Symmetry breaking and cross-streamline migration of three-dimensional vesicles in an axial Poiseuille flow

We analyze numerically the problem of spontaneous symmetry breaking and migration of a three-dimensional vesicle [a model for red blood cells (RBCs)] in axisymmetric Poiseuille flow. We explore the three relevant dimensionless parameters: (i) capillary number, $C_a$, measuring the ratio between the flow strength over the membrane bending mode, (ii) the ratio of viscosities of internal and external liquids, $\lambda$, and (iii) the reduced volume, $\nu =[V/(4/3)\pi ]/{(A/4\pi )}^{3/2}$ ($A$ and $V$ are the area and volume of the vesicle). The overall picture turns out to be quite complex. We find that the parachute shape undergoes spontaneous symmetry-breaking bifurcations into a croissant shape and then into slipper shape. Regarding migration, we find complex scenarios depending on parameters: The vesicles either migrate towards the center, or migrate indefinitely away from it, or stop at some intermediate position. We also find coexisting solutions, in which the migration is inwards or outwards depending on the initial position. The revealed complexity can be exploited in the problem of cell sorting out and can help understanding the evolution of RBCs' in vivo circulation.

Figure 1

The final parachute shape; $\nu =0.9,$ $C_a=100,$ $\lambda =1.$ (a) Side view; (b) rear view showing the axial symmetry. Colors refer to the mean curvature.

Figure 2

The sections by symmetry plane for slipper (S, $C_a=10$), croissant (C, $C_a=20$), and parachute (P, $C_a=100$) shapes. $\nu =0.9,$ $\lambda =1.$ Colors refer to the mean curvature (same scale for all sections). The concavity at the back of the parachute shape is clearly seen. The black line is the symmetry axis of the flow and of the parachute and croissant shapes.

Figure 3

Instantaneous migration velocity (in units of $\alpha R_0^2$) of a vesicle in Poiseuille flow as a function of the distance from the flow axis (in units of $R_0$). Insets show instantaneous shapes adopted by the vesicle at chosen distances from the flow axis. The arrows point to the exact distances where the shape is observed. The terminal shape is shown in the bottom left corner without an arrow. $\nu =0.9,$ $C_a=100,$ $\lambda =1.$ Color by mean curvature (with the scale shared by all insets). Note the very large absolute value of the mean curvature at the tail region of some of the intermediate shapes; $H\simeq -10/R_0.$

Figure 4

The final croissant shape; $\nu =0.9,$ $C_a=20,$ $\lambda =1.$ (a) Side view; (b) rear view showing the two planes of symmetry (horizontal and vertical planes). A shape obtained by a rotation with an arbitrary angle about the flow axis is also a solution. Colors refer to the mean curvature.

Figure 5

The final slipper shape; $\nu =0.9,$ $C_a=10,$ $\lambda =1.$ (a) Side view; (b) rear view showing the plane symmetry (that plane of symmetry is vertical on the right image). A shape obtained by a rotation with an arbitrary angle about the flow axis is also a solution. Colors refer to the mean curvature.

Figure 6

Different scenarios depending on $C_a$ for $\lambda =1$. Symbols show stable lateral position (in units of $R_0$). P, parachute; C, croissant; S, slipper.

Figure 7

Different scenarios depending on $C_a$ for $\lambda =1$. Symbols show stable lateral position (in units of $R_0$) multiplied by capillary number. The arrows indicate the direction of migration. P, parachute; C, croissant; S, slipper.

Figure 8

Phase diagram of steady states of vesicles in axial Poiseuille flow for $\lambda =1.$ P, parachute; C, croissant; S, slipper. No metastability is observed for $\lambda =1.$

Figure 9

Outward migration of a freely moving vesicle with high viscosity contrast ($\lambda =6,$ $\nu =0.9,$ $C_a=100).$ Migration velocity (in units of $\alpha R_0^2$) is plotted against the position of the center of mass (in units of $R_0$). The overall migration direction is outward but periods of inward migration are observed due to the transient behavior. The oscillations of the migration velocity decay after a very long transient.

Figure 10

Quasistatic approximation, $C_a=100.$ Migration velocity of a vesicle (in units of $\alpha R_0^2$) as a function of distance from the flow axis (in units of $R_0$) for several values of viscosity contrast $\lambda .$ Approximate positions of stationary points [$y_G,$ such that ${\dot{y}}_G\left(y_G\right)=0$] are marked by vertical lines. Arrows show the direction of migration; $\nu =0.9.$ Solid lines show the instantaneous migration velocity for a freely migration vesicle, where unambiguous measurements were possible.

Figure 11

Quasistatic approximation, $\nu =0.9,$ $\lambda =6.$ Migration velocity of a vesicle (in units of $\alpha R_0^2$) as a function of distance from the flow axis (in units of $R_0$) for several values of $C_a.$

Figure 12

Migration of a vesicle far from the flow axis; $C_a=10.$ The limit $y_G\to \infty$ is approximated by setting $y_G/R_0=10.$ Migration velocity (in units of $\alpha R_0^2$) is plotted as a function of $\lambda$ for $\nu =0.95$ and $\nu =0.9.$

Figure 13

The solid curve is the critical line separating the region where migration is inward from that where it is outward, $C_a=10.$ The limit $y_G\to \infty$ is approximated by setting $y_G/R_0=10.$ The dashed curve marks the region beyond which flow alignment or unsteady dynamics are observed.

Figure 14

Instantaneous bending energy (in units of $\kappa$) of the vesicle during its migration towards the centerline or the terminal slipper state as a function of the displacement from the center (in units of $R_0$); $\nu =0.9,$ $\lambda =1.$

Figure 15

The energies of the parachute and croissant states (in units of $\kappa$) as a function $C_a$ for $\nu =0.9$ and $\lambda =1.$ The parachute state is unstable for $C_a\lesssim 45.$ The croissant state is unstable for $C_a\lesssim 15$ with respect to bifurcation to the slipper state (not shown here). Only the parachute state exists for $C_a\gtrsim 50.$