Vesicle dynamics in confined steady and harmonically modulated Poiseuille flows

We present a numerical study of the time-dependent motion of a two-dimensional vesicle in a channel under an imposed flow. In a Poiseuille flow the shape of the vesicle depends on the flow strength, the mechanical properties of the membrane, and the width of the channel as reported in the past. This study is focused on the centered snaking (CSn) shape, where the vesicle shows an oscillatory motion like a swimmer flagella even though the flow is stationary. We quantify this behavior by the amplitude and frequency of the oscillations of the vesicle's center of mass. We observe regions in parameter space, where the CSn coexists with the parachute or the unconfined slipper. The influence of an amplitude modulation of the imposed flow on the dynamics and shape of the snaking vesicle is also investigated. For large modulation amplitudes transitions to static shapes are observed. A smaller modulation amplitude induces a modulation in amplitude and frequency of the center of mass of the snaking vesicle. In a certain parameter range we find that the center of mass oscillates with a constant envelope indicating the presence of at least two stable states.

Figure 1

Phase diagram showing the different shapes of a vesicle in a steady Poiseuille flow as a function of $C_{k}W$ and $C_n$ (compare Refs. [33] and [30]). Note that $W$ is scaled by $R_0$ as explained in Sec. 2. The figure reproduces the regions of Figure 9(b) of Ref. [30] not showing the details of the off-centered snaking region. In addition to our earlier studies we find that the region containing the centered snaking shapes (CSn) consists of three different parts: two regions of coexistence ($①$: CSn and parachutes, $②$: CSn and unconfined slippers) and the region $③$, where only CSn are found.

Figure 2

Centered snaking shape in a steady Poiseuille flow for $C_{k}W=5$ and $C_n=0.55$ (cf. point B in Fig. 1). (a) Sketch of the system. (b) Vertical component of the vesicle's center of mass ($y_{\text{cm}}$) as a function of time. (c) Zoom of Fig. 2. (d) Poincaré map and (e) Fourier transform of the signal of Fig. 2. All variables in Figs. 2 are scaled as described in Sec. 2.

Figure 3

Scaled amplitude (triangles) and frequency (circles) of the oscillation of the CSn (a) for a fixed confinement $C_n=0.55$ and (b) for a fixed capillary number $C_k=1.37$. As an orientation you can find the points A, B, and C in the phase diagram, Fig. 1, together with the two dashed curves corresponding to $C_n=0.55$ and $C_k=1.37$, respectively.

Figure 4

(a) Vertical component of the vesicle's center of mass from the simulation as a function of time in a harmonically modulated Poiseuille flow of amplitude ${\varepsilon }_m=0.1$ and frequency $f_m=0.01$ around the point B. For this case $C_k^0=1.37$, $y_{\text{cm}}^0=0.19$, and $f_0=0.04$ [compare Figs. 1 and 3]. The dashed line represents the time at which the modulation of the Poiseuille flow is switched on. (b) Zoom of Fig. 4. (c) Corresponding mixed modulation signal $y_{\text{cm}}^{\text{MM}}$ [according to Eq. (8)] with indices of modulation $m=0.09$ and $\beta =0.4$ together with envelope (solid red) and frequency (dashed green). (d) Comparison of the Fourier transforms of theory (black) and simulation (red). All variables are scaled as described in Sec. 2.

Figure 5

Fourier transform of $y_{\text{cm}}$ from the simulation for a fixed modulation amplitude ${\varepsilon }_m=0.1$ and modulation frequencies $f_m$ ranging from 0.009 to 0.08. The frequency $f$ of the Fourier transform is depicted on the vertical axis and the amplitude $Y_{\text{cm}}$ is shown using a heat map. $Y_{\text{cm}}$ is scaled by its maximum value whereas all other variables are scaled as described in Sec. 2.

Figure 6

Vertical component of the vesicle's center of mass, $y_{\text{cm}}$, as a function of time for a fixed modulation amplitude ${\varepsilon }_m=0.7$ and frequencies (a) $f_m=0.04$, (b) $f_m=0.056$, (c) $f_m=0.092$, (d) $f_m=0.093$. The dashed line represents the time at which the modulation of the Poiseuille flow is switched on. All variables are scaled as described in Sec. 2.

Figure 7

Region of oscillations with constant envelope (hashed) as a function of the imposed amplitude ${\varepsilon }_m$ and frequency $f_m$ of the harmonically modulated flow. The dashed lines indicate an analytical estimate of the boundaries of this region for small ${\varepsilon }_m$ (see text).