Thomson-Einstein's Tea Leaf Paradox Revisited: Aggregation in Rings

Abstract

A distinct particle focusing spot occurs in the center of a rotating fluid, presenting an apparent paradox given the presence of particle inertia. It is recognized, however, that the presence of a secondary flow with a radial component drives this particle aggregation. In this study, we expand on the examination of this "Thomson-Einstein's tea leaf paradox" phenomenon, where we use a combined experimental and computational approach to investigate particle aggregation dynamics. We show that not only the rotational velocity, but also the vessel shape, have a significant influence on a particle's equilibrium position. We accordingly demonstrate the formation of a single focusing spot in a vessel center, as has been conclusively demonstrated elsewhere, but also the repeatable formation of stable ring-shaped particle arrangements.

Keywords: computational fluid dynamics; hydrodynamic interactions; micromanipulation; stokes drag; tea leaf paradox.

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 $\theta =-20°,-10°,-5°,0°,+5°,+10°,+20°,+40°$ 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 $\overline{J_x}$ ($nN·\mathrm{s}$) 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) $\theta =-20°$, (b,e) $\theta =0°$, (c,f) $\theta =20°$, 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.