Lateral vesicle migration in a bounded shear flow: Viscosity contrast leads to off-centered solutions

The lateral migration of a suspended vesicle (a model of red blood cells) in a bounded shear flow is investigated numerically at vanishing Reynolds number (the Stokes limit) using a boundary integral method. We explore, among other parameters, the effect of the viscosity contrast $\lambda ={\eta }_{\mathrm{in}}/{\eta }_{\mathrm{out}}$, where ${\eta }_{\mathrm{in}},{\eta }_{\mathrm{out}}$ denote the inner and the outer fluids' viscosities. It is found that a vesicle can either migrate to the center line or towards the wall depending on $\lambda$. More precisely, below a critical viscosity contrast ${\lambda }_c$, the terminal position is at the center line, whereas above ${\lambda }_c$, the vesicle can be either centered or off-centered depending on initial conditions. It is found that the equilibrium lateral position of the vesicle exhibits a saddle-node bifurcation as a function of the bifurcation parameter $\lambda$. When the shear stress increases the saddle-node bifurcation evolves towards a pitchfork bifurcation. A systematic analysis is first performed in two dimensions (due to numerical efficiency), and the overall picture is confirmed in three dimensions. This study can be exploited in the problem of cell sorting and can help understand the intricate nature of the dynamics and rheology of confined suspensions.

Figure 1

The evolution of the equilibrium lateral position ${\mathrm{h}}_{\mathrm{f}}$ and the dimensionless tank treading velocity $V_{tt}/(R_0\dot{\gamma }/2)$ (a and b, respectively) according to the bifurcation parameter $\lambda$. Here $\tau =0.7,C_{\kappa }=1$, and $W/R_0=5$. $R_0\dot{\gamma }/2$ is the velocity of membrane of a circular vesicle under shear flow in unbounded geometry.

Figure 2

The evolution of the lateral position of the vesicle ${\mathrm{h}}_{\mathrm{f}}$ as a function of time (in units of $t_c)$ for several $\lambda$. The center is at 0 on the vertical axis, and the lower wall is at $-0.5$. In this case, below ${\lambda }_c=16$, the vesicle is centered. For very low $\lambda$ (TT motion), it reaches the center of the channel faster, while for larger lambda, but still below ${\lambda }_c$, TB motion is observed, and the time to reach the center becomes significantly longer. Beyond ${\lambda }_c$ the vesicle is off-centered. A typical sequence of snapshots are shown. Here we set $C_{\kappa }=1,\tau =0.7$ and $W/R_0=5$. We note that in panel (c) $\left(\lambda =16\right)$, the final value of ${\mathrm{h}}_{\mathrm{f}}$ is computed as a time average over more than $4000t_c$.

Figure 3

The evolution of the equilibrium lateral position ${\mathrm{h}}_{\mathrm{f}}$ as a function of the bifurcation parameter $\lambda$ for several channel width $W$. Here we set $C_{\kappa }=1$ and $\tau =0.7$ and the initial position of the vesicle is close enough to the lower wall. Beyond ${\lambda }_c$ the vesicle is initially set either at the center or at a small enough distance $\left(0.72R_0\right)$ from the lower wall. TT, TB, and FA are shown.

Figure 4

Phase diagram showing regions where migration is towards the center or towards the wall depending on $\lambda$ and $W$. Migration towards the center line is favored by confinement (small $W)$. The solid line is a guide for the eyes. The simulations data are shown as dots. Here we set $\tau =0.7$ and $C_{\kappa }=1$.

Figure 5

(a) Decomposition of the linear shear flow into pure rotational and elongational components. (b)–(e) Schema showing the link between the orientation angle of the vesicle with the direction of the flow (horizontal axis) and the lift of the vesicle. We show here the pressure profile given in Ref. [8] as an illustration.

Figure 6

Angle $\mathrm{\Psi }$ vs lateral position of the vesicle ${\mathrm{h}}_{\mathrm{f}}$. Here $\lambda =16,C_{\kappa }=1$, and $W=5R_0$.

Figure 7

Phase diagram showing regions where migration is towards the center (if initial position is within yellow area) or towards the wall (if initial position is within orange area). The saddle-node bifurcation occurs at ${\lambda }_c\simeq 16$ for a reduced area $\tau =0.7$, a capillary number $C_{\kappa }=1$, and a channel width $W=5R_0$. The channel center is at zero, and the wall is at $-0.5$ on the vertical axis. The simulations data are shown as dots.

Figure 8

Lateral position of the vesicle ${\mathrm{h}}_{\mathrm{f}}$ vs time before and after a small destabilization. From this figure we see clearly that the final solution (i.e., final position) is stable. Here $\lambda =50,C_{\kappa }=1,W=5R_0$, and $\tau =0.7$.

Figure 9

The evolution of the equilibrium lateral position ${\mathrm{h}}_{\mathrm{f}}$ according the bifurcation parameter $\lambda$ for several capillary numbers $C_{\kappa }$. The initial position of the vesicle is close enough to the lower wall. Beyond ${\lambda }_c$ the vesicle is initially set at the center line or at a small enough distance $\left(0.72R_0\right)$ from the lower wall. TT, TB, and FA are shown. Beyond a certain value of $C_{\kappa }$, the TB motion is suppressed.

Figure 10

Phase diagram showing regions where migration is towards the center or towards the wall depending on $\lambda$ and $C_{\kappa }$. We see here that the capillary number $\left(C_{\kappa }\right)$ has a nonmonotonic effect on the lateral position of the vesicle; see the text for explanation. The solid line is a guide for the eye. The simulations are shown as dots. Here we set $\tau =0.7$ and $W=5R_0$. Initially the vesicle is close enough to the one of the walls.

Figure 11

The evolution of the equilibrium lateral position ${\mathrm{h}}_{\mathrm{f}}$ according to the bifurcation parameter $\lambda$ for several reduced area $\tau$. Here we set $C_{\kappa }=1$ and $W=5R_0$. Initially the vesicle is close enough to one of the walls. Beyond ${\lambda }_c$ the vesicle is initially set at the center line or at a small enough distance $\left(0.72R_0\right)$ from the lower wall. TT, TB, and FA are shown.

Figure 12

Phase diagram showing regions where migration is towards the center or towards the wall depending on ${\lambda }_c$ and reduced area $\tau$. Migration towards the center line is favored with increasing $\tau$. The solid line is a guide for the eyes. The simulations data are shown as dots. Here we set $W=5R_0$ and $C_{\kappa }=1$.

Figure 13

Average steady-state $z$ position ${\mathrm{h}}_{\mathrm{f}}$ for a 3D vesicle as a function of the viscosity ratio $\lambda$. Three different initial positions are used: in the center $\left(z_0=0\right)$ and near the walls $\left(z_0\lessgtr 0\right)$. Here we set $W=5R_0,{\tau }^{3D}=0.7$, and $C_{\kappa }=1$.