Shear viscosity of bimodal capsule suspensions in simple shear flow

We present a numerical analysis of the shear viscosity of bimodal suspensions in Stokes flow. The dynamics of capsules whose membrane follows Skalak law is simulated for the total volume fraction $\varphi =0.4$. A suspension of two types of capsules with different sizes and the same deformability (capillary number) has a decreased shear viscosity compared to monomodal suspensions. Increasing the deformability of the capsules amplifies the extent of the viscosity reduction. The viscosity reduction is mainly caused by a decrease in the shear component of the stresslet of the large capsules, due to a decrease in their deformation and orientation angle with respect to the flow direction. In the case of a suspension of two types of capsules with the same size and different deformabilities, the shear viscosity is nearly the same as the weighted average of monomodal suspensions. An increase in the stresslet of the floppy capsules is counterbalanced by a decrease in the stresslet of the stiff capsules. For a suspension of two types of capsules with different sizes and the same membrane stiffness, the effects of the size difference and deformability difference coexist. The interplay of these effects decreases the stresslet for both the large and small capsules, resulting in a further viscosity reduction.

Figure 1

Three categories of bimodal capsule suspensions. (a) $R_a\ne 1$ and $R_{Ca}=1$, (b) $R_a=1$ and $R_{Ca}\ne 1$, (c) $R_a\ne 1$ and $R_{Ca}=R_a$.

Figure 2

(a) The temporal value of the specific viscosity, where we take only the spatial average. Three cases with different initial positions, represented by different line colors, are simulated for each mesh size. The transient phase is finished within $\dot{\gamma }t<$ 10. (b) Effects of the mesh size (the number of elements per capsule $N_E$) and the ending time $t_e$ on the ensemble average of the specific viscosity. Different line colors show different initial capsule positions. (c) Effect of the number of capsules on the specific viscosity, where the error bars show the standard deviations of the three cases. Parameters are $R_{\varphi }=0.5$, $R_a=0.5$, $R_{\text{Ca}}=1$, and ${\text{Ca}}^A=0.4$.

Figure 3

(a) Specific viscosity ${\mu }_{sp}$ and (b) its ratio to that of monomodal suspensions ${\mu }_{sp}^{*}$ for $R_a=0.5$ and $R_{\text{Ca}}=1$.

Figure 4

Shear component of the stresslet $S_{12}$ for $R_a=0.5$ and $R_{\text{Ca}}=1$. Ensemble average for (a) large capsules (type A) and (b) small capsules (type B). The values are normalized by those for monomodal suspensions, denoted by the superscript “mono.”

Figure 5

Principal stresslet difference $S_1-S_2$ for $R_a=0.5$ and $R_{\text{Ca}}=1$. Ensemble average for (a) large capsules (type A) and (b) small capsules (type B).

Figure 6

The value of $sin2{\theta }_S$ for $R_a=0.5$ and $R_{\text{Ca}}=1$. Ensemble average for (a) large capsules (type A) and (b) small capsules (type B).

Figure 7

Effect of the size ratio $R_a$ on the specific viscosity, where $R_{\varphi }=0.5$, $R_{\text{Ca}}=1$, and ${\text{Ca}}^A=0.4$.

Figure 8

(a) Specific viscosity ${\mu }_{sp}$ and (b) its ratio to the weighted average for monomodal suspensions ${\mu }_{sp}^{*}$ for $R_a=1$ and $R_{\text{Ca}}=0.5$.

Figure 9

Shear component of the stresslet $S_{12}$ for $R_a=1$ and $R_{\text{Ca}}=0.5$. Ensemble average for (a) floppy capsules (type A) and (b) stiff capsules (type B).

Figure 10

Principal stresslet difference $S_1-S_2$ for $R_a=1$ and $R_{\text{Ca}}=0.5$. Ensemble average for (a) floppy capsules (type A) and (b) stiff capsules (type B).

Figure 11

The value of $sin2{\theta }_S$ for $R_a=1$ and $R_{\text{Ca}}=0.5$. Ensemble average for (a) floppy capsules (type A) and (b) stiff capsules (type B).

Figure 12

Effect of the capillary number ratio $R_{\text{Ca}}$ on the specific viscosity, where $R_{\varphi }=0.5$, $R_a=1$, and ${\text{Ca}}^A=0.4$.

Figure 13

(a) Specific viscosity ${\mu }_{sp}$ and (b) its ratio to the weighted average of monomodal suspensions ${\mu }_{sp}^{*}$ for $R_a=0.5$ and $R_G=1$ ($R_{\text{Ca}}=0.5$).

Figure 14

Shear component of the stresslet $S_{12}$ for $R_a=0.5$ and $R_G=1$ ($R_{\text{Ca}}=1$). Ensemble average for (a) large capsules (type A) and (b) small capsules (type B).

Figure 15

Principal stresslet difference $S_1-S_2$ for $R_a=0.5$ and $R_G=1$ ($R_{\text{Ca}}=0.5$). Ensemble average for (a) large capsules (type A) and (b) small capsules (type B).

Figure 16

The value of $sin2{\theta }_S$ for $R_a=0.5$ and $R_G=1$ ($R_{\text{Ca}}=0.5$). Ensemble average for (a) large capsules (type A) and (b) small capsules (type B).

Figure 17

Effect of the size ratio (the capillary number ratio) on the specific viscosity, where $R_{\varphi }=0.5$, $R_G=1$, and ${\text{Ca}}^A=0.4$.