Stresslet in a dilute suspension of rigid spheres in an Oldroyd-B fluid

The stresslet in a dilute suspension of rigid spheres in an Oldroyd-B fluid under imposed shear and uniaxial exentional flow is studied using a combination of analytical and numerical techniques. Previous work has focused on the stresslet contribution to the per-particle viscosity of a suspension. This contribution has been shown to decrease with increasing suspending fluid elasticity in shear and decrease with increasing strain in transient uniaxial extension. By considering the surface tractions in the Newtonian limit, we propose analytical scalings for the three components of the stresslet—pressure, viscous stress, and polymer stress—as functions of the appropriate Weissenberg or Deborah number. We conduct direct numerical simulations of a dilute sphere in viscoelastic fluid flow to compare against analytical scalings and see good agreement for weak elasticity, with increasing deviations as the fluid becomes increasingly elastic. Finally, we discuss the consequences of these trends on the tractions and stresses experienced by the suspended particles.

Figure 1

Comparison of results from theory [Eq. (25)] and simulations for the stresslet and component contributions to the effective shear viscosity with changing Wi, $\beta =0.68$.

Figure 2

Comparison of results from theory [Eq. (25)] and simulations for the stresslet and component contribution to the effective shear viscosity, as a function of $\beta$. (a) Wi $=0.5$; (b) Wi $=2.2297$.

Figure 3

Comparison of results from theory [Eq. (27)] and simulations for extensional stresslet and components at $\beta =0.68$, with time nondimensionalized by the polymer relaxation time $\lambda$. (a) Wi $=0.2$; (b) $\mathrm{Wi}=0.4$.

Figure 4

Comparison of results from theory [Eq. (28)] and simulation for extensional stresslet contributions to the effective viscosity at $\beta =0.68$, with time nondimensionalized by the polymer relaxation time $\lambda$. (a) $\mathrm{Wi}=0.2$; (b) Wi $=0.4$.

Figure 5

Comparison of results from theory [Eq. (27)] and simulations for extensional stresslet and components at Wi $=0.2$, with time nondimensionalized by the polymer relaxation time $\lambda$. (a) $\beta =0.50$; (b) $\beta =0.25$.

Figure 6

Comparison of results from theory [Eq. (28)] and simulation for extensional stresslet contributions to the effective viscosity at Wi $=0.2$, with time nondimensionalized by the polymer relaxation time $\lambda$. (a) $\beta =0.50$; (b) $\beta =0.25$.