Fused Filament Fabrication (FFF) has emerged as a prominent Additive Manufacturing (AM) technique due to its cost-effectiveness, versatility in producing intricate geometries, and widespread availability of AM machines. Understanding the melting behavior of the polymer filament in the FFF process has been a subject of extensive research, leading to various theoretical models. While the prevailing model involves the formation of a polymer melt pool pushed out by the solid filament acting as a piston, this work explores an alternative model based on the formation of small melt film driven by the force exerted by the solid filament. Referred to as the “Fused Filament Fabrication melting model,” this approach has so far assumed Newtonian fluid behavior, disregarding the complex rheological nature of polymeric melts. To enhance the model’s accuracy, this study focuses on incorporating the shear thinning behavior of the material into the equations. Furthermore, preliminary results for the generalisation of this work and new solutions for the Poiseuille-Couette flow will presented.