Transport of spherical microparticles in a 3D vortex flow

TL;DR

The paper analyzes how neutrally buoyant spherical particles move in a steady 3D vortex generated in a microfluidic cross-slot at moderate $Re$. By combining controlled experiments with immersed-boundary DNS, it isolates finite-size and inertial effects on particle trajectories. Small particles follow Burgers-vortex-like self-similar spirals around the core, while larger particles are progressively repelled from the vortex core due to inertia, with a cross-stream migration velocity scaling as $U_L \\sim (a/w)^3 Re$. These findings advance understanding of particle transport and separation in vortical microflows and provide quantitative benchmarks for particle-resolved and point-particle methods in complex vortical fields.

Abstract

Particles are common in biological and environmental flows and are widely used in industrial and pharmaceutical applications. Their motion and flow dynamics are strongly affected by interactions with the surrounding flow structure. While particle-flow interactions have been extensively studied in low Reynolds number (Re) flows as well as in fully developed turbulence, the transport mechanisms of these particles in intermediate flow regimes remain less explored. Here, we investigate the response of neutrally buoyant spherical particles to a single vortex flow field. Using a microfluidic cross-slot geometry, we generate a well-characterized, stationary, three-dimensional streamwise vortex at moderate $\text{Re}$ ($\sim 50$). Our experimental results, supported by numerical simulations, show that with increasing particle diameter, they are progressively excluded from the vortex core. Initially, small particles follow a Burgers vortex-like self-similar motion, but for larger particle diameters, deviations from this trend emerge due to fluid inertia and finite-size effects. These findings enhance our understanding of particle dynamics in vortical flows and have implications for microfluidic applications involving particle sorting and separation.

Figures (8)

• Figure 1: Particle flow in the vortical field formed in the cross-slot geometry; Particle inflows from two opposing directions interact with a 3D vortex that is stretched downstream in the opposing outlet directions. Insets show superposition of experimental images of $20~\mu m$ particles in the flow ($Re = 56$) in both orthogonal planes of interest: the Core Vortex Plane (CVP) at $x = 0$ and the Stretched Vortex Plane (SVP) at $z = 0$.
• Figure 2: Base vortical flow field at $Re = 56$. (a) Simulated 3D streamlines of the base flow wrap and spiral around the vortex core. (b) Streamlines superimposed to the normalized velocity magnitude of the base flow from experiments (left) and simulations (right). Red lines mark the border of the streamlines interacting with the central vortex. (c) Normalized vorticity field of the base flow from experiments (left) and simulations (right). (d) Dimensionless velocity profiles along $z/w$ at $y/w=0$ and $x/w=0$. (e) Dimensionless vorticity profiles along $z/w$ at $y/w=0$ and $x/w=0$ and $x/w = 0.5$. The core vorticity is also fitted with the profile $\omega(z) = \omega_0 \exp(-\gamma z^2 / 4\nu)$ (red line, with fitting parameters $\omega_0 = 28$, $\gamma/4 \nu = 0.01$), matching the vorticity profile of a Burgers vortex.
• Figure 3: Stacks of experimental and simulated particle images in the cross-slot geometry at $Re = 56$. Particle diameters are 20, 40 and 80 $\mu$m for experiments and 140 $\mu$m for simulations. Top row: visualization of the Core Vortex Plane (CVP), obtained with a microfluidic glass device. Bottom row: Stretched Vortex Plane (SVP), imaged using a PDMS device. Examples of the reconstruction of particle trajectories are obtained from experimental data and overlaid as magenta lines in the 20 $\mu$m panel. Trajectories obtained from simulations for 140 microns particles are color-coded with time with $t_{in}$ the entry time of the particles and $t_{out}$ the exit time.
• Figure 4: Effect of particle size on trajectories and density distribution in the CVP. Left panels: Particle trajectories extracted from experiments in the $x=0$ (CVP) plane, color-coded by normalized velocity magnitude. Experiments were conducted at $Re=56$ for particle diameters: (a) $a=20$$\mu$m; (b) $a=40$$\mu$m; and (c) $a=80$$\mu$m. Data are averaged and mirrored around the axis $z = -y$. Right panels: Corresponding normalized particle density profiles along $z/w$ measured at $x/w=y/w=0$. Particle density is computed as $n_{bin}/n_{tot}*100$ where $n_{bin}$ is the number of particles within each bin (bin width = 0.04) and $n_{tot}$, is the total number of particles in the observed plane.
• Figure 5: Types of particle trajectories from simulations for 80 $\mu$m particles (top) and 140 $\mu$m (bottom) at $Re = 56$. Two distinct behaviors are observed: particles that spiral around the stagnation point while being advected into the outlets and particles that are advected without interacting with the vortex core. All panels are color-coded based on the number of spirals N a particle completes around the vortex axis. (a) 3D Visualization of particle trajectories in the cross-slot and then the outlets; Particle trajectories projection: (b) in the SVP; and (c) in the CVP (volume $|x| < 0.5$). (d) Initial positions of particle trajectories (lines indicate separatix streamlines at the cross-slot entry)