Rheological properties of a vesicle suspension

The rheological behavior of a dilute suspension of vesicles in linear shear flow at a finite concentration is analytically examined. In the quasispherical limit, two coupled nonlinear equations that describe the vesicle orientation in the flow and its shape evolution were derived [Phys. Rev. Lett. 96, 028104 (2006)] and serve here as a starting point. Of special interest is to provide, for the first time, an exact analytical prediction of the time-dependent effective viscosity ${\eta }_{\mathrm{eff}}$ and normal stress differences $N_1$ and $N_2.$ Our results shed light on the effect of the viscosity ratio $\lambda$ (defined as the inner over the outer fluid viscosities) as the main controlling parameter. It is shown that ${\eta }_{\mathrm{eff}},N_1$, and $N_2$ either tend to a steady state or describe a periodic time-dependent rheological response, previously reported numerically and experimentally. In particular, the shear viscosity minimum and the cusp singularities of ${\eta }_{\mathrm{eff}},N_1$, and $N_2$ at the tumbling threshold are brought to light. We also report on rheology properties for an arbitrary linear flow. We were able to obtain a constitutive law in a closed form relating the stress tensor to the strain rate tensor. It is found that the resulting constitutive markedly contrasts with classical laws known for other complex fluids, such as emulsions, capsule suspensions, and dilute polymer solutions (Oldroyd B model). We highlight the main differences between our law and classical laws.

Figure 1

Time-dependent reduced viscosity $\left[\eta \right]$ during TB and VB regimes for different values of $\Gamma .$ Parameters are $\Delta =0.437,h=0.3,\Gamma =0.91$ (solid blue line), and $\Gamma =1.02$ (dashed red line).

Figure 2

Reduced average effective viscosity $\langle \left[\eta \right]\rangle$ as a function of $h$ for $\Delta =1.$ The cusp singularity is due to the transition from TB or VB to TT.

Figure 3

Reduced viscosity for capsules with a RBC-type membrane as a function of the viscosity ratio for different values of $\mathcal{C}a$: $\mathcal{C}a=0.06$ (solid red line), 0.1 (doted black line), 0.15 (dotted-dashed green line), 0.2 (dashed blue line), and 0.26 (circle black line).

Figure 4

Time evolution of the first normal stress difference for $\Delta =0.5,h=0.3$, and $\Gamma =0.95$ (tumbling regime). Parameters $\phi ,{\eta }_{\mathrm{ext}}$, and $\dot{\gamma }$ are such that $\phi {\eta }_{\mathrm{ext}}\dot{\gamma }=0.2.$