This talk is based on joint work with Fredrik Viklund. In the classical Hele-Shaw flow a domain in the complex plane grows according to the gradient of its Green's function, thus modelling the propagation of a viscous fluid trapped in a thin layer. We introduce a competitive version of the flow where several domains in the complex plane (or more generally in a Riemann surface of finite type) similarly strive to expand but at the same time hinder each other. Interestingly, stationary flows correspond to a special class of quadratic differentials whose associated half-translation surfaces have a simple description. We also introduce a discrete model, closely related to Propp's competitive erosion model, which conjecturally allows us to simulate the flow.