Since, many of the mixing devices in the polymer processing field (e.g. internal mixers and twin-screw extruders) are operated in a partially filled condition, a new method is developed to evaluate the flow field parameters (the velocity components and the pressure values) and predict the melt/air interface, considering the existence of the rotary parts (blades or rotors) in the mixing chamber. For this purpose, the shape functions of the finite element method (FEM) are enhanced with moving least squares (MLS) interpolation functions, to consider the area occupied by air, the melt filled zone and the solid parts (the rotors) as a unique but discontinuous flow field. By employing the MLS aided finite element method (MLS-FEM) there will be no need to generate a new mesh for the flow domain, as the internal parts rotate and the melt/air interface changes. On the other hand, the flow field parameters are evaluated on a unique background mesh to avoid the time consuming procedure of the re-meshing. Then, the outcomes of the flow field calculation alongside with the particle tracking are used to evaluate the distributive mixing in the cross-section of an internal mixer, using the mean strain function criterion. Finally, it is shown that the existence of the air can considerably affect the mixing quality.