# Particle Coherent Structures in Confined Oscillatory Switching Centrifugation

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**u**yield

**v**the particle velocity, and

**g**the gravity acceleration. The left-hand side of the Maxey–Riley equation represents the rate of change of the particle momentum, while the right-hand side of (2) consists of the Basset history force, the added mass, the Stokes drag, the buoyancy force and the force exerted on the particle by the undisturbed flow. The Faxén correction [24] is included by the ${a}^{2}{\nabla}^{2}\mathit{u}$ terms and it is identically zero since ${\nabla}^{2}\mathit{u}\equiv 0$ for the flow (1). The two notations used for the Lagrangian derivative, i.e., ${\mathrm{d}}_{t}$ and ${\mathrm{D}}_{t}$, refer to the material derivative along the particle trajectory

**A**denotes an arbitrary vector field transported by

**v**and

**u**, respectively, and ${\partial}_{t}$ is the Eulerian derivative.

**g**directed along the cylinder axis, the dimensionless form of the Maxey–Riley equation reads

## 3. Results

## 4. Discussion and Conclusions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic of a small single particle (light blue) moving in an incompressible fluid flow confined in a hollow cylinder and driven by the rotation of the cylinder surface.

**Figure 2.**Radial coordinate ${r}_{\mathrm{p}}$ of a particle trajectory for $\varrho =0.5$, $\mathrm{St}=1$ and $\mathrm{Str}=0.005$, 0.002, 0.005, and 0.0055. The trajectories are integrated for $t=10$ and they are denoted by the light-blue line. The darkblue solid line shows the last 10% of the particle orbit and the purple dot its initial position. The red dashed line identifies the sliding surface of the PSI model, which surface is at distance $\Delta =a$ from the driving boundary.

**Figure 3.**Radial coordinate ${r}_{\mathrm{p}}$ of a particle trajectory for $\varrho =0.8$, $\mathrm{St}=1$ and $\mathrm{Str}=0.062$, 0.0625, 0.097, and 0.1. The trajectories are integrated for $t=10$ and they are denoted by the light-blue line. The darkblue solid line shows the last 10% of the particle orbit and the purple dot its initial position. The red dashed line identifies the sliding surface of the PSI model, which surface is at distance $\Delta =a$ from the driving boundary.

**Figure 4.**Radial coordinate ${r}_{\mathrm{p}}$ of a particle trajectory for $\varrho =0.9$, $\mathrm{St}=1$ and $\mathrm{Str}=0.0935$, 0.094, 0.0945, and 0.1. The trajectories are integrated for $t=10$ and they are denoted by the light-blue line. The darkblue solid line shows the last 10% of the particle orbit and the purple dot its initial position. The red dashed line identifies the sliding surface of the PSI model, which surface is at distance $\Delta =a$ from the driving boundary.

**Figure 5.**Radial coordinate ${r}_{\mathrm{p}}$ of a particle trajectory for $\varrho =1.1$, $\mathrm{St}=1$ and $\mathrm{Str}=0.0115$, 0.012, 0.0125, and 0.032. The trajectories are integrated for $t=10$ and they are denoted by the light-blue line. The darkblue solid line shows the last 10% of the particle orbit and the purple dot its initial position. The red dashed line identifies the sliding surface of the particle–surface interaction (PSI) model, which surface is at distance $\Delta =a$ from the driving boundary. The orange background denotes the particle attraction phase to a quasi-steady unstable attractor, the yellow background highlights the transition phase and the green background shows the orbit attraction to the stable traveling attractor.

**Figure 6.**Radial coordinate ${r}_{\mathrm{p}}$ of a particle trajectory for $\varrho =1.1$, $\mathrm{St}=1$ and $\mathrm{Str}=0.0745$, 0.0765, 0.077, and 0.0875. The trajectories are integrated for $t=10$ and they are denoted by the light-blue line. The darkblue solid line shows the last 10% of the particle orbit and the purple dot its initial position. The red dashed line identifies the sliding surface of the PSI model, which surface is at distance $\Delta =a$ from the driving boundary. The orange background denotes the particle attraction phase to a quasi-steady unstable attractor, the yellow background highlights the transition phase and the green background shows the orbit attraction to the stable traveling attractor.

**Figure 7.**Radial coordinate ${r}_{\mathrm{p}}$ of a particle trajectory for $\varrho =1.2$, $\mathrm{St}=1$ and $\mathrm{Str}=0.0395$, 0.054, 0.0765, and 0.085. The trajectories are integrated for $t=10$ and they are denoted by the light-blue line. The darkblue solid line shows the last 10% of the particle orbit and the purple dot its initial position. The red dashed line identifies the sliding surface of the PSI model, which surface is at distance $\Delta =a$ from the driving boundary.

**Figure 8.**Radial coordinate ${r}_{\mathrm{p}}$ of a particle trajectory for $\varrho =1.5$, $\mathrm{St}=1$ and $\mathrm{Str}=0.035$, 0.0355, 0.034, and 0.0345. The trajectories are integrated for $t=10$ and they are denoted by the light-blue line. The darkblue solid line shows the last 10% of the particle orbit and the purple dot its initial position. The red dashed line identifies the sliding surface of the PSI model, which surface is at distance $\Delta =a$ from the driving boundary.

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**MDPI and ACS Style**

Romanò, F. Particle Coherent Structures in Confined Oscillatory Switching Centrifugation. *Crystals* **2021**, *11*, 183.
https://doi.org/10.3390/cryst11020183

**AMA Style**

Romanò F. Particle Coherent Structures in Confined Oscillatory Switching Centrifugation. *Crystals*. 2021; 11(2):183.
https://doi.org/10.3390/cryst11020183

**Chicago/Turabian Style**

Romanò, Francesco. 2021. "Particle Coherent Structures in Confined Oscillatory Switching Centrifugation" *Crystals* 11, no. 2: 183.
https://doi.org/10.3390/cryst11020183