  Various microfluidic systems, such as chemical and biological lab-on-a-chip
devices, involve motion of multiple droplets within an immersing fluid in
shallow micro-channels. Modeling the dynamics of such systems requires
calculation of the forces of interaction between the moving droplets. These
forces are commonly approximated by superposition of dipoles solutions, which
requires an assumption of sufficiently large distance between the droplets. In
this work we obtain exact solutions (in the Hele-Shaw limit) for two moving
droplets, and a droplet within a droplet, located within a moving immersing
fluid, without limitation on the distance between the droplets. This is
achieved by solution of the pressure field in a bi-polar coordinate system and
calculation of the force in a Cartesian coordinate system. Our results are
compared with numerical computations, experimental data and the existing
dipole-based models. We utilize the results to calculate the dynamics of a
droplet within a droplet, and of two close droplets, located within an
immersing fluid with oscillating speed. The obtained results may be used to
study the dynamics of dense droplet lattices, common to many microfluidic
systems.
