Average stress in a dilute suspension of rigid spheroids in a second-order fluid in a linear flow field

The stress response of a suspension of rigid particles in a non-Newtonian fluid is a key variable of interest in studying the rheology of various biological suspensions. While such investigations have been theoretically studied for suspensions of rigid spheres in weakly viscoelastic fluids, the effect of the non-sphericity of particles on the stress remains relatively unexplored. The interplay between the response of the polymeric fluid and the particle orientation yields rich physics. We determine the average extra stress in a dilute suspension of rigid, non-Brownian spheroids in a second-order fluid subject to shear and extensional flows by performing a perturbation expansion of the extra stress up to order Wi (Weissenberg number). We obtain the particle contribution to the first and second normal stress differences, the effective shear viscosity, and the effective extensional viscosity. This analysis also elucidates the time dependence of the average stress for short-term and long-term orientations of the particle.