Investigation into the rheology of a solid sphere suspension in second-order fluid using a cell model

We consider a suspension of solid spheres in a viscoelastic suspending fluid. We use the second-order fluid model to capture the first effects of viscoelasticity in the suspension and a mean-field cell model to express the influence of the solid particles. We create a semianalytical constitutive equation for the whole suspension, a second-order fluid with modified material parameters. We find that, in simple shear, the ratio of first normal stress difference to shear stress is independent of volume fraction, but that the second normal stress difference can change sign as the volume fraction of solids increases. Our model is valid for all flows for which the second-order fluid expansion is applicable.

Figure 1

Schematic of the cell model. The outer boundary is a virtual surface outside which the flow is unaffected by the presence of the solid sphere. It remains fixed relative to the particle and does not deform with the flow.

Figure 2

Using the cell model for dense suspensions. There may be some fluid which is not contained within any cell; the radius $b$ of the sphere of influence now becomes a semiempirical measure of how close neighboring cells may be.

Figure 3

Solid line: The empirical function $b\left(\varphi \right)$ defining the cell radius as a function of the solid volume fraction $\varphi$ to mimic the Krieger–Dougherty viscosity. We have taken ${\varphi }_m$ as 0.64, the maximum random packing fraction for monodisperse spheres. The dotted line shows the standard assumption $b\left(\varphi \right)={\varphi }^{-1/3}$.

Figure 4

Solid line: The function $f\left(\varphi \right)$ is defined as the ratio between $b\left(\varphi \right)$ and ${\varphi }^{-1/3}$. The empirical function $b\left(\varphi \right)$ defines the cell radius as a function of the solid volume fraction $\varphi$ to mimic the Krieger–Dougherty viscosity. We have taken ${\varphi }_m$ as 0.64, the maximum random packing fraction for monodisperse spheres. The dotted line shows the same function $f\left(\varphi \right)$ under the standard assumption $b\left(\varphi \right)={\varphi }^{-1/3}$.

Figure 5

A comparison of normalized effective viscosity ${\eta }_{\mathrm{eff}}/{\eta }_0$ in suspension between the numerical results of Yang and Shaqfeh [43], the dilute theory [30] and our own results using $b\left(\varphi \right)$. The numerical results are given by the square and triangular data points, dilute theory by the solid and long dashed lines and our cell model approach by the short dashed lines. The matching process which results in the empirical function $b\left(\varphi \right)$ guarantees that ${\eta }_{\mathrm{eff}}$ is given precisely by the Krieger–Dougherty Eq. (3). We show results at semidilute values of solid volume fraction $\varphi =0.05$ and 0.1.

Figure 6

The normalized effective material functions ${\alpha }_{0,\mathrm{eff}}/{\alpha }_0$ and ${\alpha }_{1,\mathrm{eff}}/{\alpha }_1$ as a function of solid volume fraction $\varphi$ using our Krieger–Dougherty matching $b\left(\varphi \right)$ shown in panels (a) and (b), respectively. We have taken ${\varphi }_m$ as 0.64, the maximum random packing fraction for monodisperse spheres. In panel (a), we see that the graph coincides exactly with the Krieger–Dougherty form (which we imposed for viscosity). We have taken two choices of parameter in panel (b) to show the divergent behavior: $\epsilon =-7/14$, the solid line and $\epsilon =-9/14$, the dashed line.

Figure 7

The individual components to first and second bulk normal stress differences at solid volume fraction $\varphi =0.01$ from our analysis and from Yang et al. [47] are shown in panels (a) and (b), respectively. The normalized $N_1$ is presented as a cumulative total, whereas $N_2$ directly shows the sizes of each individual term. The northwest shaded bar shows the stress contribution from the fluid phase in the absence of particles, the northeast shaded shows the extra mean polymer stress contribution due to the solid phase and the crosshatched bar shows the stresslet term. We note here that the bulk stress contribution (northwest shaded) in panel (a) is the dominant term, with its value given directly from the $y$ axis. The actual values of the extra mean polymer stress and stresslet are shown in the labels to the right of their respective bars. All components are normalized by ${\dot{\gamma }}^2\mathrm{Wi}$ with $\epsilon =-0.3198$ and $-0.500977$ for $N_1$ and $N_2$, respectively. The inertial stress terms present in Ref. [47] have been omitted from both charts.

Figure 8

The second normal stress difference $N_2$ scaled by $\dot{\gamma },\mathrm{Wi},$ and ${\sigma }_{xy}$, the shear rate, Weissenberg number, and shear stress, respectively. The four choices of parameter $\epsilon$ are $-7/14$ the solid line, $-23/42$ the dashed line, $-25/42$ the dash dotted line and $-9/14$ the dotted line. It shows the scaled $N_2$ as a function of solid volume fraction $\varphi$ using our Krieger–Dougherty matching $b\left(\varphi \right)$, having taken ${\varphi }_m=0.64$ as the maximum random packing fraction for monodisperse spheres.

Figure 9

The second normal stress difference $N_2$ scaled by $\dot{\gamma },\mathrm{Wi},$ and ${\sigma }_{xy}$, the shear rate, Weissenberg number, and shear stress, respectively. The results of Escott and Wilson are shown against the data points from Yang et al. [47] and Koch and Subramanian [30] for three values of solid volume fraction, along with their respective linear regression lines. The value of $\epsilon$ which matches that of [47] is $\epsilon =-0.500977$.

Figure 10

The first and second normal stress differences from our analytics and the experiments of Dai et al. [15] are given in panels (a) and (b), respectively. $N_1$ and $N_2$ are scaled with ${\dot{\gamma }}^2$, and shown over a range of solid volume fraction up to $\varphi =0.4$. We consider the choice of parameter $\epsilon =-1/2$ based on the ratio $N_2/N_1=0$ at $\varphi =0$ found experimentally [15]. Our Weissenberg number here is $\mathrm{Wi}=0.116$ based on the value of $N_1$ in the absence of a solid phase.