Bifurcations in inertial focusing of a particle suspended in flow through curved rectangular ducts

Abstract

Particles suspended in a fluid flow through a curved duct can focus to specific locations within the duct cross-section. This particle focusing is a result of a balance between two dominant forces acting on the particle: (i) the inertial lift force arising from small but non-negligible inertia of the fluid, and (ii) the secondary drag force due to the cross-sectional vortices induced by the curvature of the duct. By adopting a simplified particle dynamics model developed by Ha et al.~[1], we investigate both analytically and numerically, the particle equilibria and their bifurcations when a small particle is suspended in low-flow-rate fluid flow through a curved duct having a $2\times1$ and a $1\times2$ rectangular cross-section. In certain parameter regimes of the model, we analytically obtain the particle equilibria and deduce their stability, while for other parameter regimes, we numerically calculate the particle equilibria and stability. Moreover, we observe a number of different bifurcations in particle equilibria such as saddle-node, pitchfork and Hopf, as the model parameters are varied. These results may aid in the design of inertial microfluidic devices aimed at particle separation by size.

Figures (7)

• Figure 1: Schematic showing the theoretical setup. A particle (black filled circle) of radius $a$ with its center located at $\mathbf{x}_p=\mathbf{x}(\theta_p,r_p,z_p)$ is suspended in a fluid flow through a curved duct of radius $R$ having a rectangular cross-section of width $W$ and height $H$ with aspect ratio defined as $\text{AR}=W/H$. Enlarged view of the two cross-sections ($\text{AR}=2$ and $\text{AR}=1/2$) show the local cross-sectional $(r,z)$ co-ordinate system, and the secondary flow (gray closed curves) induced by the curvature of the duct.
• Figure 2: Contour plot showing the distribution of dimensionless inertial lift and secondary drag forces inside a $2\times1$ rectangular cross-section. (a) $L^{2\times1}_r(r,z)$, (b) $L^{2\times1}_z(r,z)$, (c) $D^{2\times1}_r(r,z)$ and (d) $D^{2\times1}_z(r,z)$ are shown. The black curves in each panel show the zero level contours of the corresponding force field.
• Figure 3: Contour plot showing the distribution of inertial lift and secondary drag forces inside a $1\times2$ rectangular cross-section. (a) $L^{1\times2}_r(r,z)$, (b) $L^{1\times2}_z(r,z)$, (c) $D^{1\times2}_r(r,z)$ and (d) $D^{1\times2}_z(r,z)$ are shown. The black curves in each panel show the zero level contours of the corresponding force field.
• Figure 4: Particle equilibria (filled circles) and particle trajectories (gray curves) in the limit of $\tilde{R} \xrightarrow{} \infty$ (left) and $\tilde{a} \xrightarrow{}0$ (right) for a $2\times1$ rectangular cross-section. The color of the filled circles indicates the type of particle equilibria obtained from linear stability analysis: unstable node in red, saddle point in yellow, stable node in green and a center in blue.
• Figure 5: Bifurcations in particle equilibria inside a $2\times1$ rectangular cross-section as a function of the dimensionless bend radius $\tilde{R}$ for a fixed dimensionless particle size $\tilde{a}=0.05$. The (a) radial $r$ and (b) vertical $z$ location of the particle equilibria as well as the (c) real and (d) imaginary parts of the eigenvalues $\lambda$ are shown as a function of $\tilde{R}$ (Note that $\tilde{R}$ decreases from left to right). (e)-(j) show the particle equilibria and particle trajectories (gray curves) in the cross-section for $\tilde{R}=10^5$, $3500$, $2200$, $2050$, $1900$ and $100$, respectively. The filled circles denote the fixed points with the size matched with the size of the particle and the color denoting the type of equilibrium point: unstable node (red), stable node (green), saddle point (yellow), unstable spiral (purple) and a stable spiral (cyan). The eigenvalue curves in panels (c) and (d) corresponding to particle equilibria have also been color-coded using the same convention.