Inertial migration of a rigid sphere in three-dimensional Poiseuille flow

Abstract

Inertial lift forces are exploited within inertial microfluidic devices to position, segregate, and sort particles or droplets. However the forces and their focusing positions can currently only be predicted by numerical simulations, making rational device design very difficult. Here we develop theory for the forces on particles in microchannel geometries. We use numerical experiments to dissect the dominant balances within the Navier-Stokes equations and derive an asymptotic model to predict the lateral force on the particle as a function of particle size. Our asymptotic model is valid for a wide array of particle sizes and Reynolds numbers, and allows us to predict how focusing position depends on particle size.

Direct measurement of particle inertial migration in rectangular microchannels

“…Thus, our method provides the first direct measurement of inertial migration velocities. In addition to verifying the existence of a slow-focusing manifold, our position measurements show that significant inertial focusing occurs while particles are funneled into the channel, and that once this contribution is accounted for, inertial migration velocities agree fully with an asymptotic theory [5]. Reconstructed probability density function (PDF) of particle distributions across the channel cross-section for 10 µm particles at Re = 30 shows that within the first 1 mm of the channel particles are initially focused to two narrow bands of streamlines (density shown in grayscale).…”

“…A text file giving the values of c 4 (x 0 ) and c 5 (x 0 ) for a grid of particle locations is included in the supplemental materials. The method above, which adapts the results from Hood et al [5] for a channel with aspect ratio two, gives only the focusing force on a particle that is not free to migrate across streamlines. The particles in our experiments are free to migrate under inertial focusing forces.…”

“…Though there are many theories for the magnitudes of inertial focusing forces, direct experimental measurement of these forces remains an unmet challenge. Indeed existing theory [2][3][4][5], numerical simulations [6][7][8], and indirect experimental measurements [9] have produced contradictory scalings for the dependence of forces on particle size and velocity. In this paper, we directly measure inertial migration velocities by tracking the motion of particles in a rectangular channel over Reynolds numbers ranging from 30 to 180, and find that their measured migration velocities agree well with existing asymptotic theory [5].…”

“…Rather, the numerical data suggested F L ∼ a 3 . Hood et al [5] extended the asymptotic analysis of Ho & Leal [3]. The resulting scaling law F L ∼ c 4 a 4 + c 5 a 5 , reconciles the asymptotic scaling F L ∼ a 4 in the limit a H with the numerical data of Di Carlo et al [6] up to experimentally used particle sizes, in which a ∼ H. By contrast, Saffman's asymptotic study of inertial lift force assumes that the particle experiences an external force in the direction of flow in addition to the inertial lift force [2].…”

Particle migration and sorting in microbubble streaming flows

“…33,34 Even the behavior of common microparticles in the transport flow through ordinary, ubiquitous microfluidic channels still reveals novel insights today. 35,36 Because the bubbles are driven acoustically in our experiments, one may suspect acoustic radiation forces at work; however, using our typical parameters to evaluate these forces 37 and translating them to particle displacements during passage near the bubble, we find that even our largest (a p ¼ 5 lm) particles would not be displaced perpendicular to streamlines by more than $100 nm if the particle and fluid densities were not matched, and far less under the density-matched conditions of our experiment.…”