The objective of this work(1) is to present a new strategy for the transient simulation of fluid flow in geometries involving moving parts and small gaps. It is based upon a fictitious domain method and a mesh refinement technique that relies upon one single reference mesh. With this technique, at each time iteration, the reference mesh may be adapted locally according for instance to the position of the gaps in the computational domain. The method will be discussed in detail and applied to the two-dimensional (2D) simulation of fluid flow in twin-screw extruders where one of the key issues is the ability to predict accurately the shear rates in the gaps formed by the rotating screws.