Einstein viscosity with fluid elasticity

We give the first correction to the suspension viscosity due to fluid elasticity for a dilute suspension of spheres in a viscoelastic medium. Our perturbation theory is valid to $O\left(\varphi {\mathrm{Wi}}^2\right)$ in the particle volume fraction $\varphi$ and the Weissenberg number $\mathrm{Wi}=\dot{\gamma }\lambda$, where $\dot{\gamma }$ is the typical magnitude of the suspension velocity gradient, and $\lambda$ is the relaxation time of the viscoelastic fluid. For shear flow we find that the suspension shear-thickens due to elastic stretching in strain “hot spots” near the particle, despite the fact that the stress inside the particles decreases relative to the Newtonian case. We thus argue that it is crucial to correctly model the extensional rheology of the suspending medium to predict the shear rheology of the suspension. For uniaxial extensional flow we correct existing results at $O\left(\varphi \mathrm{Wi}\right)$, and find dramatic strain-rate thickening at $O\left(\varphi {\mathrm{Wi}}^2\right)$. We validate our theory with fully resolved numerical simulations.

Figure 1

Comparison of the theoretical results (lines) to fully resolved numerical simulations (markers) of the particle contribution to the suspension viscosities, as function of $\mathrm{Wi}$. Top row shows suspension shear viscosity, bottom row extensional viscosity in uniaxial strain. Left panels show stresslet contribution, right panels particle-induced fluid stress. In all panels markers are numerical results for ${\mu }_r=0.68$ (red crosses), ${\mu }_r=0.5$ (green squares), and ${\mu }_r=0.01$ (blue circles). The solid lines show the $O\left({\mathrm{Wi}}^2\right)$ theoretical results from Eqs. (16) and (20). The dotted lines show the $O\left(\mathrm{Wi}\right)$ terms of Eq. (20).

Figure 2

Contributions to the $O\left(\varphi {\mathrm{Wi}}^2\right)$ suspension shear viscosity for ${\mu }_r=1$. The leftmost six bars represent contributions from gradient correlations in Eq. (12) with $\mathit{a}=\mathit{e}+\mathit{o}$. The seventh bar represents the effect of the convective term. These seven bars add up to ${\alpha }_S^{\mathrm{fluid}}$.

Figure 3

Section in flow-shear plane of two fields: (a) Flow type, represented by the gradient eigenvalue discriminant $\mathrm{\Delta }={\left(\mathrm{Tr}{\mathit{a}}^2\right)}^3-6{\left(\mathrm{Tr}{\mathit{a}}^3\right)}^2$. Positive values of $\mathrm{\Delta }$ imply strain-dominated flow where $\mathit{a}$ has three real eigenvalues. (b) Integrand for $O\left(\varphi {\mathrm{Wi}}^2\right)$ contribution to ${\alpha }_S^{\mathrm{fluid}}$ [the $xy$ component of the second bracket in Eq. (12)]. The shear-thickening contributions to the suspension shear viscosity comes from regions of strain-dominated flow.

Figure 4

Comparison of theory (solid lines) to data (markers) presented in Figs. 1 and 2 in Koch et al. [21]. Left panel: Surface integral component of stresslet in reciprocal theorem, Eq. (17). Center panel: Volume integral component of stresslet in reciprocal theorem, Eq. (18). Right panel: Particle-induced fluid stress.