Traction reveals mechanisms of wall effects for microswimmers near boundaries

The influence of a plane boundary on low-Reynolds-number swimmers has frequently been studied using image systems for flow singularities. However, the boundary effect can also be expressed using a boundary integral representation over the traction on the boundary. We show that examining the traction pattern on the boundary caused by a swimmer can yield physical insights into determining when far-field multipole models are accurate. We investigate the swimming velocities and the traction of a three-sphere swimmer initially placed parallel to an infinite planar wall. In the far field, the instantaneous effect of the wall on the swimmer is well approximated by that of a multipole expansion consisting of a force dipole and a force quadrupole. On the other hand, the swimmer close to the wall must be described by a system of singularities reflecting its internal structure. We show that these limits and the transition between them can be independently identified by examining the traction pattern on the wall, either using a quantitative correlation coefficient or by visual inspection. Last, we find that for nonconstant propulsion, correlations between swimming stroke motions and internal positions are important and not captured by time-averaged traction on the wall, indicating that care must be taken when applying multipole expansions to study boundary effects in cases of nonconstant propulsion.

Figure 1

A three-sphere swimmer and its image system with respect to an infinite planar wall. Spheres with radius $a$ are labeled $\alpha =1,2,3$ from left to right. The two arms $l^{12}$ and $l^{23}$ perform periodic motions. Initially, the swimmer is placed parallel to the wall ($\theta =\pi /2$).

Figure 2

The change in swimming velocity due to the wall (solid line), $v_2^0-v^{0,\mathrm{free}}$, and the wall-induced velocity (dashed line), $u_2^{0*}$, versus the (nondimensional) initial distance from the wall at (a) $t=\pi /2$ and (b) $t=3\pi /2$. Note that both lines coincide.

Figure 3

The log-log plots of the magnitude of velocities versus the (nondimensional) initial distance from the wall at (a) $t=\pi /2$ and (b) $t=3\pi /2$. The solid blue line represents the exact wall-induced velocity of (i) sphere 1, (ii) sphere 2, (iii) sphere 3, and (iv) the swimmer. The dashed green line represents the corresponding dipolar model. The dotted red line represents the corresponding combination of the dipole and quadrupole model.

Figure 4

The instantaneous traction in the $x_1$ direction (left) and in the $x_2$ direction (right) exerted by a three-sphere swimmer at $t=\pi /2$ when the (nondimensional) initial distance from the wall is (a) $h=2.0$, (b) $h=1.5$, (c) $h=0.8$, (d) $h=0.6$, (e) $h=0.4$, and (f) $h=0.2$. The instantaneous traction exerted by individual spheres at $h=0.2$ is shown in (g). In each panel, the values are scaled by the largest magnitude traction in that panel.

Figure 5

(a) The traction plot in the $x_1$ direction (left) and in the $x_2$ direction (right) of a Blakeslet dipole when the only nonzero component of dipole strength is $f_2{d}_2=1$. The dipole is placed at a (nondimensional) distance $h=1$ from the wall. In each panel, the values are scaled by the largest magnitude traction in that panel. (b) Instantaneous traction correlation coefficient, which quantifies the correlation between the instantaneous traction due to the swimmer and a Stokeslet dipole with corresponding dipole strength ${\sum }_{\beta }{\mathbf{f}}^{\beta }{\mathbf{d}}^{\beta }$ at $t=\pi /2$ (solid line) and $t=3\pi /2$ (dashed line).

Figure 6

(a) Semilog and (b) absolute log-log scale of the normalized time-averaged wall-induced velocity of a three-sphere swimmer in the $x_2$ direction $〈u_2^{*}\left({\mathbf{x}}^0\right)〉/〈v^{0,\mathrm{free}}〉$ (solid blue line) and a Stokeslet quadrupole model (dotted red line) versus the (nondimensional) initial distance from the wall.

Figure 7

The time-averaged traction in the $x_1$ direction (left) and in the $x_2$ direction (right) exerted by a three-sphere swimmer at the (nondimensional) initial distance from the wall (a) $h=20.0$, (b) $h=5.0$, and (c) $h=0.4$. In each panel, the values are scaled by the largest magnitude traction in that panel.

Figure 8

(a) The traction plot in the $x_1$ direction (left) and in the $x_2$ direction (right) of a Blakeslet quadrupole when the only nonzero component of quadrupole strength is $f_2{d}_2{d}_2=1$. The quadrupole is placed at a (nondimensional) distance $h=1$ from the wall. In each panel, the values are scaled by the largest magnitude traction in that panel. (b) Traction correlation coefficient quantifying the correlation between the time-averaged traction due to the swimmer and a Stokeslet quadrupole with corresponding quadrupolar strength ${\sum }_{\beta }{\mathbf{f}}^{\beta }{\mathbf{d}}^{\beta }{\mathbf{d}}^{\beta }$.

Figure 9

(a) Velocity correlation coefficient and (b) stress correlation coefficient quantifying the correlation between the unbounded swimmer and a Stokeslet dipole at $t=\pi /2$ (solid lines) and $t=3\pi /2$ (dashed lines).

Figure 10

A Stokeslet at ${\mathbf{x}}^0$ and its image system at ${\mathbf{x}}^{0*}$ with the infinite wall located at $x_1=0$ plane. An infinitesimal area $dS$ on the wall is centered at ${\mathbf{x}}^{\mathrm{w}}$.

Figure 11

(a) Instantaneous ($t=\pi /2$) and (b) time-averaged traction correlation coefficient by varying the square of side length $Nh$ on the wall.