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This paper evaluates the behavior of a single rigid ellipsoidal particle suspended in homogeneous viscous flow with a power-law generalized Newtonian fluid rheology using a custom-built finite element analysis (FEA) simulation. The combined effects of the shear-thinning fluid rheology, the particle aspect ratio, the initial particle orientation, and the shear-extensional rate factor in various homogeneous flow regimes on the particles dynamics and surface pressure evolution are investigated. The shear-thinning fluid behavior was found to modify the particle’s trajectory and alter the particle’s kinematic response. Moreover, the pressure distribution over the particle’s surface is significantly reduced by the shear-thinning fluid rheology. The FEA model is validated by comparing results of the Newtonian case with results obtained from the well-known Jeffery’s analytical model. Furthermore, Jeffery’s model is extended to define the particle’s trajectory in a special class of homogeneous Newtonian flows with combined extension and shear rate components typically found in axisymmetric nozzle flow contractions. The findings provide an improved understanding of key transport phenomenon related to physical processes involving fluid–structure interaction such as that which occurs within the flow field developed during material extrusion–deposition additive manufacturing of fiber reinforced polymeric composites. These results provide insight into important microstructural formations within the print beads.