Shape transition and hydrodynamics of vesicles in tube flow

The steady motion and deformation of a lipid-bilayer vesicle translating through a circular tube in low Reynolds number pressure-driven flow are investigated numerically using an axisymmetric boundary element method. This fluid-structure interaction problem is determined by three dimensionless parameters: reduced volume (a measure of the vesicle asphericity), geometric confinement (the ratio of the vesicle effective radius to the tube radius), and capillary number (the ratio of viscous to bending forces). The physical constraints of a vesicle—fixed surface area and enclosed volume when it is confined in a tube—determine critical confinement beyond which it cannot pass through without rupturing its membrane. The simulated results are presented in a wide range of reduced volumes [0.6, 0.98] for different degrees of confinement; the reduced volume of 0.6 mimics red blood cells. We draw a phase diagram of vesicle shapes and propose a shape transition line separating the parachutelike shape region from the bulletlike one in the reduced volume versus confinement phase space. We show that the shape transition marks a change in the behavior of vesicle mobility, especially for highly deflated vesicles. Most importantly, high-resolution simulations make it possible for us to examine the hydrodynamic interaction between the wall boundary and the vesicle surface at conditions of very high confinement, thus providing the limiting behavior of several quantities of interest, such as the thickness of lubrication film, vesicle mobility and its length, and the extra pressure drop due to the presence of the vesicle. This extra pressure drop holds implications for the rheology of dilute vesicle suspensions. Furthermore, we present various correlations and discuss a number of practical applications. The results of this work may serve as a benchmark for future studies and help devise tube-flow experiments.

Figure 1

Schematic illustration of a vesicle flowing along the centerline of a circular tube of radius $R$ in a pressure-driven flow. The system is rotationally symmetric about the $x$ axis. The boundaries of the control volume ($D$) are the inlet and outlet sections $I$ and $O$, the solid surface of the tube wall $W$, and the membrane/medium interface $\mathrm{\Gamma }$, i.e., $\partial D\equiv I\cup O\cup W\cup \mathrm{\Gamma }$. The vesicle shape and the gap size between the membrane and the tube wall are denoted by $\overline{h}$ and $h(\equiv R-\overline{h})$, respectively. The vesicle enclosed volume is denoted by $\mathrm{\Omega }$.

Figure 2

Steady-state vesicle profile as a function of the bending capillary number ${\mathrm{Ca}}_{\mathrm{B}}$ for $\nu =0.84$ and $\lambda =1$.

Figure 3

Steady-state vesicle shapes (solid lines) as a function of the confinement $\lambda$ for a wide range of reduced volumes $\nu$ (${\mathrm{Ca}}_{\mathrm{B}}=50)$: (a) $\nu =0.98$, (b) $\nu =0.9$, (c) $\nu =0.8$, and (d) $\nu =0.6$. The dashed lines in (b) and (c) represent the 3D BEM simulations of Ref. [25], Fig. 9(b) ($\nu =0.9, {\lambda }_{\mathrm{A}}=1.2$, 1, 0.8) and Fig. 9(c) ($\nu =0.8, {\lambda }_{\mathrm{A}}=1.4$, 1.2, 1), respectively.

Figure 4

Parachute-bullet phase diagram in the ($\nu , \lambda$) space. Filled symbols denote the present numerical results ($•$ for bulletlike shape, $\blacksquare$ for parachutelike shape), crosses represent experimental data of Ref. [36] ($\times$ for bullet-like shape, $*$ for parachutelike shape). Capillary number varies over four orders of magnitude (${10}^1<{\mathrm{Ca}}_{\mathrm{B}}<{10}^4$).

Figure 5

(a) Shape transition line (blue) from parachute (low limit in error bars) to bullet (up limit in error bars) in the ($\nu , \lambda$) space (${\mathrm{Ca}}_{\mathrm{B}}=50$). Vesicles are flowing from left to right. The two typical vesicles are characterized, respectively, by $\nu =0.95, \lambda =0.8$ and ${\mathrm{Ca}}_{\mathrm{B}}=5$ for the bullet-like shape, and $\nu =0.83, \lambda =0.67$ and ${\mathrm{Ca}}_{\mathrm{B}}=15$ for the parachute-like shape. Insets show the comparison of the computed shapes (red line) with the reported ones in an experimental study [5]. The rupture line (red) represents the critical confinement ${\lambda }_c$ Eq. (22). Membrane lysis occurs when $\lambda >{\lambda }_c$. (b) The shape at the parachute-bullet transition of a vesicle with the same typical deflation of red blood cells ($\nu =0.6$) flowing through a narrow capillary tube ($\lambda \simeq 1.8$).

Figure 6

The dimensionless thickness of the lubrication layer $\delta$ plotted as a function of the reduced radius ratio $\lambda /{\lambda }_c$ for a wide range of reduced volumes $\nu$ (${\mathrm{Ca}}_{\mathrm{B}}=50$), together with the asymptotic prediction Eq. (24).

Figure 7

The dimensionless thickness of the lubrication layer $\delta$ plotted as a function of the capillary number ${\mathrm{Ca}}_{\mathrm{v}}=\eta V/{\gamma }_{\mathrm{F}}$ for a wide range of reduced volumes $\nu$, together with two scalings.

Figure 8

Normalized vesicle length $\ell /{\ell }_c$ versus the reduced radius ratio $\lambda /{\lambda }_c$ for a wide range of reduced volumes $\nu$ (${\mathrm{Ca}}_{\mathrm{B}}=50$), together with the linear scaling $\ell /{\ell }_c=\lambda /{\lambda }_c$ [Eq. (26)].

Figure 9

Variation of the relative velocity $V/U$ as a function of the confinement $\lambda$ for a wide range of reduced volumes $\nu$ (${\mathrm{Ca}}_{\mathrm{B}}=50$). The filled symbols represent the present numerical results with the open symbols of the same shape and color for the 3D numerical results of BS2018: [25] (Fig. 10, $\nu =0.9$, 0.8, and 0.7). Also shown are the asymptotic prediction for a small rigid sphere moving along the centerline of a tube Eq. (27a) (dotted line), the asymptotic predictions of Eq. (28a) (dashed lines colored differently according to the appropriate reduced volume), and the prediction of a lubrication model for red blood cells [34] (RBC theory).

Figure 10

Variation of the dimensionless pressure drop $\mathrm{\Delta }{p}^{+}{R}_0/\left(\eta U\right)$ as a function of the confinement $\lambda$ for a wide range of reduced volumes $\nu$ (${\mathrm{Ca}}_{\mathrm{B}}=50$). The filled symbols represent the present numerical results with the open symbols of the same shape and color for the 3D numerical results of BS2018: Ref. [25] (Fig. 11, $\nu =0.9$, 0.8, and 0.7). Also shown are the asymptotic prediction for a small rigid sphere moving along the centerline of a tube Eq. (27b) (dotted line), the asymptotic predictions of Eq. (28b) (dashed lines), and the prediction of a lubrication model for red blood cells [34] (RBC theory).

Figure 11

Dimensionless pressure drop $\mathrm{\Delta }{p}^{+}{R}_0/\left(\eta U\right)$ plotted as a function of $\alpha (=\lambda \ell /\delta )$ for highly confined vesicles (i.e., $\lambda \to {\lambda }_c$). The dashed line shows the scaling $3\alpha /2$.

Figure 12

Dimensionless pressure drop $\mathrm{\Delta }{p}^{+}{R}_0/\left(\eta U\right)$ versus the reciprocal of the relative velocity $V/U$. The dotted curve shows the analytical prediction for a small rigid sphere Eq. (29). The dashed line is the best fitting Eq. (30) to $\nu =0.6$ in the range of $0.6<{(V/U)}^{-1}<0.995$.

Figure 13

Variation of relative apparent blood viscosity ${\eta }_{\mathrm{rel}}$ with tube diameter $d$ in $\mu \mathrm{m}$ for a hematocrit $H_T$ of 0.45. Curve –$▾$– represents simulation results based on single-file fluid model, and curve –$▴$– represents lubrication model of RBCs [34]. Dashed line in blue shows asymptotic theory for dimensionless extra pressure drop [32] [Eq. (28b)]. Solid curve in cyan represents a fitting empirical equation to in vitro experimental data [1]. Also shown is the spectral 3D BEM simulations of Ref. [28] (for $H_T=0.30$). The vertical line (black) indicates a lower limit ($d_c\simeq 2.8\mu \mathrm{m}$) to the diameter of tubes beyond which normal red blood cells cannot pass through without rapture.

Figure 14

Schematic of lubrication theory analysis for a steady, spherocylindrical vesicle moving along the axis of a circular tube of radius $R$. In a coordinate frame moving with the vesicle, there can be no flow and pressure gradient inside the vesicle. The vesicle has a cylindrical main body with a length $L_0$, assumed large compared to $R$. Plotted vesicle profile corresponds to a vesicle shape for $\nu =0.6, \lambda =1.92$, and ${\mathrm{Ca}}_{\mathrm{B}}=50$ [shown in Fig. 3]; the tube wall is placed at more than $20\%$ away from its actual place to amplify the width of the gap between the vesicle and the tube wall.