Vesicle dynamics in a confined Poiseuille flow: From steady state to chaos

Red blood cells (RBCs) are the major component of blood, and the flow of blood is dictated by that of RBCs. We employ vesicles, which consist of closed bilayer membranes enclosing a fluid, as a model system to study the behavior of RBCs under a confined Poiseuille flow. We extensively explore two main parameters: (i) the degree of confinement of vesicles within the channel and (ii) the flow strength. Rich and complex dynamics for vesicles are revealed, ranging from steady-state shapes (in the form of parachute and slipper shapes) to chaotic dynamics of shape. Chaos occurs through a cascade of multiple periodic oscillations of the vesicle shape. We summarize our results in a phase diagram in the parameter plane (degree of confinement and flow strength). This finding highlights the level of complexity of a flowing vesicle in the small Reynolds number where the flow is laminar in the absence of vesicles and can be rendered turbulent due to elasticity of vesicles.

Figure 1

Stationary shape and history of the center of mass vertical position as a function of time. (a and b) Slipper solution ($C_{k}W/R_0=6.25$ and $C_n=0.8$). (c and d) Parachute solution ($C_{k}W/R_0=6.25$ and $C_n=0.44$). The $x$ axis codes are shown for the lateral position of the mass center $y_{\text{CM}}$ of the cell scaled by the effective radius of the cell $\left(R_0\right)$.

Figure 2

The center of mass vertical position as a function of time. (a) Slipper solution ($C_{k}W/R_0=5$ and $C_n=0.769$). (b) Snaking dynamics ($C_{k}W/R_0=5$ and $C_n=0.733$). (c) Period-doubling dynamics ($C_{k}W/R_0=5$ and $C_n=0.729$). (d) Period-quadrupling dynamics ($C_{k}W/R_0=5$ and $C_n=0.727$).

Figure 3

The center of mass vertical position as a function of time. An apparently chaotic regime is found for $C_{k}W/R_0=5$ and $C_n=0.689$.

Figure 4

Period-doubling bifurcation diagram. The capillary number is fixed $\left(C_{k}W/R_0=5\right)$, and only the confinement $C_n$ is changed. The $x$ and $y$ axes stand for the confinement and the amplitude of the oscillations, respectively.

Figure 5

Period-tripling motion of the mass center. (a) Temporal behavior for a period-tripling dynamics ($C_{k}W/R_0=4.611$ and $C_n=0.71$). (b) Bifurcation diagram $\left(C_{k}W/R_0=4.611\right)$.

Figure 6

Poincaré sections of different oscillations observed by decreasing the confinement when the capillary number $C_{k}W/R_0$ has been fixed at 5. The $y$ axis stands for the $y$ component of the mass center of the cell, and the $x$ axis stands for its derivative with respect to time. (a) $C_n=0.733$. (b) $C_n=0.729$. (c) $C_n=0.727$. (d) $C_n=0.689$.

Figure 7

Snapshots of a cycle of period-doubling dynamics ($C_{k}W/R_0=5$ and $C_n=0.729$). The cell seems to move like a spermatozoon, using its tail as a flagellum. The straight solid blue line indicates the centerline of the channel, and the blue point indicates the mass center of the cell.

Figure 8

Fourier spectrum of different dynamics in linear (a, c, e, and g) and semi-logarithmic (b, d, f, and h) coordinates: (a and b) Snaking dynamics ($C_{k}W/R_0=5$ and $C_n=0.733$). (c and d) Period-doubling dynamics ($C_{k}W/R_0=5$ and $C_n=0.729$). (e and f) Period-quadrupling dynamics ($C_{k}W/R_0=5$ and $C_n=0.727$). (g and h) Chaotic dynamics ($C_{k}W/R_0=5$ and $C_n=0.689$). The semi-logarithmic scale allows one to see more easily the continuum spectrum characteristic of chaotic regimes. The $x$ axis codes are shown for the frequency $F$ scaled by the characteristic time ${\tau }_c$.

Figure 9

Phase diagram summarizing the different morphologies and dynamics of a single vesicle with a viscosity contrast set to unity $\left(\lambda =1\right)$. (a) Overview of the phase diagram. (b) Zoom on the region where oscillations occur. The combined effect of the confinement and flow strength leads to nine distinct regions represented by different colors in the phase diagram: peanutlike shape (purple), unconfined slipperlike shape (orange), parachutelike shape (dark red), confined slipperlike shape (gray), centered oscillations (cyan), multiple periodic oscillations (dark blue), chaotic oscillations (white), off-centered oscillations (pink), and pinlike shape (green). Note that three regions are not represented in the legend, namely, the multiple periodic, chaotic, and off-centered oscillations.

Figure 10

Stationary shapes exhibited by a RBC-like vesicle under the same conditions of flow and confinement ($C_k=120$ and $C_n=0.7$): $\lambda =5$, cytoplasmic viscosity $\approx 5$ cP (top) and $\lambda =10$, cytoplasmic viscosity $\approx 10$ cP (bottom).

Figure 11

Phase diagram of a red blood cell-like vesicle $\left(\lambda =5\right)$ in a Poiseuille flow showing the existence of three different regimes at very high capillary number.

Figure 12

Phase diagram for $\lambda =10$. The red and blue dots indicate, respectively, parachutelike and slipperlike shapes.