Pattern formation of spherical particles in an oscillating flow.

We study the self-organization of spherical particles in an oscillating flow through experiments inside an oscillating box. The interactions between the particles and the time-averaged (steady streaming) flow lead to the formation of either one-particle-thick chains or multiple-particle-wide bands, depending on the oscillatory conditions. Both the chains and the bands are oriented perpendicular to the direction of oscillation with a regular spacing between them. For all our experiments, this spacing is only a function of the relative particle-fluid excursion length normalized by the particle diameter, A_{r}/D, implying that it is an intrinsic quantity that is established only by the hydrodynamics. In contrast, the width of the bands depends on both A_{r}/D and the confinement, characterized by the particle coverage fraction ϕ. Using the relation for the chain spacing, we accurately predict the transition from one-particle-thick chains to wider bands as a function of ϕ and A_{r}/D. Our experimental results are complemented with numerical simulations in which the flow around the particles is fully resolved. These simulations show that the regular chain spacing arises from the balance between long-range attractive and short-range repulsive hydrodynamic interactions, caused by the vortices in the steady streaming flow. We further show that these vortices induce an additional attractive interaction at very short range when A_{r}/D≳0.7, which stabilizes the multiple-particle-wide bands. Finally, we give a comprehensive overview of the parameter space where we illustrate the different regions using our experimental data.