Abstract This paper presents applications of a recently proposed Immersed Boundary (IB) method to the solution of the flow around moving and complex shaped surfaces, in particular inside twin-screw extruders. Solving the flow around rotating screw elements implies significant changes in the computational topology at every time step. Using multiple meshes or adaptive methods to tackle these would require extensive meshing and interpolation work that has to be repeated each time step. Mesh generation and solution interpolation between successive grids may be costly and may introduce errors if the geometry changes significantly during the course of the computation. These drawbacks are avoided when the solution algorithm can tackle grids that do not fit the shape of immersed objects. In this work a fixed mesh is used covering both the fluid and solid regions, and the boundary of immersed objects is defined using a time dependent level-set function. The Body Conformal Enrichment (BCE) method is used to accurately impose boundary conditions on the surface of immersed bodies. The proposed algorithm enriches the finite element discretization of interface elements with additional degrees of freedom, the latter being eliminated at element level. Numerical applications are shown in which the flow inside twin-screw extruders is computed for multiple screw elements. A generalized non-Newtonian fluid is used to model molten polymer. Solutions will be shown for various rotation velocities of the screw as the viscosity depends on the shear rate.