Shear thinning and shear thickening of a confined suspension of vesicles

Widely regarded as an interesting model system for studying flow properties of blood, vesicles are closed membranes of phospholipids that mimic the cytoplasmic membranes of red blood cells. In this study we analyze the rheology of a suspension of vesicles in a confined geometry: the suspension, bound by two planar rigid walls on each side, is subject to a shear flow. Flow properties are then analyzed as a function of shear rate $\dot{\gamma }$, the concentration of the suspension $\varphi$, and the viscosity contrast $\lambda ={\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}}$, where ${\eta }_{\mathrm{in}}$ and ${\eta }_{\mathrm{out}}$ are the fluid viscosities of the inner and outer fluids, respectively. We find that the apparent (or effective viscosity) of the suspension exhibits both shear thinning (decreasing viscosity with shear rate) or shear thickening (increasing viscosity with shear rate) in the same concentration range. The shear thinning or thickening behaviors appear as subtle phenomena, dependant on viscosity contrast $\lambda$. We provide physical arguments on the origins of these behaviors.

Figure 1

The normalized effective viscosity $\left[\eta \right]$ and the normalized normal stress difference $\left[N\right]$ as a function of capillary number $C_{\kappa }$ for different concentrations. For all explored concentrations, the suspension exhibits a shear thinning behavior.

Figure 2

Typical configurations of a suspension of vesicles for several concentration $\varphi$ and typical capillary numbers $C_{\kappa }$. (a) $C_{\kappa }=3$. (b) $C_{\kappa }=80$. (c) $C_{\kappa }=10$.

Figure 3

Streamlines inside and outside a single vesicle of the suspension for the viscosity contrast $\lambda =1$, for two capillary numbers: $C_{\kappa }=1$ (left) and $C_{\kappa }=200$ (right). In this case the inclination angle of the vesicles is almost constant with $C_{\kappa }$, whereas the shape undergoes some moderate change. Owing to this, the streamlines also deform following the deformation of vesicles. We can see in this case that the streamlines inside and outside of the vesicles are more deformed for lower $C_{\kappa }$ than for higher $C_{\kappa }$, in agreement with the shear thinning behavior. The same set of values of the stream function is used for streamlines in left and right panels.

Figure 4

Steady contours of a vesicle as a function of the capillary number $C_{\kappa }$ for a viscosity contrast $\lambda =1$. The capillary number $C_{\kappa }$ increases in the direction of the arrows.

Figure 5

The Taylor deformation index $D$ as a function of capillary number $C_{\kappa }$ for $\lambda =1$ and $\lambda =10$ ($\varphi =5.58\%$).

Figure 6

The normalized effective viscosity $\left[\eta \right]$ and the normalized normal stress difference $\left[N\right]$ as a function of capillary number $C_{\kappa }$ for $\varphi =5.58\%$ and $\varphi =19.55\%$. For all explored concentrations, the suspension exhibits a shear thickening behavior.

Figure 7

Typical configurations of a vesicle suspension. Top: $\varphi =5.58\%$, $C_{\kappa }=1$, and $\lambda =10$. Bottom: $\varphi =19.55\%$, $C_{\kappa }=1$, and $\lambda =10$.

Figure 8

Inclination angle of vesicles ($\varphi =5.58\%$) as a function of $C_{\kappa }$ for a viscosity contrast $\lambda =10$. The angle increases with $C_{\kappa }$.

Figure 9

Absolute difference of the inclination angle corresponding to two capillary numbers ($C_{\kappa }=1$ and $C_{\kappa }=80$) as a function of the viscosity contrast $\lambda$.

Figure 10

Streamlines inside and outside a vesicle for the viscosity contrast $\lambda =10$, for two capillary numbers: $C_{\kappa }=1$ (left) and $C_{\kappa }=200$ (right). Here ($\lambda =10$), the inclination angle increases with $C_{\kappa }$. The same set of values of the stream function is used for streamlines in the left and right panels.

Figure 11

Streamlines associated with the induced flow inside and outside a vesicle for $\lambda =10$, and for two capillary numbers: $C_{\kappa }=1$ (left) and $C_{\kappa }=200$ (right). The same set of values of the stream function is used for streamlines in the left and right panels.

Figure 12

The normalized effective viscosity $\left[\eta \right]$ as a function of capillary number $C_{\kappa }$ for a single vesicle for two values of reduced area: $\nu =0.7$ and $\nu =0.9$. For $\lambda =1$ ($\lambda =10$) the suspension exhibits a shear thinning (shear thickening) behavior regardless of the reduced area $\nu$.

Figure 13

The evolution of the Taylor deformation index $D$ and the inclination angle of the vesicle $\mathrm{\Psi }$ as a function of capillary number $C_{\kappa }$ for reduced area $\nu =0.9$, for two values of viscosity contrasts: $\lambda =1$ and $\lambda =10$.

Figure 14

Shear thinning to shear thickening crossover for a suspension with concentration $\varphi =5.58\%$.

Figure 15

The evolution of the Taylor deformation index $D$ and the inclination angle $\mathrm{\Psi }$ of the vesicle as a function of capillary number $C_{\kappa }$ for different values of the viscosity contrast $\lambda$.