  The self-organization of clusters of particles is a fundamental phenomenon
across various physical systems, including hydrodynamic and colloidal systems.
One example is that of dense spherical particles submerged in a fluid and
subjected to horizontal oscillations. The interaction of the particles with the
oscillating flow leads to the formation of one-particle-thick chains or
multiple-particle-wide bands. In this study, we model the hydrodynamic
interactions between such particles and parallel chains using simplified
potentials. We first focus on the hydrodynamic interactions between chains,
which we characterize using data from numerical simulations. Based on these
interactions, we propose a simplified model potential, called the Siren
potential, which combines the representative hydrodynamic interactions:
short-range attraction, mid-range repulsion, and long-range attraction. Through
one-dimensional Monte Carlo simulations, we successfully replicate the
characteristic patterns observed in hydrodynamic experiments and draw the phase
diagram for the model potential. We extend our analysis to two-dimensional
systems, introducing a dipole-capillary model potential that accounts for both
chain formation and Siren-like chain interactions. This potential is based on
colloidal particles at an interface, where chain formation is driven by an
external electric field inducing a dipole moment parallel to the interface in
each particle. The capillary force contributes the long-range attraction.
Starting with parallel chains, the patterns in the two-dimensional Monte Carlo
simulations of this colloidal system are similar to those observed in the
hydrodynamic experiments. Still, the model potentials help clarify the dynamic
behavior of the particles and chains due to the complex interactions
encountered in both hydrodynamic and colloidal systems, drawing parallels
between them.
