The subject of this talk involves the motion of fluid droplets between two (narrowly spaced) parallel plates driven by electric fields that modulate surface tension (i.e. this is a moving boundary problem). The effect is called Electrowetting On Dielectric (EWOD) and may be useful for biomedical applications, such as automated biochemical testing and microfluidic mixing. The fluid dynamics are modeled using Hele-Shaw type equations (in 2-D), with a modified boundary condition to account for contact line pinning or `sticking' of the droplet's liquid-gas interface. Our analysis first focuses on the time-discrete (continuous in space) problem and is presented in a mixed variational framework; in particular, the contact line pinning is captured by a variational inequality. We discuss the well-posedness of the semi-discrete problem, and also for the fully discrete problem when discretized with finite elements. We further discuss our numerical method, as well as our solution method for solving the system with variational inequality. Finally, simulations are presented and compared to experimental videos of EWOD driven droplets. These experiments exhibit droplet pinching and merging events and are reasonably captured by our simulations.