Inertial forces for particle manipulation near oscillating interfaces

Due to the inherent nonlinearity of fluid dynamics, a large class of oscillating flows gives rise to rectified effects of steady motion. It has recently been shown that particle transport in such flows leads to differential displacement and efficient sorting of microparticles. Here we present a model that generalizes a Maxey-Riley-like equation for particle motion, incorporating important viscous and inviscid effects near oscillating interfaces and efficiently bridging the acoustofluidic and microfluidic approaches. Resulting in direct predictions for particle motion on slower timescales, the model predicts a richer and qualitatively different behavior from that expected from simplified radiation-force formalisms: depending on experimental control parameters, the net effect of interfacial oscillation can be either an attraction to or a repulsion from the interface, and particles can be captured at a fixed distance or released. These results are verified in comparison with experiments.

Figure 1

Problem setup and nomenclature for a spherical particle of radius $a_p$ near an oscillating interface (curvature length scale $a_b$).

Figure 2

Comparison of slow-time [steady, Eq. (6)] and oscillatory [unsteady, Eq. (4)] numerical solution for $\epsilon =0.1$, $\gamma =0.026$, $r_p\left(0\right)=1.36$, and $r_p^{\prime }\left(0\right)=\epsilon u_0(r_p\left(0\right),0)=0.054$ (with $u_L=0$, $n_B=4$): (a) $\widehat{\kappa }=-0.06$ (particle lighter than fluid), $\lambda =10$; (b) $\widehat{\kappa }=0$ (density matched), $\lambda =10$; (c) total steady force on the particle as a function of interface separation, showing no fixed points for the repulsive case (a) and a stable fixed point for the attractive case (b).

Figure 3

Phase diagrams of particle behavior as a function of $\gamma$ and $\widehat{\kappa }$; panels (a) and (b) have fixed $\delta =0.022$. (a) Attractive case: contours indicate the fixed-point equilibrium distance $h_s$ between particle and interface; the red line is the boundary for existence of fixed points from (8). The black dot identifies the experimental parameters of Sec. 4. (b) Repulsive case: contours indicate the time $T_{2a_p}$ for a particle near touching the interface to traverse a distance $2a_p$. (c) Boundary between the attractive and repulsive cases for different $\delta$. The dot-dashed and dashed lines give the analytical predictions based on the sign change of $F_{\rho }$ and the balancing of the terms $F_{\rho }$ and $F_{i,2}$, respectively (see the text for more details).

Figure 4

Experimental setup (figure modified from Ref. [8]); (a) a spherical bubble is exposed to the microchannel flow $u_L$ and driven to volumetric oscillations by a piezoelectric transducer; (b) a spherical particle is captured at a close distance to the bubble interface.

Figure 5

(a) Computed flow field $u_L$ around the quiescent bubble, indicating the drag $F_D$ on the particle; (b) with volumetric bubble oscillations, the force balance from (6) determines particle position.

Figure 6

Force contributions for $\lambda =55$, $\gamma =0.05$, $\widehat{\kappa }=0.033$ corresponding to the experiments with $\epsilon =0.012$. The distance between bubble and particle surfaces $h_0$ is normalized by $a_b=150\mu \mathrm{m}$. (a) Forces from (6); (b) $F_{SR}$ and $u_L$; (c) sum of forces in (a) showing two fixed points at $h_s$ and $h_u$; (d) sum of forces in (b) resulting in only one unstable fixed point.

Figure 7

Particle release. (a) Sum of forces from (6): as $\epsilon$ is decreased, the stable fixed point is lost at a finite distance from the bubble surface (${\epsilon }_c\approx 0.007$ in agreement with experiment); (b) sum of $F_{SR}$ and $F_D$: the unstable fixed point is lost at the bubble surface, and ${\epsilon }_c$ does not agree with the measured value.

Figure 8

Normalized force on a particle for large distances from the oscillation source, graphed as a function of ${\delta }_p=\sqrt{2\nu /\left(a_p^2\omega \right)}$ for $\widehat{\kappa }=0.033$ (corresponding to a polystytrene particle in water). The present work [Eq. (12)] predicts a sign change of the force as viscous effects become important, in agreement with Ref. [6] and contradicting Ref. [5].