The Schottky-Klein prime function is a special transcendental function which plays a central role in problems involving multiply connected domains. This function can be used to great advantage in many varied applications. In this talk, we will explore two different free boundary problems (arising in fluid mechanics) defined over two distinct multiply connected geometries. For both problems, we will show that it has been expedient to employ the Schottky-Klein prime function and its associated function theory in order to construct analytical solutions.