The design of an axial cyclone for the removal of liquid droplets from a liquid stream is considered. The design constraints are that the swirl in the separation section is to be maximized and that an adverse pressure gradient in the flow is to be avoided everywhere in order to prevent droplet breakup. First, a simplified expression is derived for the separation efficiency, defined as the maximum length needed to remove droplets or particles from the flow. The result shows that it is advantageous to position the vanes that generate the swirl at a radius larger than the section where actual separation takes place. This leads to a design in which the swirl generator is mounted on a central body positioned in a wider section of the pipe than the separation section. To calculate the flow and the pressure field in this cyclone geometry, we develop a stream function method which uses a superposition of axisymmetric sources and sinks. We show by means of an example that we can indeed design a geometry in which the undesirable adverse pressure gradients is avoided.