Shape stability of deflated vesicles in general linear flows

The dynamics of vesicles in simple shear or extensional flows has been extensively studied, but the conditions where vesicles experience more complex flow types, such as those seen in microfluidic devices or industrial processing conditions, warrant greater investigation. In this study, we use the boundary element method to investigate the shape stability of deflated vesicles in a general linear flow, i.e., linear combinations of extensional and rotational flows. We model the vesicles as a droplet with an incompressible interface with a bending resistance. We simulate a range of flow types from purely shear to extensional at viscosity ratios ranging from 0.01 to 5.0 and reduced volumes from 0.60 to 0.70. The vesicle's viscosity ratio appears to play a minimal role in describing its shape and stability for many mixed flows, even in cases when significant flows are present in the vesicle interior. We find in these cases that the critical capillary number for shape instabilities collapses onto similar values if the capillary number is scaled by an effective extensional rate. These results contrast with droplet studies where both viscosity ratio and flow type have significant effects on breakup. Our simulations suggest that if the flow type is not close to pure shear flow, one can accurately quantify the shape and stability of vesicles using the results from an equiviscous vesicle in pure extension. When the flow type is nearly shear flow, we start to see deviations in the observations discussed above. In this situation, the vesicle's stationary shape develops asymmetric cusps, which introduces a stabilizing effect and makes the critical capillary number depend on the viscosity ratio.

Figure 1

(a) Steady-state vesicle shapes and the corresponding external velocities. The arrows correspond to the exit streamline. (b) Simulated asymmetrical instability and surface velocities.

Figure 2

Odd Legendre polynomials for a stable and an unstable simulation. The stable (unstable) simulation decreases (increases) exponentially after some transient initial effects. Mixed and extensional flows show qualitatively similar results for this analysis. The dips in the coefficients are some transient behavior and were also seen by Spann et al. [20].

Figure 3

Predicted stability boundaries for several reduced volumes $\nu$ and viscosity ratios $\lambda$. Several of the boundaries overlap exactly. All viscosity ratios were simulated for a reduced volume of 0.65. Simulations of viscosity ratios of 1.0 and 5.0 are shown for a reduced volume of 0.60. The boundaries for the reduced volume 0.70 runs are larger due to higher $\text{Ca}$ requiring longer simulation times.

Figure 4

Orientation angle of vesicle simulations compared to the exit streamline for vesicles near the stability boundary. The inset focuses on the low-$\alpha$ regime for $\nu =0.65$.

Figure 5

Rescaled stability boundaries by $\sqrt{\alpha }$, corresponding to the effective extension along the exit streamline.

Figure 6

Comparison of the stationary vesicle shape for vesicles in a purely extensional flow to those seen in mixed flows. Both plots have the parameters $\lambda =1.00$ and ${\text{Ca}}_s=3.00$. (a) The $\alpha =0.500$ flow results in a stationary shape that closely follows the extensional case. (b) The $\alpha =0.125$ stationary shape deviates significantly from the extensional case. The $y$-axis scaling for (b) has been increased to make the shape deviation more noticeable.

Figure 7

Maximum tension values along the major axis of the vesicle stationary shape. Note that these are the tensions nondimensionalized by the bending modulus, equivalent to $\sigma \text{Ca}$. Tension is maximized on the $z=0$ plane of the vesicle in the planar flow.

Figure 8

Upper half of the vesicle stationary shape comparison for ${\text{Ca}}_s=2.25$, showing the effect of viscosity ratio. The stationary shape for $\alpha =0.125$ is shown to deviate from the purely extensional shape (solid line). The increased viscosity ratio vesicle ($\lambda =5.00$) shows less deviation than the decreased viscosity ratio case ($\lambda =0.01$).

Figure 9

Difference in pressure profiles between the stationary and perturbed configurations along the major axis of the vesicle. The ${\text{Ca}}_{\text{eff}}$ is defined in Eq. (14); ${\text{Ca}}_{\text{eff}}$ is approximately 3.00 because the orientation angle of the vesicles is not known a priori. The exact ${\text{Ca}}_{\text{eff}}$ for the $\alpha =[0.125,0.500,1.00]$ are $[2.982,3.006,3.000]$, respectively. Note that ${\text{Ca}}_{\text{eff}}\approx {\text{Ca}}_s$ for the $\alpha =0.500$ and 1.000 simulations. The inset shows the exaggerated perturbed shape for reference.

Figure 10

Stationary shape maximum tension profile for the pressure difference simulations. Note also the nondimensionalization by the bending modulus (see Fig. 7). Here ${\text{Ca}}_{\text{eff}}\approx 3.00$, $\lambda =1.00$, and $\nu =0.65$.