Inertial migration of spherical and oblate particles in a triangular microchannel

The. inertial migration of both spherical and oblate particles within an equilateral triangular channel is studied numerically. Our study primarily focuses on the effects of fluid inertia, quantified by the Reynolds number (Re) and particle size ($\beta$). Our observations reveal two distinct equilibrium positions: the corner equilibrium position (CEP) is situated along the angle bisector near the corner, while the face equilibrium position (FEP) is located on a segment of the line perpendicular from the triangle's center to one of its sides. Spherical particles with varying initial positions predominantly reach the FEP. For oblate particles initially positioned along the angle bisector with a specific orientation, meaning the particle's evolution axis is inside the plane bisecting the angle, they will migrate along the angle bisector to reach the CEP while rotating in the tumbling mode. Conversely, for particles with different initial orientations and positions, they will employ the log-rolling mode to reach the FEP. Notably, we identify a dual-stage particle migration process to the FEP, with trajectories converging to an equilibrium manifold, which bears a resemblance to the cross section of the channel. To further illustrate the transition between FEP and CEP under general initial conditions, except for those along the angle bisector, we construct a phase diagram in the (Re, $\beta$) parameter space. This transition is often triggered by the size of larger particles (as the FEP cannot accommodate them) or the influence of inertia for smaller particles. For the FEP, especially for medium- or small-size particles, we notice an initial outward movement of the FEP from the center of the cross section as Re increases, followed by a return towards the center. This behavior results from the interplay of three forces acting on the particle. This research holds potential implications for the design of microfluidic devices, offering insights into the behavior of particles within equilateral triangular channels.

Figure 1

Sketch of an oblate particle with the symmetric axis ($x^{\prime }$ axis). $Oxyz$ and $O{x}^{\prime }{y}^{\prime }{z}^{\prime }$ denote the space-fixed and body-fixed frames, respectively. $(\varphi ,\theta ,\psi )$ represents the Euler angles.

Figure 2

(a) Sketch of a sphere in a tube Poiseuille flow. Dimensionless steady-state values of velocities (b) $u^{*}$, (c) ${\omega }^{*}$, and (d) the lift force $F^{*}$ with different radial positions. $F$ is normalized by $\rho u_m^2{R}^2$.

Figure 3

Sketch of a particle in a triangular channel Poiseuille flow. Here, $h$ and $w$ are the height and width of the equilateral triangular cross section, respectively. $L_z$ is the channel length.

Figure 4

The sketches of small, medium, and large spherical particles (from left to right, $\beta =[0.23,0.35,0.46]$, respectively).

Figure 5

Effects of (a) grid size and (b) channel length on the migration of an oblate particle ($\mathrm{Ar}=0.5, \beta =0.23$ and $\mathrm{Re}=200$). For simulations in (a), $L_z=20d_e$ is used, and in (b), $\delta x/w=1/150$ is used.

Figure 6

Inertial migration of particles in an equilateral triangle channel ($\mathrm{Re}=20, \beta =0.23$). Solid lines with different colors represent particle trajectories. The hollow circles indicate the initial positions of the particles, and the hollow triangles represent the equilibrium positions. The black dash-dotted lines denote the contours of the shear rate at the equilibrium position. Contours traversing the CEP are not depicted for the sake of clarity. The blue dashed line corresponds to the particle equilibrium manifold.

Figure 7

The particle experiences the shear-induced lift force ${\mathit{F}}_{\mathrm{S}}$ and the wall-induced force ${\mathit{F}}_W$ during the first stage, and the rotation-induced lift force ${\mathit{F}}_{\mathrm{\Omega }}$ becomes significant in the second stage.

Figure 8

(a) The migration trajectories of nonrotating and rotating cases. Point $A$ is the initial position. The blue and red solid lines represent the trajectories of rotating and nonrotating particles, respectively. Points $B$ and $C$ represent their final equilibrium positions. $\mathrm{Re}=20, \beta =0.23$. (b) Forces on the rotating particle during its migration toward the FEP. The red curve represents the Magnus force and the blue dashed line represents the lift force.

Figure 9

The effect of Re and $\beta$ on the CEP: (a) Dimensionless vertical equilibrium position $y_p^{*}$ as a function of blockage ratio $\beta$ under different Re. The initial position of each point is $(x_p^{*},y^{*}p)=(0,0.28)$. (b) Dimensionless vertical position $y_p^{}$ as a function of focusing length (dimensionless stream length $z_p^{*}$) under different Re with $\beta =0.29$.

Figure 10

Net lift force coefficient ${\mathit{C}}_{\mathrm{L}}$ over triangular channels. The arrows represent the direction, and the colors denote the magnitude of ${\mathit{C}}_{\mathrm{L}}$. The background represents the shear rate contour. The hollow triangle represents the saddle point, and the solid represents the stable equilibrium position. $\mathrm{Re}=20$ and $\beta =0.23$ are fixed.

Figure 11

(a) Directions of the rotations and forces acting on particles in the triangular channel. These two particles migrate to the CEP and FEP, respectively. (b) Trajectories of particles with increased $\beta$ and Re. Point $A$ represents the equilibrium position of the particle at $\beta =0.23$ and $\mathrm{Re}=20$. Point $B$ represents the equilibrium position when $\beta$ is increased to $\beta =0.35$. Point $C$ represents the equilibrium position when Re is increased to $\mathrm{Re}=100$.

Figure 12

The effect of Re and $\beta$ on the face equilibrium position: (a) Dimensionless vertical equilibrium position $y_p^{*}$ as a function of blockage ratio Re under different $\beta$. The initial position of each point is $(x_p^{*},y_p^{*})=(0,-0.18)$. (b) Dimensionless vertical position $y_p^{*}$ as a function of focusing length under different Re with $\beta =0.35$.

Figure 13

Phase diagram of particle equilibrium positions in the (Re, $\beta$) parameter plane. The red diamond represents the corner equilibrium position (CEP), and the blue inverted triangle represents the center-edge equilibrium position (FEP). The initial position of each point is $(x_p^{*},y_p^{*})=(0.23,-0.2)$.

Figure 14

Evolution of an oblate particle from its initial state to equilibrium states: (a) Log-rolling mode. (b) Tumbling mode. (c) Variation of equilibrium positions and $x,y$ components of the orientation vector ($n_x, n_y$) ($n_z=0$) with particle migration for different initial orientations.

Figure 15

(a) Inertial migration of oblate particles in an equilateral triangular channel from different initial positions. The dashed ellipses represent the initial states of the particles, while the solid ellipses represent typical snapshots at the final states. $\mathrm{Re}=20, \beta =0.23, ({\varphi }_0,{\theta }_0,{\psi }_0)=({80}^{\circ },0^{\circ },0^{\circ })$. (b) The $x$ component of the orientation vector, $n_x$, as a function of time for the different initial positions. (c) The nondimensional position $x_p^{*}$ of the particle during the migration process as a function of the nondimensional focusing length. The black dashed line represents the $x_p^{*}$ at the equilibrium position.

Figure 16

(a)–(c) The lateral migration trajectories (thick solid lines) of oblate particles with different sizes. The solid ellipses represent the final states of the particles. In all cases, $\mathrm{Re}=20$. (d) The nondimensional position $y_p^{*}$ of the particle during the migration process as a function of the nondimensional focusing length. (e) The $x$ component of the orientation vector, $n_x$, as a function of time for the different $\beta$. In all figures, pink, yellow, and blue represent the cases of $\beta =0.23$, 0.35, and 0.46, respectively.

Figure 17

(a) The nondimensional equilibrium position $y_p^{*}$ of the particle as a function of Re. The red curve represents the oblate particles. The blue represents the sphere. (b) The $x$ component of the orientation vector, $n_x$, as a function of time for $\mathrm{Re}=[40,120,200]$.

Figure 18

Phase diagram of particle equilibrium positions in (Re, $\beta$) parameter plane, $\mathrm{Ar}=0.5$. The initial position of each point is $(x_p^{*},y_p^{*})=(0.23,-0.2)$, which is a typical general initial location.