Shape Diagram of Vesicles in Poiseuille Flow

Soft bodies flowing in a channel often exhibit parachutelike shapes usually attributed to an increase of hydrodynamic constraint (viscous stress and/or confinement). We show that the presence of a fluid membrane leads to the reverse phenomenon and build a phase diagram of shapes—which are classified as bullet, croissant, and parachute—in channels of varying aspect ratio. Unexpectedly, shapes are relatively wider in the narrowest direction of the channel. We highlight the role of flow patterns on the membrane in this response to the asymmetry of stress distribution.

Figure 1

Vesicle cross sections in a channel of increasing aspect ratio. (a) $\nu =0.913$, $\widehat{R}=0.35$, and $C_a=59$. (b) $\nu =0.973$, $\widehat{R}=0.38$, and $C_a=81$. $\widehat{R}$ and $C_a$ are given for the square section.

Figure 2

Cross sections for two vesicles in symmetric flow (experiment, $\widehat{R}=0.38$; simulation and theory, $\widehat{R}=0$). Left, $\nu =0.979$ and $C_a=4191$; right, $\nu =0.946$ and $C_a=3776$. No fitting parameters.

Figure 3

Shapes for $\alpha =1$. Dots refer to experimental data and the full line to theory and simulations. (a) Bullet-parachute phase diagram in the $(\nu ,C_a)$ plane for not too confined vesicles ($0.12\le \widehat{R}\le 0.5$). Pictures in the inset ($\nu =0.964$, $\widehat{R}=0.24$, $C_a=20$ and $\nu =0.959$, $\widehat{R}=0.25$, $C_a=667$) illustrate the concavity change as $C_a$ increases. (b) Bullet-parachute phase diagram in the $(\nu ,\widehat{R})$ plane. For $\widehat{R}\ge 0.4$, $C_a$ goes from 15 to 70 000, while for $\widehat{R}\le 0.4$, values are limited to the $C_a<100$ range due to shape dependency on $C_a$ in the unbounded case. Pictures in the inset ($\nu =0.938$ and $\widehat{R}=0.49$, 0.69, and 0.89) illustrate the concavity change as $\widehat{R}$ increases.

Figure 4

Shape diagram in asymmetric channel at low constraint. Experiment: Each point corresponds to a vesicle with $\widehat{R}\le 0.5$ and $C_a\lesssim 500$ when $\alpha =1$. Theory: $\widehat{R}=0$ and $C_a\le 1000$.

Figure 5

Cross sections for the same vesicle ($\nu =0.985$) in three channels. (a) $d_y=d_z=83\mu \mathrm{m}$; (b) $d_y=83\mu \mathrm{m}$, $d_z=92\mu \mathrm{m}$; (c) $d_y=d_z=92\mu \mathrm{m}$. (a) and (b) give indications on what would be seen in the $xz$ plane of Fig. 1b.

Figure 6

Velocity field in the vesicle comoving frame. Side (a) and rear (b) view. Simulations, $\nu =0.95$, $C_a=100$, and $\alpha =1.2$ (croissant shape). The norm of velocity is color-coded. Maximum velocity is about 5 times lower than vesicle velocity. See also 27.