Hydrodynamic mobility of a solid particle near a spherical elastic membrane. II. Asymmetric motion

In this paper, we derive analytical expressions for the leading-order hydrodynamic mobility of a small solid particle undergoing motion tangential to a nearby large spherical capsule whose membrane possesses resistance toward shearing and bending. Together with the results obtained in the first part [Daddi-Moussa-Ider and Gekle, Phys. Rev. E 95, 013108 (2017)], where the axisymmetric motion perpendicular to the capsule membrane is considered, the solution of the general mobility problem is thus determined. We find that shearing resistance induces a low-frequency peak in the particle self-mobility, resulting from the membrane normal displacement in the same way, although less pronounced, to what has been observed for the axisymmetric motion. In the zero-frequency limit, the self-mobility correction near a hard sphere is recovered only if the membrane has a nonvanishing resistance toward shearing. We further compute the in-plane mean-square displacement of a nearby diffusing particle, finding that the membrane induces a long-lasting subdiffusive regime. Considering capsule motion, we find that the correction to the pair-mobility function is solely determined by membrane shearing properties. Our analytical calculations are compared and validated with fully resolved boundary integral simulations where a very good agreement is obtained.

Figure 1

The configuration of the system. A small solid particle of radius $b$ situated at ${\mathit{x}}_2={\mathrm{Re}}_z$ near a large spherical capsule of undeformed radius $a$. In an asymmetric situation, the force is directed perpendicularly to $\mathit{d}$ shown here along the $x$ direction.

Figure 2

Scaled particle self-mobility correction versus scaled frequency $\beta$ for various values of the small particle radius $b$ for a membrane with pure shearing. The real and imaginary parts are shown as dashed and solid lines, respectively. Dashed lines on the vertical axis at small $\beta$ represent the hard-sphere limit given by Eq. (25). The curve in gray corresponds to the self-mobility correction for a planar membrane given by Eq. (27). Here the solid particle is set at $h=2b$.

Figure 3

Log-log plot of the rescaled peak-frequency ${\beta }_{\mathrm{peak}}$ versus $b$ for different particle-to-membrane distance $h$.

Figure 4

Scaled self-mobility correction versus ${\beta }_{\mathrm{B}}$ for various values of the capsule radius, for a membrane with pure bending. The dashed and continuous lines represent the real and imaginary parts, respectively. The horizontal dashed lines are the vanishing frequency limits approximated by Eq. (28). The curve shown in gray is the solution given by Eq. (29) for a planar membrane. Here we take $h=2b$.

Figure 5

(a) Scaled particle mobility correction versus $\beta$ for various values $b$ for a membrane possessing both shearing and bending rigidities. The real and imaginary parts are shown as dashed and solid lines, respectively. Horizontal dashed lines are the hard-sphere limit given by Eq. (25). The curve shown in gray corresponds to the mobility correction for a planar elastic membrane [13] as obtained by linear superposition of Eqs. (27) and (29). Here the solid particle is set at $h=2b$ and the membrane has a reduced bending modulus $E_{\mathrm{B}}=1$. (b) Scaled mobility correction versus $\beta$ for various values of $E_{\mathrm{B}}$. The horizontal dashed line in black is the hard-sphere limit given by Eq. (25), whereas the gray dashed line corresponds to the infinite bending rigidity limit predicted for a bending-only membrane as given by Eq. (28). Here we take $b=1/10$ and $h=2b$.

Figure 6

Mean-square displacement versus time for Brownian motion of a 100-nm particle parallel to a planar and a spherical red blood cell membrane with curvature radius $a=1\mu \mathrm{m}$. Dotted and dashed lines represent the corresponding MSDs near a hard wall or sphere, respectively. The inset shows the variation of the excess MSD as defined by Eq. (32).

Figure 7

Scaled frequency-dependent particle self-mobility correction versus the scaled frequency $\beta$ nearby a membrane endowed with only shearing (green), only bending (red), and both rigidities (black). The small particle of radius $b=1/10$ is at a distance $h=2b$. Here we take a Skalak ratio $C=1$ and a reduced bending modulus $E_{\mathrm{B}}=2/3$. The theoretical predictions are presented as dashed lines for the real parts and as solid lines for the imaginary parts. Symbols refer to boundary integral simulations where the real and imaginary parts are shown as squares and circles, respectively. The horizontal dashed lines are the vanishing frequency limits predicted by Eqs. (25) and (28).

Figure 8

The scaled pair-mobility correction versus the scaled frequency nearby a membrane possessing only shearing (green), only bending (red), and both rigidities (black). The analytical prediction given by Eq. (33) is shown as dashed line for the real part and as solid line for the imaginary part. Simulation results are shown as squares and circles for the real and imaginary parts, respectively. The horizontal long-dashed line is the hard-sphere limit predicted by Eq. (34), where the short-dashed line is the viscous drop limit given by Eq. (35).

Figure 9

The membrane displacement versus the polar angle $\theta$ in the plane of maximum displacement (the plane $\varphi =0$ for $u_r$ and $u_{\theta }$ and the plane $\varphi =\pi /2$ for $u_{\varphi }$) for three scaled forcing frequencies $\beta$ at quarter period for $t{\omega }_0=\pi /2$. Solid lines are the theoretical predictions obtained from Eqs. (36) and (37) and symbols are boundary integral simulations.