2.1. Computational Fluid Dynamics
The study employs Navier–Stokes equations to model the motion of viscous fluid numerically. Specifically, for an incompressible single-phase Newtonian fluid, the momentum and continuity equations are expressed as follows:
where
denotes the fluid velocity,
is the pressure,
is the density, and
is the dynamic viscosity [
14] (see
Table S1). The computational domains are shown in
Figure S1 where the diameter is
(mm), and the height is
(mm); a positive inclination angle
indicates a concave shape, and a negative angle indicates a convex shape (
Figure S2a,b). The axisymmetric flow can be characterized by a rotational (circumferential) Reynolds number:
Implying the initial rotational rate , the Reynolds number ranges from 800 to 2100 at the rotational rates 30–80 (RPM).
In the initial state (
s), the fluid is assumed to rotate with a constant angular velocity
around the axis of symmetry (Z)
where
is the distance from the rotational axis; and
denotes the initial angular velocity. The vessel boundaries at the initial frame (
s) have an identical rotational rate as the fluid. Over a period of 0.2 s (from −0.1 s to 0.1 s), the domain boundaries decelerated and came to a stop. A no-slip boundary condition is applied to the side and bottom boundaries of the fluid domain (
Figure S2c)
where
is a smoothed unit step function, enabling a smooth transition from the vessel rotation to a state of rest
At the top of the computational domain, a slip boundary condition is applied to define a free surface
where
denotes the unit vector normal to the plane. In general, when a liquid rotates with a constant angular velocity
, it creates a parabolic-shaped free surface. The maximum elevation of this surface for a vessel with diameter
can be calculated as
where
is the acceleration of gravity. However, in our study, the highest angular velocity used,
RPM, which results in a relatively small elevation
, which is negligible compared to the height of the domain. Therefore, for simplicity, we assume a flat top surface for the computational domain.
The Computational Fluid Dynamics (CFD) simulations were performed using the COMSOL Multiphysics 5.5 software, which utilizes the Finite Element Method (FEM) for solving the governing equations. The model was implemented using a time-dependent single-phase laminar flow (SPF) module. The simulations compute the system evolution from −0.1 s to 90 s, where the smoothed rotation termination is centered at the 0-time frame. A MATLAB routine is used to execute the parametric solver and process the resulting data.
2.2. Computational Mesh Study
The structured mesh employed in this study is depicted in
Figure S4a. To address the issue of highly skewed cells, an O-grid technique was implemented to discretize circular cross-sections. To ensure accuracy and capture the viscous boundary layer, the mesh was refined near the side and bottom walls. The color table in
Figure S2b represents the skewness of the mesh elements
where
represents a perfectly regular element and
represents a degenerated element,
is the angle over an edge in the element, and the maximum is taken over all edges of the element.
To assess the mesh convergence, this study follows the approach of Devendran et al. [
15] and Muller et al. [
16].
Figure S2b plots the convergence function
where
is the current solution; and
is a reference solution with a maximum mesh element size of
(mm). The analysis was performed at a rotational velocity of
(RPM) and a base inclination angle of
. All components of the speed were analyzed independently. The threshold for mesh convergence was set
, corresponding to a maximum mesh size of
(mm).
2.3. Particle Modeling
The particle sedimentation process is modeled based on the results of a time-dependent CFD analysis. Since we study particles that are heavier than water, we examine the velocity profiles in the bottom 5 mm thick region of the domain (see
Figure S2c). The velocity components in the
-planes are averaged over 40 equidistant φ values. To predict particle aggregation, their motion was decoupled into out-of-plane (circumferential) and in-plane (
plane) motion.
To estimate the forces acting on the particle, we consider it as a spherical particle, quasi-static in the rotating
-frame (
Figure S2d). The local coordinate system rotates around the vessel axis with an angular velocity corresponding to the out-of-plane fluid velocity
at the particle location in the
-plane. This setup simulates a particle with a circular trajectory around the axis. The forces along
z and
x axes are decoupled since particle displacement in these coordinates has different order of length scales.
In the first step, we calculate particle separation distance from the bottom surface (along
z-axis). We assume that the particles are located in a close proximity to the bottom of the vessel. Due to the forces counteracting the gravitational sediment, particles can move along the vessel bottom. The total force on a particle along
z-axis is the combination of the lift force, drag force, centrifugal force, and gravitational force projections, with
For a particle with an effective particle radius in a shear velocity field, this experiences a lift force according to the following approximated expression [
17]:
where
is the kinematic viscosity of the water,
is the viscosity of the fluid,
is the fluid velocity at the particle center, and the
function extracts the sign of a quantity. The drag force acting on the particle due to
fluid component is given by
Further, the projection of the centrifugal force projection can be derived as follows, with
where
R is the distance from the rotational axis,
is the out-of-plane fluid velocity, and
is the bottom inclination angle. The gravitational force projection on the z-axis is defined as
where
is the fluid density,
is the particle density,
is the effective particle radius, and
is gravitational acceleration. The total force acting on the particle is evaluated at a given point above the surface. If the total force is pointing downwards everywhere (
), the particle is considered to be in contact with the surface. Otherwise, by imposing the condition
, the equilibrium particle elevation can be calculated. As the lift force rapidly decays away from the surface, the particle equilibrium position is
.
In the second step, the x-force component is evaluated taking into account the equilibrium particle elevation along z-axis. When a particle is placed in a shearing flow near a plane wall, the classic Stokes drag is no longer valid since the nearby wall induces alterations in the fluid flow field around the particle. We therefore employ a modified Stokes formula [
18], with
where
is a compensation factor depending on the distance from the surface (see
Table S3) and reaches
for a sphere adjacent to the surface. The centrifugal and gravitational force components are given by
Particles in contact with the boundary experience the solid friction force which is proportional to the normal force component
where
is the solid friction constant. The total force along the local
x-axis is determined as follows, with
Figure S3a–c shows the dimensionless velocity components in the local coordinate system 5 s after the termination. The fluid flow has a dominant out-of-plane
component which governs the circumferential particle motion. In-plane components of the fluid flow, however, exhibit lower amplitude in general, though with highest velocities in the adjacent layer to the bottom surface.
Figure S3d plots the evolution of the resulting force
, acting on a quasi-static particle (in the rotating frame) at different positions along the axis. The modeled particle experiences a focusing (non-zero) force towards the rotational axis up to a 12 s timepoint, promoting particle aggregation.
2.4. Tea Leaf Settling Analysis
A schematic of the experimental setup is presented in
Figure 1b. The setup consists of a vertical cylinder with a radius of
(mm), where the base inclinations
range from −20 to 40 degrees. The cylinders were fabricated using an SLA 3D Printer (AnyCubic Photon, ANYCUBIC Technology, Shenzhen, China). Prior to the experiment, the cylinder is filled with a suspension of tea leaves to a height of
(mm), which corresponds to the numerical study domain depicted in
Figure S2. The rotation of the vessel is driven by a stepper motor controlled by an Arduino microcontroller (Arduino, Ivrea, Italy). The cylinder with quiescent fluid is spun for 15 s at a rate ranging from 30 to 80 (RPM). After rotation, the cylinder is immobilized, and the particle evolution is studied. The motion and spatial distribution of particles are recorded using a digital Dino-Lite microscope (Dino-Lite, Hsinchu, Taiwan) mounted vertically above the rotational cell, as shown in
Figure 1b. The grayscale video files are processed using a Processing Toolbox in MATLAB to analyze the post-rotation distribution of particles in the cell.
2.5. Tea Leaf Particle Characterization
An experimental setup was utilized to evaluate the size and density of tea leaves, which are crucial parameters for the numerical model. The size of the wetted tea leaf particles is estimated using a gauged microscope slide. Subsequently, the sedimentation time of the particles
was measured in a graduated cylinder with a known height of
(m). Assuming particle sedimentation in the Stokes drag regime, the effective density of particles can be deduced using
where
is the fluid viscosity,
is the fluid density,
is gravitational acceleration, and
is the effective particle radius. The average particle Reynolds number is 4.8 in the experimental conditions, suggesting that this equation is valid for these conditions. The measured and processed values for effective hydrodynamic particle radius and density are shown in
Table S2.
Figure S4 subsequently plots the effective density measured for particles of different radii. A linear trendline is fitted to the data as shown in
Figure S4 to estimate particle density used in the numerical model (
Table S1).
To estimate the coefficient of friction, the angle of friction
was measured. The friction angle represents the maximum angle at which a particle will begin sliding. In the experiment, tea leaves are released above a submerged inclined surface. It is observed that when the angle exceeds
, most particles start to slide. The solid friction constant (sliding friction) can thus be approximated from the friction angle, with