Simulations of Steady Streaming Flows and the Time-Averaged Motion of Inertial Particles

Abstract

Applications of microfluidic systems often involve microparticle or cell trapping. Hydrodynamic tweezers are an emerging technology that utilizes an oscillating primary flow to induce microscale steady streaming eddy flows in the vicinity of engineered obstructions. Particles with sufficient inertia, as described by the particle Stokes number (<italic>St</italic>), are collected near the center of eddies, while particles with negligible inertia tend to follow the fluid motion. Device fabrication and testing has generally led our understanding of trapping physics in hydrodynamic tweezers. Here, we describe a major advance in our understanding of the physics of particle trapping in these devices. We explore the weak inertia regime (<italic>St</italic> « 1) for particle motion using the well-known Maxey-Riley equation. To accelerate our computation of particle trajectories and fluid motion, we employ a leading-order analytical approximation for the time-periodic motion and superimpose a second-order steady motion. The time-averaged formalism of Stokes drift calculations (originally for non-inertial tracer particles) is extended to a weakly inertial particle to account for the relative velocity between fluid and particle. The resulting analytical formulation is combined with flow field simulations to reveal key physics and parameters that influence particle trapping. We demonstrate the versatility of our fast analytic-numeric flow field computations via a comparison with 12 device geometries, including circular, triangular, and rectangular wall protrusions, wall cavities, and mid-channel features, as well as high symmetry arrays of cylinders. Instantaneous particle trajectories show that inertial deviations scale as (<italic>η</italic> - 1)*<italic>St</italic> when tracking the fast motion over a single fluid oscillation period, where <italic>η</italic> is the ratio of fluid-to-particle densities. Time-averaged inertial Stokes drift analysis shows that leading order inertial effects in the mean motion of a particle scales as <italic>η*St</italic>. The difference in scaling between the instantaneous and time-averaged motion means that a neutrally buoyant inertial particle (<italic>η</italic> = 1) follows the fast motion in a manner identical to a non-inertial tracer (to leading order) independent of <italic>St</italic>, whereas the time averaged velocity remains dependent on <italic>St</italic>. We use this analysis to predict trapping locations, i.e., the spatial locations where the mean particle velocity goes to zero. Our analytic-numeric approach has advanced the computational tools for hydrodynamic tweezers design by laying the groundwork needed to link device geometry and operation (flow frequency and amplitude) to the physics linked to trapping strength and particle shear stress.