# Thomson–Einstein’s Tea Leaf Paradox Revisited: Aggregation in Rings

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Computational Fluid Dynamics

#### 2.2. Computational Mesh Study

#### 2.3. Particle Modeling

#### 2.4. Tea Leaf Settling Analysis

#### 2.5. Tea Leaf Particle Characterization

## 3. Results and Discussion

#### 3.1. Experimental Study

#### 3.2. Computational Analysis

#### 3.3. Discussion and Outlook

## 4. Conclusions

## Supplementary Materials

_{0}, and height $H$ in cylindrical coordinates (R, Z, φ). (a) Base geometries with concave inclinations (pointing upwards) are defined with a positive angle θ, (b) geometries with convex inclinations (tapered downwards) are defined with a negative angle θ. (c) Schematic diagram of the cross-section. (d) Forces acting on a swirling particle. Figure S2. (a) Computational mesh convergence study (b) Computational mesh. Figure S3. (a–c) Velocity components in the xz-plane near the bottom of the vessel 5 s after rotation termination. (d) The resulting force acting on the 0.375 um particle over time. The initial rotation rate is 80 RPM, θ = 0. Figure S4. Effective tea-leaf density for particles of different radii. The data is fitted with a least-squares linear trendline overlaid. Figure S5. Radial density distribution of settled tea leaf particles in vessels with various bottom surface inclinations and rotational rates. Figure S6. In-plane fluid velocity 5 s after the rotation stopped. The streamlines visualize local flow direction. Initial angular velocity is 30 RMP (

**a**–

**c**), 80 RPM (

**d**–

**f**). The base inclinations here are $\theta =20\xb0$ (a,d), $\theta =0\xb0$ (b,e), $\theta =-20\xb0$ (c,f). Video S1: Experimental results for distinct tea leaf patterns including dot, ring, and edge particle aggregation governed by the vessel shape and rotation parameters. Video S2: Tea leaf focusing performance of vessels of different bottom inclination and rotation rate.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Tea leaf paradox for particle manipulation. (

**a**) Tea leaf focusing in a cup after being stirred. (

**b**) Schematic figure of the experimental setup. (

**c**–

**e**) Experimental results depicting distinct tea leaf patterns (top-down view), including (

**c**) dot, (

**d**) ring, and (

**e**) edge particle aggregation governed by the vessel and rotation parameters. The subfigures illustrate the temporal evolution of the aggregation process: (

**c1**–

**e1**) initial distribution of tea leaves, (

**c2**–

**e2**) particles accumulating at the vessel’s edge during constant rotation, (

**c3**–

**e3**) particle movement induced by secondary flow immediately after the rotation ceases, (

**c4**–

**e4**) final distribution of tea leaves.

**Figure 2.**Tea leaf concentration profiles, settled in rotating vessels with bottom inclination $\theta =-20\xb0,-10\xb0,-5\xb0,0\xb0,+5\xb0,+10\xb0,+20\xb0,+40\xb0$ and initial angular velocity ${\omega}_{0}=30,40,50,60,70,80$ (RPM).

**Figure 3.**Vertical (

**a**–

**c**) and radial (

**d**–

**f**) components of the fluid flow in containers at ${\omega}_{0}=80$ RPM and 5 s after rotation termination.

**Figure 4.**Numerical modeling results for (

**a**) 0.25 and (

**b**) 0.5 mm spherical particle settlement. The colormap indicates the average impulse $\overline{{J}_{x}}$ ($nN\xb7\mathrm{s}$) along the local coordinate x. The size of the circle corresponds to the stagnation radius, providing an estimate of the aggregation size.

**Figure 5.**Calculated focusing spot radius $r/{R}_{0}$ as a function of particle density and size for vessels with bottom inclination angles (

**a**,

**d**) $\theta =-20\xb0$, (

**b**,

**e**) $\theta =0\xb0$, (

**c**,

**f**) $\theta =20\xb0$, and rotation rates (

**a**–

**c**) ${\omega}_{0}=30$, (

**d**–

**f**) ${\omega}_{0}=80$ (RPM). The pink boxes schematically illustrate the vessel shape. The dashed line corresponds to the experimentally estimated tea leaf properties. The inserts in (

**f**) schematically show the particle aggregation modes: (*) focusing in the center, (**) ring-shaped aggregation, and (***) aggregation at the edge of the vessel.

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**MDPI and ACS Style**

Kolesnik, K.; Pham, D.Q.L.; Fong, J.; Collins, D.J.
Thomson–Einstein’s Tea Leaf Paradox Revisited: Aggregation in Rings. *Micromachines* **2023**, *14*, 2024.
https://doi.org/10.3390/mi14112024

**AMA Style**

Kolesnik K, Pham DQL, Fong J, Collins DJ.
Thomson–Einstein’s Tea Leaf Paradox Revisited: Aggregation in Rings. *Micromachines*. 2023; 14(11):2024.
https://doi.org/10.3390/mi14112024

**Chicago/Turabian Style**

Kolesnik, Kirill, Daniel Quang Le Pham, Jessica Fong, and David John Collins.
2023. "Thomson–Einstein’s Tea Leaf Paradox Revisited: Aggregation in Rings" *Micromachines* 14, no. 11: 2024.
https://doi.org/10.3390/mi14112024