Bifurcation phenomena on the inertial focusing of a neutrally buoyant spherical particle suspended in square duct flows

Spherical particles in dilute suspension flows through a square duct are known to focus at certain points in the downstream cross-section. When the blockage ratio of a particle diameter to the duct width equals approximately 0.1, previous experimental and numerical studies showed three types of particle equilibrium points: channel-face equilibrium positions located at the center of the duct faces, channel-corner equilibrium positions near the corners on diagonals, and intermediate equilibrium positions located at intermediate positions symmetric with respect to the diagonals. Several patterns of the particle focusing over the duct cross-section were observed according to the appearance or disappearance of these equilibrium points depending on the Reynolds number. In the present study, we numerically investigate the inertial focusing patterns of a neutrally buoyant spherical particle suspended in square duct flows. We describe the transitions between the particle focusing patterns in terms of bifurcation of the particle equilibrium positions. Saddle-node bifurcations occur at two different critical Reynolds numbers corresponding to the appearance and disappearance of the intermediate equilibrium positions, respectively. Pitchfork bifurcation occurs at another Reynolds number where stability of the channel-corner equilibrium positions changes from saddle to stable. Channel-face equilibrium positions are always stable for all the investigated Reynolds numbers at this blockage ratio.

Figure 1

(a) Schematic of inertial particle focusing in a duct flow. (b)–(d) Particle focusing patterns for the blockage ratio $\sim 0.1$: locations where particles flow through the downstream cross-section in square ducts at (b) low, (c) intermediate, and (d) relatively high Reynolds numbers. Particle focusing points are named channel-face equilibrium position (CFE), intermediate equilibrium position (IME), and channel-corner equilibrium position (CCE), respectively.

Figure 2

Flow configuration.

Figure 3

(a) Collocation points ${\mathit{Y}}_{ij}$. (b) Lateral force map ${\mathit{F}}_{ij}$ (gray thick arrows). Equilibrium positions are shown with circles (black, white, and gray circles represent stable, saddle, and unstable equilibrium positions, respectively). Particle trajectories estimated from $\tilde{\mathit{F}}$ in the duct cross-section are plotted using blue thin curves with arrows. $\beta =0.125$, $\mathrm{Re}=280$, $N=14$.

Figure 4

Verification for the interpolated lateral forces. $\beta =0.125$, $\mathrm{Re}=280$. (a) Profile of the lateral force $F_r$ acting on a particle located on the $y$ ($z$) axis. (b) $F_r$ along the diagonal. (c) Comparison between the equilibrium radial position $r^{\mathrm{tra}}$ obtained from trajectory calculations [14] and the corresponding equilibrium radial position $r^{\mathrm{lat}}$ estimated from the lateral forces (present study). ${\varepsilon }_{\text{CFE}}$ (circles) and ${\varepsilon }_{\text{CCE}}$ (crosses) show the relative difference between $r^{\mathrm{lat}}$ and $r^{\mathrm{tra}}$ for the CFE and CCE, respectively, i.e., ${\varepsilon }_i=|r_i^{\mathrm{lat}}-r_i^{\mathrm{tra}}|/r_i^{\mathrm{tra}}$, $i=\mathrm{CFE},\mathrm{CCE}$.

Figure 5

Bifurcation diagram for the azimuthal angle of equilibrium positions in the duct cross-section. $\beta =0.125$. Equilibrium positions in the first quadrant of the cross-section are plotted. Solid and dashed lines show stable and saddle equilibrium positions, respectively. ${\mathrm{Re}}_{\mathrm{ci}}$ ($i=1,2,3$) are critical Reynolds numbers.

Figure 6

Equilibrium positions and particle trajectories in the duct cross-sections at $\beta =0.125$. (a) $\mathrm{Re}=100$ in Regime A, (b) $\mathrm{Re}=280$ in Regime B, (c) $\mathrm{Re}=320$ in Regime C, and (d) $\mathrm{Re}=400$ in Regime D. Circles show the equilibrium positions. The black, white, and gray circles indicate stable nodes, saddle points, and unstable nodes, respectively. Trajectories of the particle center are plotted using gray lines with arrows. The black solid line represents a separatrix and the dashed line represents a slow manifold, called the pSS ring. The center positions of particles observed experimentally at $\beta =0.125$ are plotted by small red dots and the peak positions of their probability density function (particle focusing position) are shown by green crosses.

Figure 7

Additional experimental results at $\beta =0.125$. (a) Regimes determined numerically and experimentally. The azimuthal angle of the particle position determined numerically is plotted with solid lines. Red squares and circles show particle focusing points observed in the previous experiment [14] and the present experiment, respectively. (b) Representative particle distributions over the duct cross-section in Regime A, B, C, and D. A dot represents the center of a particle observed. Each distribution consists of at least 250 particles.

Figure 8

Schematic directional map of the radial and azimuthal components of the lateral force acting on a particle. $\beta =0.125$. (a) $\mathrm{Re}=100$ in Regime A, (b) $\mathrm{Re}=280$ in Regime B, (c) $\mathrm{Re}=320$ in Regime C, and (d) $\mathrm{Re}=400$ in Regime D. The black chain curve, named $r$-nullcline, is the zero-level set contour of the radial component of the lateral force ($F_r$). The zero-level set contours of the azimuthal component of the lateral force ($F_{\theta }$) consist of the $y$ and $z$ axes, the diagonal and red solid curve, named $\theta$-nullcline, forming a ring between the duct center and the duct wall. The signs of $F_r$ and $F_{\theta }$ are plotted in black and red arrows, respectively. $\mathrm{Q}$ is the intersection between $\theta$-nullcline and the diagonal. Blue dashed curves are particle trajectories joining equilibrium positions on the pSS ring, corresponding to the dashed lines in Fig. 6.

Figure 9

Detailed directional map of the radial and azimuthal components of the lateral force acting on a particle corresponding to Figs. 8. $\beta =0.125$. (a) $\mathrm{Re}=100$ in Regime A, (b) $\mathrm{Re}=280$ in Regime B, (c) $\mathrm{Re}=320$ in Regime C, and (d) $\mathrm{Re}=400$ in Regime D. Figures $\mathrm{i})$ or $\mathrm{ii})$ in (b)–(d) show magnified views of the corresponding areas surrounded by dashed lines. Point $\mathrm{P}$ is the intersection between the red curve ($\theta$-nullcline in Fig. 8) and the $y$ or $z$ axis and point $\mathrm{Q}$ is the intersection between the red curve and the diagonal. See the legend of Fig. 8 for more explanation.

Figure 10

Bifurcation diagram showing the radial location of equilibrium positions in the duct cross-section. $\beta =0.125$. Solid and dotted lines show stable and saddle equilibrium positions, respectively. ${\mathrm{Re}}_{\mathrm{ci}}$ ($i=1,2,3$) are critical Reynolds numbers. $\mathrm{P}$ (red chain line) and $\mathrm{Q}$ (red solid line) are the boundaries between the light and dark red regions in Figs. 8 and 9 on the $y$ ($z$) axis and on the diagonal, respectively.