Swirling of vesicles: Shapes and dynamics in Poiseuille flow as a model of RBC microcirculation

We report on a systematic numerical exploration of the vesicle dynamics in a channel, which is a model of red blood cells in microcirculation. We find a spontaneous transition, called swirling, from straight motion with axisymmetric shape to a motion along a helix with a stationary deformed shape that rolls on itself and spins around the flow direction. We also report on a planar oscillatory motion of the mass center, called three-dimensional snaking, for which the shape deforms periodically. Both emerge from supercritical pitchfork bifurcation with the same threshold. The universality of these oscillatory dynamics emerges from Hopf bifurcations with two order parameters. These two oscillatory dynamics are put in the context of vesicle shape and dynamics in the parameter space of reduced volume $v$, capillary number, and confinement. Phase diagrams are established for $v=0.95$, $v=0.9$, and $v=0.85$ showing that oscillatory dynamics appears if the vesicle is sufficiently deflated. Stationary shapes (parachute/bullet/peanut, croissant, and slipper) are fixed points, while swirling and snaking are characterized by two limit cycles.

Figure 1

Phase diagram of a confined vesicle with $v=0.95$ (left) and $v=0.9$ (right) in a Poiseuille flow in the parameter space $(Ca,C_n)$. Only stationary shapes exist.

Figure 2

Examples of stationary shapes with $v=0.85$. (a) Peanut shape (axisymmetric): $Ca=1$ and $C_n^{-1}=1.8$. (b) Slipper shape: $Ca=3$ and $C_n^{-1}=5$. (c) Croissant shape: $Ca=20$ and $C_n^{-1}=2$. See text for details on the properties of each shape and on parachute. The color code corresponds to the mean curvature $H$.

Figure 3

3D snaking and swirling: $v=0.85$, $Ca=1$, and $C_n^{-1}=2.75$. (a) Snaking: ${\theta }_y\left(0\right)=0$. The center of mass oscillates in the plane $(x,y)$: blue curve. The shape deforms. (b) Swirling: ${\theta }_y\left(0\right)\ne 0$. The center of mass moves along a helix (blue curve) centered on the microcapillary's axis, the $x$ axis. The shape does not deform and turns around the helix. The black arrows point outward to the center of mass, with solid and dashed lines in the planes $(x,y)$ and $(x,z)$, respectively. Color code corresponds to interfacial velocity; see movies in Ref. [44].

Figure 4

Swirling: $v=0.85$, $Ca=1$, $C_n^{-1}=2.75$, and ${\theta }_y\left(0\right)={10}^{\circ }$. (a) Temporal variation of the radial position of the mass center $R_{CM}$. (b) Temporal variation of the Cartesian positions $(Y_{CM},Z_{CM})$. (c) The angles ${\theta }_{xy}$ and ${\theta }_{xz}$ of the longest semiaxis $L_1$ oscillate in quadrature in the planes $(x,y)$ and $(x,z)$, respectively. The angle ${\theta }_{yz}$ of $L_1$ in the plane $(y,z)$ varies linearly with time. The inset refers to the shape and its rotation (yellow arrow) during one period seen from the rear at four times 1–4 (color code). (d) The lengths $L_j$ of three semiaxes of the equivalent ellipsoid tend to a constant. (e) Cross sections of the shapes in the plane $(x,y)$ at the times 1–4 defined in (b). (f) Cross sections of the shapes in the plane $(x,z)$. The color code corresponds to numbers 1–4 defined in (b).

Figure 5

(Left panel) 3D snaking: $v=0.85$, $Ca=2.3$, $C_n^{-1}=2.5$, and ${\theta }_y\left(0\right)=0^{\circ }$. (a) The same initial shape evolves to slipper (orange, $H=0.03$) or snaking (blue, $H=0.0005$) following the initial position $H$; see the blue and orange arrows for the path. Snaking is an oscillation of the mass center position in a plane, here $(x,y)$. (b) Phase portrait of (a) showing the fixed point (slipper) and limit cycle (snaking). (c) The 3D shape oscillates during snaking, temporal variations of the lengths $L_j$ of the three semiaxes of the equivalent ellipsoid. (Right panel) (d) Transition from 3D snaking to swirling after a rotation ${\theta }_y={10}^{\circ }$ of the long axis angle around the $y$ axis: $v=0.85$, $Ca=1$, and $C_n^{-1}=2.75$. Insert: In the phase space $(Y_{CM},Z_{CM})$, the black line corresponds to 3D snaking and the red circle to the limit cycle of swirling. (e) Phase diagram in the parameter space $(C_n^{-1},H)$ with $Ca=1$. For steric reasons, the domain above the dotted line is not relevant.

Figure 6

Analysis of stability. (a) Diagram of bifurcation: $v=0.85$ and $Ca=1$. Order parameters are the position of the mass center ($Y_{CM}$) if the mode is stationary and the amplitudes ($A_{CM}$) if oscillating. Above $C_n^{-1}=2.95\pm 0.05$, the vesicle transits to the slipper branch whatever the perturbation. This branch is extended to zero by considering initially $H=0.1$. Below $C_n^{-1}=2.95\pm 0.05$, the initial shape is axisymmetric and $H=0.0005$. The swirling (snaking) branch is reached with ${\theta }_y\left(0\right)={10}^{\circ }$ ($0^{\circ }$). Blue arrows correspond to the snaking-to-swirling transition when a perturbation breaking the mirror symmetry (${\theta }_y={10}^{\circ }$) is applied to snaking. Below $C_n^{-1}=1.98\pm 0.015$, only the axisymmetric branch exists whatever the perturbation. (b) Phase diagram in the plane $(Ca,C_n)$.