Interaction forces between microfluidic droplets in a Hele-Shaw cell

Various microfluidic systems, such as chemical and biological lab-on-a-chip devices, involve motion of multiple droplets within an immersing fluid in shallow microchannels. Modelling the dynamics of such systems requires calculation of the forces of interaction between the moving droplets. These forces are commonly approximated by superposition of dipole 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 bipolar coordinate system and calculation of the force in Cartesian coordinates. 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. Overall, the obtained results establish the solid base for the rather important future extensions for modelling the complex, long-range interdroplet interactions in the limit of dense droplet media.