Lateral migration of a spherical capsule near a plane wall in Stokes flow

Lateral migration is the motion of a particle perpendicular to the direction of the surrounding flow. One of the factors leading to the lateral migration of a deformable particle in Stokes flow is the presence of a nearby wall. We numerically investigate the lateral migration of a capsule in a near-wall simple shear flow using a boundary integral method coupled with a finite element method. We find that asymmetrical deformation of the capsule induced by the wall is correlated with a reduction in the lift velocity relative to the lift velocity predicted by a far-field analytical solution. A combination of this asymmetrical deformation, which decreases the lift velocity, and an increase in the value of the capsule stresslet near the wall, which works to increase the lift velocity, leads to a migration velocity that is nearly independent of capillary number and membrane constitutive law at large deformation near the wall.

Figure 1

Schematic showing initial conditions and parameters of computation.

Figure 2

Illustration of (a) symmetrical and (b) asymmetrical deformation of a capsule with positions of volume and surface centroids marked.

Figure 3

Example of lateral migration of a capsule with neo-Hookean membrane at $\text{Ca}=0.5$. Snapshots are shown for (a) $t^{*}=0$, (b) $t^{*}=2$, (c) $t^{*}=7$, (d) $t^{*}=10$, and (e) $t^{*}=15$.

Figure 4

Lift velocity of neo-Hookean capsules (solid lines, continuous simulation results; points, instantaneous quasisteady values) compared to corresponding far-field analytical solutions [Eq. (17), shown with dotted lines].

Figure 5

Slip velocity of neo-Hookean capsules (solid lines, continuous simulation results; points, instantaneous quasisteady values) compared to corresponding far-field analytical solutions [Eq. (19), shown with dotted lines].

Figure 6

(a) Lift velocity and (b) slip velocity of neo-Hookean capsules (solid lines, continuous simulation results; points, instantaneous quasisteady values) compared with the results of Singh et al., Figs. 7 and 7, respectively.

Figure 7

Change in the $S_{33}$ component of the stresslet for neo-Hookean capsules. Solid lines show continuous simulation results; points show instantaneous quasisteady values.

Figure 8

Taylor parameter $D_{12}$ for neo-Hookean capsules. Solid lines show continuous simulation results; points show instantaneous quasisteady values. Dotted horizontal lines represent steady-state values of $D_{12}$ in an infinite shear flow.

Figure 9

$x_3$ component of the asymmetry parameter for neo-Hookean capsules. Solid lines show continuous simulation results; points show instantaneous quasisteady values.

Figure 10

Components of lift velocity vs. distance for neo-Hookean capsules with varying Ca. Solid lines show continuous simulation results; points show instantaneous quasisteady values.

Figure 11

Absolute value of self term vs. distance for neo-Hookean capsules with varying Ca. Solid lines show continuous simulation results; points show instantaneous quasisteady values.

Figure 12

(a) The image system of a stresslet of magnitude $S_{ij}^{\infty }$ acting at the position of the capsule centroid. (b) The image system of a stresslet with magnitude $S_{ij}\left(h\right)>S_{ij}^{\infty }$ acting at the position of the capsule centroid. (c) The image system of a stresslet with magnitude $S_{ij}\left(h\right)$ acting on capsule.

Figure 13

(a) Normal and (b) log-log graph of components of $U_3^{\text{wall}}$ for a neo-Hookean capsule at $\text{Ca}=0.5$.

Figure 14

Results of analysis for Skalak capsules with $C=1$ for $\text{Ca}=0.2,\text{Ca}=0.8$, and $\text{Ca}=2.4$. Solid lines show continuous simulation results; points show instantaneous quasisteady values.

Figure 15

Results of analysis for Skalak capsules with $C=10$ for $\text{Ca}=0.4,\text{Ca}=1.2$, and $\text{Ca}=4.8$. Solid lines show continuous simulation results; points show instantaneous quasisteady values.

Figure 16

(a) Maximum lift velocity and (b) lift velocity at quasisteady state $\left(h/a=2.1\right)$ vs. Ca for various constitutive laws.