Stress analysis of dilute particle suspensions in non-Newtonian fluids with efficient evaluation in the weakly non-Newtonian limit

We present a semi-analytical framework to compute the suspension stress in dilute particle-laden non-Newtonian fluids, separating Newtonian and non-Newtonian contributions. The ensemble-averaged stress includes both the particle-induced non-Newtonian stress (PINNS) and an interaction stresslet arising from surface tractions due to the non-Newtonian stress and its induced Newtonian flow. Using a generalized reciprocal theorem, we express this interaction stresslet entirely in terms of the non-Newtonian stress, for a general constitutive model. For weakly non-Newtonian fluids, regular perturbation expansion combined with the method of characteristics yields all leading-order stress contributions from the Newtonian velocity field alone, avoiding the need to solve coupled partial differential equations. This generalizes the method of Koch et al. [Phys. Rev. Fluids 1, 013301 (2016)] beyond polymeric fluids to any weakly non-Newtonian medium driven by velocity and its gradients. We apply the method to two systems: (i) spheres suspended in a fluid of smaller spheroids, where the interaction stress becomes negative for sufficiently anisotropic shapes due to orientation misalignment of the spheroids; and (ii) suspensions in weakly anisotropic nematic liquid crystals. In the latter, assuming a uniform director field fixed by an external field, PINNS vanishes while interaction stresslets remain, either opposing or enhancing background anisotropic stress. These results demonstrate the utility of our framework in capturing first-order particle–microstructure interactions across a broad class of non-Newtonian fluids.