Pattern formation of inertial particles in oscillatory flow

Particles exposed to oscillatory flows interact through complex inertial forces, which can lead to self-assembly. We develop a particle-dynamics framework to study the organization and pattern formation of spherical particles subjected to a uniform oscillating background flow. The framework is based on semi-analytical pairwise interactions that capture the time-averaged inertial dynamics of multiple particles with a density contrast relative to the surrounding fluid. The collective dynamics are governed by two dimensionless parameters: a frequency that characterizes the ratio of inertial to viscous forces over an oscillation, and an oscillation amplitude that quantifies the ratio of inertial to gravitational forces.

When gravity acts along the oscillation axis, low frequencies produce coarsely distributed hexagonal lattices. As the frequency increases, particles form densely packed hexagonal domains perpendicular to the oscillation axis, and the number of layers shows an inverse correlation with amplitude. With gravity perpendicular to the axis, the particles organize into one or more chains depending on the frequency, and the number of chains shows a positive correlation with frequency. These behaviors are mapped on a frequency–amplitude phase plane, demonstrating that the framework reproduces several experimental observations.