LL$ | Landau & Lifshitz |

$NSE$ | Navier-Stokes equations |

$ODE$ | ordinary differential equation |

S | Stokes |

$SN$ | Schiller & Naumann |

$STP$ | standard temperature and pressure |

Indices | |

D | drag |

I | inertia |

p | particle |

f | fluid |

0 | initial state |

## Appendix A. Derivation of the Slip Velocity Amplitude Calculated with the Stokes Drag Model

## Appendix B. Derivation of Slip Velocity Amplitude Calculated with SN and Deviation from the Stokes Model

## Appendix C. Derivation and Solution of Particle Motion with the Landau & Lifshitz Model

## References

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**Figure 1.**Drag models considered for various combinations of Reynolds number $Re$, amplitude parameter $\u03f5$, and Womersley number $W{o}^{2}$. (

**left**) Various drag models displayed in the $\u03f5$-$Re$ plane. The striped patterns indicate the areas of validity, while the solid colors mark the preferable model in the area. Note that the origin has the coordinates $\u03f5=1,Re=1$. (

**right**) Various drag models with their limits of validity.

**Figure 2.**Vector representation of fluid velocity amplitude ${U}_{f}$, particle velocity amplitude ${U}_{p}$, and slip velocity amplitude U, including phase shift angles in reference to the fluid oscillation of the particle velocity ${\varphi}_{p}$ and slip velocity $\varphi $, respectively.

**Figure 3.**Normalized slip velocity amplitude, calculated with various drag models. (

**left**) $\gamma $ = 25,000 (corresponds to iron particles in hot air); (

**right**) $\gamma =8$ (corresponds to iron particles in water at STP).

**Figure 4.**Critical density ratio ${\gamma}_{crit}$ depending on the Womersley number $W{o}^{2}$ for cut off criteria of $U/{U}_{f}>90\%$, $U/{U}_{f}>95\%$, and $U/{U}_{f}>99\%$. Above the respected line, the particle motion is not affected by the flow and the slip velocity can be set equal to the fluid velocity (no relaxation case).

**Figure 5.**Areas where the deviation between Stokes and the respective drag model is above (blue—Schiller & Naumann, purple—Basset, green—Landau & Lifshitz) and below (red) 5%. In the red areas, the Stokes model can be applied in the validity range of the respective model.

**Table 1.**Amplitude ratio of slip to fluid velocity $U/{U}_{f}$ for several drag models dependent on the Womersley number $W{o}^{2}$ and the density ratio $\gamma $.

Name | Drag Force ${\mathit{F}}_{\mathit{D}}$ | Slip Velocity Amplitude Ratio $\mathit{U}/{\mathit{U}}_{\mathit{f}}=$ | |
---|---|---|---|

Stokes | $3\pi \eta du$ | ${\left[1+{\left(\frac{18}{W{o}^{2}\gamma}\right)}^{2}\right]}^{-1/2}$ | |

Schiller & Naumann | $3\pi \eta duSN$ | ${\left[1+{\left(\frac{18\phantom{\rule{4pt}{0ex}}SN}{W{o}^{2}\gamma}\right)}^{2}\right]}^{-1/2}$ | $SN=1+0.158R{e}^{2/3}$ |

Basset | $\begin{array}{l}3\pi \eta du-\frac{\pi}{6}{d}^{3}\frac{\partial p}{\partial x}+\frac{\pi}{12}{d}^{3}{\rho}_{f}\frac{du}{dt}\\ +\frac{3}{2}{d}^{2}\sqrt{\pi {\rho}_{f}\eta}{\int}_{{t}_{0}}^{t}\frac{du/dt}{\sqrt{t-{t}^{\prime}}}d{t}^{\prime}\end{array}$ | ${\left[1-{(1+{f}_{1})}^{2}-{f}_{2}^{2}\right]}^{-1/2}$ | ${f}_{1}=\frac{\left(1-\gamma \right)\left(0.5+\gamma +\frac{9}{\sqrt{2}Wo}\right)}{{\left(\frac{18}{W{o}^{2}}+\frac{9}{\sqrt{2}Wo}\right)}^{2}+{\left(0.5+\gamma +\frac{9}{\sqrt{2}Wo}\right)}^{2}}$ ${f}_{2}=\frac{\left(1-\gamma \right)\left(\frac{18}{W{o}^{2}}+\frac{9}{\sqrt{2}Wo}\right)}{{\left(\frac{18}{W{o}^{2}}+\frac{9}{\sqrt{2}Wo}\right)}^{2}+{\left(0.5+\gamma +\frac{9}{\sqrt{2}Wo}\right)}^{2}}$ |

Landau & Lifshitz | $\begin{array}{l}3\pi \eta d\left(1+\frac{d}{2\delta}\right)u+\\ \frac{3}{4}\pi {d}^{2}\sqrt{\frac{2\eta \rho}{w}}\left(1+\frac{d}{9\delta}\right)\frac{du}{dt}\end{array}$ | ${\left[{\left(\frac{18}{\gamma W{o}^{2}}+\frac{9}{\gamma Wo}\right)}^{2}+{\left(\frac{9}{\sqrt{2}\gamma Wo}+\frac{1}{\sqrt{2}\gamma}+1\right)}^{2}\right]}^{-1/2}$ |

**Table 2.**Relations of the drag models listed in Table 1 to the Stokes model, expressed with the oscillation Stokes number $Stk$ and the density ratio $\gamma $. Additionally, the limit is given up to which the deviation stays below 5%, in which case the Stokes model can be applied in the validity range of the respective model. Multiple criteria are set in relation to each other via the logic operator ∧—‘and’.

Name | Relation to Stokes ${\mathit{U}}_{\mathit{i}}/{\mathit{U}}_{\mathbf{Stk}}$ | $|{\mathit{U}}_{\mathit{i}}-{\mathit{U}}_{\mathbf{Stk}}|/{\mathit{U}}_{\mathbf{Stk}}<5\%$ | |
---|---|---|---|

Schiller & Naumann | ${\left\{\frac{St{k}^{2}+1}{St{k}^{2}+S{N}^{2}}\right\}}^{1/2}$ | $SN=1+0.158R{e}^{2/3}$ | $\frac{\sqrt{3Re}}{Stk}<1$ |

Basset | ${\left\{\left[1+\frac{1}{St{k}^{2}}\right]\left[1-{(1+{f}_{1})}^{2}-{f}_{2}^{2}\right]\right\}}^{1/2}$ | ${f}_{1}=\frac{2\left(1-\gamma \right)\left(1+2\gamma +3\sqrt{\frac{\gamma}{Stk}}\right)}{{\left(2\frac{\gamma}{Stk}+3\sqrt{\frac{\gamma}{Stk}}\right)}^{2}+{\left(1+2\gamma +3\sqrt{\frac{\gamma}{Stk}}\right)}^{2}}$ ${f}_{2}=\frac{2\left(1-\gamma \right)\left(2\frac{\gamma}{Stk}+3\sqrt{\frac{\gamma}{Stk}}\right)}{{\left(2\frac{\gamma}{Stk}+3\sqrt{\frac{\gamma}{Stk}}\right)}^{2}+{\left(1+2\gamma +3\sqrt{\frac{\gamma}{Stk}}\right)}^{2}}$ | $(\gamma St{k}^{6/5}>370)$∧$\left(\gamma >5\right)$ |

Landau & Lifshitz | ${\left\{\frac{St{k}^{2}+1}{St{k}^{2}\left[{\left(\frac{3}{\sqrt{2\gamma Stk}}+\frac{1}{Stk}\right)}^{2}+{\left(\frac{3}{2\sqrt{\gamma Stk}}+\frac{1}{\sqrt{2}\gamma}+1\right)}^{2}\right]}\right\}}^{1/2}$ | $(\gamma >1000)$ |

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**MDPI and ACS Style**

Heidinger, S.; Unz, S.; Beckmann, M. Simple Particle Relaxation Modeling in One-Dimensional Oscillating Flows. *Processes* **2022**, *10*, 1322.
https://doi.org/10.3390/pr10071322

**AMA Style**

Heidinger S, Unz S, Beckmann M. Simple Particle Relaxation Modeling in One-Dimensional Oscillating Flows. *Processes*. 2022; 10(7):1322.
https://doi.org/10.3390/pr10071322

**Chicago/Turabian Style**

Heidinger, Stefan, Simon Unz, and Michael Beckmann. 2022. "Simple Particle Relaxation Modeling in One-Dimensional Oscillating Flows" *Processes* 10, no. 7: 1322.
https://doi.org/10.3390/pr10071322