A generalized formula for inertial lift on a sphere in microchannels

Inertial microfluidics has been widely used in high-throughput manipulation of particles and cells by hydrodynamic forces, without the aid of externally applied fields. The performance of inertial microfluidic devices largely relies on precise prediction of particle trajectories that are determined by inertial lift acting on particles. The only way to accurately obtain lift forces is by direct numerical simulation (DNS); however, it is burdensome when applied to practical microchannels with complex geometries. Here, we propose a fitting formula for inertial lift on a sphere drawn from DNS data obtained in straight channels. The formula consists of four terms that represent the shear-gradient-induced lift, the wall-induced lift, the slip-shear lift, and the correction of the shear-gradient-induced lift, respectively. Notably, as a function of the parameters of a local flow field, it possesses good adaptability to complex channel geometries. This generalized formula is further implemented in the Lagrangian particle tracking method to realize fast prediction of particle trajectories in two types of widely used microchannels: a long serpentine and a double spiral microchannel, demonstrating its ability to efficiently design and optimize inertial microfluidic devices.