# 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

## References

- Aref, H. Stirring by chaotic advection. J. Fluid Mech.
**1984**, 143, 1–21. [Google Scholar] [CrossRef] - Lasheras, J.C.; Tio, K.K. Dynamics of a small spherical particle in steady two-dimensional vortex flows. Appl. Mech. Rev.
**1994**, 47, S61–S69. [Google Scholar] [CrossRef] - Raju, N.; Meiburg, E. Dynamics of small, spherical particles in vortical and stagnation point flow fields. Phys. Fluids
**1997**, 9, 299–314. [Google Scholar] [CrossRef] - Kynch, G.J. A theory of sedimentation. Trans. Faraday Soc.
**1952**, 48, 166–176. [Google Scholar] [CrossRef] - Babiano, A.; Cartwright, J.H.E.; Piro, O.; Provenzale, A. Dynamics of a small neutrally buoyant sphere in a fluid and targeting in Hamiltonian systems. Phys. Rev. Lett.
**2000**, 84, 5764–5767. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Brenner, H. The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Eng. Sci.
**1961**, 16, 242–251. [Google Scholar] [CrossRef] - Kuehn, C.; Romanò, F.; Kuhlmann, H.C. Tracking particles in flows near invariant manifolds via balance functions. Nonlinear Dyn.
**2018**, 92, 983–1000. [Google Scholar] [CrossRef] [Green Version] - Romanò, F.; des Boscs, P.E.; Kuhlmann, H. Forces and torques on a sphere moving near a dihedral corner in creeping flow. Eur. J. Mech. B Fluids
**2020**, 84, 110–121. [Google Scholar] [CrossRef] - Aref, H.; Blake, J.R.; Budišić, M.; Cardoso, S.S.; Cartwright, J.H.; Clercx, H.J.; El Omari, K.; Feudel, U.; Golestanian, R.; Gouillart, E.; et al. Frontiers of chaotic advection. Rev. Mod. Phys.
**2017**, 89, 025007. [Google Scholar] [CrossRef] [Green Version] - Sapsis, T.; Haller, G. Inertial particle dynamics in a hurricane. J. Atm. Sci.
**2009**, 66, 2481–2492. [Google Scholar] [CrossRef] - Haller, G. Lagrangian coherent structures. Ann. Rev. Fluid Mech.
**2015**, 47, 137–162. [Google Scholar] [CrossRef] [Green Version] - Kuhlmann, H.C.; Romanò, F.; Wu, H.; Albensoeder, S. Particle-Motion Attractors due to Particle-Boundary Interaction in Incompressible Steady Three-Dimensional Flows. In Proceedings of the The 20th Australasian Fluid Mechanics Conference, Perth, Australia, 5–8 December 2016; Ivey, G., Zhou, T., Jones, N., Draper, S., Eds.; Australasian Fluid Mechanics Society: Perth, Australia, 2016; p. 102. [Google Scholar]
- Romanò, F.; Kuhlmann, H.C. Finite-size Lagrangian coherent structures in thermocapillary liquid bridges. Phys. Rev. Fluids
**2018**, 3, 094302. [Google Scholar] [CrossRef] - Romanò, F.; Wu, H.; Kuhlmann, H.C. A generic mechanism for finite-size coherent particle structures. Int. J. Multiph. Flow
**2019**, 111, 42–52. [Google Scholar] [CrossRef] - Romanò, F.; Kunchi Kannan, P.; Kuhlmann, H.C. Finite-size Lagrangian coherent structures in a two-sided lid-driven cavity. Phys. Rev. Fluids
**2019**, 4, 024302. [Google Scholar] [CrossRef] - Romanò, F.; Kuhlmann, H.C. Finite-size coherent structures in thermocapillary liquid bridges: A review. Int. J. Microgravity Sci. Appl.
**2019**, 36, 360201-1–360201-17. [Google Scholar] - Wu, H.; Romanò, F.; Kuhlmann, H.C. Attractors for the motion of a finite-size particle in a two-sided lid-driven cavity. J. Fluid Mech.
**2021**, 906, A4. [Google Scholar] [CrossRef] - Xu, S.; Nadim, A. Oscillatory counter-centrifugation. Phys. Fluids
**2016**, 28, 021302. [Google Scholar] [CrossRef] - Verhille, G.; Le Gal, P. Aggregation of fibers by waves. In Nonlinear Waves and Pattern Dynamics; Springer: Berlin/Heidelberg, Germany, 2018; pp. 127–136. [Google Scholar]
- Romanò, F. Oscillatory switching centrifugation: Dynamics of a particle in a pulsating vortex. J. Fluid Mech.
**2018**, 857, R3. [Google Scholar] [CrossRef] [Green Version] - Ducrée, J.; Haeberle, S.; Lutz, S.; Pausch, S.; Stetten, F.V.; Zengerle, R. The centrifugal microfluidic Bio-Disk platform. J. Micromech. Microeng.
**2007**, 17, S103–S115. [Google Scholar] [CrossRef] - Mark, D.; Haeberle, S.; Roth, G.; Stetten, F.V.; Zengerle, R. Microfluidic lab-on-a-chip platforms: Requirements, characteristics and applications. In Microfluidics Based Microsystems; Springer: Berlin/Heidelberg, Germany, 2010; pp. 305–376. [Google Scholar]
- Maxey, M.R.; Riley, J.J. Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids
**1983**, 26, 883–889. [Google Scholar] [CrossRef] - Faxén, H. Der Widerstand gegen die Bewegung einer starren Kugel in einer zähen Flüssigkeit, die zwischen zwei parallelen ebenen Wänden eingeschlossen ist. Ann. Phys.
**1922**, 373, 89–119. [Google Scholar] [CrossRef] [Green Version] - Hofmann, E.; Kuhlmann, H.C. Particle accumulation on periodic orbits by repeated free surface collisions. Phys. Fluids
**2011**, 23, 0721106. [Google Scholar] [CrossRef] - Romanò, F.; Kuhlmann, H.C. Numerical investigation of the interaction of a finite-size particle with a tangentially moving boundary. Int. J. Heat Fluid Flow
**2016**, 62 Pt A, 75–82. [Google Scholar] - Romanò, F.; Kuhlmann, H.C. Particle-boundary interaction in a shear-driven cavity flow. Theor. Comput. Fluid Dyn.
**2017**, 31, 427–445. [Google Scholar] [CrossRef] [Green Version] - Romanò, F.; Kuhlmann, H.C.; Ishimura, M.; Ueno, I. Limit cycles for the motion of finite-size particles in axisymmetric thermocapillary flows in liquid bridges. Phys. Fluids
**2017**, 29, 093303. [Google Scholar] [CrossRef] - Barmak, I.; Romanò, F.; Kuhlmann, H.C. Particle accumulation in high-Prandtl-number liquid bridges. PAMM
**2019**, 19, e201900058. [Google Scholar] [CrossRef] [Green Version] - Romanò, F.; Kuhlmann, H.C. Interaction of a finite-size particle with the moving lid of a cavity. PAMM
**2015**, 15, 519–520. [Google Scholar] [CrossRef] - Smith, J.P.; Barbati, A.C.; Santana, S.M.; Gleghorn, J.P.; Kirby, B.J. Microfluidic transport in microdevices for rare cell capture. Electrophoresis
**2012**, 33, 3133–3142. [Google Scholar] [CrossRef] [Green Version] - Pertoft, H. Fractionation of cells and subcellular particles with Percoll. J. Biochem. Biophys. Methods
**2000**, 44, 1–30. [Google Scholar] [CrossRef] - Tondreau, T.; Lagneaux, L.; Dejeneffe, M.; Delforge, A.; Massy, M.; Mortier, C.; Bron, D. Isolation of BM mesenchymal stem cells by plastic adhesion or negative selection: Phenotype, proliferation kinetics and differentiation potential. Cytotherapy
**2004**, 6, 372–379. [Google Scholar] [CrossRef] [PubMed]

**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